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WebAssembly | https://ncatlab.org/nlab/source/WebAssembly | See also [[virtual machine]], [[zoranskoda:hyperledger]], [[zoranskoda:EOS]], [[Rust]].
__WebAssembly__ (wasm) is a modern low level language (mimicking assembly code, but independent of a machine) intended for execution a [[virtual machine]]. It is represented in one of the three common forms. Virtual machine accepts the bytecode version. The corresponding assembly like representation is also used where the commands are given the names as usual. The third is lisp-like bracketed notation, the so called S-expression representation (which is complemented by a little bit of additional information not explicit in assembly like representation).
It is optimized for small compiling time and near native execution time, at least when implemented on a virtual machine on one of major [computer architectures](https://en.wikipedia.org/wiki/Instruction_set_architecture) (like the x86 or ARM series). It is created as a new VM standard for web browsers, backed by major internet companies; it is also used or planned on a number of [[blockchain]] projects, most notably Parity [[zoranskoda:substrate|Substrate]]. On web browsers it is highly interoperable with JavaScript.
* WebAssembly [wikipedia](https://en.wikipedia.org/wiki/WebAssembly), [spec](https://webassembly.github.io/spec), [webassembly.org](https://webassembly.org), [msdn](https://developer.mozilla.org/en-US/docs/WebAssembly) docs, [rust-to-wasm](https://developer.mozilla.org/en-US/docs/WebAssembly/Rust_to_wasm), [github](tps://github.com/WebAssembly)
* [WebAssembly-Links](https://wiki.parity.io/WebAssembly-Links) in Parity Tech Documentation
* AssemblyScript (maps a subset of javascript code to wasm) [github](https://github.com/AssemblyScript/assemblyscript), [news](https://www.infoworld.com/article/3224006/assemblyscript-compiles-typescript-to-webassembly.html)
* How does WASM get interpreted by the EOS virtual machine? [eosio.stackexchange](https://eosio.stackexchange.com/questions/167/how-does-wasm-get-interpreted-by-the-eos-virtual-machine)
[[Rust]] language has small runtime, which is desirable in common applications of WebAssembly. Thus Rust commonly compiles either to native code or to wasm.
* [www.rust-lang.org/what/wasm](https://www.rust-lang.org/what/wasm)
* Rust and wasm official [rustwasm.github.io/docs/book](https://rustwasm.github.io/docs/book)
* K. Hoffman, _Programming Webassembly with Rust_, book
* Nick Fitzgerald, _Rust & WebAssembly_, [yt](https://www.youtube.com/watch?v=ZiiTRxWk8gA)
General support for wasm outside of browsers is not yet standardized. One has to complement sandboxed wasm with some system calls to have reasonable functionality. WASI is a generic term for the wasm system interface.
* Lin Clark, _Standardizing WASI: A system interface to run WebAssembly outside the web_ [2019/03](https://hacks.mozilla.org/2019/03/standardizing-wasi-a-webassembly-system-interface); _WebAssembly’s post-MVP future: A cartoon skill tree_ [2018/10](https://hacks.mozilla.org/2018/10/webassemblys-post-mvp-future))
* wasmer is a standalone for wasm applications, see [github](https://github.com/wasmerio/wasmer), [wasmer.io](https://wasmer.io), [this](https://changelog.com/podcast/341) podcast of S. Akbary and article _WebAssembly & CloudABI_, [Medium](https://medium.com/wasmer/webassembly-cloudabi-b573047fd0a9)
* github/CraneStation/[wasmtime](https://github.com/CraneStation/wasmtime)
* medium/[running-webassembly-from-any-language](https://medium.com/wasmer/running-webassembly-from-any-language-5741f6320ccd)
Efficiency of wasm and its standardization and programming tools support make wasm VM ideal for distributed ledger applications. Ethereum flavoured version of wasm VM specification is at github/[ewasm](https://github.com/ewasm), see also github/[ewasm/design](https://github.com/ewasm/design). eWasm has a [testnet](http://ewasm.ethereum.org/explorer). According to article _[ewasm explained](https://www.mycryptopedia.com/ewasm)_,
> The ewasm specification consists of a subset of WebAssembly components suitable for Ethereum’s needs, namely determinism and relevant features. It also includes a number of system smart contracts that provide access to Ethereum platform features.
[[zoranskoda:EOS]] is a [[high performance distributed ledger]] using wasm VM.
* Shankar Nathan, _How to deploy and run a smart contract on the EOS blockchain_, [web](https://hackernoon.com/how-to-deploy-and-run-a-smart-contract-on-the-eos-blockchain-from-zero-to-hero-72ca592803ba)
* gurghet, [Write EOS contracts in Rust instead of C++](https://steemit.com/crypto/@gurghet/write-eos-contracts-in-rust-instead-of-c)
category: computer science |
Wedderburn-Artin theorem | https://ncatlab.org/nlab/source/Wedderburn-Artin+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
In [[algebra]], the *Wedderburn--Artin theorem* gives a clean characterization of those [[rings]] that are [[matrix algebras]] over [[division rings]].
Concretely, the theorem says that any [[semisimple ring]] is a [[finite limit|finite]] [[direct sum]] of [[matrix algebras]] over [[division rings]]. (Here a [[ring]] is called *[[semisimple ring|semisimple]]* if each of its left or equivalently right [[modules]] is a finite [[direct sum]] of [[simple modules]].)
## Statement ##
\begin{proposition}\label{TheTheorem}
**(Wedderburn--Artin Theorem)** \linebreak
Every [[semisimple ring]] is [[isomorphism|isomorphic]] to a finite [[direct sum]] of [[matrix algebras]] over [[division rings]]. A semisimple ring is [[simple ring|simple]] if and only if it is a matrix algebra over a division ring.
\end{proposition}
Beware: a simple ring may not be semisimple! A ring is simple iff it has no two-sided ideals, and in the absence of further hypotheses this does not imply that all of its left (or equivalently right) modules are direct sums of simple modules. For example, the [[Weyl algebra]] is simple but not semisimple, and not isomorphic to a matrix algebra over a division ring. However, a simple ring that is left or right [[Artinian ring|artinian]] is semisimple, so we have:
\begin{corollary}\label{Corollary}
\linebreak
Every [[simple ring]] that is left or right artinian is [[isomorphism|isomorphic]] to a matrix algebra over a division algebra.
\end{corollary}
There is also a version of the Wedderburn--Artin theorem for [[associative algebras]] over [[fields]]:
\begin{proposition}
**(Wedderburn--Artin Theorem for Algebras over Fields)**\linebreak
Every [[semisimple algebra]] over a [[field]] $k$ is [[isomorphism|isomorphic]] to a finite [[direct sum]] of [[matrix algebras]] over [[division algebras]] over $k$. A semisimple algebra over $k$ is simple if and only if it is a matrix algebra over a division algebra over $k$.
\end{proposition}
There is also a version for [[endomorphism rigs]] in a [[semiadditive category]]:
\begin{proposition}
**(Wedderburn--Artin Theorem for Endomorphism rigs)**\linebreak
If $R$ is a [[semisimple object]] in a [[semiadditive category]] $\mathsf{A}$, then the [[endomorphism rig]] $End(R)$ is a [[finite product]] of matrix rigs over [[division rigs]].
\end{proposition}
## Proofs ##
There are many [[proofs]] of the Wedderburn--Artin theorem (Prop. \ref{TheTheorem}).
### Common proof
A common modern approach proceeds as follows.
Suppose $R$ is semisimple. Then by definition any right $R$-module is a direct sum of [[simple module|simple $R$-modules]], and any finitely generated $R$-module must be a finite direct sum of such. Thus, the right $R$-module $R_R$ is isomorphic to a finite direct sum of simple $R$-modules, which will be [[minimal ideal|minimal]] right [[ideals]] of $R$. Write this direct sum as
$$
R_R
\;\cong\;
\bigoplus_i I_i^{\oplus m_i}
\,,
$$
where the $I_i$ are mutually nonisomorphic [[simple module|simple]] right $R$-modules, the $i$th one appearing with multiplicity $m_i$. Then we have for the [[endomorphisms]]
$$
End(R_R)
\;\cong\;
\bigoplus_i End\big(I_i^{\oplus m_i}\big)
$$
and we can identify $End\big(I_i^{\oplus m_i}\big)$ with a ring of [[matrices]]
$$
End\big(I_i^{\oplus m_i}\big)
\;\cong\;
M_{m_i}\big(End(I_i)\big)
\,,
$$
where the [[endomorphism ring]] $End(I_i)$ of the right $R$-module $I_i$ is a division ring by [[Schur's lemma]] because $I_i$ is simple. Since $R \cong End(R_R)$ we conclude
$$
R \;\cong\;
\bigoplus_i M_{m_i}\big(End(I_i)\big)
\,.
$$
(We use right modules because $R \cong End(R_R)$; if we used left modules we would have $R \cong End({}_R R)^{op}$, but the proof would still go through.)
### Nicholson's proof
A different style of proof can be found in [Nicholson (1993)](#Nicholson93). A key step in this approach is "Brauer's Lemma".
\begin{lemma}\label{BrauerLemma}
**(Brauer's Lemma)** \linebreak
Suppose $R$ is a ring and $K \subseteq R$ is a [[minimal ideal|minimal left ideal]] with $K^2 \ne 0$. Then $K = R e$ for some $e \in K$ with $e^2 = e$, and $e R e$ is a [[division ring]].
\end{lemma}
Here a **[[minimal ideal|minimal left ideal]]** is a nonzero left ideal for which the only smaller left ideal is $\{0\}$.
For example, suppose $R$ is the ring of $n \times n$ matrices over some division ring $D$ This has a minimal ideal $K$ consisting of matrices with just one nonzero column, say the $j$th column. You can see $K^2 \ne 0$. To write $K = R e$, we can take $e$ to be the matrix that's zero everywhere except for a 1 in the $j$th row and $j$th column. Clearly $e^2 = e$, and $e R e$ consists of matrices that are zero except in the $j$th row and $j$th column, so $e R e$ is isomorphic to $D$.
No deep techniques or preliminary lemmas are required to prove Brauer's Lemma.
**Proof of Brauer's Lemma (\ref{BrauerLemma})**
Since $0 \ne K^2$, we must have $K u \ne 0$ for some $u \in K$. Of course $u \ne 0$. But $K u$ is a left ideal contained in $K$, so by minimality
$$ K u = K.$$
Thus $u = e u$ for some $e \in K$. Now, let
$$ L = \{a \in K \colon a u = 0 \} $$
$L$ is a left ideal since for any $r \in R$ we have
$$
a \in L
\; \implies \;
a u = 0
\; \implies \;
r a u = 0
\; \implies \;
r a \in L
\,.
$$
Note $L \subseteq K$ by definition, so by the minimality of $K$ we must have either $L = 0$ or $L = K$. But $e$ is in $K$ but not in $L$, since $e u = u \ne 0$, so we must have $L = 0$. In other words
$$
a \in K
\; \text{and} \;
a u = 0
\quad \implies \quad
a = 0
\,.
$$
To use this implication, recall that $u = e u$ so $e u = e^2 u$ so $(e - e^2)u = 0$, and take $a$ above to be $e - e^2$. We conclude that $e - e^2 = 0$, so $e$ is [[idempotent]]:
$$ e^2 = e \,.$$
Now we claim that $K = R e$. The reason is that $e$ is in the left ideal $K$, so $R e \subseteq K$. Since $K$ is minimal this implies either $R e = 0$ or $R e = K$. But $e \ne 0$ so $R e \ne 0$.
Why is $e R e$ a division ring? Its unit is $e$, of course. Suppose $b \in e R e$ is nonzero. To show $b$ has an inverse, note that $R b$ is a nonzero ideal contained in $R e = K$ so by minimality $R b = R e$. So, we must have $e = r b$ for some $r \in R$. But this implies $e r e$ is the left inverse of $b$ in $e R e$:
$$ (e r e) b = e r (e b) = e r b = e^2 = e.$$
Finally, in a ring where every nonzero element has a *left* inverse, every element has a two-sided inverse, so it's a division ring! To see this, say $b \ne 0$. It has a left inverse, say $x$ for short. To see that $x$ also the right inverse of $b$ note that $x$ must also have its own left inverse, say $y$. Thus we have $x b = 1$ and $y x = 1$, giving
$$ y = y (x b) = (y x) b = b .$$
Thus $b$ is the left inverse of $x$. So $x$ is the right inverse of $b$. ▮
## Related concepts
* [[central simple algebra]]
## References
* {#Nicholson93} William K. Nicholson, *A short proof of the Wedderburn--Artin theorem*, *New Zealand J. Math* **22** (1993) 83-86 [[pdf](https://www.thebookshelf.auckland.ac.nz/docs/NZJMaths/nzjmaths022/nzjmaths022-01-010.pdf)]
* Tsiu-Kwen Lee, *A short proof of the Wedderburn-Artin theorem*, Communications in Algebra **45** 7 (2017) [[doi:10.1080/00927872.2016.1233242](https://doi.org/10.1080/00927872.2016.1233242)]
* [[John Baez]], *The Wedderburn–Artin Theorem*, n-Category Café, June 14, 2013 ([web](https://golem.ph.utexas.edu/category/2023/06/the_wedderburnartin_theorem.html))
See also:
* Wikipedia, _[Wedderburn--Artin theorem](https://en.wikipedia.org/wiki/Wedderburn%E2%80%93Artin_theorem)_
[[!redirects Wedderburn's theorem]]
[[!redirects Artin-Wedderburn theorem]]
[[!redirects Wedderburn–Artin theorem]]
[[!redirects Wedderburn–Artin Theorem]] |
Wedderburn–Artin theorem > history | https://ncatlab.org/nlab/source/Wedderburn%E2%80%93Artin+theorem+%3E+history |
see at *[[Wedderburn-Artin theorem]]* |
wedge | https://ncatlab.org/nlab/source/wedge |
* [[wedge product]]
* [[wedge sum]]
* [[end#explicit_definition|wedge]] as in the definition of an end.
category: disambiguation
[[!redirects wedges]] |
wedge axiom | https://ncatlab.org/nlab/source/wedge+axiom |
For a [[generalized cohomology theory]]-functor $E^\bullet \colon Ho(Top^{\ast/})^{op} \longrightarrow Ab^{\mathbb{Z}}$, the _wedge axiom_ says that it takes small [[coproducts]] ([[wedge sums]]) to [[products]] ([[direct sums]]).
[[!redirects wedge axioms]] |
wedge sum | https://ncatlab.org/nlab/source/wedge+sum |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The __wedge sum__ $A \vee B$ of two [[pointed sets]] $A$ and $B$ is the [[quotient set]] of the [[disjoint union]] $A \uplus B$ where both copies of the basepoint (the one in $A$ and the one in $B$) are identified. The wedge sum $A \vee B$ can be identified with a [[subset]] of the [[cartesian product]] $A \times B$; if this subset is collapsed to a point, then the result is the [[smash product]] $A \wedge B$.
The wedge sum can be generalised to [[pointed objects]] in any category $C$ with [[pushouts]], and is the [[coproduct]] in the category of pointed objects in $C$ (which is the [[coslice category]] $*/C$). A very commonly used case is when $C=$[[Top]] is a category of [[topological spaces]].
In particular, if $C$ itself is a [[pointed category]], then every object is uniquely a pointed object, so that the coproduct in $C$ itself may be called a _wedge sum_. A commonly used case is when $C=$[[Spectra]] is a category of [[spectra]].
Also, the wedge sum also makes sense for any [[family]] of pointed objects, not just for two of them, as long as $C$ has pushouts of that size.
## Definition
+-- {: .num_defn }
###### Definition
For $\{x_i \colon * \to X_i\}_i$ a set of [[pointed objects]] in a [[category]] $\mathcal{C}$, their _wedge sum_ $\bigvee_i X_i$ is the [[pushout]] in $\mathcal{C}$
$$
\bigvee_i X_i \coloneqq (\coprod_i X_i) \coprod_{\coprod_{i} *} *
$$
in
$$
\array{
\coprod_{i} * &\stackrel{(x_i)}{\to}& \coprod_i X_i
\\
\downarrow && \downarrow
\\
* &\to& \bigvee_i X_i
}
\,,
$$
if this exists.
=--
Equivalently (see at [overcategory -- limits and colimits](overcategory#LimitsAndColimits)) this is just the [[coproduct]] in the [[undercategory]] $\mathcal{C}\backslash\ast$ of [[pointed objects]].
## Examples
* A wedge sum of pointed [[circles]] is also called a **bouquet** of circles. See for instance at _[[Nielsen-Schreier theorem]]_.
* For $X$ a [[CW complex]] with [[filtered topological space]] structure $X_0 \hookrightarrow \cdots \hookrightarrow \X_k \hookrightarrow X_{k+1} \hookrightarrow \cdots \hookrightarrow X$ the quotient topological spaces $X_{k+1}/X_k$ are wedge sums of $(k+1)$-spheres.
## Related concepts
* [[wedge sum type]]
* [[loop space of a wedge of circles]]
## References
Texbook accounts:
* {#Munkres75} [[James Munkres]], §71 of: _Topology_, Prentice Hall (1975, 2000) $[$[pdf](http://mathcenter.spb.ru/nikaan/2019/topology/4.pdf)$]$
* {#tomDieck2008} [[Tammo tom Dieck]], p. 31 of: _Algebraic topology_, European Mathematical Society, Zürich (2008) $[$[doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf)$]$
See also:
* Wikipedia *[Wedge sum](https://en.wikipedia.org/wiki/Wedge_sum)*
[[!redirects wedge sum]]
[[!redirects wedge sums]]
|
wedge sum type | https://ncatlab.org/nlab/source/wedge+sum+type | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _wedge sum type_ is an [[axiom|axiomatization]] of the [[wedge sum]] in the context of [[homotopy type theory]].
## Definition
The wedge sum of two pointed types $(A,a)$ and $(B,b)$ can be defined as the [[higher inductive type]] with the following constructors:
* Points come from the [[sum type]] $in : A + B \to A \vee B$
* And their base point is glued $path : inl(a) = inr(b)$
Clearly this is pointed.
The wedge sum of two types $A$ and $B$, can also be defined as the [[pushout type]] of the span
$$A \leftarrow \mathbf{1} \rightarrow B$$
where the maps pick the base points of $A$ and $B$. This pushout is denoted $A \vee B$ and has basepoint $\star_{A \vee B} \equiv \mathrm{inl}(\star_A)$
## See also
* [[higher inductive type]]
* [[wedge sum]]
## References ##
* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013) |
Wei Cui | https://ncatlab.org/nlab/source/Wei+Cui |
## Selected writings
On [[QFT with defects|defects]] in the [[KK-compactification]] of the [[D=6 N=(2,0) SCFT]] on [[4-manifolds]]:
* [[Jin Chen]], [[Wei Cui]], [[Babak Haghighat]], [[Yi-Nan Wang]], *SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds*, JHEP **2023** 208 (2023) [[arXiv:2305.09734](https://arxiv.org/abs/2305.09734), <a href="https://doi.org/10.1007/JHEP11(2023)208">doi:10.1007/JHEP11(2023)208</a>]
category: people
|
Wei-Liang Chow | https://ncatlab.org/nlab/source/Wei-Liang+Chow |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Wei-Liang_Chow)
* *Wei-Liang Chow*, Notices of the AMS (Oct 1996) [[pdf](https://www.ams.org/notices/199610/chow.pdf)]
## Selected writings
Early discussion of the [[braid group]]:
* [[Wei-Liang Chow]], *On the Algebraical Braid Group*, Annals of Mathematics Second Series, **49** 3 (1948) 654-658 [[doi:10.2307/1969050](https://doi.org/10.2307/1969050)]
Introducing *[[Chow's theorem]]*:
* [[Wei-Liang Chow]], *On compact complex analytic varieties*, American Journal of Mathematics **71** 4 (1949) 893-914 [[doi:10.2307/2372375](https://doi.org/10.2307/2372375)]
## Related entries
* [[Chow group]]
category: people |
Weierstrass elliptic function | https://ncatlab.org/nlab/source/Weierstrass+elliptic+function |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Weierstrass elliptic function_ $\wp$ is a doubly periodic [[meromorphic function]] on the [[complex numbers]] $\mathbb{C}$ (with the periods typically normalized to $1$ and $\tau$ satisfying $Im(\tau) \gt 0$, so that $\wp(z) = \wp(z + 1)$ and $\wp(z + \tau) = \wp(z)$) that exhibits an explicit parametrization of the form
$$(\wp, \wp'): \mathbb{C}/L \to C$$
where $C \subset \mathbb{P}^2(\mathbb{C})$ is the set of solutions to the cubic [[Weierstrass equation]], and $L \subset \mathbb{C}$ is the [[lattice]] $\langle 1, \tau \rangle$. In other words, we have a cubic relation of type
$$(\wp')^2 = 4\wp^3 + a\wp + b$$
for some constants $a, b$, providing an explicit parametrization of an [[elliptic curve]] (a nonsingular projective [[cubic curve]] $C$ considered over $\mathbb{C}$) by a [[complex manifold|complex]] [[torus]] $\mathbb{C}/L$.
See at _[[elliptic curve]]_ and at _[[Möbius transformation]]_ for more.
## Related concepts
* [[elliptic curve]]
* [[elliptic function]]
* [[pillowcase orbifold]]
## References
Named after [[Karl Weierstrass]].
Lecture notes:
* {#Mukase04} Motohico Mulase, Section 1.3 of: _Lectures on the combinatorial structure of the moduli spaces of Riemann surfaces_, 2004 ([[MulaseLecturesModuliRiemannSurfaces.pdf:file]])
See also
* Wikipedia, _[Weierstrass's elliptic functions](https://en.wikipedia.org/wiki/Weierstrass's_elliptic_functions)_
[[!redirects Weierstrass elliptic functions]]
[[!redirects Weierstrass elliptic functions]]
[[!redirects Weierstrass's elliptic functions]]
[[!redirects Weierstrass elliptic curve]]
[[!redirects Weierstrass elliptic curves]]
|
Weierstrass preparation theorem | https://ncatlab.org/nlab/source/Weierstrass+preparation+theorem |
#Contents#
* table of contents
{:toc}
##Overview##
We are interested in the local structure of zeros of [[analytic function]]s in $\mathbb{C}^n$ as well as in analogues, e.g. in [[rigid analytic geometry]].
In one variable, a holomorphic function $f$, locally holomorphic around $z_0$, can be represented as $f(z)=(z-z_0)^n u(z)$ where $u(z_0)\neq 0$, $u$ is holomorphic and $n$ is a [[natural number|nonnegative integer]]; therefore the solution set is discrete. In many variables, these zero sets are more complicated but far from arbitrary; in fact the analytic sets are often pretty close to [[algebraic varieties]]: for example, analytic subsets of the [[projective space]] are algebraic.
The Weierstrass preparation theorem and related facts (Weierstrass division theorem and Weierstrass formula) provide the most basic relations between [[polynomial]]s and holomorphic functions.
Let $n\geq 2$; then we separate the first $(n-1)$ complex coordinates $z = (z_1,\ldots,z_{n-1})$ and the $n$-th coordinate which will be denoted by $w$. We consider an analytic function $f = f(z_1,\ldots, z_{n-1},w)$ vanishing at origin $f(0,\ldots, 0)=0$, and such that it is not identically zero on the $w$-axis.
##Weierstrass polynomial##
The __Weierstrass polynomial__ of $w$ is a polynomial of the form
$$
w^d + a_1(z) w^{d-1}+\ldots+a_d(z),\,\,\,\,\,a_i(0)=0.
$$
The integer $d$ is called *the degree of the Weierstrass polynomial*.
## Weierstrass preparation theorem in $\mathbb{C}^n$##
Let $f$ be a function which is holomorphic in some neighborhood of origin $0\in\mathbb{C}^n$ and not identically equal to zero on the $w$-axis. Then there is a neighborhood of origin such that $f$ is *uniquely* representable in the form
$$
f = P\cdot h
$$
where $P$ is a Weierstrass polynomial of degree $d$ of $w$ and $h(0) \neq 0$.
##Weierstrass division theorem##
Let $\mathcal{O}_{n,a}$ be the [[local ring]] of [[germs]] of holomorphic functions at $a\in\mathbb{C}^n$ and $\mathcal{O}_n:=\mathcal{O}_{n,0}$. Let $g=g(z,w)\in\mathcal{O}_{n-1}[w]$ be a Weierstrass polynomial of degree $k$ of $w$. Then every holomorphic function $f\in\mathcal{O}_n$ can be represented as
$$
f = g\cdot h+r
$$
where $r = r(z,w)$ is a polynomial of degree $\lt k$.
As a corollary, if another function $h$ vanishes on the zero set of $f$, then $f$ divides $h$ in $\mathcal{O}_n$.
##Ingredients of proofs##
Weierstrass's original result considered the ring of holomorphic functions, and therefore used analytic methods such as the [[residue theorem]] and [[Cauchy integral formula]] are used. Analogous results are now known in the ring of formal power series, as well as for power series over non-Archimedean fields; in these settings, the proof can be made fully algebraic.
## References
Named after [[Karl Weierstraß]].
* Wikipedia, _[Weierstrass preparation theorem](https://en.wikipedia.org/wiki/Weierstrass_preparation_theorem)_
category: analysis, geometry |
Weierstrass sigma-function | https://ncatlab.org/nlab/source/Weierstrass+sigma-function |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[modular form]].
## Properties
### Relation to Eisenstein series
Let $G_{2k}$ be the [[Eisenstein series]], then
$$
\frac{x}{e^{x/2} - e^{-x/2}}
\prod_{n\geq 1}
\frac{(1-q^n)^2}{(1-q^n e^x)(1-q^n e^{-x})}
=
\exp\left(
\sum_{k \geq 2} 2 G_k \frac{x^k}{k!}
\right)
$$
([Ando-Hopkins-Rezk 10, prop. 10.9](#AndoHopkinsRezk10))
## Related concepts
* The Weierstrass $\sigma$-function is proportional to the (inverse of the) characteristic series of the [[Witten genus]] ([Ando-Basterra 00, section 5.1](#AndoBasterra00))
## References
Named after [[Karl Weierstrass]].
An introductory review is in
* {#Hain08} Richard Hain, section 5.1 of _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))
A textbook account includes for instance
* Joseph Silverman, _The arithmetic of elliptic curves_, volume 106 of Graduate Texts in Mathematics. Springer, 1986
Relation to the [[Witten genus]] is discussed for instance in
* {#AndoBasterra00} [[Matthew Ando]], Maria Basterra, _The Witten genus and equivariant elliptic cohomology_ ([arXiv:0008192](http://arxiv.org/abs/math/0008192))
* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))
|
weight | https://ncatlab.org/nlab/source/weight |
* [[weight (in representation theory)]]
* [[weighted limit]]
* [[weighted colimit]]
[[!redirects weights]] |
weight (in representation theory) | https://ncatlab.org/nlab/source/weight+%28in+representation+theory%29 |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Let $G$ be a [[Lie group]] which is [[compact Lie group|compact]] and [[connected topological space|connected]]. Write $T \hookrightarrow G$ for the [[maximal torus]] [[subgroup]].
+-- {: .num_defn }
###### Definition
A **weight** on $G$ is an [[irreducible representation]] of the [[maximal torus]] [[subgroup]] $T \hookrightarrow G$.
=--
+-- {: .num_defn }
###### Definition
For $\rho : G \to Aut(V)$ a [[representation]] of $G$, and for $\alpha : T \to Aut(\mathbb{C})$ a weight, the **weight space** of $\rho$ with respect to $\alpha$ is the [[subspace]] of $V$ which as a representation of $T$ is a [[direct sum]] of $\alpha$-s.
=--
+-- {: .num_remark }
###### Remark
In other words, the weight space of a $G$-representation for a weight $\alpha$ is the corresponding [[eigenspace]] under the action of $T$.
=--
## Properties
### For connected compact Lie groups
For connected [[compact Lie groups]] the
## Related concepts
* [[group character]]
* [[root (in representation theory)]]
* [[maximal torus]], [[Cartan subalgebra]]
## References
* [[Howard Georgi]], §6 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]
> with an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]
* Peter Woit, _Topics in representation theory: Roots and weights_ ([pdf](http://www.math.columbia.edu/~woit/notes6.pdf))
* Wikipedia, _[Weight (representation theory)](http://en.wikipedia.org/wiki/Weight_%28representation_theory%29)_
[[!redirects weight lattice]]
[[!redirects weight lattices]]
|
weight filtration | https://ncatlab.org/nlab/source/weight+filtration |
A _[[mixed Hodge structure]]_ consists of a [[filtration]] of [[cohomology groups]] which in addition to the [[Hodge filtration]] has a filtration by "weight" (...).
* [[Mikhail V. Bondarko]], _Weights for relative motives; relations with mixed complexes of sheaves_, [arXiv:1007.4543](http://arxiv.org/abs/1007.4543).
[[!redirects weight filtrations]]
|
weight system | https://ncatlab.org/nlab/source/weight+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[knot theory]] a _framed weight system_ is an assignment of [[numbers]] to [[chord diagrams]] that is [[invariant]] under the [[4T-relation]].
If the assignment is in addition invariant under the [[1T-relation]], then it is called an _unframed weight system_, or just _weight system_, for short.
Similarly, in [[braid group]]-theory, a _horizontal weight system_ is an assignment of [[numbers]] to [[horizontal chord diagrams]] that is invariant under the horizontal [[2T relation]] and horizontal [[4T relation]].
## Definition
### Round weight systems
+-- {: .num_defn #LinearSpanOfChordDiagramsModulo4T}
###### Definition
**([[linear span]] of [[chord diagrams]] [[quotient vector space|modulo]] [[4T-relations|4T]])**
Let $k$ be a [[field]] (or just a [[commutative ring]]). Write
1. $\mathcal{D}^c$ for the [[set]] of [[chord diagrams]],
1. $k\langle \mathcal{D}^c\rangle$ for its $k$-[[linear span]],
1. $\mathcal{A}^c \;\coloneqq\; k\langle \mathcal{D}^c\rangle/4T$ for the [[quotient vector space]] by the [[4T-relations]].
=--
+-- {: .num_defn #FramedWeightSystems}
###### Definition
**([[framed weight system]])**
A _$k$-valued framed weight system_ is a [[linear function]] on the [[linear span]] of [[chord diagrams]] [[quotient vector space|modulo]] [[4T-relations]] (Def. \ref{LinearSpanOfChordDiagramsModulo4T})
$$
w
\;\colon\;
\mathcal{A}^c \longrightarrow k
\,,
$$
hence the $k$-[[vector space]] of framed weight systems is the [[dual vector space]]
\[
\mathcal{W}
\;\coloneqq\;
(\mathcal{A}^c)^\ast
\]
=--
([Bar-Natan 95, Def. 1.6](#BarNatan95), see [Chmutov-Duzhin-Mostovoy 11, Def. 4.1.1](#ChmutovDuzhinMostovoy11), [Jackson-Moffat 19, Section 11.7](#JacksonMoffat19))
+-- {: .num_remark}
###### Remark
Since [[chord diagrams modulo 4T are Jacobi diagrams modulo STU]], framed weight systems equivalently form the [[dual vector space]]
\[
\label{SpaceOfWeightSystemsAsLinearDual}
\begin{aligned}
\label{FramedWeightSystemsAsDualOfChordAndJacobiDiagrams}
\overset{
\mathclap{
\color{blue}
{
{framed}
\atop
{{weight}\atop{systems}}
}
}
}{
\mathcal{W}
}
&
\;\coloneqq\;
\big(
\overset
{
\color{blue}
{
{
{chord}
\atop
{diagrams}
}
\atop
{
{modulo}\,{4T}
}
}
}
{
\mathcal{A}^c
}
\big)^\ast
\\
&
\;\simeq\;
\big(
\underset{
\color{blue}
{
{
{Jacobi}
\atop
{diagrams}
}
\atop
{
{modulo}\,{STU}
}
}
}{
\mathcal{A}^t
}
\big)^\ast
\end{aligned}
\]
of the [[quotient vector space]] $\mathcal{A}^t \coloneqq k\langle \mathcal{D}^t \rangle/STU$ of the [[linear span]] of [[Jacobi diagrams]] by the [[STU-relation]].
=--
\linebreak
### Horizontal weight systems
+-- {: .num_defn #LinearSpanOfHorizontalChordDiagramsModulo2TAnd4T}
###### Definition
**([[linear span]] of [[horizontal chord diagrams]] [[quotient vector space|modulo]] [[2T relation|2T]]/[[4T-relations|4T]])**
Let $k$ be a [[field]] (or just a [[commutative ring]]). Write
1. $\mathcal{D}^{pb}$ for the [[set]] of [[horizontal chord diagrams]],
1. $k\langle \mathcal{D}^{pb}\rangle$ for its $k$-[[linear span]],
1. $\mathcal{A}^{pb} \;\coloneqq\; k\langle \mathcal{D}^c\rangle/(2T,4T)$ for the [[quotient vector space]] by the [[2T relations]] and [[horizontal 4T-relations]].
=--
(The superscript "${}^{pb}$"in Def. \ref{LinearSpanOfHorizontalChordDiagramsModulo2TAnd4T} is for _[[pure braids]]_, alluding to the fact that [[horizontal weight systems are associated graded of Vassiliev braid invaraints]].)
+-- {: .num_defn #HorizontalWeightSystems}
###### Definition
**([[horizontal weight system]])**
A _$k$-valued horizontal weight system_ is a [[linear function]] on the [[linear span]] of [[horizontal chord diagrams]] [[quotient vector space|modulo]] [[2T relation|2T]]&[[4T-relations]] (Def. \ref{LinearSpanOfHorizontalChordDiagramsModulo2TAnd4T})
$$
w
\;\colon\;
\mathcal{A}^{pb} \longrightarrow k
\,,
$$
hence the $k$-[[vector space]] of horizontal weight systems is the [[dual vector space]]
\[
\mathcal{W}_{pb}
\;\coloneqq\;
(\mathcal{A}^{pb})^\ast
\]
=--
## Constructions of weight systems
### Lie algebra weight systems
{#LieAlgebraWeightSystems}
A large class of [[weight systems]] arises from reading a ([[horizontal chord diagram|horizontal]]) [[chord diagram]] as a [[string diagram]] in the evident way, and then labelling it by the structure morphisms of a [[Lie algebra object]] equipped with a [[Lie algebra representation]] [[internalization|internal to]] a suitable [[tensor category]]. This does yield weight systems because the required relations translate exactly to the structural equations satisfied by [[Lie modules]] ([[Jacobi identity]] and Lie action property).
The weight systems arising this way are called _[[Lie algebra weight systems]]_. See there for more.
Examples of [[weight systems]] which are _not_ [[Lie algebra weight systems]] are rare. Originally it was conjectured that none exist ([Bar-Natan 95, Conjecture 1](#BarNatan95), [Bar-Natan & Stoimenow 97, Conjecture 2.4](Lie+algebra+weight+system#BarNatanStoimenow97)).
Eventually, a (counter-)example of a weight system which at least does not arise from any [[finite dimensional vector space|finite-dimensional]] [[super Lie algebra]] was given in [Vogel 11](Lie+algebra+weight+system#Vogel11).
### Stringy weight systems
[[stringy weight systems]]
### Rozansky-Witten weight systems
[[Rozansky-Witten weight systems]]
## Properties
### As the associated graded space of Vassiliev invariants
+-- {: .num_prop #WeightSystemsAreAssociatedGradedOfVassilievInvariants}
###### Proposition
**([[weight systems are associated graded of Vassiliev invariants]])**
For [[ground field]] $k = \mathbb{R}, \mathbb{C}$ the [[real numbers]] or [[complex numbers]], there is for each [[natural number]] $n \in \mathbb{N}$ a canonical [[linear isomorphism]]
$$
\mathcal{V}_n/\mathcal{V}_{n-1}
\underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow}
\big(
\mathcal{A}_n^u
\big)^\ast
$$
from
1. the [[quotient vector space]] of order-$n$ [[Vassiliev invariants]] of [[knots]] by those of order $n-1$
1. to the space of [[unframed weight systems]] of order $n$.
In other words, in [[characteristic zero]], the [[graded vector space]] of [[unframed weight systems]] is the [[associated graded vector space]] of the [[filtered vector space]] of [[Vassiliev invariants]].
=--
([Bar-Natan 95, Theorem 1](#BarNatan95), following [Kontsevich 93](#Kontsevich93))
### As cohomology of loop spaces of configuration spaces
{#AsCohomologyOfLoopSpacesOfConfigurationSpaces}
[[weight systems are cohomology of loop space of configuration space]]:
#### For horizontal chord diagrams
+-- {: .num_prop}
###### Proposition
**(integral [[horizontal weight systems]] are [[integral cohomology]] of [[based loop space]] of [[ordered configuration space of points]] in [[Euclidean space]])**
For [[ground ring]] $R = \mathbb{Z}$ the [[integers]], there is, for each [[natural number]] $n$, a canonical [[isomorphism]] of [[graded abelian groups]] between
1. the integral [[weight systems]]
$$
\mathcal{W}_{pb}
\;\coloneqq\;
Hom_{\mathbb{Z} Mod}
\big(
\underset{
\mathcal{A}^{pb}
}{
\underbrace{
\mathbb{Z}
\langle
\mathcal{D}^{pb}
\rangle
/(2T,4T)
}
}
,
\mathbb{Z}
\big)
$$
on [[horizontal chord diagrams]] of $n$ strands (elements of the set $\mathcal{D}^{pb}$)
1. the [[integral cohomology]] of the [[based loop space]] of the [[ordered configuration space of n points]] in 3d [[Euclidean space]]:
$$
H \mathbb{Z}^\bullet
\big(
\underset{n \in \mathbb{N}}{\sqcup}
\Omega
\underset{
{}^{\{1,\cdots,n\}}
}{Conf}
(\mathbb{R}^3)
\big)
\;\simeq\;
(\mathcal{W}_{pb})^\bullet
\;\simeq\;
Gr^\bullet( \mathcal{V}_{pb})
\,.
$$
(the second equivalence on the right is the fact that [[weight systems are associated graded of Vassiliev invariants]]).
=--
This is stated as [Kohno 02, Theorem 4.1](#Kohno02)
#### For round chord diagrams
+-- {: .num_prop}
###### Proposition
**([[weight systems]] are inside [[real cohomology]] of [[based loop space]] of [[ordered configuration space of points]] in [[Euclidean space]])**
For [[ground field]] $k = \mathbb{R}$ the [[real numbers]],
there is a canonical [[injection]] of the [[real vector space]] $\mathcal{W}$ of [[framed weight systems]] ([here](weight+system#eq:SpaceOfWeightSystemsAsLinearDual)) into the [[real cohomology]] of the [[based loop spaces]] of the ordered [[configuration spaces of points]] in 3-[[dimension|dimensional]] [[Euclidean space]]:
$$
\mathcal{W}
\;\overset{\;\;\;\;}{\hookrightarrow}\;
H\mathbb{R}^\bullet
\Big(
\underset{n \in \mathbb{N}}{\sqcup}
\Omega
\underset{{}^{\{1,\cdots,n\}}}{Conf}
\big(
\mathbb{R}^3
\big)
\Big)
$$
=--
This is stated as [Kohno 02, Theorem 4.2](#Kohno02)
\linebreak
Combining the above two propositions
1. [[weight systems are associated graded of Vassiliev invariants]],
1. [[weight systems are cohomology of loop space of configuration space]]
we get this situation:
<img src="https://ncatlab.org/nlab/files/VassilievInvInCohOfLoopSpacesOfConfigSpaces.jpg" width="600">
### As cohomology of the knot graph complex
[[cohomology of knot graph complex is weight systems on chord diagrams]]
### All horizontal weight systems are partitioned Lie algebra weight systems
* [[all horizontal weight systems are partitioned Lie algebra weight systems]]
## Related concepts
[[!include chord diagrams and weight systems -- table]]
* [[Kontsevich integral]]
## References
### General
Original articles:
* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))
Textbook accounts:
* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 4 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))
* {#JacksonMoffat19} [[David Jackson]], [[Iain Moffat]], Section 11.7 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))
### Lie algebra weight systems
Discussion of [[Lie algebra weight systems]]
From the construction given in [Bar-Natan 95, Section 2.4](#BarNatan95) the interpretation of [[Lie algebra weight systems]] in terms of [[string diagrams]] for [[Lie algebra objects]] in [[tensor categories]] is evident, but standard textbooks in [[knot theory]]/[[combinatorics]] do not pick this up:
* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Chapter 6 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))
* [[David Jackson]], [[Iain Moffat]], Section 14 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))
The interpretation of Lie algebra weight systems as [[string diagram]]-calculus and generalization to [[Lie algebra objects]] (motivated by generalization at least to [[super Lie algebras]]) is made more explicit in
* {#Vaintrob94} [[Arkady Vaintrob]], _Vassiliev knot invariants and Lie S-algebras_, Mathematical Research Letters1, 579–595 (1994) ([pdf](https://pdfs.semanticscholar.org/bdc3/ac1d8da476245e2408e481a70b115b3e9aab.pdf))
* {#Vogel11} [[Pierre Vogel]], _Algebraic structures on modules of diagrams_, Journal of Pure and Applied Algebra, Volume 215, Issue 6, June 2011, Pages 1292-1339 ([doi:10.1016/j.jpaa.2010.08.013](https://doi.org/10.1016/j.jpaa.2010.08.013), [pdf](https://webusers.imj-prg.fr/~pierre.vogel/diagrams.pdf))
and fully explicit in
* {#RobertsWillerton06} [[Justin Roberts]], [[Simon Willerton]], Section 3 of: _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))
See also
* [[Vladimir Hinich]], [[Arkady Vaintrob]], _Cyclic operads and algebra of chord diagrams_, Sel. math., New ser. (2002) 8: 237 ([arXiv:math/0005197](https://arxiv.org/abs/math/0005197))
* E. Kulakova, S. Lando, T. Mukhutdinova, G. Rybnikov, _On a weight system conjecturally related to $\mathfrak{sl}_2$_, European Journal of Combinatorics Volume 41, October 2014, Pages 266-277 ([arXiv:1307.4933](https://arxiv.org/abs/1307.4933))
* Alexander Schrijver, _On Lie algebra weight systems for 3-graphs_ ([arXiv:1412.6923](https://arxiv.org/abs/1412.6923))
### As the associated graded space of Vassiliev invariants
On spaces of [[weight systems]] as the [[associated graded spaces]] of [[Vassiliev invariants]]:
* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))
* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
### As cohomology of loop spaces of configuration spaces
On [[weight systems]] as the [[real cohomology]] of [[based loop spaces]] of ordered [[configuration spaces of points]]:
* {#Kohno94} [[Toshitake Kohno]], _Vassiliev invariants and de Rham complex on the space of knots_,
In: Yoshiaki Maeda, Hideki Omori and [[Alan Weinstein]] (eds.), _Symplectic Geometry and Quantization_, Contemporary Mathematics 179 (1994): 123-123 ([doi:10.1090/conm/179](http://dx.doi.org/10.1090/conm/179))
* {#CohenGitler01} [[Fred Cohen]], [[Samuel Gitler]], _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]] (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))
* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))
* [[Fred Cohen]], [[Samuel Gitler]], _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705–1748, ([jstor:2693715](https://www.jstor.org/stable/2693715), [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013))
[[!include weight systems on chord diagrams in physics]]
[[!redirects weight systems]]
[[!redirects framed weight system]]
[[!redirects framed weight systems]]
[[!redirects unframed weight system]]
[[!redirects unframed weight systems]]
[[!redirects un-framed weight system]]
[[!redirects un-framed weight systems]]
[[!redirects horizontal weight system]]
[[!redirects horizontal weight systems]]
|
weight systems are cohomology of knot graph complex | https://ncatlab.org/nlab/source/weight+systems+are+cohomology+of+knot+graph+complex |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
The [[cochain cohomology]] of the framed [[knot graph complex]] (sometimes called the "Wilson graph complex") spanned by trivalent graphs coincides with the space of framed [[weight systems]] on [[Jacobi diagrams]], [[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] on [[round chord diagrams]]:
$$
\underset{
\phantom{a}
\atop
{
{\color{blue}framed\;weight\;systems\;on}
\atop
{\color{blue}round\;chord\;diagrams}
}
}{
\mathcal{W}^\bullet
}
\;\simeq\;
\underset{
\phantom{a}
\atop
{
{\color{blue}\weight\;systems\;on}
\atop
{\color{blue}Jacobi\;diagrams}
}
}{
\mathcal{W}^\bullet
}
\underoverset
{
\;\;\;\simeq\;\;\;
}
{
(w,\mathcal{K})
\mapsto
\left\langle
Tr_{{}_{w}}
\text{P}\exp
\left(
\int_{\mathcal{K}} A
\right)
\right\rangle
}
{
\longrightarrow
}
\underset{
\mathclap{
{\phantom{a}}
\atop
{
{\color{blue}cohomology\;of\;knot\;graph\;complex}
\atop
{\color{blue}spanned\;by\;trivalent\;graphs}
}
}
}{
H^{\bullet}
\big(
\mathcal{G}
\big)_3
}
$$
This statement is made explicit as [CCRL 02, Prop. 7.6](#CattaneoCottaRamusinoLongoni02), where it is noticed that this is implicit in statement and proof of [AF 96, Theorem 1](#AF96) (where in turn the argument is attributed to [Kohno 94](#Kohno94)!)
Moreover:
If $w \in \mathcal{W}$ is a [[weight system]] and $D \in \mathcal{D}$ is a [[Jacobi diagram]] such that $w(D) \neq 0$, then its image $\left\langle Tr_{{}_{w}} \text{P}\exp \left( \int_{(-)} A \right) \right\rangle$ under the above isomorphism contains a non-vanishing multiple of $D$ as a summand.
This is made explicit as [CCRL 02, Remark 7.7](#CattaneoCottaRamusinoLongoni02) and again this is implicit in the statement of [AF 96, Theorem 1](#AF96).
What [AF 96](#AF96) explicitly construct is a [[universal Vassiliev invariant]], which they identify with the un-traced [[Wilson loop observable]]
$$
\mathcal{K}
\mapsto
\left\langle
Tr_{{}_{(w)}}
\text{P}\exp
\left(
\int_{\mathcal{K}} A
\right)
\right\rangle
$$
of [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons theory]].
\linebreak
## Ingredients
{#Ingredients}
Write
\[
\label{CSFeynmanDiagrams}
\mathcal{D}^{cs} \;\in\; Set^{\mathbb{N}}
\]
for the [[graded set]] of [[isomorphism classes]] of trivalent framed knot graphs -- [[Feynman diagrams]] for [[Chern-Simons theory]] in the presence of [[Wilson loops]], called "Wilson graphs" [AF 96, Section 1](#AF96), slightly differing from the un-framed knot graphs in [CCRL 02](#CattaneoCottaRamusinoLongoni02).
By definition, a [[graph]] $\Gamma \in \mathcal{D}^{cs}$ must have [[even number]] of [[vertices]], and its degree is half that number ([AF 96, (2.9)](#AF96))
$$
\Gamma \;\in\; \mathcal{D}^{cs}_{(\# Vert_\Gamma)/2}
\,.
$$
For any $\Gamma \in \mathcal{D}^{cs}$ write
$$
Aut(\Gamma)
\;\in\;
Grp
$$
for its [[automorphism group]], a [[finite group]] whose [[order of a group|order]] we denote by
\[
\label{AutomorphismGroup}
\left\vert Aut(\Gamma)\right\vert
\;\in\;
\mathbb{N}
\,.
\]
Write
\[
\label{KnotGraphComplex}
\mathcal{G}^\bullet \;\in\; Ch^\bullet(\mathbb{R})
\]
for the framed [[knot graph complex]] and
$$
H^\bullet(\mathcal{G})_3
\subset
H^\bullet(\mathcal{G})
$$
for the sub-vector space of its cohomology spanned by cocycles made of trivalent graphs.
Write
\[
\label{JacobiDiagramsModuloSTU}
\mathcal{A}_\bullet
\;\coloneqq\;
Span
\left(
\mathcal{D}^t
\right)/(STU)
\;
\in
\;
Vect_\bullet(\mathbb{R})
\]
for the [[graded vector space]] of [[Jacobi diagrams]] [[quotient vector space|modulo]] the [[STU-relations]].
Also write
\[
\label{LinearizationMaps}
\array{
&& \mathcal{D}^{cs}
\\
&
{}^{\mathllap{ [-]_{{}_{\mathcal{A}}} }}
\swarrow
&&
\searrow^{\mathrlap{ [-]_{\mathcal{G}} }}
\\
\mathcal{A}_\bullet
&&
&&
\mathcal{G}^\bullet
}
\]
for the [[functions]] that send a [[graph]] to the defining [[linear basis|basis]] [[vector]] that it represents in these [[vector spaces]], respectively.
The space $\mathcal{A}_\bullet$ is the graded [[linear dual]] of the space of [[weight systems]]
$$
\mathcal{W}^\bullet
\;\coloneqq\;
(\mathcal{A}_\bullet)^\ast
\,.
$$
Hence if we regard
$$
\mathcal{A}_\bullet
\;=\;
(\mathcal{A}^{-\bullet}, d= 0)
$$
as a [[cochain complex]] in non-positive degree with vanishing [[differential]], then its [[tensor product of cochain complexes]] with the knot graph complex is the [[cochain complex]] whose closed elements are the graded [[linear maps]] from $\mathcal{W}^\bullet$ to the [[cochain cohomology]] $H^\bullet(GraphCplx)$ of the [[knot graph complex]]:
\[
\label{TensorCochainsAsMaps}
H^0
\big(
\mathcal{A}_\bullet
\otimes
\mathcal{G}^\bullet
\big)
\;\simeq\;
Hom
\big(
\mathcal{W}^\bullet
,
H^\bullet(\mathcal{G})
\big)
\]
## Statement
{#Statement}
+-- {: .num_prop}
###### Proposition
The element
\[
\label{WisonLoopObservable}
\left\langle
Tr_{(-)}
\text{P}\exp
\left(
\int_{(-)} A
\right)
\right\rangle
\;\coloneqq\;
\underset{
n \in \mathbb{N}
}{\sum}
\hbar^n
\underset{
\Gamma \in \mathcal{D}^{cs}_n
}{\sum}
\left(
\frac{1}{\left\vert \Gamma\right\vert}
\,
[\Gamma]_{\mathcal{A}} \otimes [\Gamma]_{\mathcal{G}}
\right)
\;
\in
\;\;
\mathcal{A}_\bullet \otimes \mathcal{G}^\bullet
\]
(hence the [[sum]] over [[Feynman diagrams]] (eq:CSFeynmanDiagrams) of the [[tensor product of chain complexes|tensor product]] of their [[images]] (eq:LinearizationMaps) in [[Jacobi diagrams]] [[quotient vector space|modulo]] the [[STU-relations]] (eq:JacobiDiagramsModuloSTU) and in the [[knot graph complex]] (eq:KnotGraphComplex), respectively, weighted by the inverse [[order of a group|order]] of their [[automorphism group]] (eq:AutomorphismGroup) )
is closed
$$
d_{\mathcal{G}}
\left\langle
Tr_{(-)}
\text{P}\exp
\left(
\int_{(-)} A
\right)
\right\rangle
\;=\;
0
$$
=--
This is [AF 96, Theorem 1](#AF96).
Hence (eq:WisonLoopObservable) defines a [[cochain cohomology]]-[[cohomology class|class]]
$$
\begin{aligned}
\left\langle
Tr_{(-)}
\text{P}\exp
\left(
\int_{(-)} A
\right)
\right\rangle
\;\coloneqq\;
H^0
\big(
\mathcal{A}_\bullet \otimes \mathcal{G}^\bullet
\big)
\\
&
\;\simeq\;
Hom
\big(
\mathcal{W}^\bullet
,
H^\bullet(\mathcal{G})
\big)
\end{aligned}
$$
and hence, by (eq:TensorCochainsAsMaps), it defines a graded [[linear function]]
$$
\mathcal{W}^\bullet
\overset{
\left\langle
Tr_{(-)}
\text{P}\exp
\left(
\int_{(-)} A
\right)
\right\rangle
}{
\longrightarrow }
H^\bullet(\mathcal{G})_3
\hookrightarrow
H^\bullet(\mathcal{G})
$$
from [[weight systems]] on [[Jacobi diagrams]] ([[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] on [[round chord diagrams]]) to the [[cochain cohomology]] of the framed [[knot graph complex]] spanned by trivalent graphs.
According to [CCRL 02, Prop. 7.6](#CattaneoCottaRamusinoLongoni02) this map is a [[bijection]].
To see this, use 1) [AF 96, Theorem 5, Condition U2](#AF96) to find that the map is an [[injection]], and 2) the fact that [[weight systems are associated graded of Vassiliev invariants]].
In **summary** we have the following situation:
<center>
<img src="https://ncatlab.org/nlab/files/UniversalWilsonLoopObservable.jpg" width="700">
</center>
<center>
<img src="https://ncatlab.org/nlab/files/TheGrandStoryOfVassilievKnotInvariantsII.jpg" width="800">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
## Related theorems
[[!include facts about chord diagrams and weight systems -- contents]]
## References
* {#Kohno94} [[Toshitake Kohno]], _Vassiliev invariants and de Rham complex on the space of knots_,
In: Yoshiaki Maeda, Hideki Omori, [[Alan Weinstein]] (eds.), _Symplectic Geometry and Quantization_, Contemporary Mathematics 179 (1994): 123-123 ([doi:10.1090/conm/179](http://dx.doi.org/10.1090/conm/179))
* {#AF96} Daniel Altschuler, Laurent Freidel, _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. 187 (1997) 261-287 ([arXiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010))
* {#CattaneoCottaRamusinoLongoni02} [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Configuration spaces and Vassiliev classes in any dimension_, Algebr. Geom. Topol. 2 (2002) 949-1000 ([arXiv:math/9910139](https://arxiv.org/abs/math/9910139))
Computation of the perturbative [[Wilson loop observable]] ([[universal Vassiliev invariant]]) of the [[unknot]] ("[[Wheels theorem]]"):
* [[Dror Bar-Natan]], Thang T Q Le, [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))
following
* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))
[[!redirects cohomology of knot graph complex is weight systems on chord diagrams]]
|
weight systems are cohomology of loop space of configuration space | https://ncatlab.org/nlab/source/weight+systems+are+cohomology+of+loop+space+of+configuration+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Statement
### For horizontal chord diagrams
+-- {: .num_remark #HomologyOfBasedLoopSpaceIsHopfAlgebra}
###### Remark
**([[Hopf algebra]]-[[mathematical structure|structures]])**
Both the [[ordinary homology]] of a [[based loop space]] as well as the [[universal enveloping algebra]] of a [[Lie algebra]] are canonically [[Hopf algebras]], the former via the [[Pontrjagin ring]]-[[mathematical structure|structure]] (see at _[[homology of loop spaces]]_).
=--
+-- {: .num_prop #HomologyOfLoopSpaceOfConfigurationSpaceIsUniversalEnvelopingOfInfinitesimalBraids}
###### Proposition
**([[ordinary homology]] of [[based loop space]] of [[ordered configuration space of points]] is [[universal enveloping algebra]] of [[infinitesimal braid Lie algebra]])**
For $D, n \in \mathbb{N}$ [[natural numbers]] and for any [[ground field]] $\mathbb{F}$ (in fact over every [[commutative ring]]) the [[ordinary homology]] of the [[based loop space]] of the [[ordered configuration space of points]] in the [[Cartesian space]]/[[Euclidean space]] $\mathbb{R}^D$ is [[isomorphism|isomorphic]], as a [[Hopf algebra]] (Remark \ref{HomologyOfBasedLoopSpaceIsHopfAlgebra}), to the [[universal enveloping algebra]] of the [[infinitesimal braid Lie algebra]]:
$$
H_\bullet
\big(
\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D)
\big)
\;\simeq\;
\mathcal{U}
\big(
\mathcal{L}_n(D)
\big)
\,.
$$
=--
This is due to [Fadell-Husseini 01, Theorem 2.2](#FadellHusseini01), re-stated as [Cohen-Gitler 01, Theorem 4.1](#CohenGitler01), [Cohen-Gitler 02, Theorem 2.3](#CohenGitler02).
Notice also that:
+-- {: .num_prop #UniversalEnvelopingOfInfinitesimalBraidsIsHorizontalChords}
###### Proposition
**([[universal enveloping of infinitesimal braids is horizontal chord diagrams]])
The [[associative algebra]]
$$
\Big(
\mathcal{A}_n^{pb}
\;\coloneqq\;
Span
\big(
\mathcal{D}_n^{pb}
\big)/(2T, 4T)
,
\circ
\Big)
$$
of [[horizontal chord diagrams]] on $n$ strands with product given by [[concatenation]] of strands ([this Def.](horizontal+chord+diagram#AlgebraHorizontalChordDiagrams)), [[quotient vector space|modulo]] the [[2T relations]] and [[4T relations]] ([this Def.](horizontal+chord+diagram#2TAnd4TRelations)) is [[isomorphism|isomorphic]] to the [[universal enveloping algebra]] of the [[infinitesimal braid Lie algebra]] (Def. \ref{InfinitesimalBraidLieAlgebra}):
$$
\big(\mathcal{A}_n^{pb}, \circ\big)
\;\simeq\;
\mathcal{U}(\mathcal{L}_n(D))
\,.
$$
=--
The combination of Prop. \ref{HomologyOfLoopSpaceOfConfigurationSpaceIsUniversalEnvelopingOfInfinitesimalBraids} and Prop. \ref{UniversalEnvelopingOfInfinitesimalBraidsIsHorizontalChords} yields:
+-- {: .num_cor #HomologyOfLoopSpaceOfConfigurationSpaceIsAlgebraOfHorizontalChordDiagrams}
###### Corollary
For $D, n \in \mathbb{N}$ and for any [[ground field]] $\mathbb{F}$ (in fact over every [[commutative ring]]) the [[ordinary homology]] of the [[based loop space]] of the [[ordered configuration space of points]] in the [[Cartesian space]]/[[Euclidean space]] $\mathbb{R}^D$ is [[isomorphism|isomorphic]], as a [[Hopf algebra]], to the [[associative algebra]] of [[horizontal chord diagrams]] on $n$ strands with product given by [[concatenation]] of strands ([this Def.](horizontal+chord+diagram#AlgebraHorizontalChordDiagrams)), [[quotient vector space|modulo]] the [[2T relations]] and [[4T relations]] ([this Def.](horizontal+chord+diagram#2TAnd4TRelations)):
$$
H_\bullet
\big(
\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D)
\big)
\;\simeq\;
\big(\mathcal{A}_n^{pb}, \circ\big)
\,.
$$
=--
### For horizontal weight systems
+-- {: .num_prop}
###### Proposition
**(integral [[horizontal weight systems]] are [[integral cohomology]] of [[based loop space]] of [[ordered configuration space of points]] in [[Euclidean space]])**
Given any [[ground field]] $\mathbb{F}$ (in fact any [[ground ring]], notably the [[integers]]) there is, for each [[natural number]] $n$, a canonical [[isomorphism]] of [[graded abelian groups]] between
1. the [[weight systems]]
$$
\mathcal{W}_n^{pb}
\;\coloneqq\;
Hom_{\mathbb{F} Mod}
\big(
\underset{
\mathcal{A}_n^{pb}
}{
\underbrace{
Span
\big(
\mathcal{D}_n^{pb}
\big)
/(2T,4T)
}
}
,
\mathbb{F}
\big)
$$
on [[horizontal chord diagrams]] of $n$ strands (elements of the set $\mathcal{D}^{pb}$)
1. the [[ordinary cohomology]] of the [[based loop space]] of the [[ordered configuration space of n points]] in [[Euclidean space]]:
$$
H^\bullet
\big(
\Omega
\underset{
{}^{\{1,\cdots,n\}}
}{Conf}
(\mathbb{R}^D)
\big)
\;\simeq\;
(\mathcal{W}_n^{pb})^\bullet
\;\simeq\;
Gr^\bullet( \mathcal{V}_{pb} )
\,.
$$
(the second equivalence on the right is the fact that [[weight systems are associated graded of Vassiliev invariants]], for $D =3$).
=--
This appears stated as [Kohno 02, Theorem 4.1](#Kohno02); it follows immediately by Corollary \ref{HomologyOfLoopSpaceOfConfigurationSpaceIsAlgebraOfHorizontalChordDiagrams} of Prop. \ref{HomologyOfLoopSpaceOfConfigurationSpaceIsUniversalEnvelopingOfInfinitesimalBraids}.
### For round weight systems
+-- {: .num_prop}
###### Proposition
**([[weight systems]] are inside [[real cohomology]] of [[based loop space]] of [[ordered configuration space of points]] in [[Euclidean space]])**
For [[ground field]] $k = \mathbb{R}$ the [[real numbers]],
there is a canonical [[injection]] of the [[real vector space]] $\mathcal{W}$ of [[framed weight systems]] ([here](weight+system#eq:SpaceOfWeightSystemsAsLinearDual)) into the [[real cohomology]] of the [[based loop spaces]] of the ordered [[configuration spaces of points]] in 3-[[dimension|dimensional]] [[Euclidean space]]:
$$
\mathcal{W}
\;\overset{\;\;\;\;}{\hookrightarrow}\;
H\mathbb{R}^\bullet
\Big(
\underset{n \in \mathbb{N}}{\sqcup}
\Omega
\underset{{}^{\{1,\cdots,n\}}}{Conf}
\big(
\mathbb{R}^3
\big)
\Big)
$$
=--
This is stated as [Kohno 02, Theorem 4.2](#Kohno02)
## Related theorems
[[!include facts about chord diagrams and weight systems -- contents]]
## Related concepts
[[!include chord diagrams and weight systems -- table]]
## References
The statement relating the [[ordinary homology]] of the [[based loop space]] of the [[ordered configuration space of points]] to the [[universal enveloping algebra]] of the [[infinitesimal braid Lie algebra]]:
* {#FadellHusseini01} [[Edward Fadell]], [[Sufian Husseini]], Theorem 2.2 in: _Geometry and topology of configuration spaces_, Springer Monographs in Mathematics (2001) ([MR2002k:55038](http://www.ams.org/mathscinet-getitem?mr=2002k:55038), [doi:10.1007/978-3-642-56446-8](https://link.springer.com/book/10.1007/978-3-642-56446-8))
* {#CohenGitler01} [[Fred Cohen]], [[Samuel Gitler]], Theorem 4.1 of: _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]] (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))
* {#CohenGitler02} [[Fred Cohen]], [[Samuel Gitler]], Theorem 2.3 of: _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705–1748, ([jstor:2693715](https://www.jstor.org/stable/2693715), [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013))
The dual statement identifying the [[ordinary cohomology]] of the [[based loop space]] of the [[ordered configuration space of points]] with the space of [[weight systems]] on [[horizontal chord diagrams]]:
* {#Kohno02} [[Toshitake Kohno]], Theorem 4.1 in: _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))
See also:
* {#Kohno94} [[Toshitake Kohno]], _Vassiliev invariants and de Rham complex on the space of knots_,
In: Yoshiaki Maeda, Hideki Omori and [[Alan Weinstein]] (eds.), _Symplectic Geometry and Quantization_, Contemporary Mathematics 179 (1994): 123-123 ([doi:10.1090/conm/179](http://dx.doi.org/10.1090/conm/179))
[[!redirects horizontal weight systems are cohomology of loop space of configuration space]]
[[!redirects horizontal chord diagrams are homology of loop space of configuration space]]
[[!redirects horizontal chord diagrams are the homology of the loop space of configuration space]]
|
weight systems are the associated graded objects of Vassiliev invariants | https://ncatlab.org/nlab/source/weight+systems+are+the+associated+graded+objects+of+Vassiliev+invariants |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Statement
+-- {: .num_prop #WeightSystemsAreAssociatedGradedOfVassilievInvariants}
###### Proposition
**([[weight systems]] are [[associated graded vector space|associated graded]] of [[Vassiliev invariants]])**
For [[ground field]] $k = \mathbb{R}, \mathbb{C}$ the [[real numbers]] or [[complex numbers]], there is for each [[natural number]] $n \in \mathbb{N}$ a canonical [[linear isomorphism]]
$$
\mathcal{V}_n/\mathcal{V}_{n-1}
\underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow}
\big(
\mathcal{A}_n^u
\big)^\ast
$$
from
1. the [[quotient vector space]] of order-$n$ [[Vassiliev invariants]] of [[knots]] by those of order $n-1$
1. to the space of [[unframed weight systems]] of order $n$.
In other words, in [[characteristic zero]], the [[graded vector space]] of [[unframed weight systems]] is the [[associated graded vector space]] of the [[filtered vector space]] of [[Vassiliev invariants]].
=--
([Bar-Natan 95, Theorem 1](#BarNatan95), following [Kontsevich 93, Theorem 2.1](#Kontsevich93))
The **proof** proceeds via construction of a [[universal Vassiliev invariant]] identified with the un-traced [[Wilson loop observable]] of [[perturbative quantization of 3d Chern-Simons theory| perturbative Chern-Simons theory]].
## Related theorems
[[!include facts about chord diagrams and weight systems -- contents]]
## Related concepts
[[!include chord diagrams and weight systems -- table]]
## References
* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))
* Joan S. Birman, Xiao-Song Lin,
_Knot polynomials and Vassiliev's invariants_, Invent Math (1993) 111: 225 ([doi:10.1007/BF01231287](https://doi.org/10.1007/BF01231287))
* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
Review:
* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))
[[!redirects weight systems are associated graded of Vassiliev invariants]]
[[!redirects horizontal weight systems are associated graded of Vassiliev braid invaraints]]
[[!redirects weight systems are the associated graded of Vassiliev invariants]]
|
weight systems on chord diagrams in physics | https://ncatlab.org/nlab/source/weight+systems+on+chord+diagrams+in+physics |
### Chord diagrams and weight systems in Physics
{#ReferencesWeightSystemsOnChordDiagramsInPhysics}
The following is a list of references that involve ([[weight systems]] on) [[chord diagrams]]/[[Jacobi diagrams]] in [[physics]]:
1. [In Chern-Simons theory](#ReferencesWeightSystemsInChernSimonsTheory)
1. [In Dp-D(p+2) brane intersections](#ReferencesWeightSystemsInDpDp2BraneIntersections)
1. In [[AdS-CFT in condensed matter physics|quantum many body models]] for [[AdS-CFT correspondence|for holographic]] [[brane]]/[[bulk]] correspondence:
1. [In AdS2/CFT1, JT-gravity/SYK-model](#ReferencesWeightSystemsInSYKModel)
1. [As dimer/bit thread codes for holographic entanglement entropy](#ForHolographicEntanglementEntropy)
For a unifying perspective (via [[Hypothesis H]]) and further pointers, see:
* {#SatiSchreiber19c} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Differential Cohomotopy implies intersecting brane observables]]*, Adv. Theor. Math. Phys. **26** 4 (2022) [[arXiv:1912.10425](https://arxiv.org/abs/1912.10425)]
* {#CSS21} [[David Corfield]], [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Fundamental weight systems are quantum states]]* Lett. Math. Phys. **113** 112 (2023) \[<a href="https://arxiv.org/abs/2105.02871">arXiv:2105.02871</a>, [doi:10.1007/s11005-023-01725-4](https://doi.org/10.1007/s11005-023-01725-4)\]
* {#Collari2023} [[Carlo Collari]], *A note on weight systems which are quantum states*, Can. Math. Bull. (2023) $[$[arXiv:2210.05399](https://arxiv.org/abs/2210.05399), [doi:10.4153/S0008439523000206](https://doi.org/10.4153/S0008439523000206)$]$
Review:
* [[Carlo Collari]], *Weight systems which are quantum states*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) $[$[web](Center+for+Quantum+and+Topological+Systems#CollariMar2023), [[Collari-WeightSystems.pdf:file]]$]$
#### In Chern-Simons theory
{#ReferencesWeightSystemsInChernSimonsTheory}
Since [[weight systems are the associated graded of Vassiliev invariants]], and since [[Vassiliev invariants]] are [[knot invariants]] arising as certain [[correlators]]/[[Feynman amplitudes]] of [[Chern-Simons theory]] in the presence of [[Wilson lines]], there is a close relation between [[weight systems]] and [[quantization of 3d Chern-Simons theory|quantum Chern-Simons theory]].
Historically this is the original application of [[chord diagrams]]/[[Jacobi diagrams]] and their [[weight systems]], see also at _[[graph complex]]_ and _[[Kontsevich integral]]_.
\begin{imagefromfile}
"file_name": "BarNatanChernSimonsChordDiagrams.jpg",
"float": "right",
"width": 330,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 14
},
"caption": "from [Bar-Natan 1991](#BarNatan91)"
\end{imagefromfile}
* {#BarNatan91} [[Dror Bar-Natan]], _Perturbative aspects of the Chern-Simons topological quantum field theory_, thesis 1991 ([spire:323500](http://inspirehep.net/record/323500), [proquest:303979053](https://search.proquest.com/docview/303979053), [[BarNatanPerturbativeCS91.pdf:file]])
* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))
* Daniel Altschuler, Laurent Freidel, _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. 187 (1997) 261-287 ([arxiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010))
* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Configuration spaces and Vassiliev classes in any dimension_, Algebr. Geom. Topol. 2 (2002) 949-1000 ([arXiv:math/9910139](https://arxiv.org/abs/math/9910139))
* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Algebraic structures on graph cohomology_, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 ([arXiv:math/0307218](https://arxiv.org/abs/math/0307218))
Reviewed in:
* {#Volic13} [[Ismar Volić]], Section 4 of: _Configuration space integrals and the topology of knot and link spaces_, [Morfismos, Vol 17, no 2, 2013](https://fdocuments.co/amp/document/morfismos-vol-17-no-2-2013.html) ([arxiv:1310.7224](https://arxiv.org/abs/1310.7224))
Applied to [[Gopakumar-Vafa duality]]:
* Dave Auckly, Sergiy Koshkin, _Introduction to the Gopakumar-Vafa Large $N$ Duality_, Geom. Topol. Monogr. 8 (2006) 195-456 ([arXiv:0701568](https://arxiv.org/abs/math/0701568))
See also
* [[Marcos Mariño]], _Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants_, Commun. Math. Phys. 253 (2004) 25-49 ([arXiv:hep-th/0207096](https://arxiv.org/abs/hep-th/0207096))
* [[Stavros Garoufalidis]], [[Marcos Mariño]], _On Chern-Simons matrix models_ ([pdf](http://people.mpim-bonn.mpg.de/stavros/publications/matrixmodels.pdf), [[GaroufalidisMarinoChernSimonsMatrixModel.pdf:file]])
#### For single trace operators in AdS/CFT duality
{#ForSingleTraceOperatorsInAdSCFTDuality}
Interpretation of [[Lie algebra weight systems]] on [[chord diagrams]] as certain [[single trace operators]], in particular in application to [[black hole thermodynamics]]
* [[Micha Berkooz]], [[Prithvi Narayan]], [[Joan Simón]], Section 2.1 of _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))
#### In $AdS_2/CFT_1$, JT-gravity/SYK-model
{#ReferencesWeightSystemsInSYKModel}
Discussion of ([[Lie algebra weight system|Lie algebra]]-)[[weight systems]] on [[chord diagrams]] as [[SYK model]] [[single trace operators]]:
* Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, _Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$_, JHEP 04 (2018) 146 ([arXiv:1801.02696](https://arxiv.org/abs/1801.02696))
* Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: _Large $N$ expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs_, JHEP 11 (2018) 031 ([arXiv:1806.03271](https://arxiv.org/abs/1806.03271))
* [[Micha Berkooz]], [[Prithvi Narayan]], [[Joan Simón]], _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))
following:
* László Erdős, Dominik Schröder, _Phase Transition in the Density of States of Quantum Spin Glasses_, D. Math Phys Anal Geom (2014) 17: 9164 ([arXiv:1407.1552](https://arxiv.org/abs/1407.1552))
which in turn follows
* Philippe Flajolet, Marc Noy, _Analytic Combinatorics of Chord Diagrams_, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), _Formal Power Series and Algebraic Combinatorics_, Springer 2000 ([doi:10.1007/978-3-662-04166-6_17](https://doi.org/10.1007/978-3-662-04166-6_17))
\begin{imagefromfile}
"file_name": "ChordDiagramHolographic.jpg",
"float": "right",
"width": 400,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 14
},
"caption": "from [Narovlansky 2019](#Narovlansky19)"
\end{imagefromfile}
With emphasis on the [[AdS-CFT|holographic content]]:
* [[Micha Berkooz]], Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: _Towards a full solution of the large $N$ double-scaled SYK model_, JHEP 03 (2019) 079 ([arxiv:1811.02584](https://arxiv.org/abs/1811.02584))
* {#Narovlansky19} Vladimir Narovlansky, Slide 23 (of 28) of: _Towards a Solution of Large $N$ Double-Scaled SYK_, 2019 ([[NarovlanskySYK19.pdf:file]])
* [[Micha Berkooz]], Mikhail Isachenkov, Prithvi Narayan, Vladimir Narovlansky, *Quantum groups, non-commutative $AdS_2$, and chords in the double-scaled SYK model* \[<a href="https://arxiv.org/abs/2212.13668">arXiv:2212.13668</a>\]
* [[Herman Verlinde]], *Double-scaled SYK, Chords and de Sitter Gravity* \[<a href="https://arxiv.org/abs/2402.00635">arXiv:2402.00635</a>\]
and specifically in relation, under [[AdS2/CFT1]], to [[Jackiw-Teitelboim gravity]]:
* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _The Schwarzian Theory - A Wilson Line Perspective_, JHEP 1812 (2018) 022 ([arXiv:1806.07765](https://arxiv.org/abs/1806.07765))
* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _Fine Structure of Jackiw-Teitelboim Quantum Gravity_, JHEP 1909 (2019) 066 ([arXiv:1812.00918](https://arxiv.org/abs/1812.00918))
* [[Henry W. Lin]], *The bulk Hilbert space of double scaled SYK*, J. High Energ. Phys. **2022** 60 (2022) $[$[arXiv:2208.07032](https://arxiv.org/abs/2208.07032), <a href="https://doi.org/10.1007/JHEP11(2022)060">doi:10.1007/JHEP11(2022)060</a>$]$
* [[Henry W. Lin]], [[Douglas Stanford]], *A symmetry algebra in double-scaled SYK* $[$[arXiv:2307.15725](https://arxiv.org/abs/2307.15725)$]$
#### In D$p$/D$(p+2)$-brane intersections
{#ReferencesWeightSystemsInDpDp2BraneIntersections}
\begin{imagefromfile}
"file_name": "WeightSystemsAsShapeObservabesOnFuzzySphereII.jpg",
"float": "right",
"width": 330,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 14
},
"caption": "from [Sati and Schreiber 19c](#SatiSchreiber19c)"
\end{imagefromfile}
Discussion of [[weight systems]] on [[chord diagrams]] as [[single trace observables]] for the [[non-abelian DBI action]] on the [[fuzzy funnel]]/[[fuzzy sphere]] [[non-commutative geometry]] of [[Dp-D(p+2)-brane intersections]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]):
* [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))
* [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))
* [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])
* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])
#### As codes for holographic entanglement entropy
{#ForHolographicEntanglementEntropy}
\begin{imagefromfile}
"file_name": "HanUniversalHolographicCode.jpg",
"float": "right",
"width": 300,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "From [Yan 20](#Yan20)"
\end{imagefromfile}
[[chord diagram|Chord diagrams]] encoding [[Majorana dimer codes]] and other [[quantum error correcting codes]] via [[tensor networks]] exhibiting [[holographic entanglement entropy]]:
* {#JGPE19} [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Majorana dimers and holographic quantum error-correcting code*, Phys. Rev. Research 1, 033079 (2019) ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))
* {#Yan20} [[Han Yan]], *Geodesic string condensation from symmetric tensor gauge theory: a unifying framework of holographic toy models*, Phys. Rev. B 102, 161119 (2020) ([arXiv:1911.01007](https://arxiv.org/abs/1911.01007))
\begin{imagefromfile}
"file_name": "HaPPYCodesAsDimerCode.jpg",
"width": 570,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "From [Jahn and Eisert 21](holographic+entanglement+entropy#JahnEisert21)"
\end{imagefromfile}
#### For Dyson-Schwinger equations
{#ForDysonSchwingerEquations}
Discussion of [[round chord diagrams]] organizing [[Dyson-Schwinger equations]]:
* Nicolas Marie, [[Karen Yeats]], *A chord diagram expansion coming from some Dyson-Schwinger equations*, Communications in Number Theory and Physics, 7(2):251291, 2013 ([arXiv:1210.5457](https://arxiv.org/abs/1210.5457))
* Markus Hihn, [[Karen Yeats]], *Generalized chord diagram expansions of Dyson-Schwinger equations*, Ann. Inst. Henri Poincar Comb. Phys. Interact. 6 no 4:573-605 ([arXiv:1602.02550](https://arxiv.org/abs/1602.02550))
* Paul-Hermann Balduf, Amelia Cantwell, [[Kurusch Ebrahimi-Fard]], Lukas Nabergall, Nicholas Olson-Harris, [[Karen Yeats]], *Tubings, chord diagrams, and Dyson-Schwinger equations* \[<a href="https://arxiv.org/abs/2302.02019">arXiv:2302.02019</a>\]
Review in:
* Ali Assem Mahmoud, Section 3 of: *On the Enumerative Structures in Quantum Field Theory* ([arXiv:2008.11661](https://arxiv.org/abs/2008.11661))
|
weighted colimit | https://ncatlab.org/nlab/source/weighted+colimit |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A weighted colimit (also called _indexed colimit_ or _mean tensor product_ in older texts) is a concept of [[colimit]] suitable for [[enriched category theory]], dual (in the enriched sense) to the concept of [[weighted limit]].
## Motivation ##
Recall that a colimit of a diagram in a category $C$, that is, of a functor $F: J \to C$, is given by a universal cocone for $F$. A **cocone** for $F$ is a natural transformation from $F$ to a constant diagram
$$\Delta(c) = (J \to 1 \stackrel{c}{\to} C)$$
so that a cocone for $F$ is an object of a [[comma category]]
$$F \downarrow \Delta$$
where $\Delta: C^1 \to C^J$ is the diagonal functor obtained by pulling back along the unique functor $J \to 1$. A **universal cocone** is simply an initial object of $F \downarrow \Delta$.
In enriched category theory, where one considers categories $C$ enriched in a "nice" [[monoidal category]] $V$ (generally one where $V$ is [[complete category|complete]], [[cocomplete category|cocomplete]], [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]]) there is in general no $V$-enriched diagonal functor $\Delta: C \to C^J$ to speak of. For example, when $V$ is the category [[Ab]], we have $C \simeq C^I$ where $I$ is the unit $V$-category having one object $1$ for which $hom(1, 1) = \mathbb{Z}$, but then for a general [[Ab-enriched category]] $J$, there is no [[enriched functor]] $J \to I$ to pull back along (or, there may be many, but none stand out as canonical). This shows that the usual notion of colimit doesn't adapt particularly well to the general enriched setting.
The more flexible notion of weighted colimit (also called an _indexed colimit_ in some of the older accounts) was introduced by Borceux (and Kelly?) as giving the _right_ notion of colimit for enriched category theory.
## Weighted colimits in ordinary category theory ##
First we reformulate ordinary colimits in the language of [[tensor product]]s, in a way that suggests more general weighted colimits.
Assume for the moment that the receiving category $C$ has all [[coproduct]]s and [[coequalizer]]s. As is well known, it follows that $C$ has all colimits; the proof is we can write down a formula for the colimit of $F: J \to C$: as a coequalizer of a pair
$$\sum_{j, k \in Ob(J)} hom(j, k) \times F(j) \overset{\to}{\to} \sum_{k \in Ob(J)} F(k) \to colim_J F$$
where the cartesian product on the left refers to a coproduct of copies of $F(j)$ indexed over the set $hom(j, k)$. One of the two parallel arrows is induced by a collection of actions of the category $J$ on $F$, viz.
$$hom(j, k) \times F(j) \to F(k) = \langle F(f): F(j) \to F(k) \rangle_{f: j \to k}$$
and the other is induced by a collection of projections
$$hom(j, k) \times F(j) \to F(j)$$
each of which is the application of the functor $- \times F(j): Set \to C$ to the unique map $!: hom(j, k) \to 1.$
We can think of the map $hom(j, k) \to 1$ also as a component of an [[action]]: where $J$ acts on the terminal functor $1: J \to Set$. Or rather, dual to the way in which $J$ acts covariantly on $F$ (so $F$ is a _left_ $J$-module), we will think of $J$ acting _contravariantly_ on the terminal functor $1: J \to Set$ (so that $1$ becomes a _right_ $J$-module).
Then the colimit of $F$ above is precisely a [[tensor product]] of the left module $F$ with the right module $1$. More explicitly, the tensor product is the coequalizer of two arrows
$$\sum_{j, k} 1(k) \times hom(j, k) \times F(j) \overset{\to}{\to} \sum_j 1(j) \times F(j)$$
where one arrow is induced from a right action of $J$ on the functor 1, having components
$$\rho_{j, k} \times F(j): (1(k) \times hom(j, k)) \times F(j) \to 1(j) \times F(j)$$
and the other is induced from a left action of $J$ on $F$, having components
$$1(k) \times \lambda_{j, k}: 1(k) \times (hom(j, k) \times F(j)) \to 1(k) \times F(k)$$
From this standpoint, the colimit of $F$ is a rather specialized tensor product of the form
$$1 \otimes_J F$$
and the unsuitability of this notion for general enriched categories could be thought of as a case of putting all one's eggs in the $1$ basket.
A general right $J$-module $W: J^{op} \to Set$ may be called a **weight** (with $W(j)$ the weight at $j$). Thus instead of giving all objects $j$ an equal weight $W(j) = 1$, we vary the weight and get a more general notion of colimit (just as weighted averages generalize ordinary averages). More importantly, this notion of weighted colimit makes perfect sense in the context of enriched categories.
## Definition ##
Let $J$ be a [[small category]]. Given a functor $W: J^{op} \to Set$ (the _weight_) and a functor $F: J \to C$ (the _diagram_), the **weighted colimit** or tensor product is an object $W \cdot F$ of $C$ together with an isomorphism
$$C(W \cdot F, c) \cong Set^{J^{op}}(W, C(F-, c))$$
that is [[natural isomorphism|natural]] in objects $c$ of $C$. (By the [[Yoneda lemma]], such an isomorphism is induced by a uniquely determined transformation
$$W(j) \to C(F(j), W \cdot F),$$
natural in $j$, which is a weighted analogue of the universal cocone.)
The notion of weighted colimit carries over in straightforward fashion to categories enriched in a complete, cocomplete, closed symmetric monoidal category $V$. In that case, if $J$ is a small $V$-category (that is a $V$-enriched category whose object class is small), and if $F: J \to C$ and $W: J^{op} \to V$ are $V$-functors, then a colimit of $F$ with respect to the weight $W$ is an object $W \cdot F$ of $C$ together with an $V$-[[enriched natural transformation|natural]] isomorphism
$$C(W \cdot F, c) \cong V^{J^{op}}(W, C(F-, c))$$
(between $V$-functors in the argument $c$). In fact, we can dispense with the conditions that $V$ be complete, cocomplete, and closed, at the cost of not being able to refer to functor categories $V^{J^{op}}$, without which the notion is conceptually harder to express.
A leitmotif playing in the background is that the category of weights $Set^{J^{op}}$ on $J$ (or $V^{J^{op}}$ in the enriched case) is the free ($V$-enriched) cocompletion of $J$. In other words, if $C$ is a ($V$-)category which is cocomplete in the "right" sense of the word, then every ($V$-)functor $F: J \to C$ extends, uniquely up to unique ($V$-)isomorphism, to a ($V$-)cocontinuous functor
$$\widetilde{F}: V^{J^{op}} \to C$$
which is given by the weighted colimit construction $W \mapsto W \cdot F$.
## Examples
* A [[conical colimit]] is a weighted colimit where $W(j)=I$ (the monoidal unit of $V$, e.g. $W(j)=1$ in the case $V=Set$). Then $F$ is the diagram.
* A [[copower]] is a weighted colimit where $J=1$, the one object category, and $W$ picks out the object of $V$ and $F$ picks out the object of $C$.
* The [[tensor product of functors]] is a general example.
## Cocompleteness
An enriched category admits [[coends]] if it admits [[conical colimits]] and [[copowers]]. It admits all weighted colimits if it admits [[coends]] and [[copowers]]. Thus a category with all conical colimits and copowers is cocomplete.
## Related entries
* [[homotopy weighted colimit]]
* [[weighted limit]]
## References
* Very nice descriptions on the n-Cafe by:
* [John Baez](http://golem.ph.utexas.edu/category/2007/02/day_on_rcfts.html#c007688)
* [Todd Trimble](http://golem.ph.utexas.edu/category/2007/02/day_on_rcfts.html#c007723)
* Notes by Emily Riehl based on a talk by Mike Shulman [Weighted Limits and Colimits](http://www.math.jhu.edu/~eriehl/weighted.pdf)
[[!redirects weighted colimits]]
[[!redirects indexed colimit]]
[[!redirects indexed colimits]]
[[!redirects mean tensor product]]
[[!redirects mean tensor products]] |
weighted graph | https://ncatlab.org/nlab/source/weighted+graph |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--
\tableofcontents
\section{Definition}
Recall that a [[simple graph]] is a [[set]] $V$ of [[vertices]] equipped with
* a [[family]] of [[subsingletons]] $x \in V, y \in V \vdash E(x, y)$ whose elements are called [[edges]],
* a family of bijections $x \in V \vdash i(x, y):E(x, x) \cong \emptyset$,
* a family of [[bijections]] $x \in V, y \in V \vdash s(x, y):E(x, y) \cong E(y, x)$.
A **weighted graph** is a simple graph equipped with a family of functions $x \in V, y \in V \vdash w(x, y):E(x, y) \to R$ from the edge subsingleton to a [[rig]] of [[numbers]] $R$. Usually $R$ is the [[natural numbers]] $\mathbb{N}$ or the [[real numbers]] $\mathbb{R}$.
\section{References}
* Wikipedia, [Weighted graph](https://en.wikipedia.org/wiki/Weighted_graph)
* Wikipedia, <a href="https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)#Weighted_graph">Graph (discrete mathematics)</a>
[[!redirects weighted graph]]
[[!redirects weighted graphs]] |
weighted homotopy theory | https://ncatlab.org/nlab/source/weighted+homotopy+theory | ## References
* [[nLab:Marco Grandis]], The fundamental weighted category of a weighted space (From [[directed algebraic topology|directed]] to weighted algebraic topology), 2006, [arXiv:math/0604506 ](http://arxiv.org/abs/math/0604506)
"Abstract. We want to investigate 'spaces' where paths have a 'weight', or 'cost', expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to path-reversion. Thus, 'weighted algebraic topology' can be developed as an enriched version of directed algebraic topology, where illicit paths are penalised with an infinite cost, and the licit ones are measured. Its algebraic counterpart will be 'weighted algebraic structures', equipped with a sort of directed seminorm." |
weighted limit | https://ncatlab.org/nlab/source/weighted+limit | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
The notion of _weighted limit_ (also called _indexed limit_ or _mean cotensor product_ in older texts) is naturally understood from the point of view on [[limits]] as described at _[[representable functor]]_:
Weighted limits make sense and are considered in the general context of $V$-[[enriched category theory]], but restrict attention to $V=$ [[Set]] for the moment, in order to motivate the concept.
Let $K$ denote the [[small category]] which indexes [[diagrams]] over which we want to consider limits and eventually weighted limits. Notice that for
$$
F \colon K \to Set
$$
a [[Set]]-valued [[functor]] on $K$, the [[limit]] of $F$ is canonically identified simply with the [[set]] of [cones](cone#ConesOverADiagram) with tip the [[singleton]] set $pt = \{\bullet\}$:
$$
lim F \;=\; [K,Set](\Delta pt, F)
\,.
$$
This means, more generally, that for
$$
F \,\colon\, K \to C
$$
a functor with values in an arbitrary [[category]] $C$, the [[object]]-wise limit of the functor $F$ under the [[Yoneda embedding]]
$$
C\big(-,F(-)\big)
\,\colon\,
K
\overset{F}{\longrightarrow}
C
\overset{Y}{\longrightarrow}
Set^{C^{op}}
$$
can be expressed by the right hand side of:
\[
\label{ConicalLimitInWeightedLimitForm}
\underset{
\underset{k \in K}{\longleftarrow}
}{lim}
C\big(-,F(k)\big)
\;=\;
[K,Set]\Big(
\Delta pt
,\,
C\big(-,F(-)\big)
\Big)
\,.
\]
(This is the limit over the diagram $C\big(-,F(-)\big) \,\colon\, K \to Set^{C^{op}}$ which, if [[representable functor|representable]], defines the desired limit of $F$, see [this example](representable+functor#ExampleLimits) at *[[representable functor]]*).
The **idea** of weighted limits is to:
1. allow in the formula (eq:ConicalLimitInWeightedLimitForm) the particular functor $\Delta pt$ to be replaced by any other functor $W \,\colon\, K \to Set$;
2. to generalize everything straightforwardly from the [[Set]]-[[enriched category|enriched]] context to arbitrary $V$-enriched contexts (see below).
The idea is that the weight $W \colon K \to V$ encodes the way in which one generalizes the concept of a [cones](cone#ConesOverADiagram) over a diagram $F$ (that is, something with just a tip from which morphisms are emanating down to $F$) to a more intricate structure over the diagram $F$. For instance in the application to [[homotopy limits]] discussed below, with $V$ being [[SimpSet]], the weight is such that it ensures that not only [[1-morphisms]] are emanating from the tip, but that any triangle formed by these is filled by a [[2-morphism|2-cell]], every tetrahedron by a [[3-morphism|3-cell]], etc.
## Definition ##
Let $V$ be a [[closed category|closed]] [[symmetric monoidal category]]. All categories in the following are $V$-[[enriched category|enriched categories]], all functors are $V$-[[enriched functors]].
A **weighted limit** over a functor
$$
F \,\colon\, K \to C
$$
with respect to a _weight_ or _indexing type_ functor
$$
W \,\colon\, K \to V
$$
is, if it exists, the object $lim^W F \in C$ which [[representable functor|represents]] the functor (in $c \in C$)
$$
[K,V]\Big(W, C\big(c,F(-)\big)\Big)
\;\colon\;
C^{op} \to V
\,,
$$
i.e. such that for all objects $c \in C$ there is an isomorphism
$$
C\big(c, lim^W F\big)
\simeq
[K,V]\Big(W(-), C\big(c,F(-)\big)\Big)
$$
[[natural isomorphism|natural]] in $c$.
(Here $[K,V]$ denotes the $V$-[[enriched functor category]], as usual.)
In particular, if $C = V$ itself, then we get the direct formula
$$
lim^W F \;\simeq\; [K,V](W,F)
\,.
$$
This follows from the above by the [[end]] manipulation
$$
\begin{aligned}
[K,V](W(-),C(c,F(-)))
&:=
\int_{k \in K} V(W(k),V(c,F(k)))
\\
& \simeq
\int_{k \in K} V(c,V(W(k),F(k))
\\
& \simeq
V(c, \int_{k \in K} V(W(k),F(k))
\\
& =:
V(c, [K,V](W,F))
\,.
\end{aligned}
$$
### Weighted limits for $V = Set$
Let us spell out what a weighted limit looks like in ordinary category theory, to give intuition for the difference between weighted limits and ordinary limits.
Given a weight $W : K \to Set$ and a diagram $F : K \to C$, a weighted limit comprises an object $L$ together with a projection $\pi_{k, w} : L \to F(k)$ for each $k \in K$ and $w \in W(k)$ such that the following diagram commutes for $k, k \in K$, $w \in W(k)$ and $\kappa : k \to k'$:
\begin{tikzcd}
L & {F(k)} \\
& {F(k')}
\arrow["{F(\kappa)}", from=1-2, to=2-2]
\arrow["{\pi_{k, w}}", from=1-1, to=1-2]
\arrow["{\pi_{k', W(k)(w)}}"', from=1-1, to=2-2]
\end{tikzcd}
This is required to be universal in the sense that given every such diagram as above with domain $C$, there is a unique morphism $C \to L$ making the diagrams commute.
It is clear that when $W$ is the constant functor sending everything to a singleton set, this recovers the usual notion of limit for $F$.
## Motivation from enriched category theory
Let $V$ be a [[monoidal category]].
Imagine you're tasked to write down the definition of _[[limit]]_ in a category $C$ [[enriched category|enriched]] over $V$. You would start saying there is a [[diagram]] $F \colon K \to C$ and a limit is a [[universal construction|universal]] [[cone]] over it, i.e. it's the universal choice of an [[object]] $c$ together with an arrow $f_k \colon c \to F(k)$ for each object $k$ of $K$.
Here's where you stop and ask yourself: what is 'an arrow' in $C$? $C$ has no *[[hom-sets]]* — it has *[[hom-objects]]* — hence what's 'an element' of $C(c, F(k))$ in $V$?
There are two ways to specify an element of an object $X$ in a [[monoidal category]] $(V, I, \otimes)$:
1. Give an arrow $I \to X$ (think of sets, where elements of $X$ are indeed the same thing as arrows $\{*\} \to X$. These are called [[global elements]] of $X$, and are more often than not a misbehaved notion of element, since often $I$ is 'too big' to thoroughly probe $X$ (on the other hand, notice the [[enriched category#passage_between_ordinary_categories_and_enriched_categories|underlying category of an enriched category]] is defined by taking global elements of the hom-objects)
2. Give *any* arrow into $X$. These are called [[generalized elements]], and the existence of the [[Yoneda embedding]] assures us they completely capture the categorical structure of $V$.
Hence you now say: a cone over $F$ is a choice of a *generalized element* $f_k$ of $C(c, F(k))$, for every $k$ in $K$. This means specifying an arrow $W_k \to C(c, F(k))$ in $V$, for each $k$. It's now quite natural to ask for the functoriality of this choice in $k$, hence we end up defining a 'generalized cone' over $F$ as an element
$$
[K, V](W(-), C(c, F(-)))
$$
Hence $W$ is simply a uniform way to specify the sides of a cone.
A confirmation that this is indeed the right definition of limit in the enriched settings come from the fact that conical completeness (a [[conical limit]] is one where $W = \Delta I$, hence we pick only global element) is an inadequate notion, see for example Section 3.9 in [Kelly's book](#Kelly) (aptly named *The inadequacy of conical limits*).
## Examples ##
### Homotopy limits ###
For $V$ some category of higher structures, the _local_ definition of [[homotopy limit]] over a diagram $F : K \to C$ replaces the ordinary notion of [[cone]] over $F$ by a higher cone in which all triangles of 1-morphisms are filled by 2-cells, all tetrahedra by 3-cells, etc.
One can convince oneself that for the choice of [[SimpSet]] for $V$ this is realized in terms of the weighted limit $lim^W F$ with the weight $W$ taken to be
$$
W : K \to \Simp\Set
$$
$$
W : k \mapsto N(K/k)
\,,
$$
where $K/k$ denotes the [[over category]] of $K$ over $k$ and $N(K/k)$ denotes its [[nerve]].
This leads to the classical definition of homotopy limits in $\Simp\Set$-enriched categories due to
* A.K. Bousfield and D.M. Kan, _Homotopy limits, completions, and localizations_
See for instance also
* [[Jean-Marc Cordier]] and [[Timothy Porter]], _Homotopy Coherent Category Theory_, Trans. Amer. Math. Soc. 349 (1997) 1-54, ([pdf](http://www.ams.org/journals/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf))
* [[Nicola Gambino]], _Weighted limits in simplicial homotopy theory_ ([pdf](http://www.crm.cat/Publications/08/Pr790.pdf) or [pdf](http://www.math.unipa.it/%7Engambino/Research/Papers/weighted.pdf))
In some nice cases the weight $N(K/-)$ can be replaced by a simpler weight; an example is discussed at [[Bousfield-Kan map]].
### Homotopy pullback ###
For instance in the case that
$K = \{r \to t \leftarrow s\}$ is the shape of [[pullback]] diagrams we have
$$
W(r) = \{r\}
$$
$$
W(s) = \{s\}
$$
$$
W(t) = N( \{r \to t \leftarrow s\} )
$$
and $W(r \to t) : \{r\} \to \{r \to t \leftarrow s\}$ injects the vertex $r$ into $\{r \to t \leftarrow s\}$ and similarly for $W(s \to t)$.
This implies that for $F : K \to C$ a pullback diagram in the [[SimpSet]]-enriched category $C$, a $W$-weighted [[cone]] over $F$ with tip some object $c \in C$, i.e. a natural transformation
$$
W \Rightarrow C(c, F(-))
$$
is
* over $r$ a "morphism" from the tip $c$ to $F(r)$ (i.e. a vertex in the Hom-simplicial set $C(c,F(r))$);
* similarly over $s$;
* over $t$ three "morphisms" from $c$ to $F(t)$ together with 2-cells between them (i.e. a 2-[[horn]] in the Hom-simplicial set $C(c,F(t))$)
* such that the two outer morphisms over $t$ are identified with the morphisms over $r$ and $s$, respectively, postcomposed with the morphisms $F(r \to t)$ and $F(s \to t)$, respectively.
So in total such a $W$-weighted cone looks like
$$
\array{
&&& c
\\
& \swarrow &\Rightarrow& \downarrow
&\Leftarrow& \searrow
\\
F(r)
&&
\stackrel{F(r \to t)}{\to}
&
F(t)
&
\stackrel{F(s \to t)}{\leftarrow}
&&
F(s)
}
$$
as one would expect for a "homotopy cone".
# Related pages
* [[strict 2-limit]]
* [[saturated class of limits]]
* [[weighted colimit]]
## References
### General
The notion of weighted limits was introduced (under the name "*mean cotensor product*") in:
* [[Francis Borceux]], p. 10 of: *Une notion de $V$-limite*, in *Colloque sur l'algèbre des catégories. Amiens 1973. Résumés des conférences*, Cahiers de topologie et géométrie différentielle **14** 2 (1973) 153-223 [[numdam:CTGDC_1973__14_2_153_0](http://www.numdam.org/item/CTGDC_1973__14_2_153_0)]
* [[Francis Borceux]], [[Gregory Maxwell Kelly]], *A notion of limit for enriched categories*, Bulletin of the Australian Mathematical Society **12** 1 (1975) 49-72 [[doi:10.1017/S0004972700023637](https://doi.org/10.1017/S0004972700023637)]
and, independently, (under the name "*Hom (formel)*") by:
* C. Auderset, _Adjonction et monade au niveau des 2-categories_, Cahiers de Top. et Géom. Diff. XV-1 (1974), 3-20. ([numdam](http://www.numdam.org/item/CTGDC_1974__15_1_3_0/))
Textbook accounts:
* {#Kelly} [[Max Kelly]], [section 3.1, p. 37](http://www.emis.de/journals/TAC/reprints/articles/10/tr10.pdf#page=37) in: _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories, **10** (2005) 1-136 [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]
* [[Francis Borceux]], §6.6 in: *[[Handbook of Categorical Algebra]]*, Vol. 2: *Categories and Structures*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]
In
* [[Emily Riehl]], _Weighted limits and colimits_ (2008) ([pdf](http://www.math.jhu.edu/~eriehl/weighted.pdf))
is given an account of lectures by [[Mike Shulman]] on the subject. The definition appears there as [definition 3.1, p. 4](http://www.math.jhu.edu/~eriehl/weighted.pdf#page=4) (in a form a bit more general than the one above).
### Presenting homotopy limits
On weighted limits as presentations of [[homotopy limits]]:
* {#Hirschhorn02} [[Philip Hirschhorn]], _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs **99** (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]
To compare with the above discussion notice that
* The functor
$$
W \;\coloneqq\; N(K/-)
$$
is discussed there in definition 14.7.8 on p. 269.
* the $V$-enriched hom-category $[K,V]$ which on $V$-functors $S,T$ is the [[end]] $[K,V](S,T) = \int_{k \in K} V(S(k), T(k))$ appears as $hom^K(S,T)$ in definition 18.3.1 (see bottom of the page).
* for $V$ set to [[SimpSet]] the above definition of homotopy limit appears in example 18.3.6 (2).
* [[Emily Riehl]], §6.6 and Chapter 7 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]
Discussion of weighted [[(infinity,1)-limit|$(\infty,1)$-limits]]:
* [[Martina Rovelli]], _Weighted limits in an (∞,1)-category_ (2019) [[arxiv:1902.00805](https://arxiv.org/abs/1902.00805)]
[[!redirects weighted limits]]
[[!redirects indexed limit]]
[[!redirects indexed limits]]
[[!redirects mean cotensor product]]
[[!redirects mean cotensor products]]
|
Weil algebra | https://ncatlab.org/nlab/source/Weil+algebra |
> There are two different concepts called _Weil algebra_. This entry is about the notion of Weil algebra in [[Lie theory]]. For the notion in [[infinitesimal object|infinitesimal]] geometry see [[infinitesimally thickened point]]/[[local Artin algebra]].
***
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The notion of _Weil algebra_ is ordinarily defined for a [[Lie algebra]] $\mathfrak{g}$. It may be understood as the [[Chevalley-Eilenberg algebra]] of the tangent [[Lie 2-algebra]] $T \mathfrak{g}$ or $inn(\mathfrak{g})$ of $\mathfrak{g}$, generalizing the notion of [[tangent Lie algebroid]] $T X$ from a 0-[[truncated]] Lie algebroid $X$ (a [[smooth manifold]]) to the one-object [[Lie algebroid]] $\mathfrak{g}$.
Generally, for every [[Lie-∞-algebroid]] $\mathfrak{a}$ one may define the corresponding tangent Lie-$\infty$-algebroid $T \mathfrak{a}$, whose Chevalley-Eilenberg algebra may be called the Weil algebra of $\mathfrak{a}$:
$$
W(\mathfrak{a})
=
CE(T \mathfrak{a})
\,.
$$
### Weil algebra of a Lie algebra
Let $\mathfrak{g}$ be a finite-dimensional [[Lie algebra]]. The **Weil algebra** $W(\mathfrak{g})$ of $\mathfrak{g}$ is
* the graded [[Grassmann algebra]] generated from the dual [[vector space]] $\mathfrak{g}^*$ together with another copy of $\mathfrak{g}^*$ shifted in degree
$$
W(\mathfrak{g})
\coloneqq
\wedge^\bullet
(\mathfrak{g}^* \oplus \mathfrak{g}^*[1])
$$
* equipped with a [[derivation]] $d : W(\mathfrak{g}) \to W(\mathfrak{g})$ that makes this a [[dg-algebra]], defined by the fact that on $\mathfrak{g}^*$ it acts as the differential of the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ plus the degree shift morphism $\mathfrak{g}^* \to \mathfrak{g}^*$.
This Weil algebra has trivial [[chain homology and cohomology|cohomology]] everywhere (except in degree 0 of course) and sits in a sequence
$$
CE(\mathfrak{g}) \leftarrow W(\mathfrak{g})
\leftarrow inv(\mathfrak{g})
$$
with the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ and its algebra of [[invariant polynomial]]s on $\mathfrak{g}$. This may be understood as a model for the sequence of algebras of differential forms on the [[generalized universal bundle|universal G-bundle]]
$$
G \to \mathcal{E}G \to \mathcal{B}G
\,.
$$
As such, the Weil algebra plays a crucial role in the study of the [[Lie algebra cohomology]] of $\mathfrak{g}$.
## Definition
We first consider Weil algebras of [[L-∞ algebras]], then more generally of [[L-∞ algebroid]]s.
We use the notation and grading conventions that are described in detail at [[Chevalley-Eilenberg algebra]].
### For $L_\infty$-algebras
#### Plain Weil algebra
Let $\mathfrak{g}$ be an [[L-∞ algebra]] of [[finite type]]. By our grading conventions this means that the [[graded vector space]] $\mathfrak{g}^*$ obtained by degreewise dualization is in non-negative degree, and $\wedge^1 \mathfrak{g}^* = \mathfrak{g}^*[1]$ is its shift up into positive degree.
A quick abstract way to characterize the Weil algebra of $\mathfrak{g}$ is as follows. Notice that there is a [[free functor]]/[[forgetful functor]] [[adjunction]]
$$
(F \dashv U)
\colon
dgAlg
\stackrel{\overset{F}{\leftarrow}}{\underset{U}{\longrightarrow}}
Vect[\mathbb{Z}]
$$
between the [[category]] [[dgAlg]] of [[dg-algebras]] and the category of $\mathbb{Z}$-graded [[vector space]]s (all over some fixed [[field]]). Notice that a free object is unique _up to [[isomorphism]]_ .
\begin{definition}\label{PlainWeilAlgebraOfLInfinityAlgebra}
**(plain Weil algebra of $L_\infty$-algebra)**
\linebreak
The **Weil algebra** $W(\mathfrak{g})$ of an $L_\infty$-algebra is the unique representative of the [[free functor|free]] [[dg-algebra]] on $\wedge^1 \mathfrak{g}^*$ for which the projection of graded vector spaces $\wedge^1(\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \to \wedge^1 \mathfrak{g}^*$ extends to a [[dg-algebra]] [[homomorphism]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$
\end{definition}
We discuss below in the [Properties](#Properties) section that this is equivalent to the following component-wise definition
+-- {: .num_defn #WeilForLInfinitityAlgebra}
###### Definition
The **Weil algebra** $W(\mathfrak{g})$ is the [[semi-free dga]] whose underlying graded-commutative algebra is the [[exterior algebra]]
$$
\wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1])
$$
on $\mathfrak{g}^*$ and a shifted copy of $\mathfrak{g}^*$,
and whose [[differential]] is the sum
$$
d_{W(\mathfrak{g})} = d_{CE(\mathfrak{g})} + \mathbf{d}
$$
of two graded [[derivations]] of degree +1 defined by
* $\mathbf{d}$ acts by degree shift $\mathfrak{g}^* \to \mathfrak{g}^*[1]$ on elements in $\mathfrak{g}^*$ and by 0 on elements of $\mathfrak{g}^*[1]$;
* $d_{CE(\mathfrak{g})}$ acts on unshifted elements in $\mathfrak{g}^*$ as the differential of the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ and is extended uniquely to shifted generators by graded-commutativity
$$
[d_{CE(\mathfrak{g})}, \mathbf{d}] = 0
$$
with $\mathbf{d}$:
$$
d_{CE(\mathfrak{g})} \mathbf{d} \omega
\coloneqq
- \mathbf{d} d_{CE(\mathfrak{g})} \omega
$$
for all $\omega \in \wedge^1 \mathfrak{g}^*$.
=--
#### Adjusted Weil algebras
{#AdjustedWeilAlgebras}
For some purposes of [[schreiber:L-infinity algebra connections|$L_\infty$-algebra valued connection]], the [above](#WeilForLInfinitityAlgebra) definition of Weil algebra of an $L_\infty$-algebra is not quite appropriate. While Def. \ref{WeilForLInfinitityAlgebra} gives the Weil algebra up to compatible [[isomorphism]], in applications there is in fact the freedom to choose it up to compatible [[quasi-isomorphism]], and this freedom allows to find better representatives.
For more on this see at *[[adjusted Weil algebra]]*.
### For $L_\infty$-algebroids
Where the [[Chevalley-Eilenberg algebra]] of an [[L-∞ algebra]] has in degree 0 the ground field, that of an [[L-∞ algebroid]] has more generally an [[algebra over a Lawvere theory|algebra over]] a [[Lawvere theory]]. For [[L-∞ algebroid]]s over [[smooth manifold]]s this is the algebra of [[smooth function]]s on a manifolds, regarded as a [[smooth algebra]] ($C^\infty$-ring).
So let $T$ be a [[Fermat theory]]. Write $T Alg$ for the corresponding [[category]] of [[algebra over a Lawvere theory|algebra]]. There is a [[free functor]]/[[forgetful functor]] [[adjunction]]
$$
(F \dashv U) :
T Alg
\stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
CRing
$$
to the category [[CRing]] of commutative [[Ring]]s.
We need the facts that
* a [[module]] over a $T$-algebra $A$ is uniquely specified by its underlying module over $U(A)$;
* the universal [[derivation]] on a $T$-algebra $A$ is the [[de Rham differential]]
$$
d_{dR} : A \to \Omega^1(A)
$$
with values in the $A$-module of $T$-[[Kähler differential]]s.
See the corresponding entries for more details. The second point means that for $v : A \to N$ any $T$-[[derivation]] on $A$, there is a unique $A$-[[module]] [[homomorphism]]
$$
\Omega^\bullet(A) \to N
$$
such that the diagram
$$
\array{
&& \Omega^\bullet(A)
\\
& {}^{\mathllap{d_{dR}}}\nearrow & \downarrow^{\mathrlap{v}}
\\
A &\stackrel{v}{\to}& N
}
$$
commutes.
Let now $\mathfrak{a}$ be an [[L-∞ algebroid]] with [[Chevalley-Eilenberg algebra]] considered as the following data;
1. a graded commutative [[semifree dga]] $CE(\mathfrak{a})$ over the ground field;
1. the structure of a $T$-[[algebra over a Lawvere theory|algebra]] on the [[associative algebra]] $A \coloneqq CE(\mathfrak{a})_0$ (over the ground field)
such that $d_{CE(\mathfrak{a})} : CE(\mathfrak{a})_0 \to CE(\mathfrak{a})_1$ is a [[derivation]] of $T$-algebra modules.
By [[semifree dga|semi-freeness]] there exists a $\mathbb{N}$-[[graded vector space]] $(\mathfrak{a}^*)^\bullet$ and an [[isomorphism]]
$$
CE(\mathfrak{a})
\simeq
(\wedge^\bullet_{A} (\mathfrak{a}^*), d_{CE(\mathfrak{a})})
\,.
$$
+-- {: .num_defn}
###### Definition
The **Weil algebra** $W(\mathfrak{a})$ of the $L_\infty$-algebroid $\mathfrak{a}$ is the Chevalley-Eilenberg algebra of the $L_\infty$-algebroid defined as follows
* the $T$-algebra $A$ in degree 0 is the same as that of $\mathfrak{A}$;
* the underlying graded algebra is the [[exterior algebra]] on $\mathfrak{a}^*$ and a shifted copy $\mathfrak{a}^*[1]$ as well as one copy of the [[Kähler differential]] module $\Omega^1$ in lowest degree (though of as the shifted copy of $A$ itself)
$$
\wedge^\bullet (\Omega^1(A) \oplus (\mathfrak{a}^*) \oplus \mathfrak{a}^*[1])
\,.
$$
* the [[differential]] is the sum
$$
d_{W(\mathfrak{a})} = d_{CE(\mathfrak{a})} + \mathbf{d}
$$
of two degree +1 graded derivations, where $d_{CE(\mathfrak{a})}$ and $\mathbf{a}$ are defined on $\wedge^1 \mathfrak{a}^* \oplus \mathfrak{a}^*[1]$ as [above](#WeilForLInfinitityAlgebra) for $L_\infty$-algebras and on $A$ itself $d_{CE(\mathfrak{a})}$ vanishes and $\mathbf{d}$ acts as the universal derivation
$$
\mathbf{d}|_A = d_{\mathrm{dR}} : A \to \Omega^1(A)
\,.
$$
=--
## Properties
{#Properties}
### Free property
{#FreeProperty}
The main point of the definition is that the differential restricted to the original (unshifted) generators is the original differential plus the shift:
$$
d_{W(\mathfrak{a})} |_{\mathfrak{a}^*} =
d_{CE(\mathfrak{a})} + \mathbf{d}
\,.
$$
By solving the condition $d_{W(\mathfrak{a})} \circ d_{W(\mathfrak{a})} = 0$ and using that $d_{CE(\mathfrak{a})} d_{CE(\mathfrak{a})} = 0$ this already fixes uniquely the differential $d_{W(\mathfrak{a})}$. To see this we only need to show that the value of $d_{W(\mathfrak{a})}(x)$ on a generator $x=\sigma(t) \in \mathfrak{a}^*[1]$ is completely determined by $d_{W(\mathfrak{a})}\vert_{\wedge^\bullet\mathfrak{a}^*}$. One computes:
$$
\begin{aligned}
0 & =
d_{W(\mathfrak{a})}(d_{W(\mathfrak{a})} t)
\\
& =
d_{W(\mathfrak{a})}(d_{CE(\mathfrak{a})}t + \sigma t)
\\
& = \sigma d_{CE(\mathfrak{a})} t + d_{W(\mathfrak{a}) } x
\end{aligned}
$$
and hence
$$
d_{W(\mathfrak{a})} x = - \sigma d_{CE(\mathfrak{a})} \sigma^{-1} (x)
\,.
$$
This implies the following universal [[free functor|freeness property]]:
+-- {: .num_prop}
###### Proposition
Let $\mathfrak{g}$ be an $L_\infty$-algebra. Morphisms of $dg$-algebras $W(\mathfrak{g}) \to A$ are in natural bijection to morphisms of [[graded vector space]]s $\mathfrak{g}^* \to A$.
=--
+-- {: .proof}
###### Proof
Forgetting the differential, $W(\mathfrak{g})$ is the free graded-commutative algebra generated by (a shifted copy of) $\mathfrak{g}^*$ and $\mathfrak{g}^*[1]$. Therefore,
$$
Hom_{dgca}(W(\mathfrak{g}),A)\subseteq Hom_{gca}(W(\mathfrak{g}),A)=Hom_{grVect}(\mathfrak{g}^*,A)\oplus Hom_{grVect}( \mathfrak{g}^*[1],A).
$$
Projecting down to $Hom_{grVect}(\mathfrak{g}^*,A)$, one obtains a natural map
$$
Hom_{dgca}(W(\mathfrak{g}),A)\to Hom_{grVect}(\mathfrak{g}^*,A),
$$
which is a bijection.
To prove injectivity, we just have to show that the restriction of a dgca morphism $f:W(\mathfrak{g})\to A$ to $\mathfrak{g}^*$ determines the restriction of $f$ to $\mathfrak{g}^*[1]$. One has, for any $x=\sigma(t)\in \mathfrak{g}^*[1]$,
$$
\begin{aligned}
f(x)&=f(\sigma(t))=f(d_{W(\mathfrak{g})}t-d_{CE(\mathfrak{g})}t)\\
&=d_A f(t)- f(d_{CE(\mathfrak{g})}t).
\end{aligned}
$$
Since $d_{CE(\mathfrak{g})}(t)$ lies in the sub-gca of $W(\mathfrak{g})$ generated by $\mathfrak{g}^*$, the element $f(d_{CE(\mathfrak{g})}(t))$, and therefore $f(x)$, is determined by $f\vert_{\mathfrak{g}^*}$.
Next we show surjectivity, i.e. that every morphism of graded vector spaces $\phi:\mathfrak{g}^*\to A$ can be extended to a dgca morphism $f:W(\mathfrak{g})\to A$. Denote by $f_0: \wedge^\bullet \mathfrak{g}^*\to A$ the extension of $\phi$ to a graded commutative algebra morphism, and let $\psi:\mathfrak{g}^*[1]\to A$ be the graded vector space morphism defined by
$$
\psi(x)=d_A \phi(t)-f_0d_{CE(\mathfrak{g})}(t),
$$
for any $x=\sigma(t)\in \mathfrak{g}^*[1]$.
The graded vector space morphism $\phi+\psi:\mathfrak{g}^*\oplus\mathfrak{g}^*[1]\to A$ extends to a commutative graded algebra $f:W(\mathfrak{g})\to A$, whose restriction to $\mathfrak{g}^*$ is $\phi$. We want to show that $f$ is actually a dgca morphism. We only need to test commutativity with the differentials on generators $t\in \mathfrak{g}^*$ and $x=\sigma(t)\in \mathfrak{g}^*[1]$. We have
$$
d_A f(t)=d_A\phi(t)=\psi(\sigma(t))+f_0d_{CE(\mathfrak{g})}(t)=f(\sigma(t))+
f d_{CE(\mathfrak{g})}(t)=f d_{W(\mathfrak{g})}(t),
$$
which in particular implies that $d_A f\vert_{\wedge^\bullet \mathfrak{g}^*}=f d_{W(\mathfrak{g})}\vert_{\wedge^\bullet \mathfrak{g}^*}$,
and
$$
d_A f(x)= d_A \psi(x) = -d_A f_0d_{CE(\mathfrak{g})}(t)=-d_A f (d_{CE(\mathfrak{g})}(t)).
$$
Since $d_{CE(\mathfrak{g})}(t)\in \wedge^\bullet \mathfrak{g}^*$, we obtain
$$
d_A f(x)= -f d_{W(\mathfrak{g})} (d_{CE(\mathfrak{g})}(t))=
-f d_{W(\mathfrak{g})}(d_{W(\mathfrak{g})}(t)-x)=f d_{W(\mathfrak{g})}(x).
$$
=--
+-- {: .num_example}
###### Example
For $A=CE(\mathfrak{g})$ the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$, the inclusion $\mathfrak{g}^*\hookrightarrow CE(\mathfrak{g})$ induces a canonical surjective dgca morphism $W(\mathfrak{g})\to CE(\mathfrak{g})$. This is the identity on the unshifted generators, and 0 on the shifted generators.
=--
+-- {: .num_example}
###### Example
For $A = \Omega^\bullet(X)$ the [[de Rham complex]] of a [[smooth manifold]] $X$, we have that
$$
Hom_{dgAlg}(W(\mathfrak{g}), \Omega^\bullet(X))
=
(\Omega^\bullet(X) \otimes \mathfrak{g})^1
$$
is the collection of total degree 1 [[differential form]]s with values in the $\infty$-Lie algebra $\mathfrak{g}$.
A morphism of
$$
(A, F_A) : W(\mathfrak{g}) \to \Omega^\bullet(X)
$$
sends the unshifted generators $t^a$ to differential forms $A^a$, which one thinks of as local connection forms, and sends the shifted generators $\sigma t^a$ to their [[curvature]]. The respect for the differential on the shifted generators is the [[Bianchi identity]] on these curvatures.
A morphism $W(\mathfrak{g}) \to \Omega^\bullet(X)$ encodes a collection of _flat_ $L_\infty$-algebra valued forms precisely if it factors by the canonical morphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$ from above through the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$.
=--
The freeness property of the Weil algebra can be made more explicit by exhibiting a concrete [[isomorphism]] to the free dg-algebra on $\mathfrak{g}^*$.
+-- {: .num_defn}
###### Definition
The _canonical free dg-algebra_ on $\mathfrak{g}^*$ is
$$
F(\mathfrak{g}) \coloneqq \wedge^\bullet( \mathfrak{g}^* \oplus \mathfrak{g}^*[1], d_F )
$$
where the differential $d_f$ is on the unshifted generators $t \in \mathfrak{g}^*$ the shift isomorphism $\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1]$ extended as a derivation and vanishes on the shifted generators
$$
d_F : t \mapsto \sigma(t)
\,,
$$
$$
d_F : \sigma(t) \mapsto 0
\,.
$$
=--
Or in other words, if $\bar \mathfrak{g}$ is the $\infty$-Lie algebra whose underlying graded vector space is that of $\mathfrak{g}$, but all whose brackets vanish, then
$$
F(\mathfrak{g}) = W(\bar \mathfrak{g})
\,.
$$
Notice the evident
+-- {: .num_lemma}
###### Observation
The [[cochain cohomology]] of $F(\mathfrak{g})$ vanishes in positive degree.
=--
To see this, let $K \coloneqq \sigma^{-1} : F(\mathfrak{g}) \to F(\mathfrak{g})$ be the degree down-shift isomorphism $\mathfrak{g}^*[1] \to \mathfrak{g}^*$ extended as a graded derivation of degree -1, then
$$
[d_{F(\mathfrak{g})}, K] = Id : F(\mathfrak{g}) \to F(\mathfrak{g})
$$
and hence for any $\omega \in F(\mathfrak{g})$ such that $d_{F(\mathfrak{g})} \omega = 0$ we have $\omega = d_{F(\mathfrak{g})} K \omega$.
+-- {: .num_prop}
###### Lemma
Given $\mathfrak{g}$, there is an [[isomorphism]] of [[dg-algebra]]s
$$
f : F(\mathfrak{g}) \to W(\mathfrak{g})
$$
given by
$$
f : t \mapsto t
$$
$$
f : \sigma(t) \mapsto d_{W(\mathfrak{g})} t = d_{CE(\mathfrak{g})} t + \sigma(t)
\,.
$$
=--
+-- {: .proof}
###### Proof
It is clear that $f$ is a dg-algebra homomorphism. The inverse dg-algebra morphism is given on generators by
$$
f^{-1} : t \mapsto t
$$
$$
f^{-1} : \sigma(t) \mapsto \sigma(t) - d_{CE(\mathfrak{g})}(t)
\,.
$$
Note that the isomorphism $f$ is precisely the dgca isomorphism induced between $W(\overline\mathfrak{g})$ and $W(\mathfrak{g})$ by the identity of $\mathfrak{g}^*$ as a graded vector spaces morphism $\overline{\mathfrak{g}}^*\to\mathfrak{g}^*$.
=--
+-- {: .nun_cor}
###### Corollary
The [[cochain cohomology]] of the Weil algebra of an $L_\infty$-algebra is trivial.
=--
+-- {: .num_remark}
###### Remark
This means that [[homotopy theory|homotopy-theoretically]] the Weil algebra is the point. Dually, the $\infty$-Lie algebra $inn(\mathfrak{g})$ is a model for the point. In fact, one can see that $inn(\mathfrak{g})$ is the [[universal principal ∞-bundle]] over $\mathfrak{g}$ in the canonical [[model category|model]] for the [[(∞,1)-topos]] [[SynthDiff∞Grpd]]. In fact, it is a [[groupal model for universal principal ∞-bundles]]. This is discussed at [[∞-Lie algebra cohomology]].
=--
### Characterization in the smooth $\infty$-topos
{#CharacterizationInSmoothTopos}
The Weil algebra of a Lie algebra is naturally identified with the
de Rham algebra of differential forms on the "universal $G$-principal bundle with connection" in its stacky incarnation ([Freed-Hopkins 13](#FreedHopkins13)):
Write $\mathbf{B}G_{conn}\simeq \mathbf{\Omega}(-,\mathfrak{g})//G$ for the universal [[moduli stack]] of $G$-[[principal connections]] (as discussed there), a [[smooth groupoid]]. The quotient projection may be regarded as the universal $G$-connection:
$$
\array{
&& \mathbf{\Omega}_{flat}(-,\mathfrak{g})
\\
&& \downarrow
\\
\mathbf{E}G_{conn} &\coloneqq & \mathbf{\Omega}(-,\mathfrak{g})
\\
\downarrow && \downarrow
\\
\mathbf{B}G_{conn} &\coloneqq &\mathbf{\Omega}(-,\mathfrak{g})//G
}
$$
(After forgetting the connection/form data this is just the [[universal principal bundle]] $\mathbf{E}G \to \mathbf{B}G$)
The differential $k$-forms on a [[smooth groupoid]] $X$ are just homs $X \to \mathbf{\Omega}^k(-)$ into the sheaf of $k$-forms. (See at [[geometry of physics -- differential forms]]). These $\Omega^k(X)$ inherit the de Rham differential and hence form the de Rham complex of the stack. (Notice that this is very different from the hom of $X$ into a shift of the full de Rham complex regarded as a sheaf of complexes. The latter is instead a model for the real [[ordinary cohomology]] of $X$, see at _[[smooth infinity-groupoid -- structures]]_ for more on this).
{#FreedHopkinsResult} One finds ([Freed-Hopkins 13](#FreedHopkins13)) that the de Rham complex, in this sense, of $\mathbf{E}G_{conn}$ is the Weil algebra:
$$
\Omega^\bullet(\mathbf{E}G_{conn})
\coloneqq
\Omega^\bullet( \mathbf{\Omega}(-,\mathfrak{g}) )
\simeq
W(\mathfrak{g})
\,.
$$
[[!include Weil algebra abstractly -- table]]
Turning this around, this motivates to algebraically _define_ the [[connection on a principal ∞-bundle]], [via Lie integration](connection+on+a+smooth+principal+infinity-bundle#ByLieIntegration), as discussed there.
### Relation to Cartan model for equivariant de Rham cohomology
The Weil algebra may be identified with the
[[Cartan model]] for [[equivariant de Rham cohomology]] for the special
case of the Lie group $G$ acting on itself by right multiplication. Concersely, the [[Cartan models]] form a generalization of the Weil algebra. See at _[equivariant de Rham cohomology -- Cartan model](equivariant+de+Rham+cohomology#TheCartanModel)_ for more.
## As the CE-algebra of the $L_\infty$-algebra of inner derivations {#AsInnerDer}
By the discussion at [[∞-Lie algebra]] and [[Chevalley-Eilenberg algebra]], we may _identify_ the [[full subcategory]] of the [[opposite category]] [[dgAlg]] on commutative [[semi-free dga]]s in non-negative degree with that of [[∞-Lie algebra]]s/[[∞-Lie algebroid]]s.
That means that the Weil algebra $W(\mathfrak{g})$ of some [[L-∞ algebra]] $\mathfrak{g}$ is the Chevalley-Eilenberg algebra of _another_ $\infty$-Lie algebra.
+-- {: .num_defn}
###### Definition
For any $\infty$-Lie algebra $\mathfrak{g}$ write $inn(\mathfrak{g})$ for the $\infty$-Lie algebra whose CE-algebra is $W(\mathfrak{g})$:
$$
CE(inn(\mathfrak{g}))
\coloneqq
W(\mathfrak{g})
\,.
$$
=--
In the following we discuss these _inner automorphism $\infty$-Lie algebras_ in more detail. (See section 6 of ([SSSI](#SSSI))).
### For an ordinary Lie algebra
+-- {: .num_lemma}
###### Observation
For $\mathfrak{g}$ an ordinary [[Lie algebra]] the [[inner derivation Lie 2-algebra]] is the [[strict Lie 2-algebra]] given by the [[dg-Lie algebra]]
$$
inn(\mathfrak{g}) = ( \mathfrak{g} \stackrel{d}{\to} \mathfrak{g}, [-,-])
$$
whose
* elements in degree -1 are the elements $x \in \mathfrak{g}$, thought of as inner degree-(-1) [[derivation]]s
$\iota_x : CE(\mathfrak{g}) \to CE(\mathfrak{g})$
given by contraction with $x$;
* elements in degree 0 are the derivations of degree 0 that are of the form
$ \mathcal{L}_X
\coloneqq
[d_{CE(\mathfrak{g})}, \iota_x]
\colon
CE(\mathfrak{g}) \to CE(\mathfrak{g})$;
* the differential $d = [d_{CE}, -] : \mathfrak{g} \to \mathfrak{g}$ is the commutator of derivations with the differential $d_{CE(\mathfrak{g})}$;
* the bracket is the graded commutator of derivations.
=--
Equivalently this is identified with the [[differential crossed module]] $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ with the action being the [[adjoint action]] of $\mathfrak{g}$ on itself.
One checks that for all $x, y \in \mathfrak{g}$ we have in $inn(\mathfrak{g})$ the brackets
* $[\iota_x, \iota_y] = 0$
* $[\mathcal{L}_x, \iota_y] = \iota_{[x,y]}$
* $[\mathcal{L}_x, \mathcal{L}_y] = \mathcal{L}_{[x,y]}$
and of course
* $ \mathcal{L}_x = [d, \iota_x] $.
These identities are known as [[Cartan calculus]]. In this context $\mathcal{L}_x$ is called a [[Lie derivative]].
In this sense one may understand $inn(\mathfrak{g})$ for general $\infty$-Lie algebras $\mathfrak{g}$ as providing an $\infty$-version of [[Cartan calculus]].
## Relation to other concepts
### $\infty$-Lie algebra valued differential forms {#LieAlgValuedForms}
For $\mathfrak{g}$ an [[∞-Lie algebra]], $X$ a [[smooth manifold]], an
[[∞-Lie algebra valued differential form]] is a morphism
$$
\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A
$$
of [[dg-algebra]]s, from the Weil algebra into the [[de Rham complex]] of $X$.
The image of the unshifted generators $A : \wedge^1 \mathfrak{g}^* \to \Omega^\bullet(X)$ are the forms themselves, the image of the shifted generators $F_A : \wedge^1 \mathfrak{g}^*[1]$ are the corresponding [[curvature]]s. The respect for the differential on the shifted generators are the [[Bianchi identity]] on the curvatures.
Precisely if the curvatures vanish does the morphism factor through the [[Chevalley-Eilenberg algebra]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$.
$$
(F_A = 0)
\;\;\Leftrightarrow
\;\;
\left(
\array{
&& CE(\mathfrak{g})
\\
& {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow
\\
\Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g})
}
\right)
\,.
$$
### Invariant polynomials and Chern-Simons elements
A [[cocycle]] in the [[∞-Lie algebra cohomology]] of the [[∞-Lie algebra]] $\mathfrak{g}$ is a closed element in the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$.
An [[invariant polynomial]] $\langle -\rangle$ on $\mathfrak{g}$ is a closed element in the Weil algebra $\langle -\rangle \in W(\mathfrak{g})$, subject to the additional condition that it its entirely in the shifted copy of $\mathfrak{g}$, $\langle - \rangle \in \wedge^\bullet (\mathfrak{g}^*[1])$.
$$
\langle -\rangle \in \wedge^\bullet( \mathfrak{g}^*[1] )
$$
$$
d_{W(\mathfrak{g})} \langle -\rangle = 0
\,.
$$
For $x \in \mathfrak{g}$ an element of the $\infty$-Lie algebra, let
$$
\iota_x : W(\mathfrak{g}) \to W(\mathfrak{g})
$$
the evident operation of contraction with $x$
$$
\iota_x : t \mapsto t(x)
$$
$$
\iota_x : \sigma(t) \mapsto 0
$$
extended as a graded derivation. Then the [[Lie derivative]]
$$
\mathcal{L}_x
\coloneqq ad_x
\coloneqq
[d_{W(\mathfrak{g})}, \iota_x]
\colon
W(\mathfrak{g}) \to W(\mathfrak{g})
$$
encodes the coadjoint action of $\mathfrak{g}$ on $\mathfrak{g}^*$. By the above definition of an [[invariant polynomial]] $\langle - \rangle$, we have
$$
\iota_x \langle - \rangle = 0
$$
and
$$
d_{W(\mathfrak{g})} \langle - \rangle = 0
$$
and hence
$$
ad_x \langle -\rangle = 0
\,.
$$
Since the cohomology of $W(\mathfrak{g})$ is trivial, there is necessarily for each invariant polynomial an element $cs_{\langle -\rangle}$ such that
$$
d_{W(\mathfrak{g})} cs_{\langle -\rangle} =
\langle -\rangle
\,.
$$
This is the [[Chern-Simons element]] of the invariant polynomial. Notice, crucially, that this is ingeneral _not_ restricted to the shifted part $\wedge^\bullet (\mathfrak{g}^*[1])$ Its restriction
$$
\mu_{\langle -\rangle}
\coloneqq
cs_{\langle - \rangle}|_{\wedge^\bullet \mathfrak{g}^*}
$$
to the unshifted copy, hence to the [[Chevalley-Eilenberg algebra]], is the cocycle that is in transgression with $\langle - \rangle$.
For
$$
(A,F_A) \colon W(\mathfrak{g}) \to \Omega^\bullet(X)
$$
a collection of $\mathfrak{g}$-valued differential forms (as [above](LieAlgValuedForms)) and $\langle -\rangle : CE(b^{n-1}\mathbb{R}) \to W(\mathfrak{g})$ an [[invariant polynomial]], the composite
$$
\langle F_A\rangle
:
CE(b^{n-1}\mathbb{R})
\stackrel{\langle - \rangle}{\to}
W(\mathfrak{g})
\stackrel{(A,F_A)}{\to}
\Omega^\bullet(X)
$$
is the corresponding [[curvature characteristic form]], a closed $n$-form on $X$. For $(\langle - \rangle, cs) : W(b^{n-1}) \to W(\mathfrak{g})$ the corresponding [[Chern-Simons element]] we have that $cs(A,F_A)$ is the corresponding [[Chern-Simons form]] on $X$.
## Examples
### Weil algebra of a Lie algebra {#WeilofLieAlg}
Let $\mathfrak{g}$ be a finite dimensional [[Lie algebra]]. This Lie algebra regarded as a [[Lie algebroid]] has as base manifold the point, $X_0 = pt$. Its algebra of functions is accordingly the ground field, and the algebra $\wedge^\bullet_{C^\infty(X_0)} \mathfrak{g}^*$ is just a [[Grassmann algebra]].
The [[Chevalley-Eilenberg algebra]] is
$$
CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^*, d_{\mathfrak{g}})
\,,
$$
where the differential acts on the elements of $\mathfrak{g}^*$ in degree 1 by the linear dual of the Lie bracket.
$$
d \mathfrak{g}|_{\mathfrak{g}^*} = [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*
\,.
$$
The corresponding Weil algebra is obtained by adding another copy of $\mathfrak{g}^*$ in degree 2
$$
W(\mathfrak{g}) = (\wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W(\mathfrak{g})})
$$
where with $\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1]$ the degree shift isomorphism, the differential acts as
$$
d_{W(\mathfrak{g})}|_{\mathfrak{g}^*}
:
[-,-]^* + \sigma
$$
$$
d_{W(\mathfrak{g})}|_{\mathfrak{g}^*[1]}
:
\sigma \circ d_{CE(\mathfrak{g})} \circ \sigma^{-1}
\,.
$$
For illustration, we spell this out in a basis.
Let $\{t_a\}_a$ be a basis for the underlying vector space of $\mathfrak{g}$ and let $\{C^a{}_{b c}\}$ be the corresponding structure constants of the Lie bracket
$$
[t_b, t_c] = C^a{}_{b c} t_a
\,.
$$
Then the Chevalley-Eilenberg algebra is generated on generators $\{t^a\}$ of degree 1, on which the differential acts as
$$
d_{CE(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
\,.
$$
The Weil algebra in turn is generated from these generators $\{t^a\}$ in degree 1 and generators $\{r^a\}$ in degree 2, with differential given by
$$
d_{W(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
$$
$$
d_{W(\mathfrak{g})} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c
\,.
$$
### Weil algebra of a 0-Lie algebroid
A 0-[[truncated]] Lie algebroid is one for which the chain complex of modules over the $T$-algebra in degree 0 vanishes:
$$
CE(\mathfrak{a}) = (\wedge^\bullet_A (\mathfrak{a}^*), d_{CE(\mathfrak{a})})
= (A, d = 0)
\,.
$$
For instance for $T$=[[CartSp]] the theory of [[smooth algebra]]s, any [[smooth manifold]] $X$ regarded as an
[[L-∞ algebroid]] is a 0-Lie algebroid with $CE(X) = C^\infty(X)$ the [[smooth algebra]] of [[smooth function]]s on $X$.
+-- {: .num_prop}
###### Observation
The Weil algebra of a 0-Lie algebroid $X$ is the [[Kähler differential|Kähler]] [[de Rham complex]] of $A = CE(X)$:
$$
W(\mathfrak{a}) =
(\Omega^\bullet(A), d_{dR})
\,.
$$
=--
This Weil algebra is the Chevalley-Eilenberg algebra of the
[[tangent Lie algebroid]] $T X$ of $X$, which is the [[de Rham algebra]]
$\Omega^\bullet(X)$ of $X$:
$$
W(X) = CE(T X) = (\Omega^\bullet(X), d_{dR})
\,.
$$
## Related concepts
* [[∞-Lie algebra cohomology]]
* [[Chevalley-Eilenberg algebra]]
* **Weil algebra**
* [[invariant polynomial]]
* [[Sullivan model of free loop space]]
## References
Among the original references on Weil algebras for ordinary Lie algebras is
* [[Henri Cartan]], _Cohomologie réelle d'un espace fibré principal diffrentielle_ I, II, Séminaire Henri Cartan,
1949/50, pp. 19-01 – 19-10 and 20-01 – 20-11, CBRM, (1950).
and
* {#Cartan50} [[Henri Cartan]], *Notions d'algébre différentielle; application aux groupes de Lie et aux variétés ou opère un groupe de Lie* , Colloque de topologie (espaces fibrs), Bruxelles, (1950), pp. 15–27.
This also explains the use of the Weil algebra in the calculation of the [[equivariant cohomology|equivariant]] [[de Rham cohomology]] of manifolds acted on by a compact group. These papers are reprinted, explained and put in a modern context in the book
A clasical textbook account of standard material is in chapter VI, vol III of
* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)
{#GHV}
Some remarks on the notation there as compared to ours: our $d_W$ is their $\delta_W$ on p. 226 (vol III). Their $\delta_E$ is our $d_{CE}$. Their $\delta_\theta$ is our $d_\rho$ ($\theta$/$\rho$ denoting the representation)..
In the context of [[equivariant de Rham cohomology]]:
* {#AtiyahBott84} [[Michael Atiyah]], [[Raoul Bott]], _The moment map and equivariant cohomology_, Topology 23, 1 (1984) (<a href="https://doi.org/10.1016/0040-9383(84)90021-1">doi:10.1016/0040-9383(84)90021-1</a>, [pdf](https://www.math.stonybrook.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf))
* {#Kalkman93} [[Jaap Kalkman]], _BRST model applied to symplectic geometry_, Ph.D. Thesis, Utrecht, 1993 ([arXiv:hep-th/9308132](https://arxiv.org/abs/hep-th/9308132), [cds:9308132](http://cds.cern.ch/record/568522), [euclid:1104252784](http://projecteuclid.org/euclid.cmp/1104252784))
and with an eye towards [[supersymmetry]]:
* {#Miettinen96} Mauri Miettinen, _Weil Algebras and Supersymmetry_ ([arXiv:hep-th/9612209](https://arxiv.org/abs/hep-th/9612209), [cds:317377](http://cds.cern.ch/record/317377), [spire:427720](http://inspirehep.net/record/427720))
* {#GuilleminSternberg99} [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, (1999) ([doi:10.1007/978-3-662-03992-2](https://link.springer.com/book/10.1007/978-3-662-03992-2))
The (obvious but conceptually important) observation that [[Lie algebra-valued 1-forms]] regarded as morphisms of graded vector spaces $\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A$ are equivalently morphisms of dg-algebras out of the Weil algebra $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$ and that one may think of as the identity $W(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id$ as the _universal $\mathfrak{g}$-connection_ appears in early articles for instance highlighted on p. 15 of
* Franz W. Kamber; Philippe Tondeur, _Semisimplicial Weil algebras and characteristic classes for foliated bundles in Cech cohomology_ , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283--294. Amer. Math. Soc., Providence, R.I., (1975).
A survey of Weil algebras for Lie algebras is also available at
* [[eom|Encyclopedia of Mathematics]]: [Weil algebra of a Lie algebra](http://eom.springer.de/W/w130050.htm)
Weil algebra for [[L-infinity algebra]]s and their role in defining [[invariant polynomial]]s and [[Chern-Simons element]]s on $\infty$-Lie algebras from [[infinity-Lie algebra cohomology|L-infinity algebra cocycle]] are considered in
* {#SSSI} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], *[[schreiber:L-infinity algebra connections|$L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport]]*, in *Quantum Field Theory*, Birkhäuser (2009) 303-424 [[arXiv:0801.3480](https://arxiv.org/abs/0801.3480), [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17)]
The abstract characterization is due to
* {#FreedHopkins13} [[Daniel Freed]], [[Michael Hopkins]], _Chern-Weil forms and abstract homotopy theory_, Bull. Amer. Math. Soc. 50 (2013), 431-468 ([arXiv:1301.5959](http://arxiv.org/abs/1301.5959))
Further discussion of Weil algebras for the [[string Lie 2-algebra]]:
* {#Schmidt19} [[Lennart Schmidt]], _Twisted Weil Algebras for the String Lie 2-Algebra_, in [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.) _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_, [Durham Symposium](http://www.maths.dur.ac.uk/lms/) 2018, Fortschritte der Physik 2019 ([arXiv:1903.02873](https://arxiv.org/abs/1903.02873))
[[!redirects Weil algebras]] |
Weil algebra abstractly -- table | https://ncatlab.org/nlab/source/Weil+algebra+abstractly+--+table |
**[[Chevalley-Eilenberg algebra]] CE $\leftarrow$ [[Weil algebra]] W $\leftarrow$ [[invariant polynomials]] inv**
[[differential forms]] on [[moduli stack]] $\mathbf{B}G_{conn}$ of [[principal connections]] ([Freed-Hopkins 13](Weil+algebra#FreedHopkins13)):
$$
\array{
CE(\mathfrak{g}) &\simeq& \Omega^\bullet_{li \atop cl}(G)
\\
\uparrow && \uparrow
\\
W(\mathfrak{g}) &\simeq & \Omega^\bullet(\mathbf{E}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g}))
\\
\uparrow && \uparrow
\\
inv(\mathfrak{g})
&\simeq&
\Omega^\bullet(\mathbf{B}G_{conn})
& \simeq &
\Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})/G)
}
$$ |
Weil cohomology theory | https://ncatlab.org/nlab/source/Weil+cohomology+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _Weil cohomology theory_ is defined to be a [[cohomology theory]] on a suitable class of [[projective varieties]] which satisfies some natural set of [[axioms]] among which is notably [[Poincaré duality]] and the existence of a [[Lefschetz fixed point theorem]].
These axioms are named after [[André Weil]], who noticed that the existence of such a [[cohomology theory]] would already imply the [[Weil conjectures]] about the behaviour of the number of points in [[algebraic varieties]].
Examples of Weil cohomology theories, hence of [[cohomology theories]] satisfying these axioms, are the variants of [[étale cohomology]] known as
_[[l-adic cohomology]]_ or better _[[pro-étale cohomology]]_.
## Related concepts
* [[basics of étale cohomology]]
* [[motive]], [[motivic cohomology]]
* [[Weil conjecture]], [[standard conjectures on algebraic cycles]]
## References
Reviews are in
* Mircea Mustaţă, _Weil cohomology theories and the Weil conjectures_ [pdf](http://www.math.lsa.umich.edu/~mmustata/lecture5.pdf)
* [[Alain Connes]], [[Matilde Marcolli]], Section 8.1 of _Noncommutative Geometry, Quantum Fields and Motives_ ([pdf](http://www.alainconnes.org/docs/bookwebfinal.pdf))
* Wikipedia, _[Weil cohomology theory](http://en.wikipedia.org/wiki/Weil_cohomology_theory)_
[[!redirects Weil cohomology theories]]
[[!redirects Weil cohomology]] |
Weil conjecture on Tamagawa numbers | https://ncatlab.org/nlab/source/Weil+conjecture+on+Tamagawa+numbers |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Let $q$ be a positive-definite [[quadratic form]] over the ring of [[integers]] $\mathbf{Z}$. The **mass** of $q$ is a weighted count of the number of quadratic forms in the [[genus]] of $q$, up to isomorphism (weighted by multiplicity). The [[Smith-Minkowski-Siegel mass formula]] gives a (complicated but computable) formula for the mass of $q$.
Over [[number fields]], ideas of Tamagawa and Weil allow a reformulation of this formula as the statement that the [[Tamagawa number]] of a certain [[algebraic group]] associated to $q$ is equal to 1. _Weil's conjecture_ is then the statement, now a theorem of [[Robert Langlands]], K. F. Lai and Robert Kottwitz, that the [[Tamagawa number]] of any semisimple simply-connected [[algebraic group]] is equal to 1.
There is [[function field analogy|analogue]] of the conjecture for [[function fields]], and it has been proved by [[Dennis Gaitsgory]] and [[Jacob Lurie]].
## The statement
### Number field case
{#NumberFieldCase}
Let $q$ be a positive-definite [[quadratic form]] over the ring of [[integers]] $\mathbf{Z}$.
+-- {: .num_defn}
###### Definition
The **mass** of $q$ is the sum
\[ \sum_{q'} \frac{1}{|O_{q'}(\mathbf{Z})|} \]
taken over the positive-definite [[quadratic forms]] $q'$ in the genus of $g$.
=--
Let $\mathbf{A}$ be the ring of [[adeles]], a [[locally compact]] [[commutative ring]] containing $\mathbf{Q}$ as a [[discrete]] subring.
+-- {: .num_defn}
###### Definition
Let $O_q(\mathbf{A})$ denote the [[automorphism group]] of $q_\mathbf{A}$, the [[base change]] of $q$ to $\mathbf{A}$. Let $SO_q(\mathbf{A}) \subset O_q(\mathbf{A})$ denote the subgroup of [[automorphisms]] with [[determinant]] 1.
=--
$SO_q(\mathbf{A})$ is a [[locally compact]] [[topological group]] containing $SO_q(\mathbf{Q})$ as a [[discrete]] subgroup and $SO_q(\hat{\mathbf{Z}} \times \mathbf{R})$ as a [[compact]] [[open]] subgroup.
+-- {: .num_theorem}
###### Theorem
**(Tamagawa-Weil reformulation of Siegel mass formula)**.
Let $\mu_{\mathrm{Tam}}$ denote the [[Tamagawa measure]]. Then
\[ \mu_{\mathrm{Tam}}(SO_q(\mathbf{Q}) \backslash SO_q(\mathbf{A})) = 2. \]
Equivalently,
\[ \mu_{\mathrm{Tam}}(Spin_q(\mathbf{Q})\backslash Spin_q(\mathbf{A})) = 1, \]
where $\Spin_q$ is the 2-fold universal cover of $SO_q$.
=--
+-- {: .num_theorem}
###### Theorem
**(Langlands-Lai-Kottwitz, "[[Weil conjecture]]")**.
Let $G$ be a semisimple simply-connected [[algebraic group]] over $\mathbf{Q}$. Then
\[ \mu_{\mathrm{Tam}}(G(\mathbf{Q})\backslash G(\mathbf{A})) = 1. \]
=--
### Function field case
{#FunctionFieldCase}
Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$. Let $K_X$ denote the [[function field]] of $X$. For $x \in X$, write $O_x$ for the [[completion]] of the [[local ring]] at $x$ and $K_x$ for its [[fraction field]].
+-- {: .num_defn}
###### Definition
The **[[ring of adeles]] of $K_X$** is defined as
\[ \mathbf{A}_X = \product^{res}_x K_x \subset \product_x K_x, \]
i.e. the subgroup consisting of elements $\{g_x\}_{x \in X}$ such that $g \in G(O_x)$ for all but finitely many $x$.
=--
$\mathbf{A}_X$ is a [[locally compact]] [[commutative ring]] with [[discrete]] subring $K_X \subset \mathbf{A}_X$.
Let $G_0$ be a semisimple simply-connected linear [[algebraic group]] over $K_X$. Then $G_0(K_X) \subset G_0(\mathbf{A})$ is a [[discrete]] subgroup of the [[locally compact]] group $G_0(\mathbf{A})$. One defines a [[Tamagawa measure]] on $G(\mathbf{A})$ in a similar way as usual, i.e. by choosing a [[differential form]] and multiplying the forms on $G_0(K_x)$ ($x \in X$). Then the [[function field]] version of [[Weil's conjecture]] is
+-- {: .num_theorem}
###### Theorem
**(Gaitsgory-Lurie, "Weil conjecture for function fields")**.
Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$. Then
\[ \mu_{\mathrm{Tam}}(G_0(K_X)\backslash G_0(\mathbf{A}_X)) = 1. \]
=--
This was proved by [[Dennis Gaitsgory]] and [[Jacob Lurie]]. They reformulated the conjecture as a statement about the [[cohomology]] of the [[moduli stack of G-principal bundles]] $Bun_G(X)$ on $X$, in view of the [[function field analogy]]:
+-- {: .num_remark #RelationToModuliStack}
###### Remark
Under the [[function field analogy]], a [[global field]] $K_X$ such as a [[function field]] or a [[number field]] is interpreted as the field of [[global sections]] of the [[rational functions]] on an [[arithmetic curve]] $X$ over a [[finite field]] $\mathbb{F}_q$ or "over $\mathbb{F}_1$" (the would-be [[field with one element]]), respectively. Moreover, under this [[analogy]]
* the [[ring of adeles]] $\mathbb{A}_X$ is the ring of functions on all punctured [[formal disks]] in $X$ subject to the condition that all but at most finitely many of them extend to the un-punctured disk;
* accordingly $G(\mathbb{A}_X)$ is the group of $G$-valued such functions;
* the [[quotient]] $K_{X}\backslash \mathbb{A}_{X}$ is hence the quotient of such functions on punctured formal disks around finitely many points by the functions on $\Sigma$ with these finitely many points removed; and similarly $G(K_X)\backslash G(\mathbb{A}_X)$ is the quotient of group-valued such function;
* the ring $\mathcal{O}$ is the ring of functions on all formal disks in $\Sigma$;
* hence the further double [[quotient stack]]
$$
Bun_G(X) = G(K_X)\backslash G(\mathbb{A}_X)//G(\mathcal{O})
$$
is the [[groupoid]] of [[Cech cohomology|Cech cocycles]] with Cech coboundaries between them for $G$-[[principal bundles]] relative to [[covers]] of $\Sigma$ with patches being the complement of finitely many points and the formal disks around these points.
For more on this see at _[moduli space of bundles -- over curves](http://ncatlab.org/nlab/show/moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_.
=--
First they proved a [[Grothendieck-Lefschetz trace formula]] for $Bun_G(X)$, generalizing work of [[Kai Behrend]]:
+-- {: .num_theorem}
###### Theorem
**(Gaitsgory-Lurie, "Grothendieck-Lefschetz trace formula for $Bun_G(X)$")**.
Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$. Then
\[ \frac{|Bun_G(X)(\mathbf{F}_q)|}{q^{\dim(Bun_G(X))}} = \sum_{i \ge 0} (-1)^i \mathrm{Tr}(Frob^{-1} \mid H^i(\overline{Bun}_G(X) ; \mathbf{Q}_\ell) \]
where $\overline{Bun}_G(X)$ denotes the [[base change]] of $Bun_G(X)$ to the [[algebraic closure]] of $\mathbf{F}_q$, where ${\vert -\vert}$ denotes [[groupoid cardinality]], and where $Frob : \overline{Bun}_G(X) \to \overline{Bun}_G(X)$ denotes the [[Frobenius map]].
=--
Then they proved the following result, via [[nonabelian Poincaré duality]] which provides a [[local-global principle]].
+-- {: .num_theorem}
###### Theorem
**(Gaitsgory-Lurie)**.
Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$. Then
\[ \sum_{i \ge 0} (-1)^i \mathrm{Tr}(Frob^{-1} \mid H^i(\overline{Bun}_G(X) ; \mathbf{Q}_\ell) = \prod_{x \in X} \frac{q^{d \cdot \deg(x)}}{|G(\kappa(x))|} \]
=--
## References
* Wikipedia, _[Weil conjecture on Tamagawa numbers](https://en.wikipedia.org/wiki/Weil_conjecture_on_Tamagawa_numbers)_
### Work of Gaitsgory-Lurie
A proof of the [[function field]] case is discussed in
* {#GaitsgoryLurie} [[Dennis Gaitsgory]], [[Jacob Lurie]], _Weil's conjecture for function fields_ (2014-2017) [["second draft" pdf](http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf), ["first volume of expanded account" pdf](https://www.math.ias.edu/~lurie/papers/tamagawa-abridged.pdf)]
The proof was announced in
* [[Jacob Lurie]], _Tamagawa Numbers via Nonabelian Poincaré Duality_, talk at [FRG Chern-Simons workshop](http://people.mpim-bonn.mpg.de/teichner/Older/FRG-3.html), Jan. 15-17, 2011
and is outlined in the lecture notes
* [[Jacob Lurie]], _Tamagawa Numbers via Nonabelian Poincare Duality (282y)_, lecture notes, 2014 ([web](http://www.math.harvard.edu/~lurie/282y.html))
See also the shorter lecture notes
* [[Jacob Lurie]], _Tamagawa numbers via nonabelian Poincare duality_, 5 lectures at [Young Topologists Meeting 2014](http://www.math.ku.dk/english/research/conferences/2014/ytm2014), notes taken by [[Aaron Mazel-Gee]], [pdf](http://math.berkeley.edu/~aaron/livetex/lurie-tamagawa-poincare.pdf).
### Previous work
The idea of the relationship between [[Tamagawa numbers]] and [[moduli spaces]] of [[vector bundles]] goes back to [[Günter Harder]], who primarily considered the case $G = SL_n$.
* [[Günter Harder]], _Eine Bemerkung zu einer Arbeit von P. E. Newstead._, Journal für die reine und angewandte Mathematik, 242 (1970): 16-25, [eudml](http://eudml.org/doc/151010).
* [[Günter Harder]], M. S. Narasimhan, _On the cohomology groups of moduli spaces of vector bundles on curves_, Mathematische Annalen, 212 (1975), Issue 3, pp 215-248.
The [[moduli stack]] of [[principal bundles]] was studied in more generality in
* [[Kai Behrend]], Ajneet Dhillon, _On the Motive of the Stack of Bundles_, 2005, [arXiv](http://arxiv.org/abs/math/0512640).
* [[Kai Behrend]], Ajneet Dhillon, _Connected components of moduli stacks of torsors via Tamagawa numbers._, Canad. J. Math, 2009, [arXiv](http://arxiv.org/abs/math/0503383).
See also
* Aravind Asok, Brent Doran, Frances Kirwan. _Yang–Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics._ Bulletin of the London Mathematical Society (2008), [arXiv](http://arxiv.org/abs/0801.4733).
[[!redirects Weil conjecture on Tamagawa number]]
[[!redirects Weil conjectures on Tamagawa numbers]]
|
Weil conjectures | https://ncatlab.org/nlab/source/Weil+conjectures | [[!redirects Weil conjecture]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Weil conjectures_ are a sequence of [[conjectures]] about counting the number of points of [[algebraic varieties]] $X$ over [[finite fields]]; $\mathbb{F}_p$ and extensions thereof. That is, the number of [[homomorphisms]]
$$
\operatorname{Spec} \mathbb{F}_{p^n} \longrightarrow X
\,
$$
for fixed $p$ but all $n \geq 1$.
Effectively the conjecture says that the [[generating function]] for the number of points as $n$ varies -- the [[Weil zeta function]] -- is a [[rational function]] with some nice properties.
It was realized that the all except one of the conjectures (the [[Riemann hypothesis]]) would follow formally from the existence of a suitable [[cohomology theory]] on [[algebraic varieties]] which behaves in essential aspects like [[ordinary cohomology]] of [[topological spaces]] and which in particular satisfies a [[Lefschetz fixed point theorem]] - what is now called a [[Weil cohomology theory]].
Later [[Alexander Grothendieck]] found that the relevant [[cohomology]] theory is [[étale cohomology]] of [[schemes]].
## Related concepts
* [[number theory]]
* [[Riemann zeta function]]
* [[Frobenius endomorphism]]
* [[Weil conjecture on Tamagawa number]]
## References
* [[James Milne]], section 26 of _[[Lectures on Étale Cohomology]]_
* [[Sophie Morel]] _The Weil conjectures, from Abel to Deligne_ ([IAS video](http://video.ias.edu/members/2013/1014-SophieMorel)). Note that the title was chosen as a joke by Morel; she clarifies that there is no known connection between Abel and the Weil conjectures.
* Wikipedia, _[Weil conjectures](http://en.wikipedia.org/wiki/Weil_conjectures)_
* [[Dennis Gaitsgory]], [[Jacob Lurie]], _Weil's Conjecture for Function Fields_ ([pdf](http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf))
[[!redirects Weil conjectures]]
[[!redirects Weil's conjecture]]
[[!redirects Weil's conjectures]]
|
Weil divisor | https://ncatlab.org/nlab/source/Weil+divisor |
#Contents#
* table of contents
{:toc}
## Idea
An [[algebraic cycle]] of [[codimension]] 1. For more see at _[[divisor (algebraic geometry)]]_.
## References
* [[James Milne]], section 13 of _[[Lectures on Étale Cohomology]]_
[[!redirects Weil divisors]] |
Weil group | https://ncatlab.org/nlab/source/Weil+group |
> Not to be confused with [[Weyl group]].
***
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Let $F$ be a [[p-adic field]], with [[residue field]] denoted $\kappa$.
The *Weil group* $W_F$ is the [[subgroup]] of the [[Galois group]] $\mathrm{Gal}(\overline{F}/F)$ defined as the [[inverse image]] of [[Frobenius automorphisms]] $\mathrm{Frob}^{\mathbb{Z}}\subset \mathrm{Gal}(\overline{\kappa}/\kappa)$ under the [[surjective]] map $\mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa)$.
## Related entries
* [[Weil-Deligne representation]]
* [[local Langlands conjecture]]
## References
* {#Tate77} [[John Tate]], Section 1 in: *Number theoretic background*, in: *Automorphic forms, representations and L-functions*, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI ([ISBN:978-0-8218-3371-1](https://bookstore.ams.org/pspum-33-2), [pdf](http://www.math.ucsd.edu/~csorense/teaching/math205/Tate_Weil.pdf), [[TateNumberTheory.pdf:file]])
See also:
* Wikipedia, *[Weil group](https://en.wikipedia.org/wiki/Weil_group)*
[[!redirects Weil groups]] |
Weil reciprocity law | https://ncatlab.org/nlab/source/Weil+reciprocity+law |
#Contents#
* table of contents
{:toc}
## Idea
The __Weil reciprocity law__ is a theorem of [[André Weil]] about the [[function field]] $K(C)$ of an [[algebraic curve]] $C$ over an [[algebraically closed field]] $K$.
* Askold Khovanskii, _Logarithmic functional and the Weil reciprocity law_, [pdf](http://www.math.toronto.edu/askold/Khovanskii-zima.pdf)
* enclyclo.co.uk [Weil reciprocity law](http://www.encyclo.co.uk/define/Weil%20reciprocity%20law)
## Properties
### Function field analogy
[[!include function field analogy -- table]]
## References
The following two articles make parallel between some notions of [[QFT]] and of number theory and in particular about the analogy between the Weil reciprocity law for [[function field]]s and the [[Takahashi-Ward identities]] of [[quantum field theory]]:
* [[Leon Takhtajan]], _Quantum field theories on algebraic curves and A. Weil reciprocity law_, [arxiv/0812.0169](http://arxiv.org/abs/0812.0169)
* Leon Takhtajan, _Quantum field theories on an algebraic curve_, [pdf](http://www.math.sunysb.edu/~leontak/Takhtajan-00.pdf), 2000
[[!redirects Weil reciprocity]] |
Weil uniformization theorem | https://ncatlab.org/nlab/source/Weil+uniformization+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _uniformization theorem_ for [[principal bundles]] over [[algebraic curves]] $X$ (going back to [[André Weil]]) expresses the [[moduli stack of principal bundles]] on $X$ as a double [[quotient stack]] of the $G$-valued [[Laurent series]] around finitely many points by the product of the $G$-valued [[formal power series]] around these points and the $G$-valued functions on the complement of theses points.
If a single point $x$ is sufficient and if $D$ denotes the [[formal disk]] around that point and $X^\ast, D^\ast$ denote the complements of this point, respectively then the theorem says for suitable [[algebraic group]] $G$ that there is an equivalence of [[stacks]]
$$
[X^\ast, G] \backslash [D^\ast, G] / [D,G]
\simeq
Bun_X(G)
\,,
$$
between the double [[quotient stack]] of $G$-valued functions ([[mapping stacks]]) as shown on the left and the [[moduli stack of G-principal bundles]] over $X$, as shown on the right.
The theorem is based on the fact that $G$-bundles on $X$ trivialize on the complement of finitely many points and that this double quotient then expresses the $G$-[[Cech cohomology]] with respect to the cover given by the complement of the points and the [[formal disks]] around them.
For details see at _[moduli stack of bundles -- over curves](http://ncatlab.org/nlab/show/moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_.
## Applications
The theorem is a key ingredient in the [[function field analogy]] where for $K$ a [[global field]] the nonabelian generalization of quotients of the [[idele class group]] by [[integral adeles]]
$$
GL_n(K) \backslash GL_n(\mathbb{A}_K) / GL_n(\mathcal{O}_K)
$$
are analogous to the moduli stack of $G$-bundles. This motivates notably the [[geometric Langlands correspondence]] as a geometric analog of the number-theoretic [[Langlands correspondence]].
## Related concepts
* [[differential cohesion and idelic structure]]
## References
Review is for instance
* {#Sorger99} [[Christoph Sorger]], _Lectures on moduli of principal $G$-bundles over algebraic curves_, 1999 ([pdf](http://users.ictp.it/~pub_off/lectures/lns001/Sorger/Sorger.pdf))
See also
* [[Jochen Heinloth]], _Uniformization of $\mathcal{G}$-Bundles_ ([pdf](https://dare.uva.nl/search?identifier=35a7a6fe-9e92-4e37-8c2c-4b357709cc6f))
For more references see at _[[moduli stack of bundles]]_.
[[!redirects Weil uniformization]] |
Weil zeta function | https://ncatlab.org/nlab/source/Weil+zeta+function |
#Contents#
* table of contents
{:toc}
## Idea
The _Weil zeta function_ is a [[zeta function]] for [[arithmetic varieties]] over [[finite fields]] which, under the [[function field analogy]], is [[analogy|analogous]] to the [[Dedekind zeta function]] for [[number fields]] $K$ (and hence of the [[Riemann zeta function]], for $K = \mathbb{Q}$).
The [[Weil conjectures]], and their proof, concern the properties of the Weil zeta function.
## Properties
### Relation to other zeta-, theta- and L-functions
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]
### Function field analogy
[[!include function field analogy -- table]]
## References
Introductions and lecture notes include
* [[Marc Hindry]], section 2.3 of _Introduction to zeta and $L$-functions from arithmetic geometry and some applications_ ([pdf](http://www.math.jussieu.fr/~hindry/Notes_rev_Brasilia.pdf))
* Daqing Wan, _Lectures on zeta functions over finite fields_, 2007 ([pdf](http://www.math.uci.edu/~dwan/gottingen.pdf))
* E. Kowalski, section 1.6 in _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))
[[!redirects Weil zeta functions]]
|
Weil-Deligne representation | https://ncatlab.org/nlab/source/Weil-Deligne+representation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Let
* $F$ be a [[p-adic field]] with $\kappa$ denoting its [[residue field]];
* $\Vert\sigma\Vert$ denotes the [[valuation]] of the corresponding element of $F^{\times}$ under the isomorphism of local [[class field theory]].
\begin{definition}\label{WeilDeligneRepresentation}
**(Weil-Deligne representation)**\linebreak
A *Weil-Deligne representation* is a [[pair]] $(\rho_{0},N)$ where
* $\rho_{0} \colon W_{F} \to GL_{n}(\mathbb{C})$ is a [[linear representation]] of the [[Weil group]] of $F$
and
* $N$ is a nilpotent _monodromy operator_
satisfying
$$
\rho_{0}(\sigma)N\rho_{0}(\sigma)^{-1}
\;=\;
\left\Vert \sigma \right\Vert N
$$
for all $\sigma\in W_{F}$.
\end{definition}
## Related entries
* [[local Langlands correspondence]]
## References
* {#Tate77} [[John Tate]], Section 4 in: *Number theoretic background*, in: *Automorphic forms, representations and L-functions*, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI ([ISBN:978-0-8218-3371-1](https://bookstore.ams.org/pspum-33-2), [pdf](http://www.math.ucsd.edu/~csorense/teaching/math205/Tate_Weil.pdf), [[TateNumberTheory.pdf:file]])
* [[Robin Zhang]], *Weil-Deligne Representations I -- Local Langlands seminar* ([pdf](http://math.columbia.edu/~rzhang/files/Weil-Deligne%20Representations%20I.pdf), [[ZhangWeilDeligneRepresentations.pdf:file]])
[[!redirects Weil-Deligne representations]]
|
Weil-Petersson metric | https://ncatlab.org/nlab/source/Weil-Petersson+metric |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A natural [[Riemannian metric]] on a [[moduli space of curves]] or more generally a [[moduli space of varieties]], such as a [[moduli space of Calabi-Yau spaces]].
## References
* Wikipedia, _[Weil-Petersson metric](https://en.wikipedia.org/wiki/Weil%E2%80%93Petersson_metric)_
[[!redirects Weil-Petersson metrics]] |
Weil-étale site | https://ncatlab.org/nlab/source/Weil-%C3%A9tale+site | [[!redirects Weil-étale topology for arithmetic schemes]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[Grothendieck topology]] (conjectured) for [[arithmetic schemes]] ([Lichtenbaum](#Lichtenbaum)).
Comparison to the standard [[étale site]] is in ([Morin 11](#Morin11)).
## References
* {#Lichtenbaum} [[Stephen Lichtenbaum]], _The Weil-étale topology for Number Rings_, Ann. of Math
* {#Morin11} [[Baptiste Morin]], _On the Weil-étale cohomology of number fields_, Trans. Amer. Math. Soc. 363 (2011), 4877-4927_ ([pdf](http://www.math.uni-muenster.de/reine/u/baptiste.morin/papers/Flask-topos.pdf))
|
Weil-étale topology for arithmetic schemes | https://ncatlab.org/nlab/source/Weil-%C3%A9tale+topology+for+arithmetic+schemes | [[!redirects Weil-étale topology for arithmetic schemes]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[Grothendieck topology]] (conjectured) for [[arithmetic schemes]] ([Lichtenbaum](#Lichtenbaum)).
Comparison to the standard [[étale site]] is in ([Morin 11](#Morin11)).
## References
* {#Lichtenbaum} [[Stephen Lichtenbaum]], _The Weil-étale topology for Number Rings_, Ann. of Math
* {#Morin11} [[Baptiste Morin]], _On the Weil-étale cohomology of number fields_, Trans. Amer. Math. Soc. 363 (2011), 4877-4927 ([pdf](http://www.math.uni-muenster.de/reine/u/baptiste.morin/papers/Flask-topos.pdf))
|
Weimin Chen | https://ncatlab.org/nlab/source/Weimin+Chen |
* [webpage](https://www.math.umass.edu/directory/faculty/weimin-chen)
## Selected writings
Introducing [[Chen-Ruan cohomology]] for [[orbifolds]]:
* {#ChenRuan00} [[Weimin Chen]], [[Yongbin Ruan]], _A New Cohomology Theory for Orbifold_, Commun. Math. Phys. 248 (2004) 1-31 ([arXiv:math/0004129](https://arxiv.org/abs/math/0004129))
On [[mapping stacks]] between [[orbifolds]]:
* [[Weimin Chen]], _On a notion of maps between orbifolds, I. Function spaces_, Commun. Contemp. Math. 8 (2006), no. 5, 569–620 ([doi:10.1142/S0219199706002246](https://doi.org/10.1142/S0219199706002246))
On [[symplectic orbifolds]]:
* [[Weimin Chen]], _Resolving symplectic orbifolds with applications to finite group actions_, Journal of Gökova Geometry Topology **12** (2018), 1-39 ([arXiv:1708.09428](https://arxiv.org/abs/1708.09428))
category: people
|
Weinstein symplectic category | https://ncatlab.org/nlab/source/Weinstein+symplectic+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
When [[symplectic geometry]] is used to model [[mechanics]] in [[physics]], then a [[symplectic manifold]] $(X,\omega)$ encodes the [[phase space]] of a [[mechanical system]] and a [[symplectomorphism]]
$$
\phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2)
$$
encodes a process undergone by this system, for instance the time evolution induced by a [[Hamiltonian vector field]].
Now the [[graph]] of a [[symplectomorphism]] $\phi$ is a [[Lagrangian submanifold]] of the [[Cartesian product]] space $X_1 \times X_2$ regarded as a [[symplectic manifold]] with symplectic form $p_1^\ast \omega_1 - p_2^\ast \omega_2$. In other words, a symplectomorphism $\phi$ as above constitutes a [[Lagrangian correspondence]] between $(X_1,\omega_1)$ and $(X_2, \omega_2)$. See for instance ([Cattaneo-Mnev-Reshetikhin 12](#CattaneoMnevReshetikhin12)) for a review.
This suggests that instead of the [[category]] whose [[objects]] are [[symplectic manifolds]] and whose [[morphisms]] are [[symplectomorphisms]], one might consider a kind of [[category of correspondences]] whose objects are symplectic manifolds, and whose morphisms include [[Lagrangian correspondences]], so that [[composition]] is given by forming the [[fiber product]] along adjacent legs of [[correspondences]].
[[Alan Weinstein]] called this would-be category the _symplectic category_ and suggested that it is the natural [[domain]] for [[geometric quantization]].
However, take at face value, symplectic manifolds with [[Lagrangian correspondences]] between them do not quite form a [[category]], since the usual [[composition]] is only well-defined when the intersection of $L_1 \times L_2 \cap X_1 \times \Delta(X_2) \times X_3$ is [transverse](http://ncatlab.org/nlab/show/transversal+maps).
Proposals for how to rectify this are in [Wehrheim & Woodward](#WehrheimWoodward) and in [Kitchloo](#Kitchloo) (by turning this into an [[(infinity,1)-category]]).
## Refinements
### Prequantum correspondences
{#PrequantumCorrespondences}
A refinement of the symplectic category to [[prequantum geometry]] is the following (see [S 13](#SyntheticQFT)).
Write $\mathbf{B}U(1)_{conn}$ for the [[moduli stack]] of smooth [[circle group]]-[[principal connections]]. Write [[Smooth∞Grpd]] for the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]], and $Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}$ for the corresponding [[slice (∞,1)-topos]]. Finally write
$$
Corr_1(Smooth\infty Grpd,\mathbf{B}U(1)_{conn})
$$
for the [[(∞,1)-category of correspondences]] in $Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}$.
An object in here is a [[prequantum geometry]] $(X,\nabla)$ given by a map
$$
\array{
X
\\
\downarrow^{\mathrlap{\nabla}}
\\
\mathbf{B}U(1)_{conn}
}
\,.
$$
Under the [[curvature]] map $F_{(-)} \colon \mathbf{B}U(1)_{conn} \to \Omega^2_{cl}$ this maps to a [[presymplectic structure]]
$$
(X,\omega) = (X, F_{\nabla})
\,.
$$
If here $\omega$ is non-degenerate, this is a symplectic structure as in Weinstein's symplectic category.
Moreover, a [[morphism]] $(X_1,\nabla_1) \to (X_2,\nabla_2)$ is a [[diagram]] of the form
$$
\array{
&& Z
\\
& {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}}
\\
X_1
&& \swArrow_{\eta} &&
X_2
\\
& {}_{\mathllap{\nabla_1}}\searrow && \swarrow_{\mathrlap{\nabla_2}}
\\
&& \mathbf{B}U(1)_{conn}
}
\,,
$$
hence a [[correspondence]] space (smooth $\infty$-groupoid) $Z$ over $X$ and $Y$ together with an [[equivalence in an (∞,1)-category]]
$$
\eta \colon i_2^\ast \nabla_2 \stackrel{\simeq}{\to} i_1^\ast \nabla_1
\,.
$$
On the underlying [[curvatures]] this implies that
$$
i_2^\ast \omega_2 = i_1^\ast \omega_1
\,.
$$
Hence if $Z \to X \times Y$ is a maximal inclusion with this property, the above diagram is a prequantization of a morphism in the Weinstein symplectic category.
### Motivic stabilization
[[Nitu Kitchloo]] defines the [[stable (infinity,1)-category|stable]] symplectic category $\mathbb{S}$, which has as [[objects]] [[symplectic manifolds]], and [[morphisms]] are certain [[Thom spectra]]
associated to [[Lagrangian correspondences]] $\overline{M} \times N$, where $\overline{M}$ denotes the conjugate with symplectic form $-\omega$. One can view this as a category of symplectic [[motives]].
Considering an oriented version of the category $\mathbb{S}$, there is a canonical [[fiber functor]] $F : M \mapsto \mathbb{S}(pt, M)$, and one may consider the [[motivic Galois group]] $G$ of monoidal [[automorphisms]] of $F$. ([Kitchloo 12, question 8.6, p. 19](#Kitchloo)).
It turns out to have a natural subgroup which is isomorphic to the quotient of the [[Grothendieck-Teichmüller group]].
## Related concepts
* [[Fukaya category]]
* [[Lagrangian correspondences and category-valued TFT]]
## References
{#References}
The way that [[Lagrangian correspondences]] encode [[symplectomorphisms]] in [[symplectic geometry]] and hence evolution in [[mechanics]] is reviewed (and put in the broader context of [[BV-BRST formalism]]) in
* [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical and quantum Lagrangian field theories with boundary_ ([arXiv:1207.0239](http://arxiv.org/abs/1207.0239))
{#CattaneoMnevReshetikhin12}
In his work on Fourier integral operators,
* [[Lars Hörmander]], _Fourier Integral Operators I._, Acta Math. 127 (1971) 79–183.
14
following
* [[Victor Maslov]], _Theory of Perturbations and Asymptotic Methods_, (in Russian) Moskov. Gos. Univ., Moscow (1965).
observed that, under a transversality assumption, the set-theoretic composition of two [[Lagrangian submanifolds]] is again a Lagrangian submanifold, and that this composition is a "[[classical limit]]" of the composition of certain [[linear operators]].
Shortly thereafter,
* [[Jędrzej Śniatycki]], W.M. Tulczyjew, *Generating forms of Lagrangian submanifolds*, Indiana Univ. Math. J. **22** (1972) 267-275
defined symplectic relations as [[isotropic submanifolds]] of products and showed that this class of relations was closed under "clean" composition. Following in part some (unpublished) ideas of [[Alan Weinstein]],
* [[Victor Guillemin]], [[Shlomo Sternberg]], _Some problems in integral geometry and some related problems in microlocal analysis_, Amer. J. Math. 101 (1979), 915–955 ([JSTOR](http://www.jstor.org/stable/2373923))
observed that the linear canonical relations (i.e., lagrangian subspaces of products of [[symplectic vector spaces]]) could be considered as the morphisms of a category, and they constructed a
partial [[quantization]] of this category (in which lagrangian subspaces are enhanced by halfdensities.)
The quantization of the linear symplectic category was part of a larger project of quantizing canonical relations (enhanced with extra structure, such as half-densities) in a functorial way, and this program was set out more formally
* [[Alan Weinstein]], _Symplectic Manifolds and Their Lagrangian Submanifolds_, Advances in Math. **6** (1971) 329346 \[<a href="https://doi.org/10.1016/0001-8708(71)90020-X">doi:10.1016/0001-8708(71)90020-X</a>\]
* [[Alan Weinstein]], _Symplectic geometry_, Bulletin Amer. Math. Soc. **5** (1981) 1-13 [[doi:10.1090/S0273-0979-1981-14911-9](http://dx.doi.org/10.1090/S0273-0979-1981-14911-9)]
See also:
* W.M.Tulczyjew, S.Zakrzewski, _The category of Fresnel kernels_, J. Geom. Phys. 1:3, 1984, 79--120 <a href="https://doi.org/10.1016/0393-0440(84)90021-4">doi</a>
Lecture notes reviewing these developments include
* {#Weinstein09} [[Alan Weinstein]], _Symplectic Categories_, proceedings of Geometry Summer School, Lisbon, July 2009 ([arXiv:0911.4133](http://arxiv.org/abs/0911.4133))
from the introduction of which parts of the commented list of references above is taken. Further review includes
* {#Canez11} Santiago Canez, _Double Groupoids, Orbifolds, and the Symplectic Category_ ([arXiv:1105.2592](http://arxiv.org/abs/1105.2592))
Further refinements in [[higher category theory]]:
* {#WehrheimWoodward} [[Katrin Wehrheim]], [[Chris T. Woodward]], _Functoriality for Lagrangian correspondences in Floer theory_, Quantum Topology **1** 2 (2010) 129-170 [[doi:10.4171/qt/4](https://doi.org/10.4171/qt/4), [arXiv:0708.2851](https://arxiv.org/abs/0708.2851)]
* {#Kitchloo} [[Nitu Kitchloo]], _The Stable Symplectic Category and Geometric Quantization_ [[arXiv:1204.5720](http://arxiv.org/abs/1204.5720)]
* [[Nitu Kitchloo]], [[Jack Morava]], _The Grothendieck--Teichmüller group and the stable symplectic category_, 2012 [[arxiv:1212.6905](http://arxiv.org/abs/1212.6905)]
* [[Damien Calaque]], _Three lectures on derived symplectic geometry and topological field theories_, Indagationes Mathematicae **25** 5 (2014) 926–947 [[doi:10.1016/j.indag.2014.07.005](https://doi.org/10.1016/j.indag.2014.07.005)]
* [[Rune Haugseng]], _Iterated spans and classical topological field theories_, Mathematische Zeitschrift **289** 3 (2018) 1427–1488 [[arXiv](https://arxiv.org/abs/1409.0837), [doi:10.1007/s00209-017-2005-x](https://doi.org/10.1007/s00209-017-2005-x)]
A closed [[symmetric monoidal category]] version of the symplectic category and the observation that this hence is a [[categorical semantics]] for [[quantum logic]] qua [[linear logic]] is in
* {#Slavnov05} [[Sergey Slavnov]], _From proof-nets to bordisms: the geometric meaning of multiplicative connectives_, Mathematical Structures in Computer Science __15__:06 (2005) 1151--1178
* [[Sergey Slavnov]], _Geometrical semantics for linear logic (multiplicative fragment)_, Theoretical Computer Science 357, no. 1--3 (2006) 215--229 [doi](https://doi.org/10.1016/j.tcs.2006.03.020)
Remarks about refinements to correspondences of smooth $\infty$-groupoids in the slice over prequantum moduli is in
* {#SyntheticQFT} [[Urs Schreiber]], _[[schreiber:Synthetic Quantum Field Theory]]_, Talk at [CMS Summer Meeting 2013](http://cms.math.ca/Events/summer13/)
String diagrams for the linear and affine Weinstein category using graphical linear algebra
* Cole Comfort, [[Aleks Kissinger]], _A Graphical Calculus for Lagrangian Relations_, In Proceedings ACT 2021. [doi](https://doi.org/10.4204/EPTCS.372.24)
[[!redirects stable symplectic category]]
[[!redirects symplectic category]]
[[!redirects motivic symplectic category]]
|
Weinstein, the geometry of Lubin-Tate spaces | https://ncatlab.org/nlab/source/Weinstein%2C+the+geometry+of+Lubin-Tate+spaces | category: reference
This entry in a note concerning the text
* [[Jared Weinstein]], the geometry of Lubi-Tate spaces, Lecture 1: Formal groups and formal modules, [pdf](http://www.math.ias.edu/~jaredw/FRGLecture.pdf)
## Table of contents
* Motivation: the local Kronecker-Weber theorem.
* Formal groups and formal $O_F$ -modules
* The invariant differential form, and the logarithm.
* Formal groups: Categorical definition.
* The de Rham complex, and the module $D(G /A)$.
* The crystalline nature of $H^1_{dR}$
* The action of Frobenius.
* The Dieudonné module.
* More on Dieudonné theory (but not quite enough).
* The Grothendieck-Messing crystal. |
Weiping Zhang | https://ncatlab.org/nlab/source/Weiping+Zhang |
* [GoogleScholar page](https://scholar.google.com/citations?user=Jx6VIsMAAAAJ)
## Selected writings
On the [[Witten genus]] in the case of [[string^c structure]]:
* {#ChenHanZhang10} [[Qingtao Chen]], [[Fei Han]], [[Weiping Zhang]], _Generalized Witten Genus and Vanishing Theorems_, Journal of Differential Geometry 88.1 (2011): 1-39. ([arXiv:1003.2325](http://arxiv.org/abs/1003.2325))
On the [[Green-Schwarz mechanism]] in [[heterotic string theory]] and [[Hořava-Witten theory]]:
* [[Fei Han]], [[Kefeng Liu]], [[Weiping Zhang]], _Anomaly Cancellation and Modularity. II: $E_8 \times E_8$ case_, Sci. China Math. 60, 985–994 (2017) ([arXiv:1209.4540](https://arxiv.org/abs/1209.4540), [doi:10.1007/s11425-016-9034-1](https://doi.org/10.1007/s11425-016-9034-1))
On [[gravitino]] [[anomaly cancellation]] in [[D=11 N=1 supergravity]]:
* {#HanHuangLiuZhang20} [[Fei Han]], [[Ruizhi Huang]], [[Kefeng Liu]], [[Weiping Zhang]], *Cubic forms, anomaly cancellation and modularity*, Advances in Mathematics **394** (2022) 108023 [[arXiv:2005.02344](https://arxiv.org/abs/2005.02344), [doi:10.1016/j.aim.2021.108023](https://doi.org/10.1016/j.aim.2021.108023)]
category: people
|
Weiss topology | https://ncatlab.org/nlab/source/Weiss+topology |
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###Context###
#### Goodwillie calculus
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[[!include Goodwillie calculus - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _Weiss topology_ is a type of [[Grothendieck topology]] that provides a [[site]] of definition for [[(infinity,1)-toposes]] as they appear in [[manifold calculus]].
## Related concepts
* [[Goodwillie calculus]]
* [[manifold calculus]]
## References
* {#BriWei} [[Pedro Boavida de Brito]], [[Michael Weiss]], _Manifold calculus and homotopy sheaves_, Homology, Homotopy and Applications, vol. 15(2), 2013, pp.361–383 ([arXiv:1202.1305](http://arxiv.org/abs/1202.1305))
* [MO discussion](http://mathoverflow.net/a/230293/381) |
Weitzenböck connection | https://ncatlab.org/nlab/source/Weitzenb%C3%B6ck+connection |
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#### Riemannian geometry
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[[!include Riemannian geometry - contents]]
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#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
On a ([[pseudo-Riemannian manifold|pseudo]])[[Riemannian manifold]] the _Weitzenböck connection_ is the [[affine connection]] on the [[tangent bundle]] which has vanishing [[curvature]] but possibly non-vanishing [[torsion]].
This is in contrast to the [[Levi-Civita connection]], for which it is the other way around.
[[!redirects Weitzenboeck connection]]
[[!redirects Weitzenböck connections]]
[[!redirects Weitzenboeck connections]] |
Weizhe Zheng | https://ncatlab.org/nlab/source/Weizhe+Zheng |
* [webpage](https://server.mcm.ac.cn/~zheng/)
## Selected writings
On gluing (descent) of [[pseudofunctors]] ([[2-functors]]):
* {#Zheng12} [[Weizhe Zheng]], prop. 1.6 of: _Gluing pseudofunctors via $n$-fold categories_, J. Homotopy Relat. Struct. **12** 189–271 (2017) ([arXiv:1211.1877](http://arxiv.org/abs/1211.1877), [doi:10.1007/s40062-016-0126-2](https://doi.org/10.1007/s40062-016-0126-2))
category: people
|
well group | https://ncatlab.org/nlab/source/well+group |
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###Context###
#### Algebraic topology
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[[!include algebraic topology - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
In the context of [[persistent homology]], the _well group_ ([EMP 11](#EMP11)) of a [[continuous function]] $f$ into a [[metric space]] is a [[homology|homological]] measure of how robust the [[level sets]] of $f$ are against [[deformations]] of $f$. Concretely, the Well group at [[radius]] $r$ of a given point in the [[codomain]] consists of all those [[Čech homology]]-classes of the domain which are reflected in the [[level sets]] of _every_ [[continuous function]] whose values differ from those of $f$ at most by $r$.
As the level varies, the collection of well groups form a [[zigzag persistence module]], also called a *well module* ([EMP 11, Sec. 3](#EMP11)).
Since a well group becomes [[trivial group|trivial]] as soon as one of the [[level sets]] is [[empty set|empty]], the non-triviality of a well group proves that the existence of a [[inhabited set|non-empty]] [[level set]] of $f$ is *robust* within deformations of size $\lt r$. In [[topological data analysis]] this may be used to detect if there are guaranteed to be any data points at all meeting a certain target of indicator values, known with limited precision, see [there for more](topological+data+analysis#StrategyOfPersistentCohomotopy).
But well groups are known not to resolve all relevant cases and are not known to be [[computability|computable]] in all relevant cases ([Franek & Krčál 2016](#FranekKRcal16)). An enhancement of well groups (from [[ordinary homology|homology]] to [[cohomotopy]]) which fixes these problems is [[persistent cohomotopy]] ([Franek & Krčál 2017](#FranekKRcal17), [2018](#FranekKRcal18)).
## Definition
Given a [[continuous function]] $f$ to a [[Euclidean space]] and a choice of [[topological subspace]] $A$ of the latter
$$
\array{
f^{-1}(A) &\subset & X
\\
\downarrow && \big\downarrow {}^{\mathrlap{f}}
\\
A &\subset& \mathbb{R}^n
}
$$
the _well groups_ at [[radius]] $r \in (0,\infty)$ are the [[intersections]] of the [[Čech homology]] [[homology groups|groups]] of the [[pre-images]] $g^{-1}(A) \subset X$ for all [[continuous functions]] $X \overset{g}{\to} \mathbb{R}^n$ whose [[maximum|maximal]] [[distance]] from $f$ is $\left \vert g-f\right \vert \leq r$.
$$
W_\bullet(f,r)
\;\coloneqq\;
\underset{
{X \overset{g}{\to} \mathbb{R}^n}
\atop
{\left \vert g-f\right \vert \leq r}
}{\bigcap}
image
\Big(
H_\bullet\big(
g^{-1}(A)
\big)
\overset{
H_\bullet\big( g^{-1}(A) \subset X \big)
}{\longrightarrow}
H_\bullet
\big(
X
\big)
\Big)
$$
(e.g. [Franek-Krčál 16, p. 2](#FranekKrcal16))
\begin{remark}\label{NeedForCechHomology}
Since the [[preimages]] $g^{-1}(A)$ need not be [[CW-complexes]], it is important to use [[Čech homology]] in the above definition. With [[singular homology]] the definition would trivialize ([FK16, p. 3 and Sec. 2](#FranekKrcal16)).
\end{remark}
## Related concepts
* [[persistence module]]
* [[zigzag persistence]]
* [[stability of persistence diagrams]]
## References
The concept of [[well groups]] was introduced in
* {#EMP11} [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *Quantifying Transversality by Measuring the Robustness of Intersections*, Foundations of Computational Mathematics, **11** 3 (2011) 345–361 $[$[arXiv:0911.2142](https://arxiv.org/abs/0911.2142)$]$
* [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *The Robustness of Level Sets*, In: M. de Berg, U. Meyer (eds.) _Algorithms – ESA 2010_. ESA 2010. Lecture Notes in Computer Science **6346** Springer (2010) $[$[doi:10.1007/978-3-642-15775-2_1](https://doi.org/10.1007/978-3-642-15775-2_1)$]$
* [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *Homology and Robustness of Level and Interlevel Sets*, Homology, Homotopy and Applications, **15** (2013) 51-72 $[$[euclid:1383943667](https://projecteuclid.org/euclid.hha/1383943667)$]$
Review in:
* [[Sara Kališnik]], Section 4.2.2 of: _Persistent homology and duality_ (2013) ([pdf](http://www.matknjiz.si/doktorati/2013/Kalisnik-14521-4.pdf), [[KalisnikPersistent.pdf:file]])
Review, computational analysis and discussion of ([[persistent Cohomotopy|persistent]]) [[Cohomotopy]] as an improvement over homology well groups:
* {#FranekKrcal16} [[Peter Franek]], [[Marek Krčál]], _On Computability and Triviality of Well Groups_, Discrete Comput Geom (2016) 56: 126 ([arXiv:1501.03641](https://arxiv.org/abs/1501.03641), [doi:10.1007/s00454-016-9794-2](https://doi.org/10.1007/s00454-016-9794-2))
* [[Peter Franek]], [[Marek Krčál]], _Persistence of Zero Sets_, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 ([arXiv:1507.04310](https://arxiv.org/abs/1507.04310), [doi:10.4310/HHA.2017.v19.n2.a16](http://dx.doi.org/10.4310/HHA.2017.v19.n2.a16))
* [[Peter Franek]], [[Marek Krčál]], [[Hubert Wagner]], _Solving equations and optimization problems with uncertainty_, J Appl. and Comput. Topology (2018) 1: 297 ([arxiv:1607.06344](https://arxiv.org/abs/1607.06344), [doi:10.1007/s41468-017-0009-6](https://doi.org/10.1007/s41468-017-0009-6))
Survey:
* [[Peter Franek]], [[Marek Krčál]], _Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory_, talk at [ACAT meeting 2015](https://www2.ist.ac.at/acat) ([pfd slides](https://www2.ist.ac.at/fileadmin/user_upload/events_pages/acat/ACAT2015_Marek_Krcal.pdf))
[[!redirects well groups]]
[[!redirects well module]]
[[!redirects well modules]]
|
well-connected space | https://ncatlab.org/nlab/source/well-connected+space | A *well-connected* [[topological space]] is one that satisfies sufficiently strong local connectivity assumptions. These are usually defined in terms of a basis of open sets or a basis of neighbourhoods of the space. Note that this term is not universally defined, but is a placeholder for possibly more complicated adjectives that depend on the situation at hand.
## Examples ##
* A [[locally path-connected space]] has the nice property that path-components are the same as components. It is also true in this case that $X^I \to X\times X$ is an open map.
* A locally path-connected and [[semi-locally simply-connected space]] admits a [[universal covering space]].
* The path fibration $P X \to X$ of a [[path-connected space]] admits local sections if and only if $X$ is [[locally contractible space|locally relatively contractible]].
* [[CW complex]]es are the \'most\' well-connected spaces, having a basis of open sets which are themselves contractible as spaces.
[[!redirects well-connected space]]
[[!redirects well-connected topological space]]
[[!redirects well connected space]]
[[!redirects well connected topological space]]
[[!redirects well-connected spaces]]
[[!redirects well-connected topological spaces]]
[[!redirects well connected spaces]]
[[!redirects well connected topological spaces]] |
well-founded coalgebra | https://ncatlab.org/nlab/source/well-founded+coalgebra |
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#### Algebra
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[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
The notion of _well-founded coalgebra_ is due to [[Paul Taylor]] (with antecedents in the work of Gerhard Osius). Our account is largely based on ([Taylor, section 6.3)](#Taylor99)), and on his paper [The General Recursion Theorem](#tTaylor96), although in some cases we work with slightly different hypotheses.
## Definition
Let $C$ be a [[finitely complete category]], and let $T$ be an [[endofunctor]] on $C$. We will suppose that $T$ is [[taut functor|taut]], i.e., preserves [[pullbacks]] of [[monomorphisms]] (preserves [[limits]] of [[cospans]] in which one of the cospan arrows is [[monomorphism|monic]]). In particular, this implies that $T$ preserves monos. A helpful example to keep in mind is the covariant power-set functor $P: Set \to Set$.
+-- {: .un_def}
######Definition
Let $\theta: X \to T X$ be a $T$-[[coalgebra for an endofunctor|coalgebra]] structure on $X$. A [[subobject]] $i: U \hookrightarrow X$ in $C$ is $\theta$-**inductive** if in the pullback
$$\array{
H & \stackrel{j}{\to} & X \\
\downarrow & & \downarrow^\mathrlap{\theta} \\
T U & \underset{T i}{\to} & T X
}$$
the map $j$ factors through $i$ (note that $j$ is monic since $T$ preserves monos).
A coalgebra $(X, \theta)$ is **well-founded** if the only inductive subobject of $X$ is $X$ itself.
=--
**Example:** Suppose $X$ is an initial algebra for the endofunctor $T$, with algebra structure map $\alpha: T X \to X$. By Lambek's theorem, $\alpha$ is invertible, so that $X$ carries a coalgebra structure $\theta = \alpha^{-1}: X \to T X$. It is easy to check that an inductive subobject gives a subalgebra $i: U \to X$, but a subalgebra of an initial algebra must be the initial algebra itself. Hence $(X, \theta)$ is well-founded.
Our goal in this article is to show that one can perform inductive arguments and recursive constructions in the abstract context of well-founded coalgebras. An interesting challenge is to make precise what is meant by a recursive construction of a map $\phi: X \to A$, where the problem is to show how to build $\phi$ from the ground up, as it were. Stages in the recursive construction of a morphism will be _partial_ maps, defined as usual as spans
$$\array{
& & D & & \\
& ^\mathllap{i} \swarrow & & \searrow^\mathrlap{f} \\
A & & & & B
}$$
for which the arrow $i$ is _monic_. Composition of partial maps is effected by span composition, for which we need only finite cocompleteness (we do not need $C$ to be a [[regular category]], as we would if we were composing general relations between $A$ and $B$). Notice that since $T$ preserves the pulling back of monos, it carries partial maps to partial maps and also preserves partial map composition.
We denote a partial map by a dashed arrow
$$f: A \dashrightarrow B$$
(generally without explicitly mentioning the domain $D$ of $f$), or sometimes by a harpoon notation.
### Connection with initial algebras
One way of constructing the [[initial algebra of an endofunctor]], $(X, \alpha: T X \to X)$, is by constructing first some [[fixed point]] of $T$, that is, an object $Y$ together with an isomorphism $\xi: Y \cong T Y$. (For example, it might be the terminal coalgebra, whose existence is sometimes easy to establish.) Then, inside $Y$ consider the system of well-founded subcoalgebras of $\xi$. The colimit of this system, assuming it exists, will be the initial algebra. (More theory to develop here.)
The connection with initial algebras goes a little further. Firstly, an initial $T$-algebra is a *Peano $T$-algebra* in the sense of the following definition:
+-- {: .num_defn}
###### Definition
A $T$-algebra $(X, \alpha: T X \to X)$ is *semi-Peano* if every $T$-subalgebra [[monomorphism|inclusion]] $i: Y \hookrightarrow X$ is an isomorphism, and *Peano* if in addition $\alpha$ is an isomorphism.
=--
If $X$ is initial and $i: Y \to X$ is a subalgebra, then there is a unique algebra map $r: X \to Y$, and $i r = 1_X: X \to X$ by initiality, whence $i r i = i$ and then $r i = 1_Y: Y \to Y$ as $i$ is monic. Hence initial algebras are semi-Peano, and Peano by Lambek's theorem.
Secondly, a functor $T: E \to E$ induces, for any object $X$ of $E$, a functor between slices $T_\ast: E/X \to E/ T X$, and so if $(X, \theta: X \to T X)$ is a coalgebra, we may form an endofunctor on $E/X$:
$$E/X \stackrel{T_\ast}{\longrightarrow} E/ T X \stackrel{\theta^\ast}{\longrightarrow} E/X$$
and of course the terminal object $1_X: X \to X$ is automatically and uniquely a $\theta^\ast T_\ast$-algebra. The next two propositions then follow immediately from the definitions of inductive subobject and well-founded coalgebra:
+-- {: .num_prop}
###### Proposition
A subobject $i: U \to X$ of a $T$-coalgebra $(X, \theta)$ is inductive precisely when $i$ is a $\theta^\ast T_\ast$-subalgebra of the terminal object $1_X$ of $E/X$.
=--
+-- {: .num_prop}
###### Proposition
A $T$-coalgebra $X$ is well-founded precisely when the terminal $\theta^\ast T_\ast$-algebra $1_X$ is semi-Peano.
=--
## Examples
### Well-founded relations
The prototype of the notion of well-founded coalgebra is a well-founded [[relation]], which is essentially the same thing as a well-founded coalgebra over the covariant power-set functor $P: Set \to Set$. In brief, a binary [[relation]] $\prec$ on a set $X$ corresponds to a coalgebra $\theta: X \to P X$, by saying $y \prec x$ if and only if $y \in \theta(x)$. See [[well-founded relation]] for more information.
Most of the constructions of this article are well-illustrated by checking them against the background of this prototypical case.
## Induction and recursion
"One _proves_ things by [[induction]]; one _defines_ things by [[recursion]]." This slogan is not mere pedantry; it is meant to underline a difference between these processes.
A [[proof]] by induction, say a proof of a [[property]] or [[predicate]] $R(x)$ where $x$ varies over a domain $X$, proceeds by showing that the subobject $i: U \hookrightarrow X$ defined by $R$ is an inductive subset with respect to a relation on $X$. It follows that $R$ is universally true on $X$, if the relation is well-founded. The same idiom applies more generally to well-founded coalgebras.
A recursive construction on the other hand involves a _dependency_ on prior stages of the construction. A typical application is to define a map $f: X \to A$ by recursion with respect to a well-founded relation, where $f(x)$ is specified in three stages:
* Consider the collection $\theta(x) = \{y: y \prec x\}$ of all elements preceding $x$;
* Pass to the values $f(y)$ defined earlier in the construction, giving a subset $P(f)(\theta(x)) = \{f(y): y \prec x\}$ of $A$;
* Apply a given operation $\phi: P(A) \to A$ to this subset $(P(f) \circ \theta)(x)$, to obtain $f(x)$.
In the last step, the operation may be only partially defined on $P(A)$. In fact, the map $f$ itself may be only partially defined; $f(x)$ is defined only if $\phi((P(f) \circ \theta) (x))$ is defined "when we call for it".
An inductive argument is used to show that $f$, so far as it is defined, is uniquely determined. The recursive equation that uniquely determines $f$ (to the extent that it exists, of course) is
$$f = \phi \circ P(f) \circ \theta.$$
Letting $\downarrow x$ denote the down-closure of an element $x \in X$ with respect to $\prec$ (and containing $x$), we may then form the set
$$\{x \in X: \exists_{g: \downarrow x \to A} (g = \phi \circ P(g) \circ \theta)\}$$
whence the existence of a map $f: X \to A$ satisfying the recursive equation might indeed be proven by appeal to an inductive argument.
However, notice that this set is defined by a construction in [[dependent type theory]]. For other categories $C$, we might not have the luxury of interpreting dependent types, so there it wouldn't be right to conflate recursion with induction.
It is nevertheless true that one can prove, in rather considerable generality, both an induction principle and recursion theorem for well-founded coalgebras; this will occupy us in the following sections.
## Related concepts
* [[well-founded relation]]
## References
* [[Paul Taylor]], _Practical Foundations of Mathematics_ , Cambridge University Press (1999).
{#Taylor99}
* [[Paul Taylor]], _Towards a unified treatment of induction_ , I: the general recursion theorem (1996). ([pdf](http://www.paultaylor.eu/ordinals/towuti.pdf))
{#Taylor96}
[[!redirects well-founded coalgebras]] |
well-founded relation | https://ncatlab.org/nlab/source/well-founded+relation |
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### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea {#idea}
A (binary) [[relation]] $\prec$ on a [[set]] $S$ is called _well-founded_ if it is valid to do [[induction]] on $\prec$ over $S$.
## Definitions {#defn}
Let $S$ be a [[set]], and let $\prec$ be a [[binary relation]] on $S$. A [[subset]] $A$ of $S$ is __$\prec$-inductive__ if
$$ \forall (x\colon S),\; (\forall (t\colon S),\; t \prec x \;\Rightarrow\; t \in A) \;\Rightarrow\; x \in A. $$
The relation $\prec$ is __well-founded__ if the only $\prec$-inductive subset of $S$ is $S$ itself.
Note that this is precisely what is necessary to validate induction over $\prec$: if we can show that a statement is true of an element $x$ of $S$ whenever it is true of everything that precedes ($\prec$) $x$, then it must be true of everything in $S$. In the presence of [[excluded middle]] it is equivalent to other commonly stated definitions; see _Formulations in classical logic_ below.
### Formulations in classical logic
While the definition above follows how a well-founded relation is generally *used* (namely, to prove properties of elements of $S$ by induction), it is complicated. Two alternative formulations are given by the following:
1. The relation $\prec$ has __no infinite descent__ (usually attributed to [[Pierre de Fermat]]) if there exists no [[sequence]] $\cdots \prec x_2 \prec x_1 \prec x_0$ in $S$. (Such a sequence is called an _infinite descending sequence_.)
2. The relation $\prec$ is __classically well-founded__ if every [[inhabited set|inhabited]] subset $A$ of $S$ has a member $x \in A$ such that no $t \in A$ satisfies $t \prec x$. (Such an $x$ is called a _minimal_ element of $A$.)
In [[classical mathematics]], both of these conditions are equivalent to being well-founded. In constructive mathematics, we may prove that a well-founded relation has no infinite descent (see Proposition \ref{empty}), but not the converse, and that a classically well-founded relation is well-founded (see Proposition \ref{classical}), but not the converse.
The classical notion of well-foundedness enforces classical logic onto us, in the following sense.
+-- {: .num_prop #LEM}
###### Proposition
If $(X, \prec)$ is an [[inhabited subset|inhabited]] well-founded relation in a classical sense, then the unrestricted [[excluded middle|law of excluded middle]] holds.
=--
For a [[topos]]-theoretic proof see [here](http://ncatlab.org/toddtrimble/published/classical+well-foundedness).
+-- {: .proof}
###### Proof
Suppose there are $x$ and $y$ such that $y\prec x$, and let $Q$ be an arbitrary proposition. Consider then a set $P \subset X$ defined as $P = \{ x \} \cup \{ a \mid a \prec x \;\wedge\; Q\}$. Clearly, the set $P$ is inhabited, thus by classical well-foundedness it has a minimal element $x_0$. By intuitionistic reasoning, either $x_0$ is in $\{ x \}$, i.e. $x_0 = x$, or $x_0 \in \{ a \mid a \prec x \;\wedge\; Q\}$, i.e. $x_0 \prec x \;\wedge\; Q$. In the latter case we immediately see that $Q$ holds. So, suppose that $x_0 = x$ is the minimal element of $P$; we will show that $\neg Q$ holds. For suppose that $Q$ holds; then $y \in P$ and $y \prec x = x_0$, violating the condition that $x_0$ is a minimal element of $P$.
Since $Q$ was an arbitrary proposition, we can deduce $\forall Q. (Q \vee \neg Q)$.
=--
+-- {: .num_remark #LEM_rem}
###### Remark
We note that classical well-foundedness is really too strong for constructive (i.e., intuitionistic) mathematics, due to Proposition \ref{LEM}. On the other hand, the infinite descent condition is too weak to be of much use in constructive mathematics. It is the inductive notion of well-foundedness that is just right.
=--
Note however that in [[predicative mathematics]], the definition of well-founded may be impossible to even state, and so either of these alternative definitions would be preferable, provided classical logic is used.
Even in constructive predicative mathematics, (1) is strong enough to establish the [[Burali-Forti paradox]] (when applied to [[linear orders]]). In [[material set theory]], (2) is traditionally used to state the [[axiom of foundation]], although the impredicative definition could also be used as an axiom scheme (and must be in constructive versions). In any case, either (1) or (2) is usually preferred by classical mathematicians as simpler.
To round out the discussion we prove the following two results, both valid in intuitionistic mathematics:
+-- {: .num_prop #empty}
###### Proposition
If $(X, \prec)$ is a well-founded relation and $A \subseteq X$ has no minimal element, then $A$ is empty.
=--
This result makes it trivial to infer (under classical logic) that classical well-foundedness is a consequence of well-foundedness. It also shows that well-foundedness rules out infinite descent (intuitionistically), since an infinite descent sequence has no minimal element.
+-- {: .proof}
###### Proof
Let $U = \{u \in X: u \notin A\; \wedge \; (\forall x: X),\; x \prec u \Rightarrow x \notin A\}$. Clearly $U \cap A = \emptyset$. We show $U$ is inductive, so that under well-foundedness $U = X$ and $A = X \cap A = U \cap A = \emptyset$, as desired.
So, suppose $z$ is an element such that $y \in U$ whenever $y \prec z$. We must show $z \in U$. Claim: $z \notin A$. For if $z \in A$, then $z$ would be a minimal element of $A$ (as $y \prec z \Rightarrow y \in U \Rightarrow y \notin A$). But this negates the assumption that $A$ has no minimal element.
Thus $z \notin A$, and $y \prec z \Rightarrow y \in U \Rightarrow y \notin A$, so that $z \in U$. This completes the proof.
=--
+-- {: .num_prop #classical}
###### Proposition
In intuitionistic set theory, classical well-foundedness implies (inductive) well-foundedness.
=--
+-- {: .proof}
###### Proof
Let $\prec$ be a classically well-founded relation on $X$, and let $U$ be an inductive subset. We must show that every element $x \in X$ belongs to $U$. Since $U$ is inductive, it suffices to show that every $u \prec x$ belongs to $U$, i.e. we may assume given a $u$ such that $u\prec x$ and show $u\in U$. But under this assumption we have that $\prec$ is inhabited, so according to Remark \ref{LEM}, the law of [[excluded middle]] follows and we might as well then argue classically. The argument is well-known but we include it for completeness: under classical well-foundedness, if an inductive subset $U$ is not the entirety of $X$, then the complement $\neg U$ has a minimal element $y$. In that case, $v \prec y$ implies $v \in \neg\neg U = U$, but then $y \in U$ since $U$ is inductive, contradiction. Hence $U = X$ and in particular $u \in U$, which is what we wanted.
=--
To bring us full circle: in classical set theory we may prove that if $(X, \prec)$ has no infinite descent, then $\prec$ is classically well-founded. For suppose an inhabited subset $P \subseteq X$ (say with an element $x \in P$) failed to have a least element. Then we can find an infinite descent sequence $x_n \in P$ with $x_0 = x$, by choosing at each stage $x_{n+1} \in P$ such that $x_{n+1} \prec x_n$. Technically this requires the use of [[dependent choice]], but generally this is felt to be a mild choice principle (that is true even for intuitionistic mathematics).
### Coalgebraic formulation
Many inductive or recursive notions may also be packaged in [[coalgebra|coalgebraic]] terms. For the concept of well-founded relation, first observe that a binary relation $\prec$ on a set $X$ is the same as a coalgebra structure $\theta\colon X \to P(X)$ for the covariant [[power-set]] endofunctor on $Set$, where $y \prec x$ if and only if $y \in \theta(x)$.
In this language, a [[subset]] $i\colon U \hookrightarrow X$ is $\prec$-inductive, or $\theta$-inductive, if in the [[pullback]]
$$\array{
H & \stackrel{j}{\to} & X \\
\downarrow & & \downarrow^\mathrlap{\theta} \\
P U & \underset{P i}{\to} & P X
}$$
the map $j$ factors through $i$. (Note that $j$ is necessarily [[monomorphism|monic]], since $P$ preserves monos.) Unpacking this a bit: for any $x \in X$, if $\theta(x) = V$ belongs to $P U$, that is if $\theta(x) \subseteq U$, then $x \in U$. This says the same thing as $\forall_{x\colon X} (\forall_{y\colon X} y \prec x \Rightarrow y \in U) \Rightarrow x \in U$.
Then, as usual, the $P$-coalgebra $(X, \theta)$ is well-founded if every $\theta$-inductive subset $U \hookrightarrow X$ is all of $X$.
Other relevant notions may also be packaged; for example, the $P$-coalgebra $X$ is [[extensional relation|extensional]] if $\theta\colon X \to P X$ is monic. See also [[well-founded coalgebra]].
### Simulations
Given two sets $S$ and $T$, each equipped with a well-founded relation $\prec$, a [[function]] $f\colon S \to T$ is a __[[simulation]]__ of $S$ in $T$ if
1. $f(x) \prec f(y)$ whenever $x \prec y$ and
1. given $t \prec f(x)$, there exists $y \prec x$ with $t = f(y)$.
Then sets so equipped form a [[category]] with simulations as [[morphisms]]. See [[extensional relation]] for more uses of simulations.
For example, a simulation $X \to Y$ between two [[well-ordered sets]] is an isomorphism of $X$ onto an [[lower set|initial segment]] of $Y$.
In coalgebraic language, a simulation $S \to T$ is simply a $P$-coalgebra homomorphism $f\colon S \to T$. Condition (1), that $f$ is merely $\prec$-preserving, translates to the condition that $f$ is a [[colax morphism]] of coalgebras, in the sense that there is an inclusion
$$\array{
X & \stackrel{\theta_X}{\to} & P X \\
^\mathllap{f} \downarrow & \swArrow & \downarrow^\mathrlap{P f} \\
Y & \underset{\theta_Y}{\to} & P Y.
}$$
+-- {: .num_prop}
###### Proposition
If $(X, \prec)$ is well-founded and $(Y, \prec)$ is (weakly) extensional, then there is at most one simulation $f: X \to Y$.
=--
+-- {: .proof}
###### Proof
It is enough to show that if $f, g: X \to Y$ are $P$-coalgebra morphisms, then $U = \{u \in X: f(u) = g(u)\}$ is an inductive subset. Suppose $y \in X$ is an element such that $f(u) = g(u)$ whenever $u \prec y$. Then $f(\{u: u \prec y\}) = g(\{u: u \prec y\})$; since $f, g$ are $P$-coalgebra maps, this is the same as the equality $\{x: x \prec f(y)\} = \{x: x \prec g(y)\}$. By extensionality, we infer $f(y) = g(y)$, so that $y \in U$. Thus $U$ is inductive.
=--
## Properties
Every well-founded relation is [[irreflexive relation|irreflexive]]; that is, $x \nprec x$. Sometimes one wants a reflexive version $\preceq$ of a well-founded relation; let $x \preceq y$ if and only $x \prec y$ or $x = y$. Then the requirement that $x$ be a minimal element of a subset $A$ states that $t \preceq x$ only if $t = x$. But infinite descent or direct proof by induction still require $\prec$ rather than $\preceq$.
A [[well-order|well order]] may be defined as a well-founded [[linear order]], or alternatively as a [[transitive relation|transitive]], [[extensional relation|extensional]], well-founded relation.
A [[well-quasi-order]] is a well-founded [[preorder]] (referring to the reflexive version of well-foundedness above) that in addition has no infinite [[antichain|antichains]].
The [[axiom of foundation]] in material [[set theory]] states precisely that the membership relation $\in$ on the proper class of all [[pure sets]] is well-founded. In structural set theory, accordingly, one uses well-founded relations in building structural models of well-founded pure sets.
## Examples
Let $S$ be a [[finite set]]. Then any relation on $S$ whose [[transitive closure]] is irreflexive is well-founded.
Let $S$ be the set of [[natural numbers]], and let $x \prec y$ if $y$ is the [[successor]] of $x$: $y = x + 1$. That this relation is well-founded is the usual principle of _mathematical induction_.
Again let $S$ be the set of natural numbers, but now let $x \prec y$ if $x \lt y$ in the usual order. That this relation is well-founded is the principle of _strong induction_.
More generally, let $S$ be a set of [[ordinal numbers]] (or even the proper class of all ordinal numbers), and let $x \prec y$ if $x \lt y$ in the usual order. That this relation is well-founded is the principle of _transfinite induction_.
Similarly, let $S$ be a set of [[pure sets]] (or even the proper class of all pure sets), and let $x \prec y$ if $x \in y$. That this relation is well-founded is the _[[axiom of foundation]]_.
Let $S$ be the set of [[integers]], and let $x \prec y$ mean that $x$ properly divides $y$: $y/x$ is an integer other than $\pm{1}$. This relation is also well-founded, so one can prove properties of integers by induction on their proper divisors.
[[!redirects well-founded relation]]
[[!redirects well-founded relations]]
[[!redirects well founded relation]]
[[!redirects well founded relations]]
[[!redirects classically well-founded relation]]
[[!redirects classically well-founded relations]]
[[!redirects classically well founded relation]]
[[!redirects classically well founded relations]]
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[[!redirects relations without infinite descent]]
[[!redirects relation with no infinite descent]]
[[!redirects relations with no infinite descent]]
[[!redirects Fermat's method of infinite descent]]
[[!redirects Fermat\'s method of infinite descent]]
[[!redirects Fermat's method of infinite descent]]
[[!redirects Fermat method of infinite descent]]
[[!redirects method of infinite descent]]
[[!redirects infinite descent]]
|
well-generated triangulated category | https://ncatlab.org/nlab/source/well-generated+triangulated+category |
#Contents#
* table of contents
{:toc}
##Idea
A well generated [[triangulated category]] is a generalization of the notion of [[compactly generated triangulated category]] which was introduced by [Neeman, 2001](#Neeman). The following definition is from ([Krause](#Krause)) and is somewhat shorter and more natural than Neeman's original definition.
##Definition
+-- {: .num_defn}
###### Definition
Let $T$ be a [[triangulated category]] with arbitrary [[coproducts]]. Then $T$ is **well-generated** in the sense of Neeman if and only if there exists a [[set]] $S_0$ of [[objects]] satisfying:
1. an object $X$ of $T$ is [[zero object|zero]] if $[S,X]=0$ for all $S\in S_0$;
1. for every set of maps $X_i\to Y_i$ in $T$, the induced map $[S,\coprod_I X_i]\to[S,\coprod_I Y_i]$ is surjective for all $S\in S_0$ whenever $[S,X_i]\to[S,Y_i]$ is surjective for all $i$ and all $S\in S_0$.
1. the objects of $S_0$ are $\alpha$-[[compact object|small]] for some [[cardinal]] $\alpha$.
=--
We recall that an object $S$ in a triangulated category is $\alpha$-small if every map $S\to\coprod_J X_j$ factors through $\coprod_I X_j$ for some $I \subseteq J$ with $\vert I\vert \lt \alpha$.
##References
* [[Henning Krause]], _On Neeman's well generated triangulated categories_, Documenta Mathematica __6__ (2001) ([pdf](https://www.math.uni-bielefeld.de/documenta/vol-06/07.pdf)).
{#Krause}
* Henning Krause, _Localization theory for triangulated categories_, [arXiv:0806.1324](https://arxiv.org/abs/0806.1324)
* [[Amnon Neeman]], _Triangulated Categories_, Annals of Mathematics Studies 148, Princeton University Press (2001).
{#Neeman}
[[!redirects well generated triangular category]]
[[!redirects well-generated triangulated categories]]
|
well-order | https://ncatlab.org/nlab/source/well-order |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
A _well-order_ on a set $S$ is a [[relation]] $\prec$ that allows one to interpret $S$ as an [[ordinal number]] $\alpha$ and $\prec$ as the relation $\lt$ on the ordinal numbers less than $\alpha$. In particular, one can do [[induction]] on $S$ over $\prec$ (although the more general [[well-founded relations]] also allow this).
The _[[well-ordering theorem]]_ states precisely that every set may be equipped with a well-order. This theorem follows from the [[axiom of choice]], and is equivalent to it in the presence of [[excluded middle]].
## Definition
A [[binary relation]] $\prec$ on a [[set]] $S$ is a __well-order__ if it is [[transitive relation|transitive]], [[extensional relation|extensional]], and [[well-founded relation|well-founded]]. A set equipped with a well-order is called a __well-ordered set__, or (following '[[partial order|poset]]') a __woset__. Actually, the term 'well-ordered' came first; 'well-order' is a back formation, which explains the strange grammar.
Other definitions of a well-order may be found in the literature; they are equivalent given [[excluded middle]], but the definition above seems to be the most powerful in [[constructive mathematics]]. Specifically:
* a well-order is precisely a [[well-founded relation|well-founded]] [[pseudo-order]] $\prec$;
* a well-order is precisely a [[total order]] $\preceq$ whose strict version $\prec$ (defined by $x \prec y$ iff $x \preceq y$ and $x \ne y$) is [[well-founded relation|well-founded]];
* (assuming also [[dependent choice]]) a well-order is precisely a pseudo-order $\prec$ with no infinite descending sequence $\cdots \prec x_2 \prec x_1 \prec x_0$;
* (assuming also [[dependent choice]]) a well-order is precisely a total order $\preceq$ such that every infinite descending sequence $\cdots \preceq x_2 \preceq x_1 \preceq x_0$ has $x_i = x_{i^+}$ for some $i$ (and hence for infinitely many $i$);
* a well-order on $S$ is precisely a [[pseudo-order]] $\prec$ with the property that every [[inhabited subset]] $U$ of $S$ has a least element (an element $\bot_U$ such that no $x \in U$ satisfies $x \prec \bot_U$;
* a well-order on $S$ is precisely a [[total order]] $\preceq$ with the property that every [[inhabited subset]] $U$ of $S$ has a least element (an element $\bot_U$ such that every $x \in U$ satisfies $\bot_U \preceq x$.
The really interesting thing here is that every well-order is a pseudo-order; it is a constructive theorem that every pseudo-order is weakly extensional (and so extensional if well-founded) and transitive. (For a [[weak counterexample]], take the set of [[truth values]] with $x \prec y$ iff $y$ is true and $x$ is false; this is a well-order that\'s a pseudo-order iff [[excluded middle]] holds.) For the other equivalences, we\'re simply using well-known classical equivalents for well-foundedness and the classical correspondence between a pseudo-order $\prec$ and its [[reflexive closure]] $\preceq$.
For reference, a __classical well-order__ is any order satisfying the last definition (a total order that is classically well-founded). A classically well-ordered set is a [[choice set]], and so if any set with at least $2$ [[denial inequality|unequal]] elements has a classical well-order, then [[excluded middle]] follows. (But there are still interesting examples of classically well-ordered sets in constructive mathematics, such as the [[Higgs object|Higgs set]].)
### Well-orders are pseudo-orders
As stated above, well-founded extensional transitive relations $\prec$ on a set $X$ are pseudo-orders, assuming [[classical logic]].
+-- {: .proof}
###### Proof
Order $X \times X$ [[lexicographic order|lexicographically]]: $(a, b) \prec (a', b')$ if either $a \prec a'$ in $X$ or $a = a'$ and $b \prec b$ in $X$. It is not hard to see that the lexicographic order is well-founded (and in fact a well-order, although we do not need this). Now let $A \subset X \times X$ be the set of pairs $(x, y)$ of non-equal elements $x$ and $y$ that are incomparable in $X$, and suppose $A$ is inhabited. Then $A$ has a minimal element $(a, b)$ (using excluded middle). Then, for every $x \prec b$, either $a \preceq x$ or $x \preceq a$. If the former holds for some $x$, then $a \prec b$ follows by transitivity, contradiction. Hence $x \prec a$ for every $x \prec b$. Now let $a'$ be minimal such that $x \prec a' \preceq a$ for every $x \prec b$.
Claim:
$$ \{x: x \prec a'\} = \{x: x \prec b\}. $$
We know already the right side is contained in the left. In the other direction, suppose $x \prec a'$. Since $x \prec a$ and $(a, b)$ was chosen minimal in the lexicographic order, $x$ and $b$ are comparable. If $b \preceq x \prec a'$, this contradicts minimality of $a'$. Thus $x \prec b$, i.e., the left side is contained in the right.
But now, by extensionality, $a' = b$, whence $b \preceq a$, contradiction. Therefore $A$ was empty, so that $X$ is [[connected relation|connected]] and therefore (being already transitive and irreflexive and using excluded middle again) a pseudo-order.
=--
## Examples
* Any [[finite set|finite]] [[pseudo-ordered set]] $\{x_1 \lt \cdots \lt x_n\}$ is well-ordered.
* The set of [[natural numbers]] is well-ordered under the usual order $\lt$.
* More generally, any set of [[ordinal numbers]] (or even the [[proper class]] of all ordinal numbers) is well-ordered under the usual order $\lt$ (which, constructively, may not be a pseudo-order).
* The [[cardinal numbers]] of well-orderable sets (the well-orderable cardinals), forming a [[retract]] of the ordinals, are well-ordered. So by the well-ordering theorem, the class of *all* cardinal numbers is well-ordered.
* A special case of the well-ordering theorem is the existence of a well-order on the set of [[real numbers]]; this is enough for many applications of the [[axiom of choice]] to [[analysis]].
## Interpretation as an ordinal number
Any well-ordered set $S$ defines an [[ordinal number]] $\alpha$ and an order isomorphism $r$ between $S$ and the set of ordinal numbers less than $\alpha$; as such, $S$ may be identified (up to isomorphism of wosets) with the von Neumann ordinal $\alpha$. The idea is that the minimal element $\bot$ of $S$ itself (if any) is mapped to the ordinal number $0$, the minimal element of $S \setminus \{\bot\}$ (if any) is mapped to $1$, and so on; after which the next element of $S$ (if any) is mapped to $\omega$, and so on; and so on.
This may be defined immediately (and constructively) as a recursively defined function from $S$ to the class of all ordinal numbers:
$$ r(x) = \sup_{t \prec x} r(t)^+ ;$$
the validity of this sort of recursive definition is precisely what the well-foundedness of $\prec$ allows. Here, $\beta^+$ is the [[successor]] of the ordinal number $\beta$, and $\sup$ is the [[supremum]] operation on ordinal numbers (which is the [[union]] of von Neumann ordinals).
Since $S$ is a set, the image of $r$ in the class of all ordinals is also a set (by the [[axiom of replacement]]), and one can now prove that $r$ is an order isomorphism between $S$ and the set of ordinals less than the next ordinal, $\alpha \coloneqq (\im r)^+$.
## Simulations
Given two well-ordered sets $S$ and $T$, a [[function]] $f\colon S \to T$ is a __[[simulation]]__ of $S$ in $T$ if
* $f(x) \prec f(y)$ whenever $x \prec y$ and
* given $t \prec f(x)$, there exists $y \prec x$ with $t = f(y)$.
Note that any simulation of $S$ in $T$ must be unique. Thus, well-ordered sets and simulations form a category that is in fact a (large) [[preorder]], whose reflection in the category of [[posets]] is in fact the poset of [[ordinal numbers]].
## Successor
A well-ordered set $S$ comes equipped with a **[[successor]]**, which is a [[partial function|partial map]]
$
succ\colon S \to S
\,
$
that sends $a \in S$ to the lowest element of the subset $S_a := \{ s \in S, a \prec s\}$, whenever this set is inhabited.
+-- {: .num_defn}
###### Definition
A **limit** well-order is a well-order $S$ whose successor map is a [[total function]].
=--
Similarly, one may define a successor [[functor]] on the [[category]] of well-ordered sets, taking $S$ to the well-order obtained by freely adjoining a (new) top element to $S$. Since this category (which is [[thin category|thin]]) can be regarded as itself a well-ordered [[proper class]], this is a special case of the successor operation above. (Hence the ordinal of all ordinals is a limit ordinal.)
## Related entries
* [[well-quasi-order]]
* [[marked extensional well-founded relation]]
[[!redirects well order]]
[[!redirects well orders]]
[[!redirects well-order]]
[[!redirects well-orders]]
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[[!redirects well ordered set]]
[[!redirects well ordered sets]]
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[[!redirects woset]]
[[!redirects wosets]]
|
well-ordering theorem | https://ncatlab.org/nlab/source/well-ordering+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **well-ordering theorem** is a famous result in [[set theory]] stating that every [[set]] may be [[well-order|well-ordered]].
Fundamental for [[Georg Cantor| G. Cantor's]] approach to [[ordinal arithmetic]] it was an open problem until [[Ernst Zermelo|E. Zermelo]] gave a [[proof]] in 1904 using the [[axiom of choice]] (to which it is in fact equivalent).
Hence the well-ordering theorem is one of the many equivalent formulations of the [[axiom of choice]] like also e.g. [[Zorn's lemma]].
## Statement and proof
Given any [[set]] $S$, there exists a [[well-order]] $\prec$ on $S$.
The first proof was given in ([Zermelo 1904](#Zermelo04)). Within the nLab, the article [[Zorn's lemma]] gives a standard informal proof that can be formalized in [[ZFC]] under [[classical logic]], as well as the easy argument that conversely, the well-ordering principle (in its classical "least element" form; see below) implies the [[axiom of choice]] and Zorn's lemma.
\begin{remark}
That the well-ordering theorem is more of a *theorem* in need of a proof, while the axiom of choice is more of an *axiom* to be assumed without proof is, of course, a matter of opinion, but it\'s reflected in Jerry Bona\'s famous quotation:
> The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about [[Zorn's lemma]]?
\end{remark}
## Consequences
If every set can be well-ordered, then the natural map from [[ordinal number]]s to [[cardinal number]]s is a [[surjection]]. Since these form proper classes, the ordinary axiom of choice will not split this surjection automatically; however, we can easily split it by assigning each cardinal number to the *smallest* ordinal number with that cardinality. (This does require [[excluded middle]], however.) In this way, a cardinal number may be *defined* to be an ordinal number that is _initial_: such that no smaller ordinal number has the same cardinality.
As in the argument above, the [[axiom of choice]] follows; given any [[surjection]] $f: A \to B$, place a well-ordering on $A$ and then split $f$ by mapping an element $y$ of $B$ to the smallest element $x$ of $A$ such that $y = f(x)$. Again, this uses [[excluded middle]] to show that such a smallest element exists, so the well-ordering principle does not (seem to) imply the axiom of choice constructively.
To get the large (or "global") axiom of choice (that any surjection between proper classes splits), we need a large well-ordering theorem: that every proper class can be well-ordered. The large principles do not follow from the small ones.
## In constructive mathematics
In [[constructive mathematics]], the well-ordering principle is also equivalent to the axiom of choice. This was [proved by Andrew Swan](https://hott.zulipchat.com/#narrow/stream/228519-general/topic/inductive.20well-ordering.20gives.20excluded.20middle.3F/near/246863644) (2021) and formalized in [Agda by Tom de Jong](https://www.cs.bham.ac.uk/~mhe/TypeTopology/WellOrderingTaboo.html) (2021) and in [Coq by Dominik Kirst](https://github.com/dominik-kirst/sierpinski-hott/blob/master/coq/orders_lem.v) (2021).
[[Zorn's lemma]] is, on the other hand, constructively weaker than the axiom of choice, as it doesn't even imply excluded middle. But together with excluded middle it implies choice. However, Zorn's lemma is not particularly useful without excluded middle.
## Related entries
* [[Zorn's lemma]]
* [[axiom of choice]]
* [[Hartogs number]]
## History
**Georg Cantor** first developed [[set theory]] in the context of studying well-ordered sets of [[real number]]s whence the validity of the well-ordering principle became important for his theory of [[ordinal numbers]]. In his 1883 paper he calls it a _'fundamental and weighty law of thought that is remarkable for his generality'_ and promised to come back to it later ([Cantor 1932](#Cantor32), p.169). In the following, he announced proofs but they failed to materialize so that he was forced to take it as an assumption.[^choice]
[^choice]:This contrasts with Cantor's attitude towards the axiom of choice which he used implicitly but never thematised explicitly. In fact, the full explicit awareness of the use of the axiom choice in mathematics had to await the controversy over Zermelo's well-ordering theorem in 1904 (with some anticipation by G. Peano and B. Levi earlier).
Consequently, the well-ordering principle ended as 'a very strange claim' second on [[Hilbert's problems|Hilbert's millennium list]] of open problems in mathematics in 1900. Then in 1904, the Hungarian mathematician J. König announced a proof that the [[continuum]] could not be well-ordered but had to retract the proof.
Soon afterwards in 1904, [[Ernst Zermelo]] finally gave a proof using the [[axiom of choice]] following a suggestion by E. Schmidt. The proof, albeit correct, was met with heavy criticism by prominent mathematicians so that Zermelo published a new proof and a defense of the contested axiom of choice in 1908.
The attempt to make explicit the set-theoretic assumptions in the proof led him to publish his axioms for set theory in the same year which became later a part of [[Zermelo-Fraenkel set theory]]. Hence the 1904ff controversy proved to become a _decisive watershed for the development of modern mathematics_: triggering the advent of set-theoretic [[foundation of mathematics]] and putting the problem of non-constructive methods of proof on the agenda.
## References
The original proof is in
* [[Ernst Zermelo]], _Beweis, daß jede Menge wohlgeordnet werden kann_, Mathematische Annalen **59** (1904) pp.514-516. ([gdz](http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=28526))
{#Zermelo04}
The second proof together with an eloquent defense of the axiom of choice can be found in
* [[Ernst Zermelo]], _Neuer Beweis für die Möglichkeit einer Wohlordnung_ , Mathematische Annalen **65** (1908) pp.107-128. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002261952)) {#Zermel08}
Cantor's texts are collected together with comments by Zermelo in
* [[Ernst Zermelo]] (ed.), _Georg Cantor - Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts_ , Springer Berlin 1932. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN237853094)) {#Cantor32}
English versions of Zermelo's papers are in
* J. van Heijenoort (ed.), _From Frege to Gödel - A Source Book in Mathematical Logic 1879-1931_ , Harvard UP 1967.
On the relation between AC and the well-ordering principle in general toposes see
* [[Peter Freyd]], _Choice and Well-ordering_ , APAL **35** (1987) pp.149-166.
* M.-M. Mawanda, _Well-ordering and Choice in Toposes_ , JPAA **50** (1988) pp.171-184.
* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. II_, Oxford UP 2002. (D4.5, pp.987-998)
* J. Todd Wilson, "An Intuitionistic version of Zermelo's proof that every choice set can be well-ordered", J. Symbolic Logic, 66:3 (2001), 1121--1126; ([JSTOR](http://www.jstor.org/stable/2695096): paywalled), [formalization in the Lean Theorem Prover] (https://github.com/lambdacalculator/lean-choice).
The proof that Zorn's Lemma doesn't imply excluded middle (and hence doesn't imply choice without assuming excluded middle):
* {#Bell} [[John Lane Bell]], *Zorn’s lemma and complete Boolean algebras in intuitionistic type theories*, The Journal of Symbolic Logic, 62(4):1265–1279, 1997 ([doi:10.2307/2275642](https://doi.org/10.2307/2275642))
[[!redirects well-ordering theorem]]
[[!redirects well-ordering principle]] |
well-pointed endofunctor | https://ncatlab.org/nlab/source/well-pointed+endofunctor |
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## Definition
\begin{definition}\label{PointedEndofunctor}
An [[endofunctor]] $S \colon \mathcal{A}\to \mathcal{A}$ is called [[pointed endofunctor|pointed]] if it is equipped with a [[natural transformation]] $\sigma \colon Id_\mathcal{A} \to S$ from the [[identity functor]].
\end{definition}
(Beware that is *not* quite the notion of a [[pointed object]] in the endo-[[functor category]], since the [[identity functor]] is not in general a [[terminal object]] there. See the remark [there](pointed+endofunctor#MeaningOfPointedness).)
\begin{definition}
A pointed endofunctor $(S, \sigma)$ (Def. \ref{PointedEndofunctor}) is called **well-pointed** if $S\sigma = \sigma S$ as natural transformations $S \longrightarrow S \circ S$.
\end{definition}
The dual notion is known as a **well-copointed** endofunctor.
## Properties
Well-pointed endofunctors have a particularly well-behaved free algebra sequence; see *[[transfinite construction of free algebras]]*.
## References
* {#Kelly} [[Max Kelly]], _A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on._ Bull. Austral. Math. Soc. 22 (1980), 1--83. doi:[10.1017/S0004972700006353](http://dx.doi.org/10.1017/S0004972700006353)
[[!redirects well-pointed endofunctors]]
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well-pointed topical dagger 2-poset | https://ncatlab.org/nlab/source/well-pointed+topical+dagger+2-poset | [[!redirects well-pointed dagger 2-posets]]
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## Definition ##
A **well-pointed topical dagger 2-poset** is an [[elementarily topical dagger 2-poset]] $C$ such that for every object $A \in Ob(C)$ and $B \in Ob(C)$ and [[map in a dagger 2-poset|maps]] $f \in Map_C(A, B)$, $g \in Map_C(A, B)$ and $x \in Map_C(\mathbb{1}, A)$, $f \circ x = g \circ x$ implies $f = g$.
## Examples ##
The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a well-pointed topical dagger 2-poset.
## See also ##
* [[elementarily topical dagger 2-poset]] |
well-pointed topological space | https://ncatlab.org/nlab/source/well-pointed+topological+space |
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## Definition
{#Definition}
A [[pointed topological space]] $(X,x)$ is called *well-pointed* if the base-point inclusion $\{x\} \xhookrightarrow{\;} X$ is a closed [[Hurewicz cofibration]] (e.g. [tom Dieck 2008, p. 102](#tomDieck2008)). If the topological space is [[Hausdorff topological space|Hausdorff]], then closedness is implied (by [this Prop.](Hurewicz+cofibration#HurewiczCofibrationsInCGWHSpacesAreClosed)) and one may require just a Hurewicz cofibration (eg. [Bredon 1993, VII, Def. 1.8](#Bredon93)).
A [[topological group]] is called well-pointed if it is so at its [[neutral element]], hence if $\{\mathrm{e}\} \xhookrightarrow{\;} G$ is a closed [[Hurewicz cofibration]].
A [[simplicial topological group]] is well-pointed if all its component groups are.
A key property of well-pointed topological groups (in the [[convenient category of topological spaces|convenient]] context of [[compactly generated weak Hausdorff spaces]]) is that the [[nerves]] of their [[delooping groupoids]] are [[good simplicial topological spaces]] (by [this Ex.](Hurewicz+cofibration#KifiedProductsOfCofibrationsWithCompactlyGeneratedSpaces)). Similarly, the underlying simplicial topological spaces of well-pointed simplicial topological groups are [[good simplicial topological space|good]] ([this Prop.](simplicial+topological+group#RealizationOfWellPointedIsWellPointed)). These facts explain the key role of well-pointedness in [[classifying space]]-theory, where it ensures that plain [[topological realization of simplicial topological spaces]] coincides, up to [[weak homotopy equivalence]], with [[fat geometric realization]] and hence with the [[homotopy colimits]].
## Examples
\begin{example}\label{ParacompactBanachManifoldsAreWellPointed}
Every [[locally Euclidean topological space|locally Euclidean]] [[Hausdorff space]] is well-pointed, in particular every [[topological manifold]] is well pointed. In fact, every [[paracompact topological space|paracompact]] [[Banach manifold]] is well-pointed.
\end{example}
(Immediate by the discussion of [examples of Hurewicz cofibrations](Hurewicz+cofibration#Examples)).
\begin{example}\label{PointInclusionIntoPUH}
**(the [[projective unitary group]] [[PU(ℋ)]])**
\linebreak
The [[projective unitary group]] [[PU(ℋ)]] on an infinite-dimensional [[separable Hilbert space|separable]] [[Hilbert space]] is:
* a [[Banach Lie group]] in its [[norm topology]], and as such [[well-pointed topological group|well-pointed]] by Ex.\ref{ParacompactBanachManifoldsAreWellPointed};
* no longer a Banach space in its weak/strong [[operator topology]], but nevertheless still [[well-pointed topological group|well-pointed]] in this case, by [this Prop.](projective+unitary+group+on+a+Hilbert+space#WellPointedInOperatorTopology).
\end{example}
## References
Textbook accounts:
* {#Bredon93} [[Glen Bredon]], Section VII.1 of: _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0), [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))
* {#tomDieck2008} [[Tammo tom Dieck]], _Algebraic topology_. European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86))
[[!redirects well-pointed space]]
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[[!redirects well sectioned simplicial topological group]]
[[!redirects well-sectioned simplicial topological groups]]
[[!redirects well sectioned simplicial topological groups]]
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well-pointed topos | https://ncatlab.org/nlab/source/well-pointed+topos |
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### Context
#### Topos Theory
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## Definition
An [[elementary topos]] $E$ is **well-pointed** if
1. the [[terminal object]] 1 is a [[separator|generator]];
* equivalently: the [[global section]] functor $\Gamma : E \to Set = E(1, -)$
is a [[faithful functor]];
* equivalently: if $f, g: a \rightarrow b$ are morphisms such that $f x = g x$ for all [[global element]]s $x: 1 \to a$, then $f = g$;
* equivalently: the family of all global elements $x: 1 \to a$ is a [[jointly epimorphic family]].
2. and $E$ is nondegenerate, i.e., 1 is not an [[initial object]].
## Examples
* The category [[Set]]. Here the [[global section]] functor is even (isomorphic to) the identity functor.
* Any model of [[ETCS]].
* Every [[coherent category]] satisfying the [[axiom of finiteness]] is a well-pointed topos equivalent to [[FinSet]].
## Properties
### Strong generation
In a well-pointed topos, the terminal object is even an [[extremal generator]],
* equivalently: the global section functor $E(1,-)$ is a [[conservative functor]].
* equivalently: if $f:A\to B$ induces a bijection $E(1,A) \cong E(1,B)$, then $f$ is an isomorphism.
* equivalently: if $m:A\rightarrowtail B$ is a [[monomorphism]] such that every global element $1\to B$ factors through $A$, then $m$ is an isomorphism.
To prove the last version, let $\chi_A : B\to \Omega$ be the classifying map of $A$, and let $\top : B\to \Omega$ be the classifying map of the maximal subobject. If every global element $b:1\to B$ factors through $A$, then $\chi_A b = \top b$ for all such $b$. Hence $\chi_A = \top$ by well-pointedness, so $A$ is isomorphic to the maximal subobject.
Note that in any category with [[equalizers]], an extremal generator is automatically a generator, since the equalizer of two parallel morphisms is the maximal subobject just when the two morphisms are equal. So the above statements are also equivalent to well-pointedness.
### Boolean properties
Assuming that one accepts [[excluded middle]] in one\'s metalogic, a well-pointed topos is also [[Boolean topos|Boolean]]. To see this, let $U\rightarrowtail A$ be a subobject and $\neg U$ its [[Heyting algebra|Heyting complement]]. Then by definition, the global elements of $\neg U$ are precisely the global elements of $A$ that do not factor through $U$. But then every global element of $A$ factors through $U \cup \neg U$, hence by strong generation $U\cup \neg U = A$.
Similarly, a well-pointed topos is [[two-valued topos|two-valued]]. In other words, that is, the only [[global element|global elements]] of the [[subobject classifier]] are $\top$ and $\bot$ (and these are distinct, by nondegeneracy) -- or equivalently, the only [[subterminal objects]] are $0$ and $1$. To see this, note by strong generation a subobject of any object is uniquely determined by the global elements that factor through it. But $1$ has only one global element, so it only has two possible global elements.
Finally, a well-pointed topos has [[split supports]]. For the support of any object $A$ must be either $0$ or $1$ by two-valuedness. If it is $0$ then $A\cong 0$ also and its support is split. Otherwise, $A\ncong 0$; now consider the two coproduct inclusions $inl, inr : A \rightrightarrows A+A$. Their pullback is $0$ by disjointness. Since $A\ncong 0$, we have $inl\neq inr$, hence by well-pointedness there is a global element $a:1\to A$ such that $inl a \neq inr a$. Thus $a$ splits the support of $A$.
Conversely, we have the following:
\begin{theorem}\label{BooleanTwovaluedSupportssplit}
If $E$ is nondegenerate (i.e. $1\ncong 0$), Boolean, two-valued, and split supports, then it is well-pointed.
\end{theorem}
\begin{proof}
This is Proposition 9.33 on p. 314 of [Johnstone, Topos Theory](#Johnstone77), attributed there to [Freyd](#Freyd70). Let $A\rightarrowtail B$ be a monomorphism such that every global element of $B$ factors through it. Then $\forall_B A$ is a subterminal object, hence either $1$ or $0$. If it is $0$, then its complement $\neg\forall_B A = \exists_B \neg A$ is $1$, which is to say that $\neg A$ is well-supported. Hence, since supports split, $\neg A$ has a global element, which is impossible since every global element of $B$ factors through $A$. Hence it must be that $\forall_B A = 1$, which implies $A=B$.
\end{proof}
Thus, in external classical logic, a topos is well-pointed if and only if it is nondegenerate, Boolean, two-valued, and has split supports. In particular, a nondegenerate two-valued topos satisfying the external [[axiom of choice]] is well-pointed.
### Logical properties
The main point of a well-pointed topos in logic is the equivalence of *external* and *internal* properties. In particular, a statement in the [[internal logic]] of the topos will be satisfied if and only if it holds for all global terms. (For the 'only if' part, it is necessary that the topos be nondegenerate.)
### Well-pointedness and concrete categories
The category [[Set]] of sets and functions is both well-pointed and concrete. However, not every [[concrete category]] is a well-pointed category: the category $Field$ of [[fields]] and field [[homomorphisms]] is concrete, but is not well-pointed because it doesn't have a [[terminal object]]. Moreover, not every well-pointed category is a concrete category, as well-pointed categories are not required to be concrete categories: most models of [[ETCS]] aren't defined to be concrete.
The distinction between well-pointed categories and concrete categories is the distinction between [[elements]] and [[global elements]] in a concrete category, as it is not true that elements and global elements (if they exist) coincide in general.
## Generalizations
{#general}
### In constructive mathematics
To maintain this logical result in [[constructive mathematics]] (that is, without excluded middle in the metalogic), one must add the following requirements:
3. the terminal object is [[indecomposable object|indecomposable]], and
4. the terminal object is [[projective object|projective]].
These are analogues, for [[disjunction]] and [[existential quantification]], of the nondegeneracy requirement (which is about [[falsehood]]). Classically, they follow from the two conditions given above.
Incidentally, a well-pointed topos in a constructive metalogic is still "two-valued" in the sense that a global element of the subobject classifier is false if and only if it is not true. However, it is not two-valued in the (classically equivalent) sense that every global element of the subobject classifier is either true or false.
### In a pretopos or coherent category
If $E$ is only a [[pretopos]] or a [[coherent category]], we have to strengthen the condition that 1 is a generator to the condition that 1 is an [[extremal generator]], i.e. for any monomorphism $m:A\to B$, if every global element $1\to B$ factors through $m$, then $m$ is an isomorphism. In a category with a subobject classifier (such as a topos), any generator is a extremal generator.
This strengthening is important in [[predicative mathematics]], where the category of sets (and in general, a [[Grothendieck topos|category of sheaves]]) is a pretopos but need not be a topos. But of course, the same applies whenever one is studying an arbitrary pretopos, not just a predicative version of $Set$.
### In a lextensive category
If $E$ is only a [[lextensive category]], such as in a [[category of sets]] without [[quotient sets]] (as commonly found in the [[syntactic category]] of an [[extensional type theory]]), then $E$ is well-pointed if $1$ is only a [[initial object|noninitial]] [[indecomposable object|indecomposable]] [[extremal generator]].
### In more general categories
Do we know what these should be in any more general situations?
[[Mike Shulman]]: Well, the pretopos version makes sense in any [[coherent category]], and I would bet that it's the right notion in that generality. In a [[regular category]] one might just want to assert that $1$ is a (regular-)projective extremal generator, which would probably be enough for regular logic. And in a category with mere finite limits, being a extremal generator is all one could ask for, and that'd probably be enough for finite-limit logic.
### Well-pointed $(\infty,1)$-toposes {#Infty1Version}
> [[Urs Schreiber|Urs]]: an attempt
One might like to say that "[[∞Grpd]] is essentially the unique [[(∞,1)-topos]] with all small limits and colimits that is well-pointed ."
Possibly one should say: an $(\infty,1)$-topos $\mathbf{H}$ is _well-pointed_ if the terminal object is not the initial one and the [[global section]] [[(∞,1)-functor]] $\Gamma : \mathbf{H} \to \infty Grpd$ is faithful...
...which should mean that on hom-$\infty$-groupoids it is a [[monomorphism in an (∞,1)-category]]...
...which should mean that for all $X,Y \in \mathbf{H}$ the image of the morphism $\Gamma_{X,Y} : \mathbf{H}(X,Y) \to Func(\Gamma(X),\Gamma(Y))$ in the [[homotopy category]] identifies $\mathbf{H}(X,Y)$ as a [[direct sum]]mand of $Func(\Gamma(X),\Gamma(Y))$.
## As an axiom schema of separation
Well-pointedness in a [[Boolean category]] (if using [[classical logic]]) or a [[Heyting category]] (if using [[intuitionistic logic]]) could also be represented, in addition to the [[terminal object]] being a [[strong generator]], by a version of the [[axiom schema of bounded separation]].
Let us define the following
* A $\Delta_0$-variable is a [[global element]] variable.
* A $\Delta_0$-context is a context only containing $\Delta_0$-variables
* An $\Delta_0$-atomic formula is an equality of global elements
* A $\Delta_0$-quantifier is a quantifier over a $\Delta_0$-variable.
* A formula whose only atomic subformulas are $\Delta_0$-atomic and whose only quantifiers are $\Delta_0$-quantifiers is a $\Delta_0$-formula.
A [[Boolean category]] or [[Heyting category]] $\mathcal{E}$ is well-pointed if
* the [[terminal object]] $1 \in \mathrm{Ob}(\mathcal{E})$ is a strong generator: given objects $A \in \mathrm{Ob}(\mathcal{E})$ and $B \in \mathrm{Ob}(\mathcal{E})$ and [[monomorphism]] $m:A \hookrightarrow B$, if for every [[global element]] $x:1 \to B$ there exists a global element $y:1 \to A$ such that $m \circ y = x$, then $m$ is an [[isomorphism]].
* the [[axiom schemata of bounded separation]] holds for [[global elements]]: for any $\Delta_0$-formula $\phi(x)$ with global element free variable $x:1 \to B$, there exists an object $A \in \mathrm{Ob}(\mathcal{E})$ and a monomorphism $m:A \hookrightarrow B$ such that for any global element $x:1 \to B$, $\phi(x)$ holds if and only if there exists a global element $y:1 \to A$ such that $m \circ y = x$.
In fact, given that $\mathcal{E}$ is a [[finitely complete category]], we could use this as the definition of a **well-pointed Boolean category** or **well-pointed Heyting category**, and thus in a foundational categorical [[set theory]]:
* Improper subsets: given a subset $A \subseteq B$ with chosen [[injection]] $m:A \hookrightarrow B$, if for every element $x \in B$ there exists an element $y \in A$ such that $m(y) = x$, then $A$ is [[generalized the|the]] [[improper subset]] of $B$ and $m$ is a [[bijection]].
* Bounded separation: for any $\Delta_0$-formula $\phi(x)$ with free variable $x \in B$, there exists a subset $A \subseteq B$ with chosen injection $m:A \hookrightarrow B$ such that for every element $x \in B$, $\phi(x)$ holds if and only if there exists an element $y \in A$ such that $m(y) = x$.
## Related entries
* [[ETCS]]
* [[concrete category]]
## References
* {#MM} [[Sheaves in Geometry and Logic]], Sections VI.1 and 10. Thm. \ref{BooleanTwovaluedSupportssplit} in this article is a strengthening of SGL's Prop. VI.1.7. SGL's Section VI.10 is a comparison of well-pointed toposes to RZC (Restricted Zermelo with Choice).
* {#Freyd70} [[Peter Freyd]], _Aspects of Topoi_, Bull. Austral. Soc. Math. no.7 pp.1-76,467-480 (1970).
* {#Johnstone77} [[Peter Johnstone]], _Topos Theory_, Academic Press New York 1977. (also available as Dover reprint, Minneola 2014). See Section 9.3.
For constructive well-pointedness of [[Heyting categories]] as a structural [[axiom schemata of separation]] in addition to the [[terminal object]] being a [[strong generator]], see:
* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).
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well-powered abelian category | https://ncatlab.org/nlab/source/well-powered+abelian+category | [[!redirects locally small abelian category]]
A __locally small abelian category__ is an abelian category which is [[well-powered category|well-powered]].
In other words, (equivalent classes of) [[subobject]]s of any set form a set.
Remark. In older literature and occasionally in some contemporary literature in the subject of [[abelian categories]] locally small signifies what is now standardly called [[well-powered category]]. As every abelian category is a [[locally small category]] in the usual sense (all Homs are sets), the terminology locally small abelian category (if the modifier is understood as nontrivial) does not lead to ambiguities (hence it denotes a well-powered abelian category).
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## Definition
A [[category]] $C$ is **well-powered** if every object has a [[small category|small]] [[partial order|poset]] of [[subobject]]s.
Assuming that by "subobject" we mean (an [[equivalence class]] of) [[monomorphism|monomorphisms]], this means that for every object $X$, the (generally [[large category|large]]) [[preorder|preordered]] set of monomorphisms with codomain $X$ is equivalent to a small poset, or equivalently that this preordered set is essentially small. Variations exist that use notions of subobject other than monomorphisms.
If $C^{op}$ is well-powered, we say that $C$ is **well-copowered** (although "cowell-powered" is also common).
## Properties
### Relation to local smallness
A well-powered category with binary [[product|products]] is always [[locally small category|locally small]], since morphisms $f: A \to B$ can be identified with particular subobjects of $A \times B$ (their [[graph of a function|graphs]]).
Conversely, any locally small category with a [[subobject classifier]] must obviously be well-powered. In particular, a [[topos]] is locally small if and only if it is well-powered.
There are interesting conditions and applications of the preorder on the sets of subobjects in well-powered categories, cf. e.g. [[property sup]].
## Examples
* Every [[Grothendieck topos]] (indeed, every [[locally small category|locally small]] [[elementary topos]]) is well-powered (by the existence of a [[subobject classifier]] and the smallness of [[hom sets]]).
* More generally, every [[locally presentable category]] is well-powered, since it is a full reflective subcategory of a presheaf topos, so its subobject lattices are subsets of those of the latter.
* Indeed, every category $C$ with a small [[dense subcategory]] $A$ is well-powered, since the [[restricted Yoneda embedding]] $C \to [A^{op},Set]$ is then fully faithful and preserves monomorphisms, so it embeds the subobject posets of $C$ as sub-posets of those of $[A^{op},Set]$. This includes in particular all [[accessible categories]].
* On the other hand, a locally small category with a [[strong generator]] can fail to be well-powered; a counterexample is [here](https://math.stackexchange.com/a/254794).
* Every locally presentable category, indeed every [[accessible category]] with [[pushouts]], is well-*copowered*. This is shown in [Adamek-Rosicky, Proposition 1.58 and Theorem 2.49](#AR). Whether this is true for all accessible categories depends on what [[large cardinal]] properties hold: by Corollary 6.8 of Adamek-Rosicky, if [[Vopenka's principle]] holds then all accessible categories are well-copowered, while by Example A.19 of Adamek-Rosicky, if all accessible categories are well-copowered then there exist arbitrarily large [[measurable cardinals]].
## References
* {#AR} [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)
[[!redirects well powered category]]
[[!redirects well-copowered category]]
[[!redirects co-well-powered category]] |
well-quasi-order | https://ncatlab.org/nlab/source/well-quasi-order |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
# Well-quasi-orders
* table of contents
{: toc}
## Definition
In [[classical mathematics]], a __well-quasi-order__ is a [[preorder]] $(P, \leq)$ such that for any infinite sequence $x_i$ in $P$, there exist $i, j$ with $i \lt j$ and $x_i \leq x_j$. Another way of expressing this condition is:
* There are no infinite [[antichain|antichains]] in $P$: if $x_i$ is a sequence in $P$, then there exist some pair of elements $x_i$, $x_j$ that are comparable, i.e., either $x_i \leq x_j$ or $x_j \leq x_i$, and
* There is no strictly decreasing sequence in $P$: if $x_0 \geq x_1 \geq \ldots$ in $P$, then there exists $n$ such that $x_i \leq x_{i+1}$ for all $i \geq n$.
One motivation for this notion is given by the following theorem:
+-- {: .un_thm}
###### Theorem
Let $(X, \leq)$ be a quasi-order (i.e., a [[preorder]]). Define a [[partial order]] on the [[power set]] $P(X)$ by $A \leq^+ B$ if $\forall_{a\colon A} \exists_{b\colon B} (a \leq b)$. Then $\leq^+$ is a [[well-founded relation]] if and only if $\leq$ is a well-quasi-ordering.
=--
## Example
The collection of [[graph|finite simple graphs]] is well-quasi-ordered by the [[graph minor]] relation. This is the celebrated _Robertson-Seymour Graph Minor Theorem_.
## Related entries
* [[well-order]]
## References
* [Wikipedia article](http://en.wikipedia.org/wiki/Well-quasi-ordering)
[[!redirects well-quasi-order]]
[[!redirects well-quasi-orders]]
[[!redirects well-quasiorder]]
[[!redirects well-quasiorders]]
[[!redirects well quasiorder]]
[[!redirects well quasiorders]]
[[!redirects well-quasi-ordering]]
[[!redirects well-quasi-orderings]]
[[!redirects well-quasiordering]]
[[!redirects well-quasiorderings]]
[[!redirects well quasiordering]]
[[!redirects well quasiorderings]]
[[!redirects well-preorder]]
[[!redirects well-preorders]]
[[!redirects well preorder]]
[[!redirects well preorders]]
[[!redirects well-quasi-ordered set]]
[[!redirects well-quasi-ordered sets]]
[[!redirects well-quasiordered set]]
[[!redirects well-quasiordered sets]]
[[!redirects well quasiordered set]]
[[!redirects well quasiordered sets]]
[[!redirects well-preorderered set]]
[[!redirects well-preorderered sets]]
[[!redirects well preorderered set]]
[[!redirects well preorderered sets]]
|
Wen-Tsun Wu | https://ncatlab.org/nlab/source/Wen-Tsun+Wu |
* [biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Wu_Wen-Tsun.html)
category: people |
Wendong Liang | https://ncatlab.org/nlab/source/Wendong+Liang |
* [personal page](https://wendongl.github.io/)
## Selected writings
On [[gs-monoidal categories]] in [[category theory|category theoretic]] [[probability theory]]:
* [[Tobias Fritz]], [[Wendong Liang]], *Free gs-monoidal categories and free Markov categories*, Appl. Categ. Structures **31** 21 (2023) [[arXiv:2204.02284](https://arxiv.org/abs/2204.02284), [doi:10.1007/s10485-023-09717-0](https://doi.org/10.1007/s10485-023-09717-0)]
category: people
[[!redirects LiangvWendong]]
|
Wendy Lowen | https://ncatlab.org/nlab/source/Wendy+Lowen | __Wendy Lowen__ is a mathematician at Universiteit Antwerpen, working mainly in algebra, categorical algebra and algebraic geometry.
Webpage: [publications](https://win.uantwerpen.be/~wlowen/publications.html)
* Wendy Lowen, [[Michel Van den Bergh]], _On compact generation of deformed schemes_, Adv. Math. __244__ (2013) 441--464 [doi](https://doi.org/10.1016/j.aim.2013.04.024)
* Wendy Lowen, _A generalization of the Gabriel-Popescu theorem_, J. Pure and Appl. Alg. __190__ (1) (2004): 197--211, [doi](https://doi.org/10.1016/j.jpaa.2003.11.016) [MR2043328](https://www.ams.org/mathscinet-getitem?mr=2043328)
* [[Dmitry Kaledin]], Wendy Lowen, _Cohomology of exact categories and (non-)additive sheaves_, Adv. Math. __272__ (2015) 652--698 [arXiv:1102.5756](https://arxiv.org/abs/1102.5756)
category: people |
Werner Burau | https://ncatlab.org/nlab/source/Werner+Burau |
* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=21539)
## Selected writings
Introducing the [[braid representation]] now known as the [[Burau representation]]:
* [[Werner Burau]], *Über Zopfgruppen und gleichsinnig verdrillte Verkettungen*, Abh. Math. Semin. Univ. Hambg. **11** (1935) 179–186 $[$[doi:10.1007/BF02940722](https://doi.org/10.1007/BF02940722)$]$
category: people |
Werner Greub | https://ncatlab.org/nlab/source/Werner+Greub |
* [website]() ?
category: people |
Werner Heisenberg | https://ncatlab.org/nlab/source/Werner+Heisenberg |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Werner_Heisenberg)
## Selected writings
Introducing [[quantum mechanics]]:
* [[Werner Heisenberg]], *Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen*, Zeitschrift für Physik **33** (1925) 879–893 $[$[doi:10.1007/BF01328377]( https://doi.org/10.1007/BF01328377), [Engl. pdf](http://users.mat.unimi.it/users/galgani/arch/heis25ajp.pdf)$]$
Introducing the [[Schwinger effect]] of [[vacuum polarization]]:
* {#HeisenbergEuler36} [[Werner Heisenberg]], [[Hans Euler]], _Folgerungen aus der Diracschen Theorie des Positrons_, Z. Physik 98, 714–732 (1936) ([doi:10.1007/BF01343663](https://doi.org/10.1007/BF01343663))
On [[quantum physics]] and the *[[potentiality]]* of [[quantum states]]:
* {#Heisenberg1958} [[Werner Heisenberg]], *Physics and Philosophy -- The Revolution in Modern Science*, Harper & Brothers (1958) [[archive](https://archive.org/details/physics-and-philosophy-the-revolution-in-modern-scirnce-werner-heisenberg-f.-s.-c.-northrop)]
category: people
[[!redirects Heisenberg]]
|
Werner Nahm | https://ncatlab.org/nlab/source/Werner+Nahm |
* [webpage](http://www.th.physik.uni-bonn.de/People/werner/)
* [wikipedia entry](http://de.wikipedia.org/wiki/Werner_Nahm)
## Selected writings
On the [[Nahm transform]]:
* [[Werner Nahm]], _The construction of all self-dual multi-monopoles by the ADHM method_, In: Craigie et al. (eds.), _Monopoles in quantum theory_, Singapore, World Scientific 1982 ([spire:178340](http://inspirehep.net/record/178340)
* [[Werner Nahm]], _Self-dual monopoles and calorons_, In: G. Denardo, G. Ghirardi and T. Weber (eds.), _Group theoretical methods in physics_, Lecture Notes in Physics 201. Berlin, Springer-Verlag 1984 ([doi:10.1007/BFb0016145](https://doi.org/10.1007/BFb0016145))
On the classification of [[supersymmetry]] [[super Lie algebras]]:
* [[Werner Nahm]], _[[Supersymmetries and their Representations]]_, Nucl.Phys. B135 (1978) 149 ([spire:120988](https://inspirehep.net/record/120988), [pdf](http://cds.cern.ch/record/132743/files/197709213.pdf))
## Related $n$Lab entries
* [[Yang-Mills instanton]]
* [[ADHM construction]]
* [[magnetic monopole]]
* [[Nahm transform]]
* [[caloron]]
* [[supersymmetry]]
category: people |
Werner Porod | https://ncatlab.org/nlab/source/Werner+Porod |
* [Institute page](https://www.physik.uni-wuerzburg.de/tp2/personen/prof-dr-werner-porod/)
* [Spire page](https://inspirehep.net/authors/992946?ui-citation-summary=true)
## Selected writins
On the [[Higgs field]] in [[holographic QCD]]:
* [[Johanna Erdmenger]], [[Nick Evans]], [[Werner Porod]], [[Konstantinos Rigatos]], *Gauge/gravity dual dynamics for the strongly coupled sector of composite Higgs models*, JHEP **58** (2021) [[arXiv:2010.10279](https://arxiv.org/abs/2010.10279), <a href="https://doi.org/10.1007/JHEP02(2021)058">doi:10.1007/JHEP02(2021)058</a>]
category: people
|
Wess-Zumino-Witten model | https://ncatlab.org/nlab/source/Wess-Zumino-Witten+model | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Wess-Zumino-Witten theory
+--{: .hide}
[[!include infinity-Wess-Zumino-Witten theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **Wess-Zumino-Witten model** (or **WZW model** for short, also called **Wess-Zumino-Novikov-Witten** model, or short **WZNW** model) is a 2-dimensional [[sigma-model]] [[quantum field theory]] whose target space is a [[Lie group]].
This may be regarded as the boundary theory of [[Chern-Simons theory]] for Lie group $G$.
The [[vertex operator algebra]]s corresponding to the WZW model are [[current algebra]]s.
## Action functional
{#ActionFunctional}
For $G$ a [[Lie group]], the [[configuration space]] of the WZW over a 2-[[dimension]]al [[manifold]] $\Sigma$ is the space of [[smooth function]]s $g : \Sigma \to G$.
The [[action functional]] of the WZW [[sigma-model]] is the sum of two terms, a kinetic term and a topological term
$$
S_{WZW} = S_{kin} + S_{top}
\,.
$$
### Kinetic term
{#KineticTerm}
The Lie group canonically carries a [[Riemannian metric]] and the kinetic term is the standard one for [[sigma-model]]s on Riemannian [[target space]]s.
### Topological term -- WZW term
{#TopologicalTerm}
#### For the 2d WZW model
{#WZWTermFor2dModel}
In [[higher differential geometry]], then given any closed [[differential n-form|differential (p+2)-form]] $\omega \in \Omega^{p+2}_{cl}(X)$, it is natural to ask for a [[prequantization]] of it, namely for a [[circle n-bundle with connection|circle (p+1)-bundle with connection]] $\nabla$ (equivalently: [[cocycle]] in degree-$(p+2)$-[[Deligne cohomology]]) on $X$ whose [[curvature]] is $F_\nabla = \omega$. In terms of [[moduli stacks]] this means asking for lifts of the form
$$
\array{
&& \mathbf{B}^{p+1}U(1)_{conn}
\\
&{}^{\mathllap{\nabla}}\nearrow& \downarrow^{\mathrlap{F_{(-)}}}
\\
X
&\stackrel{\omega}{\longrightarrow}&
\mathbf{\Omega}^{p+2}_{cl}
}
$$
in the [[homotopy theory]] of [[smooth homotopy types]].
This immediately raises the question for natural classes of examples of such prequantizations.
One such class arises in [[infinity-Lie theory]], where $\omega$ is a [[left invariant form]] on a [[smooth infinity-group]] given by a [[cocycle]] in [[L-infinity algebra cohomology]]. The [[prequantum n-bundles]] arising this way are the higher [[WZW terms]] discussed here.
In low degree of traditional [[Lie theory]] this appears as follows: On [[Lie groups]] $G$, those closed $(p+2)$-forms $\omega$ which are [[left invariant forms]] may be identified, via the general theory of [[Chevalley-Eilenberg algebras]], with degree $(p+2)$-[[cocycles]] $\mu$ in the [[Lie algebra cohomology]] of the [[Lie algebra]] $\mathfrak{g}$ corresponding to $G$. These in turn may arise, via the [[van Est map]], as the [[Lie differentiation]] of a degree-$(p+2)$-[[cocycle]] $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+2}U(1)$ in the [[Lie group cohomology]] of $G$ itself, with [[coefficients]] in the [[circle group]] $U(1)$.
This happens to be the case notably for $G$ a [[simply connected topological space|simply connected]] [[compact Lie group|compact]] [[semisimple Lie group]] such as [[special unitary group|SU]] or [[spin group|Spin]], where $\mu = \langle -,[-,-]\rangle$ is the [[Lie algebra cohomology|Lie algebra 3-cocycle]] in [[transgression]] with the [[Killing form]] [[invariant polynomial]] $\langle -,-\rangle$. This is, up to normalization, a representative of the de Rham image of a generator $\mathbf{c}$ of $H^3(\mathbf{B}G, U(1)) \simeq H^4(B G, \mathbb{Z}) \simeq \mathbb{Z}$.
Generally, by the discussion at _[[geometry of physics -- principal bundles]]_, the cocycle $\mathbf{c}$ [[modulating morphism|modulates]] an [[infinity-group extension]] which is a [[circle n-group|circle p-group]]-[[principal infinity-bundle]]
$$
\array{
\mathbf{B}^p U(1) &\longrightarrow& \hat G
\\
&& \downarrow
\\
&& G &\stackrel{\Omega\mathbf{c}}{\longrightarrow}& \mathbf{B}^{p+1}U(1)
}
$$
whose higher [[Dixmier-Douady class]] class $ \int \Omega \mathbf{c} \in H^{p+2}(X,\mathbb{Z})$ is an integral lift of the real cohomology class encoded by $\omega$ under the [[de Rham isomorphism]]. In the example of [[spin group|Spin]] and $p = 1$ this extension is the [[string 2-group]].
Such a [[Lie theory|Lie theoretic]] situation is concisely expressed by a diagram of [[smooth homotopy types]] of the form
$$
\array{
&& &\longrightarrow& \mathbf{B}^{p+1}U(1)
\\
&{}^{\mathllap{\Omega \mathbf{c}}}\nearrow& &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}}
\\
G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R}
}
\,,
$$
where $\flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \simeq \flat_{dR}\mathbf{B}^{p+2}U(1)$ is the [de Rham coefficients](cohesive+infinity-topos+--+structures#deRhamCohomology) (see also at _[[geometry of physics -- de Rham coefficients]]_) and where the homotopy filling the diagram is what exhibits $\omega$ as a de Rham representative of $\Omega \mathbf{c}$.
Now, by the very [[homotopy pullback]]-characterization of the [[Deligne complex]] $\mathbf{B}^{p+1}U(1)_{conn}$ ([here](Deligne+cohomology#TheExactDifferentialCohomologyHexagon)), such a diagram is equivalently a [[prequantization]] of $\omega$:
$$
\array{
&& \mathbf{B}^{p+1}U(1)_{conn} &\longrightarrow& \mathbf{B}^{p+1}U(1)
\\
&{}^\mathllap{\nabla}\nearrow& \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}}
\\
G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R}
}
\,.
$$
For $\omega = \langle -,[-,-]\rangle$ as above, we have $p= 1$ and so $\nabla$ here is a [[circle n-bundle with connection|circle 2-bundle with connection]], often referred to as a [[bundle gerbe]] [[connection on a bundle gerbe|with connection]]. As such, this is also known as the _WZW gerbe_ or similar.
This terminology arises as follows. In ([Wess-Zumino 71](Wess-Zumino-Witten+model#WessZumino71)) the [[sigma-model]] for a [[string]] propagating on the [[Lie group]] $G$ was considered, with only the standard [[kinetic action]] term. Then in ([Witten 84](Wess-Zumino-Witten+model#Witten84)) it was observed that for this [[action functional]] to give a [[conformal field theory]] after [[quantization]], a certain [[higher gauge theory|higher gauge]] [[interaction term]] has to the added. The resulting [[sigma-model]] came to be known as the _[[Wess-Zumino-Witten model]]_ or _WZW model_ for short, and the term that Witten added became the _WZW term_. In terms of [[string theory]] it describes the propagation of the [[string]] on the group $G$ subject to a [[force]] of [[gravity]] given by the [[Killing form]] [[Riemannian metric]] and subject to a [[B-field]] [[higher gauge field|higher gauge force]] whose [[field strength]] is $\omega$. In ([Gawedzki 87](Wess-Zumino-Witten+model#Gawedzki87)) it was observed that when formulated properly and generally, this WZW term is the [[surface holonomy]] functional of a [[connection on a bundle gerbe]] $\nabla$ on $G$. This is equivalently the $\nabla$ that we just motivated above.
Later WZW terms, or at least their curvature forms $\omega$, were recognized all over the place in [[quantum field theory]]. For instance the [[Green-Schwarz sigma-models for super p-branes]] each have an [[action functional]] that is the sum of the standard [[kinetic action]] plus a WZW term of degree $p+2$.
In general WZW terms are "[[gauged WZW model|gauged]]" which means, as we will see, that they are not defined on the give [[smooth infinity-group]] $G$ itself, but on a bundle $\tilde G$ of differential moduli stacks over that group, such that a map $\Sigma \to \tilde G$ is a pair consisting of a map $\Sigma \to G$ and of a [[higher gauge field]] on $\Sigma$ (a "tensor multiplet" of fields).
#### Generally
{#FormalizationGenerally}
The following ([FSS 12](#FSS12), [dcct](#dcct)) is a general axiomatization of WZW terms in [[cohesive (infinity,1)-topos|cohesive homotopy theory]].
In an ambient [[cohesive (∞,1)-topos]] $\mathbf{H}$, let $\mathbb{G}$ be a [[sylleptic ∞-group]], equipped with a [[Hodge filtration]], hence in particular with a chosen morphism
$$
\iota \colon
\mathbf{\Omega}^2_{cl}(-,\mathbb{G})
\longrightarrow
\flat_{dR} \mathbf{B}^2 \mathbb{G}
$$
to its [de Rham coefficients]()
+-- {: .num_defn #RefinementOfHodgeFiltration}
###### Definition
Given an [[∞-group]] object $G$ in $\mathbf{H}$ and given a [[group cohomology|group cocycle]]
$$
\mathbf{c} \colon \mathbf{B}G \longrightarrow \mathbf{B}^2 \mathbb{G}
\,,
$$
then a _refinement of the [[Hodge filtration]]_ of $\mathbb{G}$ along $\mathbf{c}$ is a completion of the [[cospan]] formed by $\flat_{dR}\mathbf{c}$ and by $\iota$ above to a [[diagram]] of the form
$$
\array{
\mathbf{\Omega}^1_{flat}(-,G)
&\stackrel{\mu}{\longrightarrow}&
\mathbf{\Omega}^2_{cl}(-,\mathbb{G})
\\
\downarrow && \downarrow^{\mathrlap{\iota}}
\\
\flat_{dR}\mathbf{B}G
&\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}&
\flat_{dR}\mathbf{B}^2 \mathbb{G}
}
\,.
$$
We write $\tilde G$ for the [[homotopy pullback]] of this refinement along the [[Maurer-Cartan form]] $\theta_G$ of $G$
$$
\array{
\tilde G &\stackrel{\theta_{\tilde G}}{\longrightarrow}& \mathbf{\Omega}^1_{flat}(-,G)
\\
\downarrow && \downarrow
\\
G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR}\mathbf{B}G
}
\,.
$$
=--
+-- {: .num_example}
###### Example
Let $\mathbf{H} = $ [[Smooth∞Grpd]] and $\mathbb{G} = \mathbf{B}^p U(1)$ the [[circle n-group|circle (p+1)-group]].
For $G$ an ordinary [[Lie group]], then $\mu$ may be taken to be the [[Lie algebra cohomology|Lie algebra cocycle]] corresponding to $\mathbf{c}$ and $\tilde G \simeq G$.
On the opposite extreme, for $G = \mathbf{B}^p U(1)$ itself with $\mathbf{c}$ the identity, then $\tilde G = \mathbf{B}^p U (1)_{conn}$ is the [[coefficients]] for [[ordinary differential cohomology]] (the [[Deligne complex]] under [[Dold-Kan correspondence]] and [[infinity-stackification]]).
Hence a more general case is a fibered product of these two, where $\tilde G$ is such that a map $\Sigma \longrightarrow \tilde G$ is equivalently a pair consisting of a map $\Sigma \to G$ and of differential $p$-form data on $\Sigma$. This is the case of relevance for WZW models of [[super p-branes]] with "tensor multiplet" fields on them, such as the [[D-branes]] and the [[M5-brane]].
=--
+-- {: .num_prop}
###### Proposition
In the situation of def. \ref{RefinementOfHodgeFiltration} there is an essentially unique [[prequantum n-bundle|prequantization]]
$$
\mathbf{L}_{WZW}
\colon
\tilde G
\longrightarrow
\mathbf{B}^2 \mathbb{G}_{conn}
$$
of the closed differential form
$$
\mu(\theta_{\tilde G})
\colon
\tilde G
\stackrel{\theta_{\tilde G}}{\longrightarrow}
\mathbf{\Omega}^1_{flat}(-,G)
\stackrel{\mu}{\longrightarrow}
\mathbf{\Omega}^2_{cl}(-,\mathbb{G})
$$
whose underlying $\mathbb{G}$-[[principal ∞-bundle]] is [[modulating morphism|modulated]] by the [[looping and delooping|looping]] $\Omega \mathbf{c}$ of the original cocycle.
This we call the _WZW term_ of $\mathbf{c}$ with respect to the chosen refinement of the Hodge structure.
=--
## Properties
{#Properties}
### Equations of motion
{#EquationsOfMotion}
The [[variational calculus|variational derivative]] of the WZW [[action functional]] is
$$
\delta S_{WZW}(g)
=
-\frac{k}{2 \pi i }
\int_\Sigma \langle (g^{-1}\delta g), \partial (g^{-1}\bar \partial g) \rangle
\,.
$$
Therefore the classical [[equations of motion]] for function $g \colon \Sigma \to G$ are
$$
\partial(g^{-1}\bar \partial g)
= 0
\;\;\;\;\;\;
\Leftrightarrow
\;\;\;\;\;\;
\bar \partial(g \partial g^{-1}) = 0
\,.
$$
The space of solutions to these equations is small. However, discussion of the [[quantization]] of the theory ([below](#HolographyAndRigorousConstruction)) suggests to consider these equations with the real [[Lie group]] $G$ replaced by its [[complexification]] to the [[complex Lie group]] $G({\mathbb{C}})$. Then the general solution to the equations of motion above has the form
$$
g(z,\bar z)
=
g_{\ell}(z) g_r(\bar z)^{-1}
$$
where hence $g_{\ell} \colon \Sigma \to G(\mathbb{C})$ is any [[holomorphic function]] and $g_r$ similarly any anti-holomorphic function.
(e.g. [Gawedzki 99 (3.18), (3.19)](#Gawedzki99))
### Holography and Rigorous construction
{#HolographyAndRigorousConstruction}
By the [[AdS3-CFT2 and CS-WZW correspondence]] (see there for more details) the 2d WZW [[CFT]] on $G$ is related to $G$-[[Chern-Simons theory]] in $3d$.
In fact a rigorous constructions of the $G$-WZW model as a [[rational 2d CFT]] is via the [[FRS-theorem on rational 2d CFT]], which constructs the model as a [[boundary field theory]] of the $G$-[[Chern-Simons theory]] as a [[3d TQFT]] incarnated via a [[Reshetikhin-Turaev construction]].
[[!include hypergeometric KZ-solutions -- references]]
### D-branes for the WZW model
{#DBranes}
The characterization of [[D-brane]] [[submanifolds]] for the [[open string]] WZW model on a [[Lie group]] $G$ comes from two consistency conditions:
1. geometrical condition:
For the open string [[CFT]] to still have [[current algebra]] [[worldsheet]] symmetry, hence for half the current algebra symmetry of the closed WZW string to be preserved, the [[D-brane]] [[submanifolds]] need to be [[conjugacy classes]] of the group manifold (see e.g. [Alekseev-Schomerus](#AlekseevSchomerus) for a brief review and further pointers). These conjugacy classes are therefore also called the **symmetric D-branes**.
Notice that these conjugacy classes are equivalently the [[leaves]] of the [[foliation]] induced by the canonical [[Cartan-Dirac structure]] on $G$, hence (by the discussion at [[Dirac structure]]), the leaves induced by the [[Lagrangian dg-submanifold|Lagrangian sub-Lie 2-algebroids]] of the [[Courant Lie 2-algebroid]] which is the [[higher gauge groupoid]] (see there) of the background [[B-field]] on $G$.(It has been suggested by [[Chris Rogers]] that such a foliation be thought of as a higher real [[polarization]].)
1. cohomological condition:
In order for the Kapustin-part of the [[Freed-Witten-Kapustin anomaly]] of the [[worldsheet]] [[action functional]] of the open WZW string to vanish, the D-brane must be equipped with a [[Chan-Paton gauge field]], hence a [[twisted unitary bundle]] ([[bundle gerbe module]]) of some rank $n \in \mathbb{N}$ for the restriction of the ambient [[B-field]] to the brane.
For [[simply connected topological space|simply connected]] Lie groups only the rank-1 Chan-Paton gauge fields and their direct sums play a role, and their existence corresponds to a trivialization of the underlying $\mathbf{B}U(1)$-[[principal 2-bundle]] ($U(1)$-[[bundle gerbe]]) of the restriction of the [[B-field]] to the brane. There is then a discrete finite collection of symmetric D-branes = [[conjugacy classes]] satisfying this condition, and these are called the **integral symmetric D-branes**. ([Alekseev-Schomerus](#AlekseevSchomerus), [Gawedzki-Reis](#GW)). As observed in [Alekseev-Schomerus](#AlekseevSchomerus), this may be thought of as identifying a D-brane as a variant kind of a [[Bohr-Sommerfeld leaf]].
More generally, on non-simply connected group manifolds there are nontrivial higher rank [[twisted unitary bundles]]/[[Chan-Paton gauge fields]] over conjugacy classes and hence the cohomological "integrality" or "Bohr-Sommerfeld"-condition imposed on symmetric D-branes becomes more refined ([Gawedzki 04](#Gawedzki04)).
In summary, the [[D-brane]] [[submanifolds]] in a Lie group which induce an [[open string]] WZW model that a) has one [[current algebra]] symmetry and b) is [[Freed-Witten-Kapustin anomaly|Kapustin-anomaly]]-free are precisely the [[conjugacy class]]-submanifolds $G$ equipped with a [[twisted unitary bundle]] for the restriction of the background [[B-field]] to the conjugacy class.
### Quantization
on [[quantization]] of the WZW model, see at
* [[quantization of loop groups]],
* [[equivariant elliptic cohomology]]
* [[quantization of Chern-Simons theory]]
## Related concepts
* [[exponentiated pion field]]
* [[geometry of physics -- WZW terms]]
* [[basic bundle gerbe]]
* [[current algebra]], [[affine Lie algebra]]
* [[Knizhnik-Zamolodchikov equation]]
* [[coset WZW model]]
* [[gauged WZW model]]
* [[parameterized WZW model]]
* [[higher dimensional WZW model]]
* [[Green-Schwarz action functional]]
* [[analytically continued Wess-Zumino-Witten theory]]
* [[Gepner model]]
## References
{#References}
### Introduction and survey
{#IntroductionsAndSurveys}
* [[Peter Goddard]], [[David Olive]], *Kac-Moody and Virasoro algebras in relation to quantum physics*, International Journal of Modern Physics A **01** 02 (1986) 303-414 [[doi:10.1142/S0217751X86000149](https://doi.org/10.1142/S0217751X86000149), [spire:18583](https://inspirehep.net/literature/18583)]
Textbook accounts:
* [[Philippe Di Francesco]], Pierre Mathieu, David Sénéchal, Part C of: *Conformal field theory*, Springer (1997) [[doi:10.1007/978-1-4612-2256-9](https://doi.org/10.1007/978-1-4612-2256-9)]
* [[Bojko Bakalov]], [[Alexander Kirillov]], *Wess-Zumino-Witten model*, chapter 7 of: *Lectures on tensor categories and modular functors*, University Lecture Series **21**, Amer. Math. Soc. (2001) [[[BakalovKirillov-WZWModel-Ch7OfTensorCat.pdf:file]], [web](http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html), [ams:ulect/21](https://bookstore.ams.org/view?ProductCode=ULECT/21)]
* [[Ralph Blumenhagen]], [[Erik Plauschinn]], Chapter 3 of: *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]
Lecture notes:
* [[Patrick Meessen]], *Strings Moving on Group Manifolds: The WZW Model* [[pdf](http://www.unioviedo.es/hepth/people/Patrick/fysica/zooi/WZW_ClassMunoz.pdf), [[Meessen-WZWModel.pdf:file]]]
* [[Lorenz Eberhardt]], *Wess-Zumino-Witten models*, lecture notes at *[YRISW 2019: A modern primer for 2D CFT](https://conf.itp.phys.ethz.ch/esi-school/)*, Vienna (2019) [[pdf](https://conf.itp.phys.ethz.ch/esi-school/Lecture_notes/WZW%20models.pdf), [[Eberhardt-WZWModels.pdf:file]]]
A basic introduction also to the super-WZW model (and with an eye towards the corresponding [[2-spectral triple]]) is in
* {#FroehlichGawedzki93} [[Jürg Fröhlich]], [[Krzysztof Gawedzki]], _Conformal Field Theory and Geometry of Strings_, extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 ([arXiv:hep-th/9310187](http://arxiv.org/abs/hep-th/9310187))
A useful account of the WZW model that encompasses both its [[action functional]] and [[path integral]] quantization as well as the [[current algebra]] aspects of the QFT is in
* {#Gawedzki99} [[Krzysztof Gawędzki]], *Conformal field theory: a case study*, in Y. Nutku, C. Saclioglu, T. Turgut (eds.) *Conformal Field Theory -- New Non-perturbative Methods In String And Field Theory*, CRC Press (2000) [[arXiv:hep-th/9904145](https://arxiv.org/abs/hep-th/9904145), [doi:10.1201/9780429502873](https://doi.org/10.1201/9780429502873)]
This starts in section 2 as a warmup with describing the 1d QFT of a particle propagating on a group manifold. The [[Hilbert space]] of [[states]] is expressed in terms of the [[Lie theory]] of $G$ and its [[Lie algebra]] $\mathfrak{g}$.
In section 4 the [[quantization]] of the 2d WZW model is discussed in analogy to that. In lack of a full formalization of the quantization procedure, the author uses the loop algebra $\mathcal{l} \mathfrak{g}$ -- the [[affine Lie algebra]] -- of $\mathfrak{g}$ as the evident analog that replaces $\mathfrak{g}$ and discusses the [[Hilbert space]] [[space of states|of states]] in terms of that. He also indicates how this may be understood as a space of [[sections]] of a ([[prequantum line bundle|prequantum]]) [[line bundle]] over the [[loop group]].
See also
* L. Fehér, L. O'Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf, _On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories_, Phys. Rep. __222__ (1992), no. 1, 64 pp. [MR93i:81225](http://www.ams.org/mathscinet-getitem?mr=1192998), <a href="http://dx.doi.org/10.1016/0370-1573(92)90026-V">doi</a>
* [[Matthias Blau]], [[George Thompson]], *Equivariant Kähler Geometry and Localization in the $G/G$ Model*, Nucl. Phys. B **439** (1995) 367-394 [<a href="https://doi.org/10.1016/0550-3213(95)00058-Z">doi:10.1016/0550-3213(95)00058-Z</a>, [arXiv:hep-th/9407042](https://arxiv.org/abs/hep-th/9407042)]
> ([[supersymmetry|supersymmetric]] and using [[group algebra]])
* [[Krzysztof Gawedzki]], Rafal Suszek, [[Konrad Waldorf]], _Global gauge anomalies in two-dimensional bosonic sigma models_ ([arXiv:1003.4154](http://arxiv.org/abs/1003.4154))
* Paul de Fromont, [[Krzysztof Gawędzki]], Clément Tauber, _Global gauge anomalies in coset models of conformal field theory_ ([arXiv:1301.2517](http://arxiv.org/abs/1301.2517))
[[!include WZW term of QCD chiral perturbation theory -- references]]
### Interpretation via CFT and gerbes
Interpretation of the 3d WZW term as defining a [[2d CFT]]
* {#Witten84} [[Edward Witten]], _Non-Abelian bosonization in two dimensions_ Commun. Math. Phys. 92, 455 (1984)
* {#KnizhnikZamolodchikov85} [[Vadim Knizhnik]], [[Alexander Zamolodchikov]], _Current algebra and Wess-Zumino model in two dimensions_, Nucl. Phys. B 247, 83-103 (1984)
and hence as part of a [[perturbative string theory vacuum]]/[[target space]]
* [[Doron Gepner]], [[Edward Witten]], _String theory on group manifolds_, Nucl. Phys. B 278, 493-549 (1986) ([spire:230076](http://inspirehep.net/record/230076))
The WZ term on $\Sigma_2$ was understood in terms of an integral of a 3-form over a cobounding manifold $\Sigma_3$ in
* [[Edward Witten]], _Global aspects of current algebra_, Nucl. Phys. B223, 422 (1983) ([spire:13234](http://inspirehep.net/record/13234), <a href="https://doi.org/10.1016/0550-3213(83)90063-9">doi:10.1016/0550-3213(83)90063-9</a>, [pdf](https://www.phys.sinica.edu.tw/~spring8/users/jychen/pub/reference/NPB_v223_422.pdf))
for the case that $\Sigma_2$ is [[closed manifold|closed]], and generally, in terms of [[surface holonomy]] of [[bundle gerbes]]/[[circle 2-bundles with connection]] in
* {#Gawedzki87} [[Krzysztof Gawedzki]], _Topological Actions in two-dimensional Quantum Field Theories_, in [[Gerard 't Hooft]] et. al (eds.) _Nonperturbative quantum field theory_ Cargese 1987 proceedings, ([web](http://inspirehep.net/record/257658?ln=de))
* [[Giovanni Felder]] , [[Krzysztof Gawedzki]], A. Kupianen, _Spectra of Wess-Zumino-Witten models with arbitrary simple groups_. Commun. Math. Phys. 117, 127 (1988)
* [[Krzysztof Gawedzki]], _Topological actions in two-dimensional quantum field theories_. In: Nonperturbative quantum field theory. 'tHooft, G. et al. (eds.). London: Plenum Press 1988
as the [[surface holonomy]] of a [[circle 2-bundle with connection]]. See also the references at _[[B-field]]_ and at _[[Freed-Witten anomaly cancellation]]_.
See also
* {#DeligneFreed99} [[Pierre Deligne]], [[Daniel Freed]], chapter 6 of _Classical field theory_ (1999) ([pdf](https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf))
this is a chapter in
[[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], J. Morgan, D.R. Morrison, [[Edward Witten|E. Witten]] (eds.) _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))
For the fully general understanding as the [[surface holonomy]] of a [[circle 2-bundle with connection]] see the references [below](#ReferencesRelationToGerbesAndCS).
See also
* [[Edward Witten]], _On holomorphic factorization of WZW and coset models_, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. ([Euclid](http://projecteuclid.org/euclid.cmp/1104249222))
### Relation to gerbes and Chern-Simons theory
{#ReferencesRelationToGerbesAndCS}
Discussion of [[circle n-bundle with connection|circle 2-bundles with connection]] (expressed in terms of [[bundle gerbes]]) and discussion of the WZW-background [[B-field]] ([[WZW term]]) in this language (cf. *[[basic bundle gerbe]]*)
* {#GW} [[Krzysztof Gawędzki]], [[Nuno Reis]], *WZW branes and gerbes*, Rev. Math. Phys. **14** (2002) 1281-1334 [[arXiv:hep-th/0205233](https://arxiv.org/abs/hep-th/0205233), [doi:10.1142/S0129055X02001557](https://doi.org/10.1142/S0129055X02001557)]
* {#SchweigertWaldorf07} [[Christoph Schweigert]], [[Konrad Waldorf]], _Gerbes and Lie Groups_, in _Trends and Developments in Lie Theory_, Progress in Math., Birkhäuser ([arXiv:0710.5467](http://arxiv.org/abs/0710.5467))
Discussion of how this 2-bundle arises from the [[Chern-Simons circle 3-bundle]] is in
* [[Alan Carey]], Stuart Johnson, [[Michael Murray]], [[Danny Stevenson]], [[Bai-Ling Wang]], _Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories_ Commun.Math.Phys. 259 (2005) 577-613 ([arXiv:math/0410013](http://arxiv.org/abs/math/0410013))
and related discussion is in
* [[Konrad Waldorf]], _Multiplicative Bundle Gerbes with Connection_ , Differential Geom. Appl. 28(3), 313-340 (2010) ([arXiv:0804.4835](http://arxiv.org/abs/0804.4835))
See also Section 2.3.18 and section 4.7 of
* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([pdf slides](http://ncatlab.org/schreiber/files/Erlangen2011Schreiber.pdf)).
### Partition functions
* [[Terry Gannon]], _Partition Functions for Heterotic WZW Conformal Field Theories_, Nucl.Phys. B402 (1993) 729-753 ([arXiv:hep-th/9209042](http://arxiv.org/abs/hep-th/9209042))
### D-branes for the WZW model
{#ReferencesDBranes}
A characterization of [[D-branes]] in the WZW model as those [[conjugacy classes]] that in addition satisfy an integrality ([[Bohr-Sommerfeld quantization|Bohr-Sommerfeld]]-type) condition missed in other parts of the literature is stated in
* {#AlekseevSchomerus} [[Anton Alekseev]], [[Volker Schomerus]], _D-branes in the WZW model_, Phys.Rev.D60:061901,1999 ([arXiv:hep-th/9812193v2](http://arxiv.org/abs/hep-th/9812193v2))
The refined interpretation of the integrality condition as a choice of trivialization of the underling [[principal 2-bundle]]/[[bundle gerbe]] of the [[B-field]] over the brane was then noticed in section 7 of
* [[Krzysztof Gawedzki]], Nuno Reis, _WZW branes and gerbes_, Rev.Math.Phys. 14 (2002) 1281-1334 ([arXiv:hep-th/0205233](http://arxiv.org/abs/hep-th/0205233))
The observation that this is just the special rank-1 case of the more general structure provided by a [[twisted unitary bundle]] of some rank $n$ on the D-brane ([[gerbe module]]) which is twisted by the restriction of the [[B-field]] to the D-brane -- the [[Chan-Paton gauge field]] -- is due to
* {#Gawedzki04} [[Krzysztof Gawedzki]], _Abelian and non-Abelian branes in WZW models and gerbes_, Commun.Math.Phys. 258 (2005) 23-73 ([arXiv:hep-th/0406072](http://arxiv.org/abs/hep-th/0406072)).
The observation that the "multiplicative" structure of the WZW-[[B-field]] (induced from it being the [[transgression]] of the [[Chern-Simons circle 3-bundle|Chern-Simons circle 3-connection]] over the [[moduli stack]] of $G$-[[principal connections]]) induces the [[Verlinde ring]] fusion product structure on symmetric D-branes equipped with [[Chan-Paton gauge fields]] is discussed in
* [[Alan Carey]], [[Bai-Ling Wang]], _Fusion of symmetric $D$-branes and Verlinde rings_, Commun. Math. Phys.277:577-625 (2008) ([arXiv:math-ph/0505040](http://arxiv.org/abs/math-ph/0505040))
The image in [[K-theory]] of these [[Chan-Paton gauge fields]] over conjugacy classes is shown to generate the [[Verlinde ring]]/the [[positive energy representations]] of the [[loop group]] in
* [[Eckhard Meinrenken]], _On the quantization of conjugacy classes_, Enseign. Math. (2) 55 (2009), no. 1-2, 33-75 ([arXiv:0707.3963](http://arxiv.org/abs/0707.3963))
Formalization of WZW terms in [[cohesive (infinity,1)-topos|cohesive homotopy theory]] is in
* {#dcct} _[[schreiber:differential cohomology in a cohesive topos]]_
### Relation to dimensional reduction of Chern-Simons
One can also obtain the WZW-model by [[KK-reduction]]
from [[Chern-Simons theory]].
E.g.
* [[Matthias Blau]], G. Thompson, _Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model_, Nucl.Phys. B408 (1993) 345-390 ([arXiv:hep-th/9305010](http://arxiv.org/abs/hep-th/9305010))
A discussion in [[higher differential geometry]] via [[transgression]] in [[ordinary differential cohomology]] is in
* _[[schreiber:Extended higher cup-product Chern-Simons theories]]_
* _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_
### Relation to extended TQFT
Relation to [[extended TQFT]] is discussed in
* [[Dan Freed]], _[[4-3-2 8-7-6]]_
For a formulation of the WZW term in the presence of D-branes as an open-closed smooth [[functorial field theory]]:
* {#BunkWaldorf21a} [[Severin Bunk]], [[Konrad Waldorf]], *Transgression of D-branes*, Adv. Theor. Math. Phys. **25** 5 (2021) 1095-1198 [[arXiv:1808.04894] (https://arxiv.org/abs/arXiv:1808.04894), [doi:10.4310/ATMP.2021.v25.n5.a1](https://dx.doi.org/10.4310/ATMP.2021.v25.n5.a1)]
* {#BunkWaldorf21b} [[Severin Bunk]], [[Konrad Waldorf]], *Smooth functorial field theories from B-fields and D-branes*, J. Homot. Rel. Struc. **16** 1 (2021) 75-153 [[doi:10.1007/s40062-020-00272-2](https://doi.org/10.1007/s40062-020-00272-2), [arXiv:1911.09990] (https://arxiv.org/abs/arXiv:1911.09990)]
The formulation of the [[Green-Schwarz action functional]] for [[superstrings]] (and other [[branes]] of [[string theory]]/[[M-theory]]) as WZW-models (and [[schreiber:∞-Wess-Zumino-Witten theory|∞-WZW models]]) on ([[super L-∞ algebra]] [[L-∞ extensions]] of) the [[super translation group]] is in
* {#FSS12} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))
### In solid state physics
The low-energy physics of a Heisenberg antiferromagnetic spin chain is argued to be described by a WZW model in
* Zheng-Xin Liu, Guang-Ming Zhang, _Classification of quantum critical states of integrable antiferromagnetic spin chains and their correspondent two-dimensional topological phases_ ([arXiv:1211.5421](http://arxiv.org/abs/1211.5421))
See also section 7.10 of Fradkin's book.
Discussion of [[symmetry protected topological order]] phases of matter in [[solid state physics]] via [[higher dimensional WZW models]] is in
* {#CGLW11} Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B 87, 155114 (2013) [arXiv:1106.4772](http://arxiv.org/abs/1106.4772); A short version in Science __338__, 1604-1606 (2012) [pdf](http://dao.mit.edu/~wen/pub/dDSPTsht.pdf)
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[[!redirects Wess-Zumino-Witten sigma models]] |
Weyl algebra | https://ncatlab.org/nlab/source/Weyl+algebra |
> To be distinguished from _[[Weil algebra]]_.
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#### Algebra
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#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
In general, the term *Weyl algebra* refers to [[noncommutative algebra|noncommutative]] [[associative algebras]] controlled by [[canonical commutation relations]] (CCR) which are the hallmark of [[quantum mechanics]].
More specifically, by the Weyl algebra with *[[Weyl relations]]* one refers to the *exponentiated* form of these CCR where the algebra generators are ([[linear representation|represented]] by) [[unitary operators]], introduced by [Weyl 1927, pp. 27](#Weyl27) and further highlighted in [von Neumann 1931](#vonNeumann31). It is this form of "Weyl algebra" that the [[Stone-von Neumann theorem]] directly applies to.
## Definition
Given a [[field]] $k$, the $n$-th __Weyl algebra__ $A_{n,k}$ is an [[associative unital algebra]] over $k$ generated by the symbols $x^1,\ldots,x^n,\partial_1,\ldots,\partial_n$ modulo the "[[canonical commutation relations]]" $x^i x^j = x^j x^i$, $\partial_i\partial_j = \partial_j\partial_i$ and $\partial_i x^j - x^j \partial_i = \delta_i^j$ (the [[Kronecker delta]]).
In [[characteristic zero]], this agrees with the algebra of [[regular differential operators]] on the $n$-dimensional [[affine space]].
Sometimes one considers the Weyl algebras over an arbitrary $k$-algebra $R$, including noncommutative $R$, when the definition is simply $A_{n,k}\otimes_k R$. Another generalization are the symplectic Weyl algebras.
In [[quantum physics]], one often studies Weyl algebras over the [[complex numbers]] (see [below](#RelationToHeisenbergLieAlgebra)); the usual notation there is $p_j$ for $- \mathrm{i} \partial_j$ (where $\mathrm{i}$ is the imaginary unit).
## Properties
### Relation to Heisenberg Lie algebra
{#RelationToHeisenbergLieAlgebra}
Consider the standard [[symplectic form]] on the [[Cartesian space]] $\mathbb{R}^{2n}$, making a [[symplectic vector space]]. This gives rise to the corresponding [[Heisenberg Lie algebra]].
Depending on conventions, the [[universal enveloping algebra]] of the [[Heisenberg Lie algebra]] either already is the [[Weyl algebra]] on $2n$ generators or else it becomes so after after forming the [[quotient algebra]] in which the central generator is identified with the [[unit element]] of the [[ground field]] -- whereas in the former case (considered eg. in [Kravchenko 2000, Def. 2.1](#Kravchenko00); [Bekaert 2005, p. 9](#Bekaert05)) the central generator plays the role of the formal [[Planck constant]] $\hbar$ with the Weyl algebra regarded as a [[formal deformation quantization]] of the [[symplectic manifold]] $\mathbb{R}^{2m}$.
Accordingly, given a [[Heisenberg Lie n-algebra|Heisenberg Lie $n$-algebra]] it makes sense to call its [[universal enveloping E-n algebra|universal enveloping $E_n$-algebra]] a _[[Weyl n-algebra|Weyl $n$-algebra]]_.
## Related concepts
* [[Weyl relations]]
* [[canonical commutation relations]]
* [[Heisenberg algebra]]
## References
The term "Weyl algebra" for algebras freely generated subject to [[canonical commutation relations]] is due to
* [[Jacques Dixmier]], *Sur les algèbres de Weyl*, Bulletin de la Société Mathématique de France, **96** (1968) 209-242 [[numdam:BSMF_1968__96__209_0](http://www.numdam.org/item/?id=BSMF_1968__96__209_0)]
(there attributed to a suggestion by [[Irving Segal]]) and referring to the original discussion in
* {#Weyl27} [[Hermann Weyl]], (36) in: *Quantenmechanik und Gruppentheorie*, Zeitschrift für Physik **46** (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]
However, beware that the invention of [Weyl 1927](#Weyl27) was not the [[canonical commutation relations]] but their exponential reformulation via the [[Weyl relations]], whose relevance was then picked up by
* {#vonNeumann31} [[John von Neumann]], *Die Eindeutigkeit der Schrödingerschen Operatoren*, Mathematische Annalen **104** (1931) 570–578 [[doi:10.1007/BF01457956](https://doi.org/10.1007/BF01457956)]
> (proving the [[Stone-von Neumann theorem]])
More on the history:
* [[Severino C. Coutinho]], Introduction to: *A primer of algebraic D-modules*, London Math. Soc. Stud. Texts **33**, Cambridge University Press (1995) [[doi:10.1017/CBO9780511623653](https://doi.org/10.1017/CBO9780511623653)]
* [[Severino C. Coutinho]], *The Many Avatars of a Simple Algebra*, The American Mathematical Monthly **104** 7 (1997) 593-604 [[doi:10.2307/2975052](https://doi.org/10.2307/2975052), [jstor:2975052](https://www.jstor.org/stable/2975052)]
Further discussion (of either notion):
* [[Alan Weinstein]], p. 392 of: *Deformation quantization*, Séminaire Bourbaki volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Talk no. 789 [[numdam:SB_1993-1994__36__389_0](http://www.numdam.org/item/?id=SB_1993-1994__36__389_0)]
* {#Kravchenko00} Olga Kravchenko, *Deformation Quantization of Symplectic Fibrations*, Compositio Mathematica **123** (2000) 131–165 [[arXiv:math/9802070](https://arxiv.org/abs/math/9802070), [doi:10.1023/A:1002452002677](https://doi.org/10.1023/A:1002452002677)]
* {#Bekaert05} [[Xavier Bekaert]], *Universal enveloping algebras and some applications in physics* (2005) [[cds:904799](https://cds.cern.ch/record/904799), [pdf](https://cds.cern.ch/record/904799/files/cer-002575006.pdf)]
* [[Markus Pflaum]], *From Weyl quantization to modern algebraic index theory*, in *Groups and Analysis -- The Legacy of Hermann Weyl*, Cambridge University Press (2008) 84-99 [[doi:10.1017/CBO9780511721410.005](https://doi.org/10.1017/CBO9780511721410.005)]
* Jason Gaddis, *The Weyl algebra and its friends: a survey* [[arXiv:2305.01609](https://arxiv.org/abs/2305.01609)]
On [[continuous field of C-star algebras|continuous fields]] of [[Weyl algebras]] as [[continuous deformation quantizations]] of [[symplectic vector spaces|symplectic]] [[topological vector spaces]]:
* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization*, Annales Henri Poincaré **5** (2004) 327–346 [[doi:10.1007/s00023-004-0171-y](https://doi.org/10.1007/s00023-004-0171-y)]
On Weyl algebras as [[groupoid algebras]] being [[strict deformation quantizations]] of [[Lie-Poisson structures]] given by [[tangent Lie algebroids]]:
* [[Nicolaas P. Landsman]], B. Ramazan, Ex. 11.3 in: *Quantization of Poisson algebras associated to Lie algebroids*, in: *Groupoids in Analysis, Geometry, and Physics*, Contemporary Mathematics **282** (2001) [[arXiv:math-ph/0001005](https://arxiv.org/abs/math-ph/0001005), [ams:conm/282](http://www.ams.org/books/conm/282)]
On [[group algebras]] of ([[underlying]] [[discrete group|discrete]]) [[Heisenberg groups]] as [[strict deformation quantizations]] of [[presymplectic manifold|pre-]][[symplectic vector spaces|symplectic]] [[topological vector spaces]] by [[continuous field of C-star algebras|continuous fields of]] [[Weyl algebras]]:
* [[Ernst Binz]], [[Reinhard Honegger]], [[Alfred Rieckers]], *Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization*, International Journal of Pure and Applied Mathematics **38** 1 (2007) [[ijpam:2007-38-1/6](https://ijpam.eu/contents/2007-38-1/6/index.html), [pdf](https://www.ijpam.eu/contents/2007-38-1/6/6.pdf)]
* [[Reinhard Honegger]], [[Alfred Rieckers]], *Heisenberg Group Algebra and Strict Weyl Quantization*, Chapter 23 in: *Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems*, World Scientific (2015) [chapter:[doi;10.1142/9789814696586_0023](https://doi.org/10.1142/9789814696586_0023), book:[doi:10.1142/9251-vol1](https://doi.org/10.1142/9251-vol1)]
See also:
* [[eom]]: J.-E. Björk, [Weyl algebra](http://eom.springer.de/w/w097670.htm)
A [[categorification]] of the Weyl algebra is introduced in operadic language in
* Nikita Markarian, _Weyl $n$-algebras_, [arxiv/1504.01931](http://arxiv.org/abs/1504.01931)
[[!redirects Weyl algebras]]
|
Weyl bundle | https://ncatlab.org/nlab/source/Weyl+bundle | If $(M,\omega)$ is a symplectic manifold then the completed symmetric power of the cotangent bundle $W = \hat{S}(T^* M)$, and sometimes also $W_h = \hat{S}(T^* M)[[h]]$ are called the __Weyl bundle__. (The same term is used for some other, quite different, notions!)
In addition to the commutative symmetric algebra structure, there is a noncommutative product due symplectic structure.
If $a,b\in \Gamma_U(W_h)$ are sections of
$W_h$ above open $U\subset M$ then their noncommutative Moyal-Weyl product is
$$
a \ast b = \left. exp \left(\frac{h}{2}\omega_{j l}(x) \frac{\partial}{\partial y}\frac{\partial}{\partial z}\right) a(y) b(z) \right|_{y=z}
$$
There is also a grading where $deg h = 2$ and $deg w = l$ for $w\in S^l(T^* M)$. So we get a bundle of noncommutative associative algebras.
[[Fedosov connection]] is a connection on $W_h$ (depending on a choice of a cocycle, the Weyl curvature $\Omega\in Z^2(M)[[h]]$). It has the property that the exponential map identifies the smooth functions on $M$ with horizontal sections of $W_h$ for the connection.
Related entries are [[deformation quantization]]
See section 2.2 of
* [[Nicolai Reshetikhin]], Milen Yakimov, _Deformation quantization of Lagrangian fiber bundles_, Conference Moshe Flato 1999, vol. 2, 269-288, Kluwer 2000, [math.QA/9907164](http://arxiv.org/abs/math/9907164)
and section 6 of
* Simone Gutt, John Rawnsley, _Natural star products on symplectic manifolds
and quantum moment maps_, [pdf](https://homepages.warwick.ac.uk/~marke/research/files/naturalstar.pdf)
|
Weyl character formula | https://ncatlab.org/nlab/source/Weyl+character+formula |
+-- {: .rightHandSide}
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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A formula for the [[character]] of [[irreducible representations]] of a [[compact Lie group]] in terms of highest [[weights]].
## Related concepts
* [[Kac character formula]]
## References
* Wikipedia, _[Weyl character formula](http://en.wikipedia.org/wiki/Weyl_character_formula)_
|
Weyl functional calculus | https://ncatlab.org/nlab/source/Weyl+functional+calculus |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What is called _Weyl quantization_ is a method of _[[quantization]]_ applicable to [[symplectic manifolds]] which are [[symplectic vector spaces]] or [[quotients]] of these by [[discrete groups]] ([[tori]]).
In Weyl quantization of the flat space $\mathbf{R}^n$, the classical observables of the form $f(x,p)$ are replaced by suitable operators which in the case when $f$ is a polynomial correspond to writing $f$ with $x$ and $p$ replaced by noncommutative variables $x$ and $i h\frac{\partial}{\partial x}$
in symmetric or Weyl ordering. This means that all possible orderings between $x$ and $i h\frac{\partial}{\partial x}$ are summed with an equal weight. More generally, one can extend this rule to more general functions via integral formulas due Weyl and Wigner. This is also useful in fundations of the theory of [[pseudodifferential operator]]s.
## Related concepts
* [[free field theory]]
## References
* [[Lars Hörmander]], _The Weyl calculus of pseudodifferential operators_, Comm. Pure Appl. Math. __32__ (1979), no. 3, 360–444. [MR80j:47060](http://www.ams.org/mathscinet-getitem?mr=517939), [doi](http://dx.doi.org/10.1002/cpa.3160320304)
* Robert F. V. Anderson, _The Weyl functional calculus_, J. Functional Analysis __4__:240-267, 1969, [MR635128](http://www.ams.org/mathscinet-getitem?mr=635128);
* _On the Weyl functional calculus_, J. Functional Analysis __6__:110–115, 1970, [MR262857](http://www.ams.org/mathscinet-getitem?mr=262857)
* E. M. Stein, _Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals_, Princeton University Press 1993
* M. W. Wong, _Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator_, Annals Global Anal. Geom. 28 (2005) 271–283
* [[Thomas L. Curtright]], [[David B. Fairlie]], [[Cosmas K. Zachos]], *A Concise Treatise on Quantum Mechanics in Phase Space*, World Scientific (2014) [[doi:10.1142/8870](https://doi.org/10.1142/8870)]
Discussion of [[quantization of Chern-Simons theory]] in terms of Weyl quantization is in
* [[Jørgen Andersen]], _Deformation quantization and geometric quantization of abelian moduli spaces_, Commun. Math. Phys., 255 (2005), 727–745
* {#GelcaUribe02} [[Razvan Gelca]], [[Alejandro Uribe]], _The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same_, Commun.Math.Phys. 233 (2003) 493-512 ([arXiv:math-ph/0201059](http://arxiv.org/abs/math-ph/0201059))
* {#GelcaUribe10a} [[Razvan Gelca]], [[Alejandro Uribe]], _From classical theta functions to topological quantum field theory_ ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf))
* {#GelcaUribe10b} [[Razvan Gelca]], [[Alejandro Uribe]], _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/1007.2010))
Discussion of the generalization to [[BV-quantization]] is in
* [[Owen Gwilliam]], [[Rune Haugseng]], _Linear Batalin-Vilkovisky quantization as a functor of ∞-categories_ ([arXiv:1608.01290](https://arxiv.org/abs/1608.01290))
[[!redirects Weyl functional calculi]]
[[!redirects Weyl calculus]]
[[!redirects Weyl calculi]]
[[!redirects Weyl quantization]]
[[!redirects Weyl quantizations]]
|
Weyl group | https://ncatlab.org/nlab/source/Weyl+group |
> Not to be confused with [[Weil group]].
***
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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
### In Lie theory
In [[Lie theory]], a _Weyl group_ is a [[group]] associated with a [[compact Lie group]] that can either be abstractly defined in terms of a [[root system]] or in terms of a [[maximal torus]]. More generally there are Weyl groups associated with [[symmetric spaces]].
The Weyl group of a [[compact Lie group]] $G$ is equivalently the [[quotient group]] of the [[normalizer]] of any [[maximal torus]] $T$ by that torus.
$$
W \simeq N_G T / T
\,.
$$
### In equivariant homotopy theory
{#InEquivariantHomotopyTheory}
In [[equivariant homotopy theory]] one uses the term _Weyl group_ more generally for the [[quotient group]]
$$
W_G H = (N_G H) / H
$$
of the [[normalizer]] of a given [[subgroup]] $H \hookrightarrow G$ by that subgroup (e.g. [May 96, p. 13](#May96)).
The relevance of the Weyl group in this sense is that it is the maximal group which canonically [[action|acts]] on $H$-[[fixed points]] of a [[topological G-space]]. (See at _[Change of equivariance group and fixed loci](topological+G-space#ChangeOfGroupsAndFixedLoci)_ for details and, at, e.g., _[[tom Dieck splitting]]_ for applications.)
This may be seen from the fact that the Weyl group of $H \subset G$ is the [[automorphism group]] of the [[coset space]] $G/H$ in the [[orbit category]] of $G$ (and in fact the [[endomorphism monoid]] of $G/H$, since the orbit category is an [[EI-category]], see [there](EI-category#OrbitCategory)):
\[
\label{AsAutomorphismGroupInOrbitCategory}
End_{G Orbits}
\big(
G/H
\big)
\;\;
=
Aut_{G Orbits}
\big(
G/H
\big)
\;\;
\simeq
\;\;
W_G(H)
\,.
\]
Notice that $W_G G = 1$ and $W_G 1 = G$.
{#WeylGroupOfNormalSubgroup} On the other hand, if $H = N \subset G$ is a [[normal subgroup]], then its [[normalizer]] is $G$ itself, in which case the Weyl group is just the plain [[quotient group]]
$$
W_G N \;\simeq\; G/N
\,.
$$
## Definition
Given a [[compact Lie group]] $G$ with chosen
[[maximal torus]] $T$, its __Weyl group__ $W(G)=W(G,T)$ is the [[group of automorphisms]] of $T$ which are restrictions of [[inner automorphisms]] of $G$.
This is the [[quotient group]] of the [[normalizer subgroup]] of $T \subset G$ by $T$
$$
W \simeq N_G(T)/T
\,.
$$
## Properties
* The [[maximal torus]] is of [[finite index subgroup|finite index]] in its [[normalizer]]; the [[quotient]] $N(T)/T$ is [[isomorphism|isomorphic]] to $W(G)$.
* The [[cardinality]] of $W(G)$ for a compact connected $G$, equals the [[Euler characteristic]] of the [[homogeneous space]] $G/T$ ("[[flag variety]]").
* An important approach to the representations of the Weyl groups is the [[Springer theory]].
## Related concepts
* [[Schubert calculus]]
## References
* [[eom]]: [Weyl group](http://eom.springer.de/W/w097710.htm); wikipedia [Weyl group](http://en.wikipedia.org/wiki/Weyl_group)
* N. Chriss, V. Ginzburg, _Representation theory and complex geometry_, Birkhäuser 1997. x+495 pp.
* Walter Borho, Robert MacPherson, _Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes_, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 15, 707–710 [MR82f:14002](http://www.ams.org/mathscinet-getitem?mr=618892)
* {#May96} [[Peter May]], _Equivariant homotopy and cohomology theory_ CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996.
With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf), [cbms-91](https://bookstore.ams.org/cbms-91))
[[!redirects Weyl groups]] |
Weyl n-algebra | https://ncatlab.org/nlab/source/Weyl+n-algebra |
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A concept of _Weyl n-algebra_ is to be a refinement to [[higher algebra]] of the concept of _[[Weyl algebra]]_.
## Definitions
Since ordinary [[Weyl algebras]] are [[universal enveloping algebras]] of [[Heisenberg Lie algebras]] of [[symplectic vector spaces]] (see [here](universal+enveloping+algebra#WeylAlgebraAndHeisenbergAlgebra)), and since there is a sensible notion of [[Heisenberg Lie n-algebra|Heisenberg Lie $n$-algebra]] for all $n \in \mathbb{N}$, it makes sense to defines _Weyl $n$-algebras_ to be the [[universal enveloping E-n algebra|universal enveloping $E_n$-algebras]] of [[Heisenberg Lie n-algebra|Heisenberg Lie $n$-algebras]].
Another definition is considered by [Markarian 2015](#Markarian15).
## References
* {#Markarian15} [[Nikita Markarian]], _Weyl $n$-algebras_ ([arXiv:1504.01931](http://arxiv.org/abs/1504.01931))
[[!redirects Weyl n-algebra]] |
Weyl relation | https://ncatlab.org/nlab/source/Weyl+relation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What are called _Weyl relations_ is the incarnation of [[canonical commutation relations]] under passing to [[exponentials]], constituting the *[[Weyl algebra]]*.
For example if $a, a^\ast$ are two elements of an [[associative algebra]] with [[commutator]]
$$
[a,a^\ast] = \hbar
$$
then the corresponding Weyl relation is, by the [[Baker-Campbell-Hausdorff formula]],
$$
e^{z a}
e^{z^\ast a^\ast}
\;=\;
e^{z^\ast a^\ast}
e^{z a}
e^{\hbar z z^\ast}
$$
for $z,z^\ast \in \mathbb{C}$.
## In the Wick algebra of free quantum fields
+-- {: .num_prop #MoyalStarProductOnMicrocausal}
###### Proposition
**([[Hadamard distribution|Hadamard]]-[[Moyal star product]] on [[microcausal observables]] -- [[abstract Wick algebra]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$. Write $\Delta$ for the [[causal propagator]] and let
$$
\Delta_H
\;=\;
\tfrac{i}{2}\Delta + H
$$
be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).
Then the [[star product]] induced by $\Delta_H$
$$
A \star_H A
\;\coloneqq\;
prod
\circ
\exp\left(
\int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \Phi^a(x_1)}
\otimes \frac{\delta}{\delta \Phi^b(x_2)} dvol_g
\right)
(P_1 \otimes P_2)
$$
on [[off-shell]] [[microcausal observables]] $A_1, A_2 \in \mathcal{F}_{mc}$ is well defined in that the [[wave front sets]] involved in the [[products of distributions]] that appear in expanding out the [[exponential]] satisfy [[Hörmander's criterion]].
Hence by the general properties of [[star products]] ([this prop.](star+product#AssociativeAndUnitalStarProduct)) this yields a [[unital algebra|unital]] [[associative algebra]] [[structure]] on the space of [[formal power series]] in $\hbar$ of [[off-shell]] [[microcausal observables]]
$$
\left(
PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H
\right)
\,.
$$
This is the _[[off-shell]] [[Wick algebra]]_ corresponding to the choice of [[Wightman propagator]] $H$.
Moreover the image of $P$ is an ideal with respect to this algebra structure, so that it descends to the [[on-shell]] [[microcausal observables]] to yield the _[[on-shell]] [[Wick algebra]]_
$$
\left(
PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H
\right)
\,.
$$
Finally, under [[complex conjugation]] $(-)^\ast$ these are [[star algebras]] in that
$$
\left(
A_1 \star_H A_2
\right)^\ast
=
A_2^\ast \star_H A_1^\ast
\,.
$$
=--
For **proof** see at _[[Wick algebra]]_ [this prop.](Wick+algebra#MoyalStarProductOnMicrocausal).
+-- {: .num_remark #WickAlgebraIsFormalDeformationQuantization}
###### Remark
**([[Wick algebra]] is [[formal deformation quantization]] of [[Poisson-Peierls bracket|Poisson-Peierls algebra of observables]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$ with [[causal propagator]] $\Delta$ and let $\Delta_H \;=\; \tfrac{i}{2}\Delta + H$ be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).
Then the [[Wick algebra]] $\left( PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ] \,,\, \star_H \right)$ from prop. \ref{MoyalStarProductOnMicrocausal} is a [[formal deformation quantization]] of the [[Poisson algebra]] on the [[covariant phase space]] given by the [[on-shell]] [[polynomial observables]] equipped with the [[Poisson-Peierls bracket]] $\{-,-\} \;\colon\; PolyObs(E,\mathbf{L})_{mc} \otimes PolyObs(E,\mathbf{L})_{mc} \to PolyObs(E,\mathbf{L})_{mc}$ in that for all $A_1, A_2 \in PolyObs(E,\mathbf{L})_{mc}$ we have
$$
A_1 \star_H A_2
\;=\;
A_1 \cdot A_2
\;mod\;
\hbar
$$
and
$$
A_1 \star_H A_2
-
A_2 \star_H A_1
\;=\;
i \hbar \{A_1, A_2\}
\;mod\;
\hbar^2
\,.
$$
=--
([Dito 90](#Dito90), [Dütsch-Fredenhagen 01](Wick+algebra#DutschFredenhagen01))
+-- {: .proof}
###### Proof
By prop. \ref{MoyalStarProductOnMicrocausal} this is immediate from the general properties of the [[star product]] ([this example](A+first+idea+of+quantum+field+theory+--+Quantization#MoyalStarProductIsFormalDeformationQuantization)).
Explicitly, consider, without restriction of generality, $A_1 = \int (\alpha_1)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ and $A_2 = \int (\alpha_2)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ be two linear observables. Then
$$
\begin{aligned}
A_1 \star_H A_2
& =
A_1 A_2
+
\hbar
\int
\left(
\tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2)
\right)
\frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)}
\frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)}
\;mod\;
\hbar^2
\\
& =
A_1 A_2
+
\hbar
\left(
\int
(\alpha_1)_{a_1}(x_1)
\left(
\tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2)
+ H^{a_1 a_2}(x_1, x_2)
\right)
(\alpha_2)_{a_2}(x_2)
\right)
\;mod\;
\hbar^2
\end{aligned}
$$
Now since $\Delta$ is skew-symmetric while $H$ is symmetric is follows that
$$
\begin{aligned}
A_1 \star_H A_2
-
A_2 \star_H A_1
& =
i \hbar
\left(
\int
(\alpha_1)_{a_1}(x_1)
\Delta^{a_1 a_2}(x_1, x_2)
(\alpha_2)_{a_2}(x_2)
\right)
\;mod\;
\hbar^2
\\
& =
i \hbar \, \left\{ A_1, A_2\right\}
\end{aligned}
\,.
$$
The right hand side is the [[integral kernel]]-expression for the [[Poisson-Peierls bracket]], as shown in the second line.
=--
+-- {: .num_example}
###### Example
**([[Hadamard vacuum state]] [[2-point function]])**
Let
$$
A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc}
$$
for $i \in \{1,2\}$ be two [[linear observable|linear]] [[microcausal observables]] represented by [[distributions]] which in [[generalized function]]-notation are given by
$$
A_i
\;=\;
\int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i)
\,.
$$
Then their Hadamard-Moyal [[star product]] (prop. \ref{MoyalStarProductOnMicrocausal}) is the [[sum]] of their pointwise product with $\tfrac{1}{2} i \hbar$ times the evaluation
$$
\begin{aligned}
\langle A_1 A_2\rangle
& \coloneqq
\int \int
(\alpha_1)_{a_1}(x_1)
\,
\left\langle \mathbf{\Phi}^{a_1}(x_1) \mathbf{\Phi}^{a_2}(x_2)\right\rangle
\,
(\alpha_2)_{a_2}(x_2)
\, dvol_\Sigma(x_1)
\, dvol_\Sigma(x_2)
\\
& \coloneqq
\tfrac{1}{2}
i \hbar \int \int
(\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2)
\,dvol_\Sigma(x_1)
\,dvol_\Sigma(x_2)
\end{aligned}
$$
of the [[Wightman propagator]] $\Delta_H$:
$$
\label{StarProductOfTwoLinearObservables}
A_1 \star_H A_2
=
A_1 \cdot A_2
+
\langle A_1 A_2\rangle
$$
Further [below](#HadamardVacuumStatesOnWickAlgebras) we see that this evaluation is the [[2-point function]] of a [[state on a star-algebra|state]] on the [[Wick algebra]].
=--
+-- {: .num_example #WeylRelations}
###### Example
**([[Weyl relations]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] and with [[Wightman propagator]] $\Delta_H$.
Then for
$$
A_1, A_2
\;\in\;
LinObs(E,\mathbf{L})_{mc}
\hookrightarrow
PolyObs(E,\mathbf{L})_{mc}
$$
two [[linear observables|linear]] [[microcausal observables]], the Hadamard-Moyal star product (def. \ref{MoyalStarProductOnMicrocausal}) of their [[exponentials]] exhibits the [[Weyl relations]]:
$$
e^{A_1}
\star_H
e^{A_2}
\;=\;
e^{A_1 + A_2} \; e^{\langle A_1 A_2\rangle}
$$
where on the right we have the [[exponential]] [[Wightman 2-point function]] (eq:StarProductOfTwoLinearObservables).
=--
(e.g. [Dütsch 18, exercise 2.3](#Duetsch18))
## Related concepts
* [[Weyl algebra]]
* [[canonical commutation relations]]
* [[Heisenberg algebra]]
## References
> For more references see at *[[Weyl algebra]]*.
The notion goes back to
* {#Weyl27} [[Hermann Weyl]], (46) in: *Quantenmechanik und Gruppentheorie*, Zeitschrift für Physik **46** (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]
* {#vonNeumann31} [[John von Neumann]], p. 571 of: *Die Eindeutigkeit der Schrödingerschen Operatoren*, Mathematische Annalen **104** (1931) 570–578 [[doi:10.1007/BF01457956](https://doi.org/10.1007/BF01457956)]
> (proving the [[Stone-von Neumann theorem]])
See also:
* {#Duetsch18} [[Michael Dütsch]], exercise 2.3 in: _[[From classical field theory to perturbative quantum field theory]]_, 2018
[[!redirects Weyl relations]]
|
Weyl ring | https://ncatlab.org/nlab/source/Weyl+ring | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Group theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
A $\mathbb{Z}$-[[Weyl algebra]].
## Definition
### Finitely generated Weyl rings
Given an [[abelian group]] $G$, the **$n$-th Weyl ring** is a [[ring]] $A_n(G)$ with
* an abelian group [[homomorphism]] $g:G \to A_n(G)$
* a function $x:[1, n] \to A_n(G)$
* a function $\partial:[1, n] \to A_n(G)$
such that
* for every number $i, j \in [1,n]$, $x(i) \cdot x(j) = x(j) \cdot x(i)$
* for every number $i, j \in [1,n]$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$
* for every number $i \in [1,n]$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$
* for every number $i, j \in [1,n]$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$
* for every other ring $R$ with abelian group homomorphism $h:G \to R$ with
* a function $x:[1, n] \to R$
* a function $\partial:[1, n] \to R$
where
* for every number $i, j \in [1,n]$, $x(i) \cdot x(j) = x(j) \cdot x(i)$
* for every number $i, j \in [1,n]$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$
* for every number $i \in [1,n]$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$
* for every number $i, j \in [1,n]$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$
there is a unique ring homomorphism $i:A_n(G) \to R$ such that $i \circ g = h$.
### General Weyl rings
Given an [[abelian group]] $G$ and a set $S$ with [[stable equality]], the **$S$-generated Weyl ring** is a [[ring]] $A(S,G)$ with
* an abelian group [[homomorphism]] $g:G \to A(S,G)$
* a function $x:S \to A(S,G)$
* a function $\partial:S \to A(S,G)$
such that
* for every number $i, j \in S$, $x(i) \cdot x(j) = x(j) \cdot x(i)$
* for every number $i, j \in S$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$
* for every number $i \in S$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$
* for every number $i, j \in S$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$
* for every other ring $R$ with abelian group homomorphism $h:G \to R$ with
* a function $x:S \to R$
* a function $\partial:S \to R$
where
* for every number $i, j \in S$, $x(i) \cdot x(j) = x(j) \cdot x(i)$
* for every number $i, j \in S$, $\partial(i) \cdot \partial(j) = \partial(j) \cdot \partial(i)$
* for every number $i \in S$, $\partial(i) \cdot x(i) - x(i) \cdot \partial(i) = 1$
* for every number $i, j \in S$, $i \neq j$ implies $\partial(i) \cdot x(j) - x(j) \cdot \partial(i) = 0$
there is a unique ring homomorphism $i:A(S,G) \to R$ such that $i \circ g = h$.
## See also
* [[ring]]
* [[tensor ring]]
* [[symmetric ring]]
* [[exterior ring]]
* [[Clifford ring]]
* [[Weyl algebra]]
[[!redirects Weyl rings]] |
Weyl semimetal | https://ncatlab.org/nlab/source/Weyl+semimetal |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A type of [[topological phase of matter]] exhibiting [[Weyl spinor|Weyl]] [[fermion]] excitations.
## Related concepts
* [[strange metal]]
* [[topological insulator]]
* [[Chern insulator]]
## References
Review:
* Binghai Yan and Claudia Felser, *Topological Materials: Weyl Semimetals*, Annual Review of Condensed Matter Physics, Vol. 8:337-354 (2017) ([doi:10.1146/annurev-conmatphys-031016-025458](https://doi.org/10.1146/annurev-conmatphys-031016-025458))
* Satyaki Kar, Arun M Jayannavar, *A Primer on Weyl Semimetals: Down the Discovery of Topological Phases*, Asian Journal of Research and Reviews in Physics, 4(1), 34-45 (2021) ([arXiv:1902.01620](https://arxiv.org/abs/1902.01620))
See also:
* Wikipedia, *[Weyl semimetal](https://en.wikipedia.org/wiki/Weyl_semimetal)*
[[!redirects Weyl semimetals]]
[[!redirects Weyl semi-metal]]
[[!redirects Weyl semi-metals]] |
Weyl spinor | https://ncatlab.org/nlab/source/Weyl+spinor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A type of [[spin representation]].
For the moment see _[here](Majorana+spinor#DiracAndWeylRepresentations)_.
## Related concepts
* [[Dirac spinor]], [[Majorana spinor]]
* [[Weyl semimetal]]
[[!redirects Weyl spinors]]
[[!redirects Weyl representation]]
[[!redirects Weyl representations]]
[[!redirects chiral spinor]]
[[!redirects chiral spinors]]
[[!redirects chiral spin representation]]
[[!redirects chiral spin representations]] |
Weyl tensor | https://ncatlab.org/nlab/source/Weyl+tensor |
## Related concepts
* [[Riemann tensor]]
* [[conformal geometry]]
## References
* Wikipedia, _[Weyl tensor](http://en.wikipedia.org/wiki/Weyl_tensor)_
[[!redirects Weyl tensors]] |
What is an elliptic object? | https://ncatlab.org/nlab/source/What+is+an+elliptic+object%3F | This entry records the article
* [[Stephan Stolz]], [[Peter Teichner]]:
\linebreak
**What is an elliptic object?**
\linebreak
in: _Topology, geometry and quantum field theory_
London Math. Soc. LNS **308**
Cambridge Univ. Press (2004) 247-343
[[pdf](https://math.berkeley.edu/~teichner/Papers/Oxford.pdf), [[Stolz-Teichner_EllipticObject.pdf:file]]]
on [[geometric models for elliptic cohomology]] with speculations about the formalization of 2-dimensional [[CFT]] as an extended ([[2-functor]]ial) [[FQFT]], motivated by the [[Witten genus]].
Related discussion of the [[stringor bundle]]:
* {#StolzTeichner2005} [[Stephan Stolz]], [[Peter Teichner]], _The spinor bundle on loop space_ (2005) [[pdf](http://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/MPI.pdf), [[Stolz-Teichner-SpinorOnLoopSpace.pdf:file]]]
In more recent developments the authors exchanged, for technical convenience, the conformal structure in favor of flat [[Riemannian metric|Riemannian structure]] in
* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric field theories and generalized cohomology_ , in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), *[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]*, Symposia in Pure Mathematics (2011) [[arXiv:1108.0189](http://arxiv.org/abs/1108.0189)]
See also:
* [[(1,1)-dimensional Euclidean field theories and K-theory]]
* [[(2,1)-dimensional Euclidean field theories and tmf]]
category: reference
[[!redirects What is an elliptic object?]]
|
what to contribute | https://ncatlab.org/nlab/source/what+to+contribute |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Contents {: .clickToReveal}
### Contents {: .clickToHide tabindex="0"}
+-- {: .hide}
[[!include contents]]
=--
+-- {: .hide}
[[!include mathematicscontents]]
=--
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
This page is supposed to give some idea about which kind of contributions to the [[HomePage|nLab]] are appropriate or desired.
* **small contributions**: While it is certainly possible that any entry could _eventually_ grow into a comprehensive discussion, everything needs to start small. If you feel like creating an entry but only have a single sentence to say or a single reference to record for the moment, please do so! One also speaks of _stub entries_ for such entries that are waiting to be expanded into something more substantial. Their existence can be very helpful to the general process of creating a good wiki. Just be sure to announce the creation of the entry on the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/).
* **large contributions**: There is no lack of virtual room. If you feel you need to add a lot of material to do your topic justice, that's fine. We have entries that are the size of lengthy research articles. However, if a single entry grows too large, it can be useful to start breaking it up into a series of separate entries, all linked to each other. One can use "floating tables of contents" to collect groups of entries into "topic clusters". Eventually at [[HowTo]] there should be some information on how to do this.
* **standard material**: While the $n$Lab is meant to eventually push in a certain direction ("[[nPOV]]") the scope of this direction is vast and developing it requires background in all possible subjects. So all standard material in [[mathematics]] and [[physics]] is suitable and welcome for inclusion in to the $n$Lab. Your contribution need not have any visible relation to [[higher algebra]], [[homotopy theory]], [[type theory]], [[category theory]], or [[higher category theory]] at all! But after you announce it on the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/) it might be that others join in and point out such a relation.
* **original research**: The $n$Lab is expressly meant to be a tool for researchers to make notes they find relevant for their work. This means that if you have an original insight and feel like recording it in context, you are welcome to do so on the $n$Lab. Just follow common good scientific practice: clearly indicate for every non-evident statement that you add where it comes from and what the available proof is (if one is available, that is; conjectures are welcome too). And be prepared to have your contributions discussed on the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/).
* **expositional as well as specialized technical material**
The $n$Lab differs from other wikis out there by containing a considerable amount of technical material and abstract mathematics. This does not mean, however, that expositions and friendly pedagogical lead-ins are discouraged. On the contrary! As far as they seem to be lacking, this is only because nobody has yet found the time to add them.
In general: if some topic that seems to be worthwhile is missing from the $n$Lab (and a lot is) this is with high probability not by intent, but by lack of manpower. Be the first to improve on the situation!
category: meta
[[!redirects What to Contribute]]
|
wheel | https://ncatlab.org/nlab/source/wheel |
# Wheels
* table of contents
{: toc}
## Idea
A wheel is an algebraic concept similar to a [[field]] but in which division is *always* defined, even division by zero. This naturally requires modifications to the usual axioms. Every [[commutative ring]] gives rise to a wheel using a simple modification of the construction of the [[field of fractions]], although there are other examples.
The terminology 'wheel' comes from the example of the wheel of [[real numbers]], which geometrically consists of a circle (the circle of [[extended real numbers]]) together with an additional point (thought of as the hub of the wheel), thus looking like $\odot$.
## Notation
In elementary algebra, it\'s traditional to write $-x$ for the opposite (or additive inverse) of $x$, although one could write $0 - x$ (or perhaps $-1x$, meaning $-1 \cdot x$) instead. This overloads the symbol $-$, with one meaning as a unary operator and another as a binary operator, related since $y - x = y + (-x)$. However, it is *not* common to write $/x$ for the reciprocal (or multiplicative inverse) of $x$, but only to write $1/x$ (or perhaps $x^{-1}$) instead. But it would work just as well to overload the symbol $/$, again with one meaning as a unary operator and another as a binary operator, related since $y/x = y \cdot /x$.
In a wheel, the unary operator $/$ is a fundamental part of the structure, and the binary $/$ is only defined using it. Although it remains true that $y/x = y \cdot /x$, so that we could reasonably write $1/x$ instead of $/x$, this puts the emphasis the wrong way. And while we could probably write $x^{-1}$ instead of $/x$, this is dangerous, since it\'s not true that $/x$ is always the multiplicative inverse of $x$ in a wheel. (But for that matter, $-x$ is not always the additive inverse of $x$ in a wheel, even though we can still write it, and indeed define it, as $-1 \cdot x$, when this makes sense.)
If we write multiplication as juxtaposition, then $y /x$ is naturally interpreted as $y \cdot /x$, and so rather than think explicitly of the derived binary operation $/$, it\'s best to think of $y /x$ as always meaning $y \cdot /x$. So only the unary version of $/$ really matters. But notice that $/x y$ means $/x \cdot y$, not $/(x y)$ (which is different, and in fact equal to $/x \cdot /y$, or $/x /y$ for short). So we are coming down on the side of $8 /2 (2 + 2) = 16$ in the great PEMDAS debate.
## Definition
A __wheel__ consists of an [[underlying set]] $W$ equipped with two binary operations written with any common notation for addition and multiplication, a unary operation written with $/$ as a prefix (called the reciprocal), and two constants $0$ and $1$, such that:
* Addition and multiplication are each associative and commutative, with $0$ and $1$ (respectively) as identities;
* The reciprocal is an [[involution]]: $//x = x$;
* Reciprocals get along with multiplication: $/(x y) = /x /y$ and $/1 = 1$;
* Multiplication sort of distributes over addition, but not quite: neither $0 x$ nor $(x + y) z$ can be simplified in general, but instead we have $(x + y) z + 0 z = x z + y z$;
* An axiom that would imply that reciprocals are multiplicative inverses if distributivity held: $(x + y z) /z = x / z + y + 0 z$;
* Special properties of $0$ that are needed since distributivity doesn\'t hold: $0 0 = 0$, $(x + 0 y) z = x z + 0 y$, and $/(x + 0 y) = /x + 0 y$; and
* Although $0 /0$ doesn\'t simplify, it\'s an [[absorbing element]] for addition: $0 /0 + x = 0 /0$.
Although these rules are somewhat odd, we need some unusual behaviour to prevent a wheel from collapsing to a triviality via $0 = 0 x = 0 /0 x = 1 x = x$. In fact, only the last step is generally valid in a wheel. The paradigmatic example is the wheel of fractions of a commutative ring, for which all of the axioms are easily checked, but for which simpler (invalid) statements have counterexamples.
We can generalize the quasi-distributive rule beyond two terms as
$$ (\sum_{i=1}^n x_i) y + \sum_{i=1}^{n-1} (0 y) = \sum_{i=1}^n (x_i y) .$$
That is, to multiply $y$ by the sum of $n$ terms, you need to add in $n-1$ copies of $0 y$ as well, and then the result is the sum of $n$ products.
Wheels are like [[rigs]] in that there is generally no notion of subtraction. However, if the wheel happens to have a (necessarily unique) additive inverse of $1$, we may write it as $-1$, define $-x$ as $-1 x$, and define $y - x$ as $y + -x$. But notice that $-x$ is *not* in general an additive inverse of $x$; rather, we have $x - x = 0 x + 0 x$ (using the modified general form of distributivity).
It is common to write $/0$ as $\infty$ and $0 \infty$ as $\bot$.
## Examples
The original example is the __wheel of real numbers__, denoted $\mathbb{R}^\odot$. An element of $\mathbb{R}^\odot$ is an equivalence class of pairs of real numbers under a certain equivalence relation. Morally, $(a,b)$ and $(c,d)$ are set equivalent iff $a d = b c$; however, this is not quite right, because it makes $(0,0)$ equivalent to every other pair (and so it\'s not even transitive). So we additionally require that $(a,b) \ne (0,0)$ iff $(c,d) \ne (0,0)$, so that $(0,0)$ is actually equivalent only to itself. We may write the equivalence class of $(a,b)$ as $a/b$, but let\'s write it as $a : b$ instead until we\'ve established what $a /b$ means in $\mathbb{R}^\odot$. Now, we can define addition and multiplication in the usual way for fractions: $(a : b) + (c : d) \coloneqq (a d + b c) : (b d)$ and $(a : b) (c : d) \coloneqq (a c) : (b d)$. Reciprocals are immediate: $/(a : b) \coloneqq b : a$. The identities $0$ and $1$ in $\mathbb{R}^\odot$ are $0 : 1$ and $1 : 1$ respectively. More generally, any real number $a$ gives us an element $a : 1$ of $\mathbb{R}^\odot$. This function from $\mathbb{R}$ to $\mathbb{R}^\odot$ is one-to-one and respects addition and multiplication; it also respects reciprocals in that $/(a : 1) = (1/a) : 1$ if $a \ne 0$. Now we\'re justified in writing $a /b$ for $a : b$. We also write $\infty$ for $1 : 0$ and $\bot$ for $0 : 0$; these are the only elements of $\mathbb{R}^\odot$ that don\'t come from $\mathbb{R}$. We have $x + \infty = \infty$ whenever $x \ne \infty, \bot$, while $\infty + \infty = \bot$ and $x + \bot = \bot$ always; and $\infty x = \infty$ whenever $x \ne 0, \bot$, while $0 \infty = \bot$ and $\bot x = \bot$ always. In particular, $-\infty = \infty$, and $-\bot = \bot$. Of course, $/0 = \infty$ and $/\infty = 0$, while $/\bot = \bot$. Then $x - x = 0$ whenever $x \ne \infty, \bot$, while $\infty - \infty, \bot - \bot = 0$; and $x/x = 1$ whenever $x \ne 0, \infty, \bot$, while $0/0, \infty/\infty, \bot/\bot = \bot$. Fortunately, $x + 0 = x$ and $1 x = x$ remain true always. One way to think of all of this is that we are using the usual arithmetic on the version of the [[extended real numbers]] in which $\infty = -\infty$ (the [[projective line]]), but whenever the result is undefined, we take it to be $\bot$ (and any calculation involving $\bot$ stays $\bot$, as in [[domain theory]]).
## References
* Jesper Carlström (2001). _Wheels: On Division by Zero._ [Web](https://www2.math.su.se/reports/2001/11/).
[[!redirects wheel]]
[[!redirects wheels]]
|
wheeled graph | https://ncatlab.org/nlab/source/wheeled+graph |
#Contents#
* table of contents
{:toc}
##Idea##
A **wheeled graph** in the sense of ([HRY](#HRY)) is a generalization of the notion of a [[directed pseudograph]]. What differentiates a wheeled graph from a [[directed pseudograph]] is the notion of an "exceptional cell," which should be thought of as a set of half edges which are not adjacent to any vertex.
The basic idea behind a wheeled graph is that it is a set of vertices with directed edges between them but also edges that leave and enter the graph. It can also have loops and even edges that are not adjacent to any vertex (called **exceptional edges**, see [[generalized graph]] for more). Moreover, a wheeled graph $G$ is equipped with a coloring, i.e. there is a function $Vt(G)\overset{\kappa}\to \mathcal{C}$ from the vertices of $G$ to a set of colors. Since wheeled operads have inputs and outputs, this makes them suitable for modeling [[PROPs]] and [[properads]].
The term "wheeled" refers to the fact that a [[generalized graph]] $G$ might have directed loops in it. These can be in the form of vertices with loops, closed directed paths in $G$, or "exceptional loops" that have no vertices at all (see the examples below). Sometimes, we will be interested in wheel-free graphs, which are wheeled graphs without wheels (though they are not, crucially, just [[graphs]].)
This entry relies on notation defined in [[generalized graph]].
## Definitions ##
The following is the fifth item of Definition 2.5 of [HRY](#HRY):
+-- {: .num_defn }
###### Definition
A wheeled graph is a [[generalized graph]] equipped with a coloring, a direction and a listing (see [here](https://ncatlab.org/nlab/show/generalized+graph#properties) for definitions of these properties).
=--
## Extra Structure
Wheeled graphs can have a number of attributes. Most of the following are self-explanatory though stating them in the terminology of [[generalized graph]] as [HRY]{#HRY} does, can be tedious:
* One of these, confusingly, is being **wheel-free**. In other words, a wheeled graph which is wheel-free is a generalized graph with a coloring, a direction and a listing, but without any loops (either ordinary or exceptional) or directed paths with identical initial and terminal vertex.
* A wheeled graph is **connected** if it is a single exceptional edge, a single exceptional loop, or has empty exceptional cell, is non-empty and between any two vertices there is an edge.
* A wheeled graph is **simply connected** if it is connected, is not an exceptional loop, and contains no cycles.
* A wheeled graph is a **unital tree** if it is simply connected and each vertex has exactly one output flag.
* A wheeled graph is a **linear graph** if it is a linear tree in which each vertex has exactly one input flag.
Using these definitions we can define the following sets:
* Define $Gr_c^{\circlearrowleft}$ to be the set of connected wheeled graphs.
* Define $Gr_c^{\uparrow}$ to be the set of connected wheel-free graphs.
* Define $Gr_{di}^{\uparrow}$ to be the set of simply connected wheel-free graphs.
* Define $UTree$ and $ULin$ to be the sets of unital trees and linear graphs, respectively.
Note that $UTree$ and $ULin$ are effectively the object sets of the category of [[trees]] and the [[simplex category]] respectively.
## Examples
1. The most elementary wheeled graph (which is also of course an ordinary [[graph]] and a [[quiver]]) is the empty wheeled graph which has the [[empty set]] as its set of flags and as such no vertices or edges.
1. Any [[quiver]] can be realized as a wheeled graph in an obvious way.
1. There is a graph $I$ with $Flag(I)=\{e_{-1},e_1\}$, $\iota(e_i)=e_i$, $\kappa(e_i)=c$, $\delta(e_i)=i$. This is the **$c$-colored exceptional edge**. It can be represented schematically by
$$
\uparrow_c.
$$
1. There is a graph $W$ which is identical to $I$ except that $\iota(f_{-1})=f_1$ and $\iota(f_1)=f_{-1}$. This is the **$c$-colored exceptional loop**. The involution $\iota$ can be thought of as "spinning" the loop. It can be represented schematically by
$$
\circlearrowleft_c.
$$
## Related concepts
* [[graph]]
* [[quiver]]
* [[generalized graph]]
* [[PROP]]
* [[properad]]
## References
* [[Philip Hackney]], [[Marcy Robertson]] and [[Donald Yau]]. _Infinity Properads and Infinity Wheeled Properads_, Lecture Notes in Mathematics, 2147. Springer, Cham, 2015. [(arxiv version)](http://arxiv.org/pdf/1410.6716v2.pdf)
{#HRY}
|
wheeledgraph > corolla1 | https://ncatlab.org/nlab/source/wheeledgraph+%3E+corolla1 | data:image/svg+xml;base64,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 |
Wheeler superspace | https://ncatlab.org/nlab/source/Wheeler+superspace |
> This entry is about the concept in [[gravity]]/[[cosmology]]. For the concept in [[supergeometry]] see at _[[superspace]]_.
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In describing [[gravity]] ([[general relativity]]) on [[globally hyperbolic spacetimes]] as evolution along a given [[foliation]] of [[spacetime]] by [[spacelike]] slices, the [[Einstein equations]] describe a 1-parameter evolution in a [[configuration space (physics)|configuration space]]/[[moduli space]] of fields on each spatial slice.
For applications to [[cosmology]] this space may be drastically reduced to retain only some very large scale feature. The resulting configuration space had been called "superspaces" by [[John Wheeler]].
On this Wheeler superspace, the [[Einstein equations]] describe the [[classical mechanics]] of a configuration point moving around in superspace. Many or most approaches to [[quantum cosmology]] proceed by taking this reduced mechanical system on Wheeler superspace at face value and applying [[quantization]] to it, hoping that the resulting [[quantum mechanics]] on Wheeler superspace is a sensible approximation to full (and unknown) [[quantum gravity]], at least for purposes of studying [[cosmic structure formation]], [[cosmic inflation]] and similar issues.
## Examples
### In 2d gravity on string worldsheets
{#In2dGravityOnStringWorldsheets}
The [[worldsheet]]-theory of [[strings]] in [[string theory]] is [[D=2 gravity]] coupled to "matter"-fields (the string's [[sigma-model]] "embedding fields"). For plain [[closed string|closed]] [[bosonic strings]] the corresponding Wheeler superspace is just the [[smooth loop space]] of the string's [[target spacetime]]: each loop in spacetime is the configuration of a closed string at a fixed value of its chosen worldsheet-"time" parameter.
For the [[type II string theory|type II]] [[superstring]] the Wheeler superspace is essentially the [[smooth loop space]] of the [[exterior bundle]] of [[target spacetime]], regarded as a [[supermanifold]].
(Beware the clash of "super"-terminology here: the Wheeler superspace of the superstring is now a [[superspace]]-[[Wheeler superspace]], a "super-superspace", where the first "super" is in the sense of [[supersymmetry]]/[[graded manifolds|graded geometry]], while the second "super" in the sense of "[[configuration space (physics)|configuration space]]".
The Wheeler-DeWitt [[quantum mechanics]] of the [[type II superstring theory|type II]] [[superstring]] on its Wheeler superspace, in this sense, is ([Witten 85, from p. 92 (32 of 39) on](supersymmetric+quantum+mechanics#Witten85)) a [[supersymmetric quantum mechanics]] of the form which for finite-dimensional leads to the relation between supersymmetry and Morse theory ([Witten 82](supersymmetric+quantum+mechanics#Witten82)): the Morse-deformed super-charge is the [[Dirac-Ramond operator]] on [[smooth loop space]] (more on this in [Schreiber 04](supersymmetric+quantum+mechanics#Schreiber04)).
## Related concepts
* [[Tomonaga-Schwinger equation]]
* [[geometrodynamics]]
* [[quantum gravity]]
* [[quantum cosmology]]
## References
Original articles:
* {#DeWitt67} [[Bryce DeWitt]], *Quantum Theory of Gravity. I. The Canonical Theory*, Phys. Rev. **160** (1967) 1113 [[doi:10.1103/PhysRev.160.1113](https://doi.org/10.1103/PhysRev.160.1113)]
The Wheeler-DeWitt equation is (5.5) in [DeWitt 1967](#DeWitt67)
Review:
* [[Domenico Giulini]], _The Superspace of Geometrodynamics_ ([arXiv:0902.3923](https://arxiv.org/abs/0902.3923))
See also:
* Wikipedia, *[Wheeler-DeWitt equation](https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation)*
“Quantum Theory of Gravity. I. The Canonical Theory”
[[!redirects Wheeler-DeWitt equation]]
[[!redirects Wheeler-DeWitt equations]]
|
Wheels theorem | https://ncatlab.org/nlab/source/Wheels+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
\tableofcontents
## Idea
The _Wheels theorem_ says that the [[quantization of 3d Chern-Simons theory|perturbative]] [[Wilson loop observable]] ([[Kontsevich integral]]) of the [[unknot]] is, as a [[universal Vassiliev invariant]] with values in [[Jacobi diagrams]], a [[series]] of wheel-shaped [[Jacobi diagrams]] with [[coefficients]] the [[modified Bernoulli numbers]].
## Related theorems
* [[Duflo isomorphism]]
* [[PBW theorem]]
* [[Rozansky-Witten Wilson loop of unknot is A-hat genus]]
## References
The full proof is due to
* [[Dror Bar-Natan]], [[Le Tu Quoc Thang]], [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))
The weaker version for [[Lie algebra weight systems]] was proven in:
* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))
Version for [[Rozansky-Witten weight systems]]:
* [[Justin Sawon]], _Rozansky-Witten invariants of hyperkähler manifold_, Cambridge 2000 ([arXiv:math/0404360](https://arxiv.org/abs/math/0404360))
* {#RobertsWillerton} [[Justin Roberts]], [[Simon Willerton]], Section 8 of: _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))
Review:
* {#Roberts01} [[Justin Roberts]], Section 8 of: _Rozansky-Witten theory_ ([arXiv:math/0112209](https://arxiv.org/abs/math/0112209))
|
whiskering | https://ncatlab.org/nlab/source/whiskering |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Whiskering
* table of contents
{: toc}
## Idea
In a [[2-category]], the [[horizontal composition]] of a [[2-morphism]] with [[1-morphisms]] is sometimes called _whiskering_.
Whiskering from the left with an [[equivalence]] and from the right with an inverse equivalence is a [[conjugation]] [[action]] of equivalences on [[2-morphisms]].
## Examples
An important use of whiskering is the usual definition of [[adjoint functor]]s via the [[triangle identities]]: in [[Cat]] whiskering is the [[composition]] of a [[functor]] with a [[natural transformation]] to produce a natural transformation.
If we identify a functor or 1-morphism with its [[identity natural transformation]] or [[identity 2-morphism]], then whiskering is a special case of [[horizontal composition]], and composition of 1-morphisms is a special case of whiskering.
In detail:
* If $F,G\colon C \to D$ and $H\colon D\to E$ are functors and $\eta\colon F \to G$ is a natural transformation whose coordinate at any object $A$ of $C$ is $\eta_A$, then __whiskering__ $H$ and $\eta$ yields the natural transformation $H \circ \eta\colon (H \circ F) \to (H \circ G)$ whose coordinate at $A$ is $H(\eta_A)$.
* If $F\colon C \to D$ and $G,H\colon D \to E$ are functors and $\eta\colon G\to H$ is a natural transformation whose coordinate at $A$ is $\eta_A$, then __whiskering__ $\eta$ and $F$ yields the natural transformation $\eta \circ F\colon (G \circ F) \to (H \circ F)$ whose coordinate at $A$ is $\eta_{F(A)}$.
## In dependent type theory
In [[dependent type theory]], whiskering is the type theoretic equivalent of the principle in [[set theory]] that given sets $A$, $B$, and $C$, and functions $f:A \to B$ and $g:A \to B$, if $f(x) = g(x)$ for all elements $x \in A$, then
* $h(f(x)) = h(g(x))$ for all functions $h:B \to C$ and elements $x \in A$
* $f(h(x)) = g(h(x))$ for all functions $h:C \to A$ and elements $x \in C$
Given types $A$, $B$, and $C$ and functions $f:A \to B$ and $g:A \to B$ there is a function
$$\mathrm{leftwhiskering}_{A, B, C}(f, g):\left(\prod_{x:A} f(x) =_B g(x)\right) \to \left(\prod_{h:B \to C} \prod_{x:A} h(f(x)) =_C h(g(x))\right)$$
called **left whiskering**, which is defined as the [[lambda abstraction]] of the composite of the [[function application to identities]] of function $h:B \to C$ with [[homotopy]] $H:\prod_{x:A} f(x) =_B g(x)$
$$\mathrm{leftwhiskering}_{A, B, C}(f, g)(H, h) \coloneqq \lambda x.\mathrm{ap}_h(H(x))$$
Left whiskering is frequently written simply as $h \circ H$ or $h \cdot H$.
Given types $A$, $B$, and $C$ and functions $f:A \to B$ and $g:A \to B$, there is a function
$$\mathrm{rightwhiskering}_{A, B, C}(f, g):\left(\prod_{x:A} f(x) =_B g(x)\right) \to \left(\prod_{h:C \to A} \prod_{x:C} f(h(x)) =_B g(h(x))\right)$$
called **right whiskering**, defined as the [[lambda abstraction]] of the composite of [[homotopy]] $H:\prod_{x:A} f(x) =_B g(x)$ with function $h:C \to A$
$$\mathrm{rightwhiskering}_{A, B, C}(f, g)(H, h) \coloneqq \lambda x.H(h(x))$$
Right whiskering is frequently written simply as $H \circ h$ or $H \cdot h$.
## Related concepts
* [[pasting diagram]]
* [[digraph|plane digraphs]]
## References
Whiskering (though not under this name) of [[natural transformations]] is first described in:
* {#Godement58} [[Roger Godement]], Appendix (pp. 269) of: *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]
For textbook accounts see most of those listed at *[[category]]*, such as:
* [[Saunders MacLane]], §II.5, p. 43 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
For whiskering in [[dependent type theory]]:
* *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))
* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))
For whiskering in [[category theory]]:
* [A MathOverflow question about whiskering](http://mathoverflow.net/questions/40813/what-is-the-name-for-the-composition-of-a-functor-with-a-natural-transformation/40814#40814)
* [[Peter Selinger]]: Introduction to categorical logic. [pdf](https://math.vanderbilt.edu/dept/conf/tacl2013/coursematerials/SelingerTACL20132.pdf), page 41
[[!redirects whiskering]]
[[!redirects whiskerings]]
[[!redirects whisker]]
[[!redirects whiskers]]
[[!redirects whiskered]]
|
whisky club | https://ncatlab.org/nlab/source/whisky+club | [[whiskypic.jpg:file]] |
white noise | https://ncatlab.org/nlab/source/white+noise |
#Contents#
* table of contents
{:toc}
## Idea
A *white noise* is a kind of [[noise]]/[[stochastic process]] where all frequencies are represented with the same probability.
## References
* T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, _White Noise. An infinite dimensional calculus_, Kluwer 1993.
* M. de Faria, J. Potthoff, L. Streit, _The Feynman integrand as a Hida distribution_, J. Math. Phys. __32__, 2123 (1991); [doi](http://dx.doi/org/10.1063/1.529184)
See also:
* Wikipedia [white noise](https://en.wikipedia.org/wiki/White_noise)
category: applications, probability
|
Whitehead | https://ncatlab.org/nlab/source/Whitehead |
* [[Alfred North Whitehead]]
* [[George Whitehead]]
* [[John Henry Constantine Whitehead]]
category: people
|
Whitehead integral formula | https://ncatlab.org/nlab/source/Whitehead+integral+formula |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An expression for [[Hopf invariants]] in terms of [[secondary characteristic classes]] induced from the [[intersection pairing]], also known as _[[functional cup products]]_ ([Steenrod 49](#Steenrod49)) or _[[homotopy periods]]_ ([Sinha-Walter 13](#SinhaWalter13)).
The original expression due to [Whitehead 47](#Whitehead47) (see [Bott-Tu 82, Prop. 17.22](#BottTu82)) is in terms of [[smooth functions]] to an [[n-sphere]], which implies that [[wedge product]]/[[cup product]] of the [[pullback of differential forms|pullback]] of the [[volume form]] with itself vanishes identically.
More generally the homotopy Whitehead formula applies to general [[cocycles]] in [[cohomotopy]]. Its existence was suggested in [Haefliger 78, p. 17](#Haefliger78), worked out for the case of maps from the [[3-sphere]] to the [[2-sphere]] in [Griffith-Morgan 81, Section 14.5](#GriffithMorgan13) and stated generally but without proof in [Sinha-Walter 13, Example 1.9](#SinhaWalter13). A transparent proof of the general expression via lifts in [[cohomotopy]] through [[Hopf fibrations]] is in [FSS 19](#FSS19), relating the expression to the [[Hopf-Wess-Zumino term]] of the [[M5-brane]].
## Related concepts
* [[functional cup product]]
## References
* {#Whitehead47} [[J. H. C. Whitehead]], _An expression of Hopf's invariant as an integral_, Proc. Nat. Acad. Sci. USA 33 (1947), 117–123 ([jstor:87688](https://www.jstor.org/stable/87688))
* {#Steenrod49} [[Norman Steenrod]], _Cohomology Invariants of Mappings_, Annals of Mathematics Second Series, Vol. 50, No. 4 (Oct., 1949), pp. 954-988 ([jstor:1969589 ](https://www.jstor.org/stable/1969589 ))
* [[Hassler Whitney]], Section 31 in _Geometric Integration Theory_, 1957 ([pup:3151](https://press.princeton.edu/titles/3151.html))
* {#Haefliger78} [[André Haefliger]], p. 3 of _Whitehead products and differential forms_, In: P.A. Schweitzer (ed.), _Differential Topology, Foliations and Gelfand-Fuks Cohomology_, Lecture Notes in Mathematics, vol 652. Springer 1978 ([doi:10.1007/BFb0063500](https://doi.org/10.1007/BFb0063500))
* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Prop. 17.22 in _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics **82**, Springer (1982) [[doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0)]
* Lee Rudolph, _Whitehead's Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number_, Pure and Applied Mathematics Quarterly Volume 6, Number 2, 2010 ([arXiv:0912.4974](https://arxiv.org/abs/0912.4974))
* {#GriffithMorgan13} [[Phillip Griffiths]], [[John Morgan]], Section 14.5 of _Rational Homotopy Theory and Differential Forms_, Progress in Mathematics Volume 16, Birkhauser (1981, 2013) ([doi:10.1007/978-1-4614-8468-4](https://doi.org/10.1007/978-1-4614-8468-4))
* {#SinhaWalter13} [[Dev Sinha]], [[Ben Walter]], _Lie coalgebras and rational homotopy theory II: Hopf invariants_, Trans. Amer. Math. Soc. 365 (2013), 861-883 ([arXiv:0809.5084](https://arxiv.org/abs/0809.5084), [doi:10.1090/S0002-9947-2012-05654-6](https://doi.org/10.1090/S0002-9947-2012-05654-6))
* [[Felix Wierstra]], *Hopf Invariants in Real and Rational Homotopy Theory* (2017) [[diva:146246](http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-146246), [pdf](https://www.diva-portal.org/smash/get/diva2:1136442/FULLTEXT02.pdf)]
* {#FSS19} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], pp. 18 of: _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]_, Comm. Math. Phys. **384** (2021) 403-432 [[arXiv:1906.07417](https://arxiv.org/abs/1906.07417), [doi:10.1007/s00220-021-03951-0](https://doi.org/10.1007/s00220-021-03951-0)]
[[!redirects Whitehead integral formulas]]
[[!redirects Whitehead's integral formula]]
[[!redirects Whitehead's integral formulas]]
[[!redirects homotopy Whitehead integral formula]]
[[!redirects homotopy Whitehead integral formulas]]
[[!redirects homotopy Whitehead's integral formula]]
[[!redirects homotopy Whitehead's integral formulas]]
[[!redirects Whitehead integral]
[[!redirects Whitehead integrals]]
[[!redirects Whitehead's integral]]
[[!redirects Whitehead's integrals]]
[[!redirects homotopy Whitehead integral]]
[[!redirects homotopy Whitehead integrals]]
[[!redirects homotopy Whitehead's integral]]
[[!redirects homotopy Whitehead's integrals]]
|
Whitehead link | https://ncatlab.org/nlab/source/Whitehead+link |
The __Whitehead link__ is a famous [[link]] which shows the difference between the notion of [[equivalence]]. If links are only allowed to move by [[isotopy|isotopies]], then the two components are linked. However, if a link is allowed to pass through itself, then they can be unlinked.
[[!include Whitehead link - SVG]]
+-- {: .un_remark}
###### Note
To include this picture in a page, write
<nowiki><code>[[!include Whitehead link - SVG]]</code></nowiki>.
=--
[[!redirects Whitehead link]]
[[!redirects Whitehead links]]
|
Whitehead link - SVG | https://ncatlab.org/nlab/source/Whitehead+link+-+SVG |
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|
Whitehead principle of nonabelian cohomology | https://ncatlab.org/nlab/source/Whitehead+principle+of+nonabelian+cohomology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What is called _[[nonabelian cohomology]]_ is the general [[cohomology|intrinsic cohohomology]] of any [[(∞,1)-topos]] $\mathbf{H}$ with coefficients in any object $A \in \mathbf{H}$, not necessarily an [[Eilenberg-MacLane object]].
But there is a general notion of [[Postnikov tower in an (∞,1)-category]] that applies in any [[locally presentable (∞,1)-category]], in particular in [[(∞,1)-topos]]es.
This implies that every object $A\in \mathbf{H}$ has a decomposition as a sequence of objects
$$
A \to \cdots \to A_3 \to A_2\to A_1 \to A_0 \to *
\,,
$$
where $A_k$ is an $k$-[[truncated]] object, in fact the $k$-truncation of $A$.
This implies that every $n$-[[truncated]] [[connected]] object $A$ is given by a possibly nonabelian 0-truncated group object $G$ and a sequence of abelian extensions of the [[delooping]] $\mathbf{B}G$ in that we have [[fiber sequence]]s
$$
\mathbf{B}^2 K_1 \to A_2 \to \mathbf{B}G = A_1
$$
etc.
(...)
It follows that any [[cocycle]] $X \to A_2$ decomposes into the [[principal bundle]] classified by $X \to \mathbf{B}G$ and an abelian $\mathbf{B}^2 K$-cocycle on its total space
(...)
## Examples
A [[string structure]] is a nonabelian cocycle with coefficients in the [[string 2-group]]. This is equivalently a $\mathbf{B}U(1)$-cocycle (a [[bundle gerbe]]) on the total space of the underlying $Spin$-[[principal bundle]]. See the section <a href="http://ncatlab.org/nlab/show/string+structure#ClassesOnTotalSpace">In terms of classes on the total space</a>.
## References
The term "Whitehead principle" for nonabelian cohomology is used in
* {#Toen02} [[Bertrand Toën]], p. 8 of: _Stacks and Non-abelian cohomology_, lecture at _[Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory](https://www.msri.org/realvideo/index04.html)_, MSRI 2002 ([slides](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/index.html), [ps](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/meta/aux/toen.ps), [pdf](https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.pdf), [[ToenStacksAndNonabelianCohomology.pdf:file]])
[[!redirects Whitehead principle of non-abelian cohomology]]
|
Whitehead product | https://ncatlab.org/nlab/source/Whitehead+product |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
### Basic definition
+-- {: .num_defn}
###### Definition
For $X$ a [[pointed space|pointed]] [[topological space]] ([[CW-complex]]), the _Whitehead products_ ([Whitehead 41, Section 3](#Whitehead41)) are the [[bilinear maps]] on the elements of the [[homotopy groups]] $\pi_\bullet(X)$ of $X$ of the form
\[
\label{WhiteheadProductMapInSingleDegrees}
[-,-]_{Wh}
\;\colon\;
\pi_{n_1}\big( X \big)
\otimes_{\mathbb{Z}}
\pi_{n_2}\big( X \big)
\longrightarrow
\pi_{n_1 + n_2 - 1}\big( X \big)
\phantom{AA}
\text{for all}
\;
n_i \in \mathbb{N}
;
\;
n_i \geq 1
\,,
\]
given by sending any [[pair]] of [[homotopy classes]]
$$
\big[
S^{n_i} \overset{\phi_i}{\longrightarrow} X
\big]
\;\in\;
\pi_{ n_i }
\big(
X
\big)
$$
to the [[homotopy class]] of the top [[composition|composite]] in the [[diagram]]
$$
\array{
S^{ n_1 + n_2 -1 }
&
\overset{
f_{n_1,n_2}
}{
\longrightarrow
}
&
S^{n_1}
\vee
S^{n_2}
&
\overset{
(\phi_1, \phi_2)
}{\longrightarrow}
&
X
\\
\big\downarrow
&
(po)
&
\big\downarrow
\\
D^{ n_1 + n_2 }
&\underset{}{\longrightarrow}&
S^{n_1} \times S^{n_2}
}
$$
where $f_{n_1, n_2}$ is [[generalized the|the]] [[attaching map]] exhibiting the [[product space]] $S^{n_1} \times S^{n_2}$ as the result of a [[cell attachment]] to the [[wedge sum]] $S^{n_1} \vee S^{n_2}$.
=--
In this form this appears for instance in [Félix-Halperin-Thomas, p. 176 with p. 177](#FelixHalperinThomas00).
### Generalized version
There is also a **generalized Whitehead product** where we can take more general homotopy classes ([[continuous maps]] up to [[homotopy]]) $\alpha\in [S^\cdot Y,X]$ and $\beta\in [S^\cdot Z,X]$ to produce a class $[\alpha,\beta]_{Wh}\in[Y\star Z,X]$. Here $S^\cdot$ denotes the [[reduced suspension]] operation on pointed spaces and $\star$ denotes the [[join of spaces|join]] of CW-complexes. Notice that $pt\star Z = C^\cdot(Z)$ and the reduced [[cone]] of a point is $C^\cdot(pt)=S^1$. Thus for $Y=Z=pt$ the generalized Whitehead product reduced to the usual Whitehead product.
## Properties
### Super Lie algebra structure
{#SuperLieAlgebraStructure}
If one assigns degree $n-1$ to the $n$th [[homotopy group]] $\pi_n$,
then the degree-wise Whitehead products (eq:WhiteheadProductMapInSingleDegrees)
organize into a single degree-0 bilinear pairing on the [[graded abelian group]] which is the [[direct sum]] of all the [[homotopy groups]]:
\[
\label{DegreeShiftedHomotopyGroups}
\array{
\pi_{\bullet + 1}(X)
=
&
\pi_1(X)
&\oplus&
\pi_2(X)
&\oplus&
\pi_3(X)
&\oplus&
\cdots
\\
deg =
&
0
&&
1
&&
2
}
\]
This unified Whitehead product is _graded_ skew-symmetric in that for $\phi_i \in \pi_{n_i}\big( X \big)$ it satisfies
$$
\big[
\phi_1,
\,
\phi_2
\big]_{Wh}
\;=\;
(-1)^{ n_1 n_2 }
\big[
\phi_2,
\,
\phi_1
\big]_{Wh}
$$
and it satisfies the corresponding graded [[Jacobi identity]] ([Hilton 55, Theorem B](#Hilton55)).
This makes the Whitehead bracket the [[Lie bracket]] of a [[super Lie algebra]] [[structure]] on $\pi_{\bullet-1}(X)$ (eq:DegreeShiftedHomotopyGroups), over the [[ring]] of [[integers]] (sometimes called, in this context, a _graded quasi-Lie algebra_, see [below](#OfAnElementWithItself)).
### Of elements with themselves
{#OfAnElementWithItself}
Beware that the skew-symmetry of [[Lie algebras]] over the [[integers]], as opposed to over a [[field]] of [[characteristic zero]], implies for any element $\phi$ of [[even number|even]] homogeneous degree --
hence here for elements of [[homotopy groups]] in [[odd number|odd]] degree -- only that the bracket with itself vanishes after multiplication by 2
$$
[\phi,\phi]_{Wh} = - [\phi,\phi]_{Wh}
\phantom{AA}
\text{hence equivalently}
\phantom{AA}
2 \cdot [\phi,\phi]_{Wh} = 0
$$
but not necessarily that $[\phi,\phi]_{Wh} = 0$ by itself -- since [[multiplication]] by 2 is not an [[isomorphism]] over the [[integers]].
But this means that the Whitehead bracket of any even-degree [[element]] with itself -- hence of any [[element]] of a [[homotopy group]] in [[odd number|odd]] degree -- has [[order of a group element|order]] at most 2, hence is in the 2-[[torsion subgroup]] of the respective [[homotopy group]].
### As primary homotopy operations
The Whitehead products form one of the [[primary homotopy operations]].
In fact, together with [[composition operations]] and [[fundamental group]]-[[actions]] they [[generators and relations|generate]] all such operations.
This is related to the definition of [[Pi-algebras]].
### Relation to Pontrjagin product
{#RelationToPontrjaginProduct}
Under the [[Hurewicz homomorphism]], the Whitehead product on homotopy groups is the [[commutator]] of the [[Pontrjagin product]] on integral [[ordinary homology|homology]] groups of a [[based loop space]].
This is due to [Samelson (1953)](#Samelson53) and for higher Whitehead brackets due to [Arkowitz (1971)](#Arkowitz71).
In fact, in [[characteristic zero]] the [[Pontrjagin ring]] is the [[universal enveloping algebra]] of the Whitehead bracket [[Lie algebra]] [[Milnor & Moore (1965) Appendix](#MilnorMoore65)].
A textbook account is in [Whitehead (1978) Thm. X.7.10](#Whitehead78).
### In terms of Samelson products
In the context of [[simplicial groups]], representing [[connected]] [[homotopy types]], there is a formula for the Whitehead product in terms of a [[Samelson product]], which in turn is derived from a [[shuffle]] product which is a sort of non-commutative version of the [[Eilenberg-Zilber map]]. These simplicial formulae come from an analysis of the structure of the [[product of simplices]].
(This formula for the Whitehead product is due to [[Dan Kan]] and can be found in the old survey article of [[Ed Curtis]]. The proof that it works was never published. For more pointers see [MO:q/296479/381](https://mathoverflow.net/q/296479/381))
### Relation to the Sullivan models
{#RelationToSullivanModels}
We discuss (Prop. \ref{CoBinarySullivanDifferentialIsWhiteheadProduct} below) how the [[rationalization]] of the [[Whitehead product]] is the co-binary part of the [[Sullivan model|Sullivan differential]] in [[rational homotopy theory]]. First we make explicit some notation and normalization conventions that enter this statement:
In the following, for $W$ a $\mathbb{Z}$-[[graded module]], we write
$$
W \wedge W
\;\coloneqq\;
Sym^2(W)
\;\coloneqq\;
\big(
W \otimes W
\big)
/
\big(
\alpha \otimes \beta
\sim
(-1)^{ n_\alpha n_\beta }
\beta \otimes \alpha
\big)
\,,
$$
where on the right $\alpha, \beta \in W$ are elements of homogeneous degree $n_\alpha, n_\beta \in \mathbb{Z}$, respectively. The point is just to highlight that "$(-)\wedge(-)$" is not to imply here a degree shift of the generators (as it typically does in the usual notation for [[Grassmann algebras]]).
Let $X$ be a [[simply connected topological space]] with [[Sullivan model]]
\[
\label{SullivanModelX}
CE( \mathfrak{l} X )
\;=\;
\big(
Sym^\bullet\big(V^\ast\big), d_X
\big)
\]
for $V^\ast$ the [[graded vector space]] of generators, which is the $\mathbb{Q}$-linear [[dual space|dual]] [[graded vector space]] of the [[graded object|graded]] $\mathbb{Z}$-[[module]] (=[[graded abelian group]]) of [[homotopy groups]] of $X$:
$$
V^\ast
\;\coloneqq\;
Hom_{Ab}\big(
\pi_\bullet(X),
\mathbb{Q}
\big)
\,.
$$
Declare the [[wedge product]] pairing to be given by
\[
\label{WedgeProductNormalization}
\array{
V^\ast
\wedge
V^\ast
&\overset{\Phi}{\longrightarrow}&
Hom_{Ab}
\big(
\pi_\bullet(X) \wedge \pi_\bullet(X)
,
\mathbb{Q}
\big)
\\
(\alpha, \beta)
&\mapsto&
\Big(
v \wedge w
\;\mapsto\;
(-1)^{ n_\alpha \cdot n_\beta }
\alpha(v)\cdot \beta(w)
+
\beta(v)\cdot \alpha(w)
\Big)
}
\]
where $\alpha$, $\beta$ are assumed to be of homogeneous degree $n_\alpha, n_\beta \in \mathbb{N}$, respectively.
(Notice that the usual normalization factor of $1/2$ is _not_ included on the right. This normalization follows [Andrews & Arkowitz 1978, above Thm. 6.1](#AndrewsArkowitz78).)
Finally, write
\[
\label{PojectionToBinary}
[-]_2
\;\colon\;
Sym^\bullet\big(V^\ast\big)
\longrightarrow
V^\ast \wedge V^\ast
\]
for the linear projection on quadratic polynomials in the graded [[symmetric algebra]].
Then:
+-- {: .num_prop #CoBinarySullivanDifferentialIsWhiteheadProduct}
###### Proposition
**([[co-binary Sullivan differential is Whitehead product]])**
Let $X$ be a [[simply connected topological space]] of rational [[finite type]], so that it has a [[Sullivan model]] with Sullivan differential $d_X$ (eq:SullivanModelX).
Then the co-binary component (eq:PojectionToBinary) of the Sullivan differential equals the $\mathbb{Q}$-[[linear dual map]] of the [[Whitehead product]] $[-,-]_X$ on the [[homotopy groups]] of $X$:
$$
[d_X \alpha]_2
\;=\;
[-,-]_X^\ast
\,.
$$
More explicitly, the following [[commuting diagram|diagram commutes]]:
$$
\array{
V^\ast
&\overset{ [-]_2\circ d_X }{\longrightarrow}&
V^\ast \wedge V^\ast
\\
\big\downarrow\mathrlap{^=}
&&
\big\downarrow\mathrlap{^\Phi}
\\
Hom_{Ab}
\big(
\pi_\bullet(X),
\mathbb{Q}
\big)
&
\underset{
Hom_{Ab}\big(
[-,-]_X
,
\;
\mathbb{Q}
\big)
}{
\longrightarrow
}
&
Hom_{Ab}
\big(
\pi_\bullet(X)
\wedge
\pi_\bullet(X),
\;
\mathbb{Q}
\big)
}
\,,
$$
where the wedge product on the right is normalized as in (eq:WedgeProductNormalization).
=--
([Andrews & Arkowitz 1978, Thm. 6.1](#AndrewsArkowitz78), following [Deligne, Griffiths, Morgan & Sullivan 1975](#DeligneGriffithsMorganSullivan75))
\begin{remark}
\label{AbsenceOfJacobiatorInWhiteheadBracket}
Prop. \ref{CoBinarySullivanDifferentialIsWhiteheadProduct} says in particular that the binary bracket of the [[L-infinity-algebra|$L_\infty$-algebra]] dual to a [[Sullivan model]] is always an actual [[super Lie bracket]] in that it satisfies its super-[[Jacobi identity]], even if there happens to also be a nontrivial trinary bracket which would serve as a "[[Jacobiator]]".
This is due to the [[minimal model|minimality]] of Sullivan models, which implies that the co-unary part of their [[differential]] vanishes, and hence that that the unary bracket of the corresponding $L_\infty$-algebra vanishes: Since the failure of the Jacobi identity on binary brackets in an $L_\infty$-algebra is measured not by the trinary bracket itself but by its composition with the unary bracket, this vanishes in the above case.
\end{remark}
### Relation to Goodwillie Calculus:
On the relation to [[Goodwillie calculus]] see e.g. [Scherer & Chorny 2011, Sec. 1](#SchererChorny11), which also gives an application of the relationship between the Whitehead and [[Samelson products]].
## Examples
+-- {: .num_example #WhiteheadProductCorrespondingToComplexHopfFibration}
###### Example
**([[Whitehead product]] corresponding to [[complex Hopf fibration]])**
For $X = S^2$ the [[2-sphere]], consider the following two [[elements]] of its [[homotopy groups]] ([[homotopy groups of spheres|of spheres]], as it were):
1. $id_{S^2} \in \pi_2\big( S^2 \big)$ (represented by the [[identity function]] $S^2 \to S^2$)
1. $h_{\mathbb{C}} \in \pi_3\big( S^2 \big)$ (represented by the [[complex Hopf fibration]])
Then the Whitehead product satisfies
$$
\big[
id_{S^2},
\;
id_{S^2}
\big]
\;=\;
2 \cdot h_{\mathbb{C}}
\,.
$$
=--
\linebreak
## Related concepts
* [[Pontrjagin product]]
* [[Samelson product]]
## References
### General
The concept is originally due to
* {#Whitehead41} [[J. H. C. Whitehead]], Section 3 of _On Adding Relations to Homotopy Groups_, Annals of Mathematics Second Series, Vol. 42, No. 2 (Apr., 1941), pp. 409-428 ([jstor:1968907](https://www.jstor.org/stable/1968907))
with early discussion in
* {#HiltonWhitehead53} [[Peter Hilton]], [[J. H. C. Whitehead]], _Note on the Whitehead Product_, Annals of Mathematics Second Series, Vol. 58, No. 3 (Nov., 1953), pp. 429-442 ([jstor:1969746](https://www.jstor.org/stable/1969746))
* {#Hilton55} [[Peter Hilton]], _On the homotopy groups of unions of spheres_, J. London Math. Soc., 1955, 30, 154–172 ([[Hilton55.pdf:file]], [doi:doi.org/10.1112/jlms/s1-30.2.154]( https://doi.org/10.1112/jlms/s1-30.2.154))
Proof that the Whitehead product is the [[commutator]] of the [[Pontrjagin product]]:
* {#Samelson53} [[Hans Samelson]], *A Connection Between the Whitehead and the Pontryagin Product*, American Journal of Mathematics **75** 4 (1953) 744–752 [[doi:10.2307/2372549](https://doi.org/10.2307/2372549)]
and in [[characteristic zero]]:
* {#MilnorMoore65} [[John Milnor]], [[John Moore]], Appendix of: _On the structure of Hopf algebras_, Annals of Math. __81__ (1965) 211-264 [[doi:10.2307/1970615](https://doi.org/10.2307/1970615), [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf)]
and for higher Whitehead brackets:
* {#Arkowitz71} [[Martin Arkowitz]], *Whitehead Products as Images of Pontrjagin Products*, Transactions of the American Mathematical Society, **158** 2 (1971) 453-463 [[doi:10.2307/1995917](https://doi.org/10.2307/1995917)]
Textbook account:
* {#Whitehead78} [[George W. Whitehead]], §X.7 in: *Elements of Homotopy Theory*, Springer (1978) [[doi:10.1007/978-1-4612-6318-0](https://link.springer.com/book/10.1007/978-1-4612-6318-0)]
See also
* Wikipedia _[Whitehead product](http://en.wikipedia.org/wiki/Whitehead_product)_
Discussion of Whitehead products specifically of [[homotopy groups of spheres]]:
* {#James56} [[Ioan Mackenzie James]], _On the Suspension Triad_, Annals of Mathematics Second Series, Vol. 63, No. 2 (Mar., 1956), pp. 191-247 ([arXiv:1969607](https://www.jstor.org/stable/1969607))
* {#James57} [[Ioan Mackenzie James]], _On the Suspension Sequence_, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 ([arXiv:1969666](https://www.jstor.org/stable/1969666))
As [[n-excisive functors]]:
* {#SchererChorny11} [[Jerome Scherer]], [[Boris Chorny]], _Goodwillie calculus and Whitehead products_, Forum Math. 27 (2015), no. 1, 119 - 130 ([arXiv:1109.2691](https://arxiv.org/abs/1109.2691))
Discussion of Whitehead products in [[homotopy type theory]]:
* [[Guillaume Brunerie]], section 3.3 of _On the homotopy groups of spheres in homotopy type theory_ ([arXiv:1606.05916](https://arxiv.org/abs/1606.05916))
### In rational homotopy theory
{#ReferencesInRationalHomotopyTheory}
Discussion of Whitehead products in [[rational homotopy theory]] ([[the co-binary Sullivan differential is the dual Whitehead product]]):
* {#Quillen69} [[Daniel Quillen]], section I.5 of _Rational Homotopy Theory_, Annals of Mathematics Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([jstor:1970725](https://www.jstor.org/stable/1970725))
* Christopher Allday, _Rational Whitehead products and a spectral sequence of Quillen_, Pacific J. Math. Volume 46, Number 2 (1973), 313-323 ([euclid:1102946308](https://projecteuclid.org/euclid.pjm/1102946308))
* Christopher Allday, _Rational Whitehead product and a spectral sequence of Quillen, II_, Houston Journal of Mathematics, Volume 3, No. 3, 1977 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.434.8821&rep=rep1&type=pdf))
* {#DeligneGriffithsMorganSullivan75} [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], _Real homotopy theory of Kähler manifolds_, Invent Math (1975) 29: 245 ([doi:10.1007/BF01389853](https://doi.org/10.1007/BF01389853))
* {#AndrewsArkowitz78} Peter Andrews, [[Martin Arkowitz]], *Sullivan's Minimal Models and Higher Order Whitehead Products*, Canadian Journal of Mathematics, **30** 5 (1978) 961-982 [[doi:10.4153/CJM-1978-083-6](https://doi.org/10.4153/CJM-1978-083-6)]
* {#FelixHalperinThomas00} [[Yves Félix]], [[Steve Halperin]], J. C. Thomas, Prop. 13.16 in: _Rational Homotopy Theory_, Graduate Texts in Mathematics **205** Springer (2000)
* Francisco Belchí, [[Urtzi Buijs]], José M. Moreno-Fernández, Aniceto Murillo, _Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL_, Linear Algebra and its Applications, Volume 520 (2017), pages 16-31 ([arXiv:1604.01478](https://arxiv.org/abs/1604.01478), [doi:10.1016/j.laa.2017.01.008](https://doi.org/10.1016/j.laa.2017.01.008))
* Takahito Naito, _A model for the Whitehead product in rational mapping spaces_ ([arXiv:1106.4080](https://arxiv.org/abs/1106.4080))
The Whitehead product of $\mathbb{C}P^1 \vee \mathbb{C}P^1 \to \mathbb{C}P^\infty \times \mathbb{C}P^\infty$ in relation to the [[Dolbeault complex]]:
* Shamuel Auyeung, Jin-Cheng Guu, Jiahao Hu, pp. 4 of: *On the algebra generated by $\overline{\mu}$, $\overline{\partial}$, $\partial$, $\mu$*, Complex Manifolds **10** 1 (2023) [[arXiv:2208.04890](https://arxiv.org/abs/2208.04890), [doi:10.1515/coma-2022-0149](https://doi.org/10.1515/coma-2022-0149)]
[[!redirects Whitehead products]]
[[!redirects Whitehead bracket]]
[[!redirects Whitehead brackets]]
[[!redirects Whitehead Lie algebra]]
[[!redirects Whitehead Lie algebras]]
[[!redirects Whitehead super Lie algebra]]
[[!redirects Whitehead super Lie algebras]]
[[!redirects Whitehead bracket super Lie algebra]]
[[!redirects Whitehead bracket super Lie algebras]]
[[!redirects Whitehead algebra]]
[[!redirects Whitehead algebras]]
[[!redirects Whitehead L-infinity algebra]]
[[!redirects Whitehead L-infinity algebras]]
[[!redirects Whitehead bracket L-infinity algebra]]
[[!redirects Whitehead bracket L-infinity algebras]]
|
Whitehead theorem | https://ncatlab.org/nlab/source/Whitehead+theorem | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### (∞,1)-Topos theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Classical case
The classical **Whitehead theorem** asserts that
+-- {: .num_theorem }
###### Theorem (Whitehead)
Every [[weak homotopy equivalence]] between [[CW-complexes]] is a [[homotopy equivalence]].
=--
(See also the discussion at [[m-cofibrant space]]).
Using the [[homotopy hypothesis]]-theorem this may be reformulated:
+-- {: .num_cor }
###### Corollary
In the [[(∞,1)-category]] [[∞Grpd]] every [[weak homotopy equivalence]] is a [[homotopy equivalence]].
=--
## Simplicial version
\begin{theorem}
A [[simplicial map]] $f\colon X\to Y$ between [[Kan complexes]]
is a [[simplicial homotopy equivalence]] if and only if
for any $a$ and $b$ that make the following square commute
$$\array{
\partial\Delta^n&\stackrel{a}{\to}&X\\
\downarrow^\iota&{}^{\exists d}\nearrow&\downarrow^f\\
\Delta^n &\stackrel{b}{\to}&Y\\
}
$$
there is a diagonal arrow $d$ that makes the upper triangle
commutative and the lower triangle commutative up to a homotopy
$h\colon \Delta^1\times\Delta^n\to Y$
that is constant on the boundary $\Delta^1\times\partial\Delta^n$.
\end{theorem}
Of course, this statement can be reformulated using
homotopy groups like the version for topological spaces,
but the above statement is more practical.
\begin{remark}
In the above criterion, the boundary inclusion
$$\partial\Delta^n\to\Delta^n$$
can be replaced by any weakly equivalent [[cofibration]].
\end{remark}
\begin{remark}
If $X$ or $Y$ is not a [[Kan complex]],
one can formulate a similar criterion
using [[barycentric subdivisions]] of $\partial\Delta^n$ and $\Delta^n$.
A [[simplicial map]] $f\colon X\to Y$ between [[simplicial sets]]
is a [[weak homotopy equivalence]] if and only if
for any $k\ge0$ and for any $a$ and $b$ that make the following square commute
$$\array{
Sd^k \partial\Delta^n&\stackrel{a}{\to}&X\\
\downarrow^{Sd^k \iota}&{}^{\exists d}\nearrow&\downarrow^f\\
Sd^k \Delta^n &\stackrel{b}{\to}&Y\\
}
$$
there is $l\ge k$ such that in the outer rectangle in the diagram
$$\array{
Sd^l \partial\Delta^n&\to&Sd^k \partial\Delta^n&\stackrel{a}{\to}&X\\
\downarrow^{Sd^l \iota}&&\downarrow^{Sd^k \iota}&&\downarrow^f\\
Sd^l \Delta^n &\to&Sd^k \Delta^n &\stackrel{b}{\to}&Y\\
}
$$
we can find a diagonal arrow
$$d\colon Sd^l \Delta^n \to X$$
that makes the upper triangle in the diagram
$$\array{
Sd^l \partial\Delta^n&\stackrel{a}{\to}&X\\
\downarrow^{Sd^l \iota}&{}^{\exists d}\nearrow&\downarrow^f\\
Sd^l \Delta^n &\stackrel{b}{\to}&Y\\
}
$$
commutative and the lower triangle commutative up to a homotopy
$$h\colon Sd^l(\Delta^1\times\Delta^n)\to Y$$
that is constant on the boundary $Sd^l(\Delta^1\times\partial\Delta^n)$.
\end{remark}
## Equivariant version
In $G$-[[equivariant homotopy theory]] the statement is that $G$-homotopy equivalences between [[G-CW complexes]] are equivalent to maps that are weak homotopy equivalences on [[fixed point]] spaces $H^H$ for all [[closed subset|closed]] [[subgroups]] $H \subset G$ (e.g. [Greenlees-May 95, theorem 2.4](#GreenleesMay95)). See at _[[equivariant Whitehead theorem]]_.
## In general $(\infty,1)$-toposes
There is a notion of [[homotopy groups]] for objects in every [[∞-stack]] [[(∞,1)-topos]], as described at [[homotopy group (of an ∞-stack)]]. Accordingly, there is a notion of [[weak homotopy equivalence]] in every [[∞-stack]] [[(∞,1)-topos]] and hence an analog of the statement of Whiteheads theorem. One finds that
**Warning** Whitehead's theorem **fails** for general [[(∞,1)-toposes]] and non-[[truncated object in an (infinity,1)-category|truncated]] objects.
The [[∞-stack]] [[(∞,1)-topos]]es in which the Whitehead theorem does hold are the [[hypercomplete (∞,1)-topos]]es. These are precisely the ones that are [[presentable (∞,1)-category|presented]] by a local [[model structure on simplicial presheaves]].
For instance the hypercomplete $(\infty,1)$-topos [[Top]] is presented by the model structure on simplicial presheaves on the point, namely the [[model structure on simplicial sets]].
## In homotopy type theory
Since [[homotopy type theory]] admits models in [[(∞,1)-toposes]] (and in particular in non-hypercomplete ones), Whitehead's theorem is not provable when regarded as a statement about types in homotopy type theory. From this perspective, the truth of Whitehead's theorem is a [[foundational axiom]] that may be regarded as a "classicality" property, akin to [[excluded middle]] or the [[axiom of choice]] --- we call it **Whitehead's principle** (not to be confused with [[Whitehead's problem]], another statement that is independent of the usual axioms of set theory).
Whitehead's principle does hold, however, for maps between [[homotopy n-types]] for any finite $n$; this is provable in homotopy type theory by [[induction]] on $n$.
## Related concepts
* [[mod p Whitehead theorem]]
* [[equivariant Whitehead theorem]]
* [equivariant stable Whitehead theorem](G-spectrum#EquivariantWhitehead)
* [[CW-approximation theorem]]
* [[cellular approximation theorem]]
## References
Originally proved by [[J. H. C. Whitehead]]:
* [[J. H. C. Whitehead]], _Combinatorial homotopy. I_, Bulletin of the American Mathematical Society 55:3 (1949), 213–246. [doi](http://dx.doi.org/10.1090/s0002-9904-1949-09175-9).
The simplicial version is due to [[Daniel M. Kan]], see Theorem 7.2 in
* [[Daniel M. Kan]], _On c.s.s. categories_, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94. [PDF](https://dmitripavlov.org/scans/kan-on-css-categories.pdf).
Review:
* {#Bredon93} [[Glen Bredon]], Cor. 11.14 in: _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0), [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))
* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 1 of _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_, 1995 Amsterdam: North-Holland, pp. 213–253, ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))
* [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], Thm. 6.3.31 in: *Algebraic topology from a homotopical viewpoint*, Springer (2008) ([doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586))
Discussion of the [[equivariant Whitehead theorem]]:
* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], _Equivariant stable homotopy theory_, in I.M. James (ed.), _Handbook of Algebraic Topology_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))
The [[(infinity,1)-topos|$(\infty,1)$-topos]] theoretic version:
* [[Jacob Lurie]], section 6.5 of [[Higher Topos Theory]]
The analogous formulation in [[homotopy type theory]]:
* {#UFP13} [[Univalent Foundations Project]], §8.8 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]
Corresponding formalization in [[Agda]]:
* [[Dan Licata]], _[Whitehead.agda](https://github.com/dlicata335/hott-agda/blob/master/homotopy/Whitehead.agda)_
category: foundational axiom
[[!redirects Whitehead's theorem]]
[[!redirects Whitehead's principle]]
[[!redirects Whitehead principle]]
[[!redirects simplicial Whitehead theorem]] |
Whitehead torsion | https://ncatlab.org/nlab/source/Whitehead+torsion |
#Contents#
* table of contents
{:toc}
## Idea
(...)
## References
* {#Milnor66} [[John Milnor]], *Whitehead torsion*, Bull. Amer. Math. Soc. **72** (1966) 358–426 [[doi:10:1090/S0002-9904-1966-11484-2](https://www.ams.org/journals/bull/1966-72-03/S0002-9904-1966-11484-2), [maths.ed.ac.uk](https://web.archive.org/web/20160529051526/http://www.maths.ed.ac.uk/~s1122173/surgerygroup12/milnorwh.pdf), [jstor:bams/1183527946](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-72/issue-3/Whitehead-torsion/bams/1183527946.full)]
* Wikipedia, _[Whitehead torsion](https://en.wikipedia.org/wiki/Whitehead_torsion)_
category: topology
|
Whitehead tower | https://ncatlab.org/nlab/source/Whitehead+tower |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The _Whitehead tower_ of a [[pointed object|pointed]] [[homotopy type]] $X$ is an interpolation of the point inclusion $* \to X$ by a sequence of homotopy types
$$
* \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)}
\simeq X
$$
that are obtained from right to left by _removing [[homotopy groups]] from below_, hence such that
* each $X^{(n)}$ is $(n-1)$-[[n-simply connected space|connected]]
* and each [[morphism]] $X^{(n+1)} \to X^{(n)}$ induces an [[isomorphism]] on all [[homotopy groups]] in degree $k \geq (n+1)$ (and the inclusion $1 \to \pi_n(X^{(n)})$ in degree $n$ as well as the identity $1 = 1$ in degree $k \lt n$).
The notion of Whitehead tower is [[duality|dual]] to the notion of _[[Postnikov tower]]_, which instead is a factorization of the terminal morphism $X \to *$ into a tower, where homotopy groups are _added_ from right to left.
In fact, the Whitehead tower may be constructed by taking each stage $X^{(n+1)} \to X$ to be the _[[homotopy fiber]]_ of the corresponding map into the $(n+1)$st stage of the [[Postnikov tower]].
## Definition
The construction of Whitehead towers is traditionally done for [[topological spaces]] regarded up to [[weak homotopy equivalence]], hence as objects of the [[(∞,1)-category]] [[Top]]. The discussion directly generalizes to any [[(∞,1)-topos]].
The **Whitehead tower** of a [[pointed object|pointed]] [[homotopy type]] $X$ is a sequence of [[homotopy types]]
$$* \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)} \simeq X$$
where [[generalized the|the]] space $X^{(n)}$ is [[generalized the|the]] [[homotopy fiber]] of [[generalized the|the]] map $X \to X_{(n+1)}$ into the item $X_{(n+1)}$ in the [[Postnikov tower]] of $X$.
Here each [[homotopy pullback]]
$$
\array{
X^{(n)} &\to& *
\\
\downarrow && \downarrow
\\
X &\to& X_{(n+1)}
}
$$
in the [[(∞,1)-category]] [[Top]] may be computed (as described at [[homotopy pullback]]) as an ordinary [[pullback]] in the 1-[[category]] [[Top]] of a fibrantly replaced diagram, for instance with the point $*$ replaced by the path fibration $P X_{(n+1)} \simeq *$, which is a [[Hurewicz fibration]] $P X_{(n+1)} \to X_{(n+1)}$. In this case also the ordinary [[pullback]] $X^{(n)}\to X$
$$
\array{
X^{(n)} &\to& P X_{(n+1)}
\\
\downarrow && \downarrow
\\
X &\to& X_{(n+1)}
}
$$
is a fibration, and this is often taken as part of the definition of the Whitehead tower.
From this perspective the _Whitehead tower_ of a [[pointed space]] $(X,x)$ is a sequence of fibrations
$$
\ldots \to X\langle n\rangle \to \ldots \to X\langle 1 \rangle \to X\langle 0 \rangle \to X
$$
where each $X\langle n\rangle \to X\langle n-1 \rangle$ induces isomorphisms on [[homotopy group]]s $\pi_i$ for $i\gt n$ and such that $X\langle n\rangle$ is $n$-[[n-connected|connected]] (has trivial [[homotopy group]]s $\pi_i$ for $i \leq n$).
The homotopy long exact sequence then shows that the fiber of $X\langle n\rangle \to X\langle n-1 \rangle$ is a $K(\pi_{n-1}(X,x),n-1)$ [[Eilenberg-Mac Lane space]]. One has a model for $K(\pi_{n-1}(X,x),n-1)$ which is an abelian topological group; this has a remarkable consequence when $(X,x)=(G,e)$ is a [[topological group]]. Indeed, in this case one sees inductively that $G\langle n\rangle$ has a model which is a topological group, which is an abelian group extension:
$$
1\to K(\pi_{n-1}(X,x),n-1) \to G\langle n\rangle \to G\langle n-1 \rangle \to 1
$$
For instance, the [[string group]] can be realized as a topological group as a $K(\mathbb{Z},2)$-extension of the [[spin group]].
For $n=0$ we require that $X\langle 0 \rangle \hookrightarrow X$ is the inclusion of the path-component of $x$. Really this is defined up to [[homotopy (as an operation)|homotopy]], but we have a canonical model. If $X$ is locally connected and semilocally path-connected, then $X\langle 1\rangle$ can be chosen as the [[universal covering space]].
In traditional models this construction is highly non-[[functor]]ial, except for nice spaces in low dimensions as remarked above.
## Constructions
### Whitehead's construction
[Whitehead 1952](#Whitehead) answered the question, posed by [[Witold Hurewicz]], of the existence of what we would now call $n$-connected \'covers\' of a given space $X$, taking this to mean a fibration $X\langle n\rangle \to X$ with $X\langle n\rangle$ $n$-connected and otherwise inducing isomorphisms on homotopy groups.
The construction proceeds as follows (using modern terminology). Given a pointed space $(X,x)$,
* Choose a representative for the [[Postnikov section]] $X_n$ such that $X \hookrightarrow X_n$ is a closed subspace (I would be tempted to make it a closed cofibration, but I don't know any reason for this to be necessary -DMR).
* Form the $\infty$-connected cover of $X_n$, i.e. the [[path fibration]] $P X_n$. This is a [[Hurewicz fibration]].
* Pull this back to $X$, to get $p\colon X\langle n\rangle \to X$, which is still a fibration. The induced maps on long exact sequences in homotopy can be compared, and show that $p$ has the desired properties.
This gives us a single $n$-connected cover, but by considering the [[Postnikov tower]]
$$
X \to (\ldots \to X_n \to X_{n-1} \to \ldots \to X_1 \to X_0)
$$
of $X$, where each map $X \to X_n$ is the inclusion of a closed subspace, it is simple to see there are induced maps $X\langle n\rangle \to X\langle n-1\rangle$ over $X$ for all $n$.
One way of obtaining a Postnikov section as above is to choose representatives $\phi_g\colon S^{n+1} \to X$ of generators $g$ of $\pi_{n+1}(X,x)$ and attaching cells: $X(1)\coloneqq B^{n+2} \cup_{\{\phi_g\}} X$. We then choose representatives for the generators of $\pi_{n+2}(X(1),x)$ and attach cells and so on. The colimit $\lim_{\to n} X(n)$ is then a Postnikov section with the properties we require.
Understandably, this process is unbelievably non-canonical, and so we are generally reduced to existence theorems using this method -- unless there is a functorial way to construct Postnikov sections. Strictly speaking we can only say _an_ $n$-connected cover (except in special cases, like when $n=1$ and $X$ is a [[well-connected space]]).
### Functorial constructions
The $n$th stage of the Whitehead tower of $X$ is the [[homotopy fiber]] of the map from $X$ to the $n$th (or so) stage of its [[Postnikov tower]], so one can use a functorial construction of the Postnikov tower plus a functorial construction of the homotopy fiber (such as the usual one using the [[path object|path space]] of the target).
The $n$th stage of the Whitehead tower of $X$ is also the cofibrant replacement for $X$ in the right [[Bousfield localization]] of [[Top]] with respect to the object $S^n$ (or so). Since [[Top]] is right proper and cellular this localization exists by the result of chapter 5 of Hirschhorn's book on [[localization]]s of [[model category|model categories]].
## Examples
### Whitehead tower of the orthogonal group
{#OfTheOrthogonalGroup}
The Whitehead tower of the [[classifying space]]/[[delooping]] of the [[orthogonal group]] $O(n)$ starts out as
$$
\array{
& \mathbf{\text{Whitehead tower}}
\\
&\vdots
\\
& B Fivebrane &\to& \cdots &\to& *
\\
& \downarrow && && \downarrow
\\
\mathbf{\text{second frac Pontr. class}} & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z}
&\to& *
\\
& \downarrow && && \downarrow && \downarrow
\\
\mathbf{\text{first frac Pontr. class}} & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & *
\\
& \downarrow && && \downarrow && \downarrow && \downarrow
\\
\mathbf{\text{second SW class}} & B S O
&\to&
\cdots
&\to&
&\to&
& \stackrel{w_2}{\to} &
\mathbf{B}^2 \mathbb{Z}_2
&\to&
*
\\
& \downarrow && && \downarrow && \downarrow && \downarrow && \downarrow
\\
\mathbf{\text{first SW class}} & B O
&\to&
\cdots
&\to&
\tau_{\leq 8 } B O
&\to&
\tau_{\leq 4 } B O
&\to&
\tau_{\leq 2 } B O
&\stackrel{w_1}{\to}&
\tau_{\leq 1 } B O
\simeq
B \mathbb{Z}_2
&
\mathbf{\text{Postnikov tower}}
}
$$
where each square and each composite rectangle is a [[homotopy pullback]] square (all controled by the [[pasting law]]),
where the stages are the deloopings of
...
$\to$
[[fivebrane group]]
$\to$
[[string group]]
$\to$
[[spin group]]
$\to$
[[special orthogonal group]]
$\to$
[[orthogonal group]],
where lifts through the stages correspond to
* [[orthogonal structure]]
* [[orientation]]
* [[spin structure]]
* [[string structure]]
* [[fivebrane structure]]
and where the [[obstruction]] classes are the [[universal characteristic classes]]
* [[first Stiefel-Whitney class]] $w_1$
* [[second Stiefel-Whitney class]] $w_2$
* [[Pontryagin class|first fractional Pontryagin class]] $\tfrac{1}{2}p_1$
* [[Pontryagin class|second fractional Pontryagin class]] $\tfrac{1}{6}p_2$
and where every possible square in the above is a [[homotopy pullback]] square (using the [[pasting law]]).
For instance $w_2$ can be identified as such by representing $B O \to \tau_{\leq 2} B O \simeq BO/_{\sim_n}$ by a [[Kan fibration]] (see at [[Postnikov tower]]) between [[Kan complexes]] so that then the [[homotopy pullback]] (as discussed there) is given by an ordinary pullback. Since $sSet$ is a [[simplicial model category]], $sSet(S^2,-)$ can be applied and preserves the pullback as well as the homotopy pullback, hence sends $ B O \to \tau_{\leq 2} B O$ to an isomorphism on connected components. This identifies $B SO \to B^2 \mathbb{Z}$ as being an [[isomorphism]] on the second [[homotopy group]]. Therefore, by the [[Hurewicz theorem]], it is also an isomorphism on the [[cohomology group]] $H^2(-,\mathbb{Z}_2)$. Analogously for the other characteristic maps.
In summary, more concisely, the tower is
$$
\array{
\vdots
\\
\downarrow
\\
B Fivebrane
\\
\downarrow
\\
B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) & \simeq B^8 \mathbb{Z}
\\
\downarrow
\\
B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& B^3 U(1) & \simeq B^4 \mathbb{Z}
\\
\downarrow
\\
B SO &\stackrel{w_2}{\to}& B^2 \mathbb{Z}_2
\\
\downarrow
\\
B O &\stackrel{w_1}{\to}& B \mathbb{Z}_2
\\
\downarrow^{\mathrlap{\simeq}}
\\
B GL
}
\,,
$$
where each "hook" is a [[fiber sequence]].
Via the [[J-homomorphism]] this corresponds to the [[stable homotopy groups of spheres]]:
[[!include image of J -- table]]
## Whitehead tower in general $(\infty,1)$-toposes
While a notion of [[Postnikov tower in an (∞,1)-category]] depends on the _categorical_ [[homotopy groups in an (∞,1)-topos|homotopy groups in an (∞,1)-category]], the notion of Whitehead tower makes good sense with respect to the _geometric_ homotopy groups.
A good notion of geometric [[homotopy groups in an (∞,1)-topos]] exist in a [[locally contractible (∞,1)-topos]]. The notion of Whitehead tower in this context is discussed at
* [[Whitehead tower in an (∞,1)-topos]].
## Related concepts
* Applying the [[Hurewicz theorem]] stagewise to a [[Whitehead tower]] yields an method for computing the [[homotopy groups]] of the original [[space]]. This process, or rather the refinement thereof for Whitehead towers generalized to [[Adams resolutions]], is formalized by the _[[Adams spectral sequence]]_, see there for more.
* _[symmetric group -- Whitehead tower](https://ncatlab.org/nlab/show/permutation#WhietheadTowerAndSupersymmetry)_
[[!include Lurie spectral sequences -- table]]
## References
The original reference is
* [[George Whitehead]], _Fiber Spaces and the Eilenberg Homology Groups_, PNAS **38**, No. 5 (1952)
{#Whitehead}
A textbook account is around example 4.20 in
* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]
A more detailed useful discussion happens to be in section 2.A, starting on p. 11 of
* [[Linus Kramer]], _Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface_, Memoirs of the American Mathematical Society number 752 ([arXiv:math/0109133] (http://arxiv.org/abs/math/0109133), [doi:10.1090/memo/0752](http://dx.doi.org/10.1090/memo/0752), [GoogleBooks](http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false))
[[!redirects Whitehead tower]]
[[!redirects Whitehead towers]]
|
Whitehead tower in an (infinity,1)-topos | https://ncatlab.org/nlab/source/Whitehead+tower+in+an+%28infinity%2C1%29-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
If an [[(∞,1)-topos]] $\mathbf{H}$ is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] it has a good notion of [[homotopy groups in an (∞,1)-topos|geometric homotopy groups]] of its objects. In terms of these, there is an analog in $\mathbf{H}$ of the notion of the classical notion of [[Whitehead tower]] in the archetypical $(\infty,1)$-topos [[Top]]:
the Whitehead tower of an object $X \in \mathbf{H}$ is the tower of $n$-[[connected]] [[homotopy fiber]]s of the canonical morphism into the [[Postnikov tower in an (∞,1)-category|(∞,1)-topos-theoretic Postnikov tower]] $\cdots \to \mathbf{\Pi}_{n+1}(X) \to \mathbf{\Pi}_n(X) \to \cdots$ of the [[schreiber:path ∞-groupoid|structured path ∞-groupoid]] $\mathbf{\Pi}(X)$ of $X$.
See also the section <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#Coverings">Universal coverings and geometric Whitehead towers</a> at [[cohesive (∞,1)-topos]].
## Definition
Let $\mathbf{H}$ be a [[locally ∞-connected (∞,1)-topos]] $\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd$. Write
$$
\mathbf{\Pi} := LConst \circ \Pi : \mathbf{H} \to \mathbf{H}
$$
for the _internal_ [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]-functor. From the adjunction relation this comes with the canonical natural morphism
$$
X \to \mathbf{\Pi}(X)
\,.
$$
For $n \in \mathbb{N}$ write
$$
\mathbf{H}_{\leq n} \stackrel{\overset{\tau_{\geq n}}{\leftarrow}}{\overset{}{\hookrightarrow}} \mathbf{H}
$$
for the [[reflective (∞,1)-subcategory]] of [[n-truncated object of an (∞,1)-category|n-truncated objects]] of $\mathbf{H}$ and write $\mathbf{\tau}_{\leq n}$ for the [[localization of an (∞,1)-category|localization]]
$$
\mathbf{\tau}_{\leq n} : \mathbf{H} \stackrel{\tau_{\leq n}}{\to}
\mathbf{H}_{\leq n} \hookrightarrow \mathbf{H}
\,.
$$
Write
$$
\mathbf{\Pi}_n : \mathbf{H} \stackrel{\mathbf{\Pi}}{\to} \mathbf{H} \stackrel{\mathbf{\tau}_{\leq n}}{\to} \mathbf{H}
$$
for the internal **homotopy $n$-groupoid**. For $X \in \mathbf{H}$ we have the [[Postnikov tower in an (∞,1)-category|(∞,1)-Postnikov tower]]
$$
\cdots \to \mathbf{\Pi}_2(X) \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X)
$$
of $\mathbf{\Pi}(X)$.
+-- {: .un_def}
###### Definition
**(Whitehead tower)**
For $X \in \mathbf{H}$, its **Whitehead tower** is the sequence of objects
$$
* \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)} \simeq X
$$
in $\mathbf{H}$, where for each $n \in \mathbb{N}$ the object $X^{(n+1)}$ is the [[homotopy fiber]] of the canonical morphism $X \to \mathbf{\Pi}_{n+1}$, i.e. the object defined by the [[pullback]] diagram
$$
\array{
X^{(n+1)} &\to& *
\\
\downarrow && \downarrow
\\
X &\to& \mathbf{\Pi}_{n+1}(X)
}
\,.
$$
=--
Here the morphisms $X^{(n+1)} \to X^{(n)}$ are induced from the universality of the pullback:
$$
\array{
X^{(n+1)}&\to&X^{(n)}&& \mathbf{\Pi}_{(n+1)}(X)
\\
&\searrow &\downarrow&\nearrow& \downarrow
\\
&&X &\to& \mathbf{\Pi}_n(X)
}
$$
+-- {: .un_remark}
###### Remark
We have that $\mathbf{\Pi}_n(X) \simeq LConst \tau_{\leq n} \Pi(X)$.
A homotopy-commuting diagram
$$
\array{
X^{(n)} &\to& {*}
\\
\downarrow && \downarrow
\\
X &\to& \mathbf{\Pi}_n(X)
}
$$
in $\mathbf{H}$ corresponds by the [[adjoint (∞,1)-functor|adjunction relation]] to diagram
$$
\array{
\Pi(X^{(n)}) &\to& {*}
\\
\downarrow && \downarrow
\\
\Pi(X) &\to& {\Pi}_n(X)
}
$$
in [[∞Grpd]]. This being universal means that $\Pi(X^{(n)})$ is $n$-connected, and universal with that property as an object over $\Pi(X)$.
=--
## Properties {#Properties}
+-- {: .un_def}
###### Definition
For $* \to X \in \mathbf{H}$ a [[pointed object]] and $n \in \mathbb{N}$, $n \geq 1 $, define the object $\mathbf{B}^n \mathbf{\pi}_n(X)$ to be the [[homotopy fiber]] of $\mathbf{\Pi}_n(X) \to \mathbf{\Pi}_{n-1}(X)$, so that we have a [[fibration sequence]]
$$
\mathbf{B}^n \mathbf{\pi}_n(X) \to \mathbf{\Pi}_n(X) \to \mathbf{\Pi}_{n-1}(X)
\,.
$$
=--
+-- {: .un_proposition}
###### Proposition
With $\Pi(X) \in \infty Grpd \simeq Top$ the underlying topological space of $X \in \mathbf{H}$ (its [geometric realization](http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#GeomReal)) we have that
$$
\mathbf{B} \mathbf{\pi}_n(X)
\simeq
LConst \mathcal{B}^n \pi_n(X)
\,,
$$
where $\mathcal{B}^n \pi_n(X)$ denotes the [[homotopy fiber]] of
$\Pi_n(X) \to \Pi_{(n-1)}(X)$ in [[∞Grpd]].
=--
+-- {: .proof}
###### Proof
> check
This follows from $\mathbf{\tau}_{\leq n} LConst \Pi(X) \simeq LConst \tau_{\leq n} \Pi(X)$. This, in turn, can for instance be checked in terms of the [[model structure on simplicial presheaves]], using that $\tau$ is objectwise the [[coskeleton]] operation. More on that at [[Postnikov tower in an (∞,1)-category]].
=--
+-- {: .un_proposition}
###### Proposition
For each $n \geq 1$ we have a [[fibration sequence]]
$$
X^{(n)} \to X^{(n-1)} \to \mathbf{B}^n \mathbf{\pi}_n(X)
\,.
$$
=--
+-- {: .proof}
###### Proof
Regard the diagram
$$
\array{
X^{(n)} &\to& *
\\
\downarrow && \downarrow
\\
X^{(n-1)} & \to & \mathbf{B}^n \mathbf{\pi}_n(X) &\to& *
\\
\downarrow && \downarrow && \downarrow
\\
X &\to& \mathbf{\Pi}_n(X) &\to& \mathbf{\Pi}_{(n-1)}(X)
}
\,.
$$
Here the right square is the defining $(\infty,1)$-pullback diagram of $\mathbf{B}^n \mathbf{\pi}_n(X)$ from above. Take also the left bottom square to be a homotopy pullback. Then from the pasting rules of pullbacks it follows that the composite bottom rectangle is also a pullback, which identifies the object $X^{(n-1)}$ on the left as indicated.
Similarly, form now the top square as a pullback. Then by the composition law of pullbacks we find that the composite vertical rectangle is a pullback, which identifies the top left object as $X^{(n)}$.
=--
## Examples
### For $\infty$-groupoids
In the archetypical $(\infty,1)$-topos $\mathbf{H} = $ [[∞Grpd]] the functors $\Pi$ and $\mathbf{\Pi}$ are the identity and so
$\cdots \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X)$ is just the standard [[Postnikov tower]] $\cdots \to X_1 \to X_0 $.
If we use the [[model structure on simplicial sets]] in order to [[presentable (infinity,1)-category|present]] $\infty Grpd$, then the Postnikov tower may be realized by the simplicial [[coskeleton]] operation
$$
\cdots \to \mathbf{cosk}_2(X) \to \mathbf{cosk}_1(X)
\,.
$$
If $X$ is [[connected]], the [[homotopy pullback]]
$$
\array{
X^{(n+1)} &\to& *
\\
\downarrow && \downarrow
\\
X &\to& X_{(n)}
}
$$
defining the Whitehead tower of an $\infty$-groupoid incarnated as a [[Kan complex]] $X$ may be computed as an ordinary [[pullback]] using a fibrant replacement diagram, such as thatr replacing the [[point]] by the [[decalage]] simplicial set, hence as the ordinary pullback
$$
\array{
X^{(n+1)} &\to& Dec \mathbf{cosk}_n X
\\
\downarrow && \downarrow
\\
X &\to& \mathbf{cosk}_n X
}
$$
in [[sSet]].
If $X$ is a pointed $\infty$-groupoid then $\mathbf{B}^n \mathbf{\pi}_n(X)$ is the [[Eilenberg-Mac Lane space]] $K(\pi_n(X),n)$.
### For topological $\infty$-groupoids {#TopInfGrpds}
It is often useful to think of [[topological space]]s as embedded into the [[(∞,1)-topos]] of [[topological ∞-groupoid]]s, i.e. the [[(∞,1)-category of (∞,1)-sheaves]]/[[∞-stack]]s on (a [[small category|small]] version of) the [[site]] of "all" topological spaces.
$$
\mathbf{H} = Sh_{(\infty,1)}(Top)
\,.
$$
#### The universal covering spaces
For $X$ a topological space regarded as a representable object in $Sh_{(\infty,1)}(Top)$, we have that $\mathbf{\Pi}_1(X)$ is the [[fundamental groupoid]] of $X$, regarded as a [[topological ∞-groupoid]] in the obvious way.
Then the [[homotopy fiber]] $X^{(1)}$ of the constant path inclusion $X \to \mathbf{\Pi}_1(X)$ is the [[universal covering space]] of $X$, as described in detail there.
#### The higher covering spaces
The analog of this statement for the higher items $X^{(n)}$ with $n \gt 1$ in the Whitehead tower of a topological spaces regarded in the $(\infty,1)$-topos of topological groupoids has been studied in
* [[David Roberts]], _[[Fundamental Bigroupoids and 2-Covering Spaces]]_
The following reviews some central ideas of this.
##### Non-traditional approach in $Top$
**The following is some semblance of current research, except the stuff about the string 2-group. All errors and stupid ideas are mine - [[David Roberts]]**
For locally nice spaces (say locally contractible), it is desirable to have a functorial construction of the Whitehead tower. A shadow of a low-dimensional case of this can be seen in the construction of the [[string 2-group]] given by Baez-Crans-Schreiber-Stevenson. Since finite-dimensional Lie groups have non-trivial third homotopy group, it is not possible to form the 3-connected cover in the [[category]] $fin.dim.LieGrp$, like it is possible to take the 0-, and 1-connected covers. While most people give up the smoothness and make do with the [[topological group]] $String_G$, The BCSS construction leaps out of that [[category]] into that of strict ([[Frechet space|Frechet]]) Lie [[2-group]]s. The construction is also functorial.
+--{: .query}
[[David Roberts]]: As an aside, I know of two classes of spaces with explicitly constructed (i.e. not via co-killing homotopy groups) 2-connected covers: the 2-sphere and its quotients by $\mathbf{Z}/n$ (lens spaces), and the [[loop group]] $\Omega G$ of a compact, simple, simply-connected Lie group $G$ (the latter is the level-1 central extension by $U(1)$, the first needs no introduction). Does anybody know of any other $n$-connected covers (other than, say, the Hopf fibrations) that are canonically given?
=--
However, without one could demand a conceptually similar approach to the $n$-connected cover of a general (locally nice) space or smooth manifold. For $n=2$, and taking only [[topological space]]s for consideration, this is contained in [[David Roberts|my]] thesis work.
The rough result is that one gets a 2-connected topological groupoid $X^{(2)}$ equipped with a map to $X$ that factors through the universal covering space $X^{(1)}$. This is functorial, and generalises to higher connected covers (at least heuristically - I don't have $n$-categorical superpowers).
But the general idea is that one would get an $(n-1)$-groupoid $X^{(n)}$ over $X$ which is $n$-connected and such that the map to $X$ factors through the $(n-2)$-groupoid $X^{(n-1)}$. The map to $X$ should induce [[isomorphism]]s on homotopy groups $\pi_i$ for $i\gt n$, as in the usual Whitehead tower.
If we want to consider arbitrary $n$ and retain some sort of local triviality on our connected covers we cannot get away from the assumption that the space in question is locally contractible. An assumption of this sort appears in
* [[Bertrand Toen]], Vers une interpretation Galoisienne de la theorie de l'homotopie, Cahiers de Top. et Geom. Diff. Cat., Vol. XLIII-4 (2002), 257-312.
in the context of locally constant $\infty$-stacks and their monodromy (I haven't got this article at the moment, and I suspect they may be $(\infty,1)$-stacks, but I'm not 100 percent certain -DMR). Technically speaking, we don't need local contractibility but the existence of a basis of open sets $U \hookrightarrow X$ such that this inclusion map is null-homotopic. But I will continue to call this local contractibilty, for lack of a better term (if there is such a term, I'd like to know -DMR).
##### Construction using topological $n$-groupoids
Consider the fundamental $n$-groupoid $\Pi_n(X)$ of the locally contractible space $X$ (as a [[Trimble n-category|Trimble n-groupoid]], say), or at least for now its underlying globular set. We can take the compact open-topology on the set of $k$-morphisms for $k \lt n$. As the space is locally contracible, in particular semi-locally $n$-connected, the space $Hom(S^{n-1},X)$ is semi-locally simply-connected (I have a fragment of a paper saying this is true for the \'absolute case\' - that is, _locally_ $n$-connected implies the mapping space locally simply connected, but I expect it to be true for the relative case -DMR). In particular, we can take the fundamental groupoid $\Pi_1(Hom(S^{n-1},X))$, which has a topology given in the [[topological fundamental groupoid|usual way]]. The arrow space of this fundamental groupoid is then non other than the space of $n$-arrows of $\Pi_n(X)$. It needs to be checked that the $n$ compositions $\#_k$, $k=0,\ldots,n-1$ are continuous, as well as a bunch of other stuff, but I think this should follow from (unique) lifting theorems for covering spaces.
The object space of $\Pi_n(X)$ is just $X$, and so there is an inclusion $X \hookrightarrow \Pi_n(X)$, and it is this that replaces the Postnikov section in the Whitehead construction outlined above. The topological fundamental $n$-groupoid, even though it contains apparently more homotopical information than the untopologised fundamental $n$-groupoid $\Pi_n(X)^\delta$ ($\delta=$discrete topology), I posit that under the assumptions on $X$, the inclusion $\Pi_n(X)^\delta \to \Pi_n(X)$ has an ana-n-functor pseudoinverse (taking the [[Grothendieck pretopology]] of open covers should be enough). On passing to the homotopy colimit this span should become a span weak homotopy equivalences, and so we can consider the topologised and the untopologised to be different representatives of the $n$-type of $X$.
Given a basepoint $x\in X$, we can form the tangent $n$-groupoid $T_x \Pi_n(X)$, which is equivalent to the trivial $n$-groupoid $*$ (even as a _topological_ $n$-groupoid), and gives us what should be in any sensible definition a fibration $T_x\Pi_n(X) \to \Pi_n(X)$.
Pull back this fibration to $X$, and call the resulting thing $X^{(n)}$. It is fairly easy to see that $X^{(n)}$ is a topological $(n-1)$-groupoid over $X$. This then should be the $n$-connected cover of $X$. For $n=1$ this is precisely the classical construction of the universal covering space of a pointed space. For $n=2$ this is treated in
* D.M. Roberts, Fundamental bigroupoids and 2-covering spaces, PhD thesis, available [[davidroberts:HomePage|here]]
and the two-dimensional homotopical tools developed there can be used to show that $X^{(2)}$ is 2-connected.
A word is probably in order about the notion of $k$-connectedness for topological $n$-groupoids. This has its usual meaning, once homotopy groups $\pi_i$ have been defined. The reader should be warned that these have nothing to do with the groups $\pi_0 Eq(1_{._{._{1_x}}})/\sim$ obtained from considering the autoequivalence $n-i+1$-group of the identity $(i-1)$-arrow on the identity $(i-2)$-arrow on ... on the object $x$. These should be defined in such a way as to agree with the homotopy colimit $hocolim X$ of $X$ considered as a truncated simplicial space. In particular, $\pi_1$ of a topological groupoid is _not_ the group of automorphisms of the basepoint, but a quotient of the set of _generalised paths_, an idea going back to Haefliger (for an intro see the early parts of chapter 2 of the above thesis, available at [[davidroberts:HomePage]], or the preprint H. Colman, _On the 1-homotopy type of Lie groupoids_, arXiv:math/0612257).
There are other interpretations of $X^{(n)}$:
* The (0-)source fibre of $\Pi_n(X)$, which is:
* The pullback of $(s_0,t_0):Hom_{\Pi_n(X)} \to X\times X$ along the inclusion $\{x\}\times X \to X\times X$, where $Hom_{\Pi_n(X)}$ is the (internalised) hom-$(n-1)$-groupoid familiar from the definition of a Trimble $n$-groupoid.
* The \'vertical fundamental $(n-1)$-groupoid\' of $PX \to X$, the path fibration.
The last can be thought of as a families version of the usual fundamental $(n-1)$-groupoid: take vertical paths, vertical homotopies between paths etc.
[[!redirects Whitehead tower in an (∞,1)-topos]]
|
Whitney embedding theorem | https://ncatlab.org/nlab/source/Whitney+embedding+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The (strong) _Whitney embedding theorem_ states that every [[smooth manifold]] ([[Hausdorff space|Hausdorff]] and [[sigma-compact]]) of [[dimension]] $n$ has an [[embedding of smooth manifolds]] in the [[Euclidean space]] of dimension $2n$.
Notice that it is easy to see that every smooth manifold embeds into the Euclidean space of _some_ dimension ([this prop.](embedding+of+smooth+manifolds#ManifoldEmbedsIntoLargeDimensionalEuclideanSpace)). The force of Whitney's strong embedding theorem is to find the lowest dimension that still works in general.
## Related concepts
* The analogous statement for [[Riemannian manifolds]] and [[isometry|isometric]] embeddings is the _[[Nash embedding theorem]]_.
## References
Named after [[Hassler Whitney]].
* {#Pontrjagin55} [[Lev Pontrjagin]], Section 2.2 of: _Smooth manifolds and their applications in Homotopy theory_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001))
See also
* Wikipedia, _[Whitney embedding theorem](https://en.wikipedia.org/wiki/Whitney_embedding_theorem)_
* Paul Rapoport, _Introduction to Immersion, Embeddingand the Whitney Embedding Theorem_, 2015 ([pdf](http://schapos.people.uic.edu/MATH549_Fall2015_files/Survey%20Paul.pdf))
[[!redirects Whitney's embedding theorem]]
[[!redirects Whitney imbedding theorem]]
[[!redirects Whitney's imbedding theorem]]
[[!redirects Whitney's strong embedding theorem]]
[[!redirects Whitney's weak embedding theorem]]
|