weakly reductive semigroup

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## Idea

A [[semigroup]] is called _left/right weakly reductive_ if it coincides with the semigroup of its left/right translations.

## Definition

We only define left weakly reductive semigroups, right weakly reductive semigroups are defined similarly.

Let $S_l$ be the set of left translations of $S$. That is, this is the set of maps $x_l:S\to S$ defined by $x_l(y) := x\cdot y$. The semigroup $(S_l, \circ)$, where $\circ$ denotes composition of maps, is called the semigroup of left translations of $S$.

The map $f:x\mapsto x_l$ is then a morphism in the category of semigroups. We call $(S, \cdot)$ _left weakly reductive_, if $f$ is an isomorphism.

Explicitly, and this is where the name comes from, if $a, b\in S$, and $x\cdot a = x\cdot b$ for all $x\in S$, then $a = b$.

A _weakly reductive semigroup_ is a semigroup that is both right and left weakly reductive.

## In terms of varieties

A left weakly reductive semigroup can be thought of as a class of structures $(S, \cdot , w, r)$, where $\cdot, w$ are binary operations, and $r$ is a ternary operation, satisfying the following axioms for all $x, y, z$: $ x\cdot (y\cdot z) = (x\cdot y)\cdot z, r(x, y, w(x, y)\cdot x) = x, r(x, y, w(x, y)\cdot y) = y $.

## Examples

Any _left monoid_, a semigroup with a left identity element, is a left weakly reductive semigroup. In particular, any [[monoid]] is weakly reductive.

Any left weakly reductive commutative semigroup is weakly reductive.

A monogenic semigroup that isn't a [[group]] is not weakly reductive.

There exists unique smallest left weakly reductive semigroup which isn't a left monoid. It can be defined as the idempotent semigroup $(\{x, y, z\}, \cdot)$ such that $a\cdot b = z$ for $a\neq b$.

## References

* A. H. Preston and G. B. Clifford, _The algebraic theory of semigroups: Volume I_, American Mathematical Society (1961)

[[!redirects weakly reductive semigroups]]


weakly separated space

weakly separated space


weakly étale morphism of schemes

[[!redirects weakly étale morphism]]

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#### Cohomology
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## Idea

A variant of [[étale morphism of schemes]] where the finiteness conditions on [[étale morphisms]] are relaxed.

Used in the definition of _[[pro-étale site]]_ and _[[pro-étale cohomology]]_.

## Definition

+-- {: .num_defn #WeaklyEtale}
###### Definition

A [[morphism]] $f \colon X \longrightarrow Y$ of [[schemes]]
is called _weakly étale_ if

1. $f$ is a [[flat morphism of schemes]];

1. its [[diagonal]] $X \longrightarrow X \times_Y X$ is also flat.

=--

([Bhatt-Scholze 13, def. 4.1.1](#BhattScholze13))

## Properties

+-- {: .num_prop}
###### Proposition

Every [[weakly étale morphism]] is a [[formally étale morphism]].

=--

([Gabber-Ramero 03, theorem 2.5.36, prop. 3.2.16](#GabberRamero03) [Bhatt-Scholze 13, prop. 2.3.3. (2)](#BhattScholze13))

+-- {: .num_remark}
###### Remark

As discussed there, an [[étale morphism]] is a [[formally étale morphism]] which is [[locally of finite presentation]].

=--

+-- {: .num_cor}
###### Corollary

A [[weakly étale morphism]] which is [[locally of finite presentation]] is an [[étale morphism of schemes|étale morphism]].

[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]

=--

In fact a weakly étale morphism is equivalently a [[formally étale morphism]] which is "locally [[pro-object|pro-finitely]] presentable" (dually locally of [[ind-object|ind]]-finite rank) in the following sense

+-- {: .num_defn #IndEtale}
###### Definition

For $A \to B$ a [[homomorphism]] of [[rings]], say that it is an **[[ind-étale morphism]]** if that $A$-[[associative algebra|algebra]] $B$ is a [[filtered colimit]] of $A$-[[étale algebras]].

=--

+-- {: .num_prop}
###### Proposition

Let $f \;\colon\; A \longrightarrow B$ be a [[homomorphism]] of [[rings]].

* If $f$ is ind-étale, def. \ref{IndEtale}, then it is weakly étale, def. \ref{WeaklyEtale}.

Almost conversely

* If $f$ is weakly étale, then there is a [[faithfully flat morphism]] $g \colon B \to C$ which is ind-étale such that the [[composition|composite]] $g\circ f$ is ind-étale.

=--

([Bhatt-Scholze 13, theorem 1.3](#BhattScholze13))

+-- {: .num_cor}
###### Corollary

The [[sheaf toposes]] over the [[sites]]  of weak étale morphisms and of [[pro-étale morphisms of schemes]] into  some base [[scheme]] are [[equivalence of categories|equivalent]], both define the  _[[pro-étale topos]]_ over the _[[pro-étale site]]_.

=--

## Related concepts

[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[pro-étale morphism of schemes|pro-étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]

## References

* {#GabberRamero03} [[Ofer Gabber]] and Lorenzo Ramero, _Almost ring theory_, volume 1800 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2003. ([arXiv:math/0201175](http://arxiv.org/abs/math/0201175))

* {#BhattScholze13} [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))

[[!redirects weakly étale morphisms]]

[[!redirects weakly etale morphism]]
[[!redirects weakly etale morphisms]]


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

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## 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

see at *[[Wedderburn-Artin theorem]]*


wedge

* [[wedge product]]

* [[wedge sum]]

* [[end#explicit_definition|wedge]] as in the definition of an end.

category: disambiguation

[[!redirects wedges]]


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

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## 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

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## 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

## 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), doi:10.1007/JHEP11(2023)208]

category: people


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

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{: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

#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

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#### Complex geometry
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#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

* [[weight (in representation theory)]]
* [[weighted limit]]
* [[weighted colimit]]

[[!redirects weights]]


weight (in representation theory)

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#### Representation theory
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[[!include representation theory - contents]]
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#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

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

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#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:

### 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 (doi:10.1016/0040-9383(95)93237-2, [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 (doi:10.1016/0040-9383(95)93237-2, [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

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#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:

> 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]]


+-- {: .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

+-- {: .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 (doi:10.1016/0040-9383(95)93237-2, [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

### 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) \[arXiv:2105.02871, [doi:10.1007/s11005-023-01725-4](https://doi.org/10.1007/s11005-023-01725-4)\]

* {#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* \[arXiv:2212.13668\]

* [[Herman Verlinde]], *Double-scaled SYK, Chords and de Sitter Gravity* \[arXiv:2402.00635\]

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), doi:10.1007/JHEP11(2022)060$]$

* [[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* \[arXiv:2302.02019\]

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

+-- {: .rightHandSide}
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###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

+-- {: .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, Graph (discrete mathematics)

[[!redirects weighted graph]]
[[!redirects weighted graphs]]


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."