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asymptotic boundary | https://ncatlab.org/nlab/source/asymptotic+boundary |
## Related concepts
* [[boundary]], [[manifold with boundary]]
* [[asymptotic isometry]]
* [[anti de Sitter spacetime]]
* [[black brane]], [[near horizon geometry]]
* [[AdS-CFT duality]]
* [[singleton representation]]
[[!redirects asymptotic boundaries]] |
asymptotic C-star-homomorphism | https://ncatlab.org/nlab/source/asymptotic+C-star-homomorphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[noncommutative topology]] the standard notion of [[homomorphism]] of [[C*-algebras]] is too restrictive for some applications, related to the fact that some noncommutative $C^\ast$-algebras correspond to "locally badly behaved" noncommutative topological spaces. The notion of _asymptotic $C^\ast$-homomorphism_ is more flexible than that of plain $C^\ast$-homomorphisms and designed to correct this problem. Homotopy classes of asymptotic $C^\ast$-homomorphisms are the [[hom-sets]] in a [[category]] called _[[E-theory]]_. See there for more details.
## Definition
+-- {: .num_defn #AsymptoticHomomorphism}
###### Definition
For $A,B$ two [[C*-algebras]], an **asymptotic** homomorphism between them is a $[0,\infty)$-parameterized collection of [[continuous functions]] $\{\phi_t \colon A \to B\}_{t \in [0, \infty)}$, such that
* for each $a \in A$, the [[function]] $t \mapsto \phi_t(a)$ is a [[continuous function]];
* in the [[limit of a sequence|limit]] $t \to \infty$, $\phi_t$ becomes a [[star-algebra]] [[homomorphism]].
=--
As for ordinary $C^\ast$-algebra homomorphisms one puts:
+-- {: .num_defn #AsymptoticHomotopy}
###### Definition
For $f_t, g_t \colon A \to B $ to asymptotic $C^\ast$-homomorphisms, def. \ref{AsymptoticHomomorphism}, a (right) [[homotopy]] between them is an asyptotic homomorphism $\eta_t \colon A \to C([0,1],B)$ which restricts to $f$ at 0 and to $g$ at $1$, hence such that it fits into a [[commuting diagram]] of the form
$$
\array{
&& B
\\
& {}^{\mathllap{f_t}}\nearrow & \uparrow^{\mathrlap{ev_0}}
\\
A &\stackrel{\eta_t}{\to}& C([0,1], B)
\\
& {}_{\mathllap{g_t}}\searrow & \downarrow^{\mathrlap{ev_1}}
\\
&& B
}
\,.
$$
Homotopy of asymptotic $C^\ast$-homomorphisms is clearly an [[equivalence relation]]. Write $[A,B]$ for the [[set]] of homotopy-[[equivalence classes]] of asymptotic homomorphisms $A \to B$.
=--
+-- {: .num_prop}
###### Proposition
For $A,B \in $ [[C*Alg]], the set $[A, C_0((0,1), B)]$ is naturally an [[abelian group]] under the composition operation which sends the homotopy classes presented by $f,g \colon A \times (0,1) \to B$ to the homotopy class of
$$
f + g \;\colon\; A \times (0,1) \stackrel{\cdot 2}{\to} A \times (0,2) \stackrel{x \mapsto \left\{ \array{f(x) & x \lt 1 \\ g(x-1) & x \gt 1 } \right. }{\to} B
\,.
$$
=--
+-- {: .num_remark}
###### Remark
The $t$-wise composition of two asymptotic $C^\ast$-homomorphisms is not in general itself an asymptotic $C^\ast$-homomorphims. However, every asympotic homomorphism is homotopic to one which is an [[equicontinuous function]], and $t$-wise composition of equicontinuous asymptotic $C^\ast$-homomorphisms is again an asymptotic homomorphism.
=--
## Examples
+-- {: .num_example }
###### Example
Two asymptotic $C^\ast$-homomorphisms which differe just by a reparameterization of $[0,\infty)$ while having the same [[limit of a sequence|limit]] can be related by a homotopy, def. \ref{AsymptoticHomotopy}.
=--
+-- {: .num_example }
###### Example
**(self-adjointification homotopy)**
For $f_t \colon A \to B $ an asymptotic $C^\ast$-homomorphism, there is a homotopy to the asymptotic morphism
$$
\tilde f_t(a) \coloneqq \tfrac{1}{2}\left(f(a) + f(a^\ast)^\ast\right)
\,.
$$
=--
## References
The notion was introduced in
* [[Alain Connes]], [[Nigel Higson]], _Déformations, morphismes asymptotiques et $K$-théorie bivariante_, C. R. Acad. Sci. Paris Sér. I Math. __311__ (1990), no. 2, 101–106, [MR91m:46114](http://www.ams.org/mathscinet-getitem?mr=1065438), [pdf](ftp://ftp.bnf.fr/578/N5781521_PDF_107_112DM.pdf)
{#ConnesHigson90}
A review is for instance around p. 23 of
* _Introduction to KK-theory and E-theory_, Lecture notes (Lisbon 2009) ([pdf slides](http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf))
{#Introduction}
[[!redirects asymptotic C-star-homomorphisms]]
[[!redirects asymptotic C*-homomorphism]]
[[!redirects asymptotic C*-homomorphisms]]
|
asymptotic dimension | https://ncatlab.org/nlab/source/asymptotic+dimension | Related $n$Lab entries: [[dimension]], [[coarse topology]]
* [[Mikhail Gromov]], _Hyperbolic groups_, Essays in group theory, Math. Sci. Res. Inst. Publ. __8__, Springer, New York, 1987, pp. 75–263. MR MR919829 (89e:20070) 61; _Asymptotic invariants of infinite groups_, in: Geometric group theory, vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser. __182__, Cambridge Univ. Press 1993, pp. 1–295. MR1253544 (95m:20041)
* G. Bell, A. Dranishnikov, _Asymptotic dimension_, [math.GR/0703766](http://arxiv.org/abs/math/0703766)
> The asymptotic dimension theory was founded by Gromov in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and applications to the theory of discrete groups.
* B. Grave, _Asymptotic dimension of coarse spaces_, New York J. Math. 12 (2006) 249–256
category: topology |
asymptotic expansion | https://ncatlab.org/nlab/source/asymptotic+expansion |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Formal geometry
+--{: .hide}
[[!include formal geometry -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
### General
An _asymptotic expansion_ of a [[function]] is a [[formal power series]] that may not [[convergence|converge]], but whose terms decrease fast enough such that the truncation of the series at any finite order still provides a controlled approximation to a given [[function]].
A key class of examples of asymptotic expansions are the [[Taylor series]] of [[smooth functions]] (example \ref{TaylorSeriesOfSmoothFunctionIsAsymptoticSeries} below) around any point. Beware that by [[Borel's theorem]] this means that _every_ [[formal power series]] is the asymptotic expansion of _some_ smooth function and of more than one smooth function (remark \ref{TheoremBorel} below).
In [[resurgence theory]] one tries to re-identify from an asymptotic expansion the corresponding [[analytic function|non-analytic]] contributions.
### In perturbative quantum field theory
The concept of asymptotic expansions plays a key role in the interpretation of [[perturbative quantum field theory]] (pQFT): This computes [[quantum observables]] as [[formal power series]] (in the [[coupling constant]] and in [[Planck's constant]]) whose [[radius of convergence]] necessarily vanishes in cases of interest ([Dyson 52](#Dyson52)).
Nevertheless, for examples such as [[quantum electrodynamics]] and [[quantum chromodynamics]] as in the [[standard model of particle physics]], the truncation of these series to the first handful of [[loop orders]] happens to agree with [[experiment]] (such as at the [[LHC]] collider) to high precision (for [[QED]]) or at least good precision (for [[QCD]]). Therefore one interprets the [[scattering matrix]] in [[perturbative quantum field theory]] as an asymptotic expansion of what should be the true [[non-perturbative field theory|non-perturbative]] result.
With [[resurgence theory]] one may try to deduce from the [[Feynman perturbation series]] regarded as an asymptotic expansion the hidden [[non-perturbative effects]].
## Definition
+-- {: .num_defn}
###### Definition
Given a [[function]] $f \colon \mathbb{R} \to \mathbb{R}$, a [[formal power series]] $\sum_{n = 0}^\infty a_n x^n$ is an
**asymptotic expansion** of $f$ at $x = 0$ if for each $n \in \mathbb{N}$ the [[limit of a sequence|limit]] of the difference between $f$ and the [[sum]] of the first $n$ terms of the series divided by $x^n$ is zero as $x$ tends to 0:
$$
\underset{x \to 0}{\lim}
\left(
\frac{1}{x^n}
\left(
f(x) - \sum_{k = 0}^n a_k x^k
\right)
\right)
\;=\;
0
\,.
$$
=--
+-- {: .num_remark}
###### Remark
This definition makes no statement about the behaviour as $n \to \infty$. In particular an asymptotic expansion may have vanishing [[radius of convergence]] (and nevertheless provide useful approximate information).
=--
## Examples
{#Examples}
+-- {: .num_example #TaylorSeriesOfSmoothFunctionIsAsymptoticSeries}
###### Example
**([[Taylor series]] of [[smooth function]] is asymptotic series)**
The [[Taylor series]] of a [[smooth function]] $f \colon \mathbb{R} \to \mathbb{R}$ at any point is always an asymptotic expansion of $f$ around that point, regardless of whether its [[radius of convergence]] vanishes or not.
=--
+-- {: .proof}
###### Proof
This follows from the [[Hadamard lemma]], which says that for each $n \in \mathbb{N}$ and each expansion point $x_0 \in \mathbb{R}$ (which we may without restrict of generality assume to be $x_0 = 0$) there exists a smooth function $h_n \colon \mathbb{R} \to \mathbb{R}$ such that
$$
f(x) = f(0) + x f^{(0)}(0) + \frac{1}{2} x^2 f^{(2)}(0)
+ \cdots +
\frac{1}{n!} x^n f^{n}(0)
+
\frac{1}{(n+1)!} x^{n+1} h_n(x)
\,,
$$
where $f^{(k)} \colon \mathbb{R} \to \mathbb{R}$ denotes the $k$th [[derivative]] of $f$.
Therefore with
$$
(a_k)_{k \in \mathbb{N}}
\coloneqq
\left(
\frac{1}{k!} f^{(k)}(0)
\right)_{k \in \mathbb{N}}
$$
the [[coefficients]] of the [[Taylor series]] of $f$ at $x_0 = 0$, we have
$$
\begin{aligned}
\underset{x \to 0}{\lim}
\left(
\frac{1}{x^n}
\left(
f(x) - \sum_{k = 0}^n a_k x^k
\right)
\right)
& =
\underset{x \to 0}{\lim}
\left(
\frac{1}{x^n}
\left(
\frac{1}{(n+1)!} x^{n+1} h_n(x)
\right)
\right)
\\
& =
\underset{x \to 0}{\lim}
\left(
x \frac{1}{(n+1)!} h_n(x)
\right)
\\
& =
0 \cdot \frac{1}{(n+1)!} h_n(0)
\\
& = 0
\end{aligned}
\,.
$$
Here in taking the [[limit of a sequence|limit]] we used from [[Hadamard's lemma]] that $h_n(x)$ and hence also $x h_n(x)$ is a [[smooth function]], hence in particular a [[continuous function]], on all of $\mathbb{R}$, hence that its limit as $x \to 0$ is just the value of the function at $x = 0$.
=--
+-- {: .num_remark #TheoremBorel}
###### Remark
**([[Borel's theorem]])**
Beware that by [[Borel's theorem]], _every_ [[formal power series]] is the [[Taylor series]] of _some_ [[smooth function]], and of more than one smooth function; hence by example \ref{TaylorSeriesOfSmoothFunctionIsAsymptoticSeries} every formal power series is the asymptotic expansion of _some_ smooth function, and of more than one smooth function.
=--
## Properties
### Optimal truncation and superasymptotics
{#OptimalTruncation}
A rule-of-thumb for where to truncate an asymptotic series so that the resulting finite [[sum]] is as close as possible to the "actual" value is to truncate at the term that gives the _smallest_ contribution. This rule-of-thumb is called _optimal truncation_, or _superasymptotics_ ([Berry-Howls 90](#BerryHowls90), see [Berry 91](#Berry91)).
For some classes of asymptotic series there are [[proofs]] that "optimal truncation" indeed works, see the references [below](#ReferencesOptimalTruncation).
## History
From [Suslov 05](#Suslov05):
> Classical books on diagrammatic techniques $[$in [[perturbative quantum field theory]]$]$ describe the construction of [[Feynman diagram|diagram]] series as if they were well defined. However, almost all important [[Feynman perturbation series|perturbation series]] are hopelessly [[divergent series|divergent]] since they have zero [[radii of convergence]]. The first argument to this effect was [given by Dyson](#Dyson52) with regard to [[quantum electrodynamics]].
> $[$...$]$
> Even though Dyson's argument is unquestionable, it was hushed up or decried for many years: the scientific community was not ready to face the problem of the hopeless divergency of perturbation series.
> $[$...$]$
> The modern status of divergent series suggests that techniques for manipulating them should be included in a minimum syllabus for graduate students in theoretical physics. However, the theory of divergent series is almost unknown to physicists, because the corresponding parts of standard university courses in calculus date back to the mid-nineteenth century, when divergent series were virtually banished from mathematics.
## Related concepts
* [[series]], [[divergent series]]
* [[Taylor series]], [[Puiseux series]]
* [[asymptotic series]], [[transseries]]
* [[resummation]], [[regular summation method]], [[Borel summability]]
* [[perturbation series]], [[Stokes phenomenon]]
* [[Feynman perturbation series]]
## References
### General
An original article is
* G. Watson, _A theory of asymptotic series_, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character
Vol. 211, (1912), pp. 279-313 ([JSTOR](http://www.jstor.org/stable/91005))
Basic introductions include
* Joel Feldman, _Taylor series and asymptotic expansions_ lecture notes [pdf](http://www.math.ubc.ca/~feldman/m321/asymptotic.pdf)
* R. Shankar Subramanian, _An Introduction to Asymptotic Expansions_ ([pdf](https://web2.clarkson.edu/projects/subramanian/ch561/notes/Asymptotic%20Expansions.pdf))
* Richard Chapling, _Asymptotic methods_, 2016 ([pdf](https://rc476.user.srcf.net/asymptoticmethods/am_notes.pdf))
* [[Gerald Dunne]], _Introduction to Resurgence, Trans-series and
Non-perturbative Physics_, 2018 ([pdf](https://www.icts.res.in/sites/default/files/NUMSTRINGS2018-2018-01-27-Gerald-Dunne.pdf))
### Non-convergence of the Feynman perturbation series
The argument that the [[S-matrix]] formal power series in all [[perturbative quantum field theories]] of interest is necessarily divergent (and hence at best an asymptotic series) is due to
* {#Dyson52} [[Freeman Dyson]], _Divergence of perturbation theory in quantum electrodynamics_, Phys. Rev. 85, 631, 1952 ([spire](http://inspirehep.net/record/29799?ln=en))
made more precise in
* {#Lipatov77} [[Lev Lipatov]], _Divergence of the Perturbation Theory Series and the Quasiclassical Theory_, Sov.Phys.JETP 45 (1977) 216–223 ([pdf](http://jetp.ac.ru/cgi-bin/dn/e_045_02_0216.pdf))
recalled for instance in
* {#Suslov05} [[Igor Suslov]], section 1 of _Divergent perturbation series_, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 ([arXiv:hep-ph/0510142](https://arxiv.org/abs/hep-ph/0510142))
* Justin Bond, last section of _Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle_ ([pdf](https://mcgreevy.physics.ucsd.edu/s13/final-papers/2013S-215C-Bond-Justin.pdf))
* {#Strocchi13} [[Franco Strocchi]], §2.2 of: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]
* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], section 4.1 of _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026))
* Marco Serone, from 2:46 on in _A look at $\phi^4_2$ using perturbation theory_ ([recording](https://www.youtube.com/watch?v=J4nxvY1rOhI))
In the example of [[phi^4 theory]] this non-convergence of the perturbation series is discussed in
* {#Helling} Robert Helling, p. 4 of _Solving classical field equations_ ([pdf](http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf))
### Optimal truncation and superasymptotics
{#ReferencesOptimalTruncation}
Discussion of "optimal truncation" of asymptotic series and of "superasymptotics" includes the following:
* {#BerryHowls90} [[Michael Berry]], C. J. Howls, _Hyperasymptotics_, Proceedings: Mathematical and Physical Sciences Vol. 430, No. 1880 (Sep. 8, 1990), pp. 653-668 ([jstor:79960](https://www.jstor.org/stable/79960))
* {#Berry91} [[Michael Berry]], _Asymptotics, superasymptotics, hyperasymptotics_, in H. Segur, S. Tanveer, and H. Levine, (eds.) _Asymptotics Beyond All Orders_, Plenum, Amsterdam, 1991, pp. 1-14 ([doi:10.1007/978-1-4757-0435-8_1](https://doi.org/10.1007/978-1-4757-0435-8_1))
* O. Costin, M. D. Kruskal, _On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing_, Proc. R. Soc. Lond. A 455, 1931-1956 (1999) ([arXiv:math/0608410](https://arxiv.org/abs/math/0608410))
[[!redirects asymptotic expansions]]
[[!redirects asymptotic series]]
|
asymptotic freedom | https://ncatlab.org/nlab/source/asymptotic+freedom |
\tableofcontents
## Idea
(...)
## Related concepts
* [[Yang-Mills theory]], [[QCD]], [[lattice QCD]]
* [[confinement]], [[mass gap]]
* [[quark-gluon plasma]]
* [[Landau pole]]
## References
* Wikipedia, _[Asymptotic freedom](http://en.wikipedia.org/wiki/Asymptotic_freedom)_
* Chris Elliott, Brian Williams, Philsang Yoo, _Asymptotic freedom in the BV formalism_, J. Geom. Phys. __123__ (2018) 246-283 [arxiv/1702.05973](https://arxiv.org/abs/1702.05973) [doi](https://doi.org/10.1016/j.geomphys.2017.08.009)
|
asymptotic isometry | https://ncatlab.org/nlab/source/asymptotic+isometry |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Considering a class of [[diffeomorphisms]] which are not [[isometries]] but change metrics a tiny bit, with some [[asymptotic boundary]] conditions.
This is useful to obtain [[conformal symmetry]] algebras in some [[theory (physics)|physical theories]] of [[gravity]], esp. in 3D.
## Related concepts
* [[asymptotic boundary]]
## References
* J. Brown, [[Marc Henneaux]], _Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity_, Comm. Math. Phys. __104__ (1986) 207 (key paper!!) [euclid](http://projecteuclid.org/euclid.cmp/1104114999)
* Coussaert, [[Marc Henneaux]], P. van Driel, Class. Quant. Grav. __12__ (1995) 2961
...Brown-Henneaux boundary condition
* [[Marc Henneaux]], [[Claudio Teitelboim]], _Asymptotically anti-de Sitter spaces_, Comm. Math. Phys. __98__, Number 3 (1985), 391-424, [euclid](http://projecteuclid.org/euclid.cmp/1103942446)
New developments
* [[Marc Henneaux]], Soo-Jong Rey, _Nonlinear W(infinity) algebra as asymptotic symmetry of 3-dimensional higher spin Anti-de Sitter gravity_, [arxiv/1008.4579](http://arxiv.org/abs/1008.4579)
[[!redirects asymptotic symmetry]]
[[!redirects asymptotic symmetries]]
[[!redirects asymptotic isometries]] |
asymptotic notation | https://ncatlab.org/nlab/source/asymptotic+notation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--
\tableofcontents
## Definition
### Big O
Let $\mathbb{R}$ denote the [[set]] of [[real numbers]]. Then there is a function
$$O:(\mathbb{R} \to \mathbb{R}) \to \mathcal{P}(\mathbb{R} \to \mathbb{R})$$
from the set of [[endofunctions]] on the real numbers $\mathbb{R} \to \mathbb{R}$ to the set of [[subsets]] of [[endofunctions]] on the real numbers $\mathcal{P}(\mathbb{R} \to \mathbb{R})$, such that given a real-valued endofunction $g:\mathbb{R} \to \mathbb{R}$, an real-valued endofunction $f:\mathbb{R} \to \mathbb{R}$ is said to be in $O(g)$ if and only if there merely exists positive real numbers $c:\mathbb{R}^+$ and $n_0:\mathbb{R}^+$ such that for all positive real numbers $n:\mathbb{R}^+$, if $n \geq n_0$, then $\vert f(n) \vert \leq c \vert g(n) \vert$:
$$f \in O(g) \coloneqq \exists c:\mathbb{R}^+.\exists n_0:\mathbb{R}^+.\exists n:\mathbb{R}^+.(n \geq n_0) \implies (\vert f(n) \vert \leq c \vert g(n) \vert)$$
If one doesn't have [[power sets]] in the foundations, one would have to define $O(g)$ as a [[family]] of structural [[subsets]]: for each $g:\mathbb{R} \to \mathbb{R}$, a set $O(g)$ and an [[injection]] $i_O(g):O(g) \hookrightarrow (\mathbb{R} \to \mathbb{R})$. Then $f \in O(g)$ if $f$ is in the [[image]] of $i_O(g)$.
### Big Omega
Let $\mathbb{R}$ denote the [[set]] of [[real numbers]]. Then there is a function
$$\Omega:(\mathbb{R} \to \mathbb{R}) \to \mathcal{P}(\mathbb{R} \to \mathbb{R})$$
from the set of [[endofunctions]] on the real numbers $\mathbb{R} \to \mathbb{R}$ to the set of [[subsets]] of [[endofunctions]] on the real numbers $\mathcal{P}(\mathbb{R} \to \mathbb{R})$, such that given a real-valued endofunction $g:\mathbb{R} \to \mathbb{R}$, an real-valued endofunction $f:\mathbb{R} \to \mathbb{R}$ is said to be in $\Omega(g)$ if and only if there merely exists positive real numbers $c:\mathbb{R}^+$ and $n_0:\mathbb{R}^+$ such that for all positive real numbers $n:\mathbb{R}^+$, if $n \geq n_0$, then $\vert f(n) \vert \geq c \vert g(n) \vert$:
$$f \in \Omega(g) \coloneqq \exists c:\mathbb{R}^+.\exists n_0:\mathbb{R}^+.\exists n:\mathbb{R}^+.(n \geq n_0) \implies (\vert f(n) \vert \geq c \vert g(n) \vert)$$
If one doesn't have [[power sets]] in the foundations, one would have to define $\Omega(g)$ as a [[family]] of structural [[subsets]]: for each $g:\mathbb{R} \to \mathbb{R}$, a set $\Omega(g)$ and an [[injection]] $i_\Omega(g):\Omega(g) \hookrightarrow (\mathbb{R} \to \mathbb{R})$. Then $f \in \Omega(g)$ if $f$ is in the [[image]] of $i_\Omega(g)$.
### Big theta
Let $\mathbb{R}$ denote the [[set]] of [[real numbers]]. Then there is a function
$$\Theta:(\mathbb{R} \to \mathbb{R}) \to \mathcal{P}(\mathbb{R} \to \mathbb{R})$$
defined for all real-valued endofunctions $g:\mathbb{R} \to \mathbb{R}$ as the intersection of the subsets $O(g)$ and $\Omega(g)$
$$\Theta(g) \coloneqq O(g) \cap \Omega(g)$$
By the properties of [[power sets]], given a real-valued endofunction $f:\mathbb{R} \to \mathbb{R}$, $f$ is said to be in $\Theta(g)$ if it is in both $O(g)$ and $\Omega(g)$.
$$f \in \Theta(g) \coloneqq (f \in O(g)) \wedge (f \in \Omega(g))$$
If one doesn't have [[power sets]] in the foundations, one would have to define $\Theta(g)$ as a [[family]] of structural [[subsets]]: for each $g:\mathbb{R} \to \mathbb{R}$, a set $\Theta(g)$ and an [[injection]] $i_\Theta(g):\Theta(g) \hookrightarrow (\mathbb{R} \to \mathbb{R})$. Then $f \in \Theta(g)$ if $f$ is in the [[image]] of $i_\Theta(g)$.
## Examples
Let $p$ be a [[real polynomial function]] with [[degree]] $n$. Then $p \in \Theta(\lambda x:\mathbb{R}.x^n)$.
## Related concepts
* [[algorithm]]
* [[analytic number theory]]
## References
* Wikipedia, [Asymptotic notation](https://en.wikipedia.org/wiki/Asymptotic_notation)
[[!redirects big O]]
[[!redirects big omega]]
[[!redirects big Omega]]
[[!redirects big theta]]
[[!redirects big Theta]]
[[!redirects big O notation]]
[[!redirects big omega notation]]
[[!redirects big Omega notation]]
[[!redirects big theta notation]]
[[!redirects big Theta notation]]
[[!redirects asymptotic notation]]
[[!redirects Bachmann-Landau notation]] |
asymptotic representation theory | https://ncatlab.org/nlab/source/asymptotic+representation+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
*Asymptotic representation theory* is [[representation theory]] in the [[limit of a sequence|limit]] of large [[dimension]] of [[linear representation]] and/or of large [[groups]] being represented.
Much of asymptotic representation theory is concerned specifically with the [[symmetric group]] and studies asymptotics of shapes of [[Young diagrams]] and of numbers of [[Young tableaux]] under [[measures]] such as the [[Plancherel measure]] and/or the [[Schur-Weyl measure]].
## References
* [[Anatoly Vershik]], *Two lectures on the asymptotic representation theory and statistics of Young diagrams*, In: Vershik A.M., Yakubovich Y. (eds) *Asymptotic Combinatorics with Applications to Mathematical Physics* Lecture Notes in Mathematics, vol 1815. Springer 2003 ([doi:10.1007/3-540-44890-X_7](https://doi.org/10.1007/3-540-44890-X_7))
* [[G. Olshanski]], *Asymptotic representation theory*, Lecture notes 2009-2010 ([webpage](https://lpetrov.cc/art), [pdf 1](https://storage.lpetrov.cc/Olshanski_ART_course_1.pdf), [pdf 2](https://storage.lpetrov.cc/Olshanski_ART_course_2.pdf))
* Piotr Śniady, *Combinatorics of asymptotic representation theory*, European Congress of Mathematics 2012 ([arXiv:1203.6509](https://arxiv.org/abs/1203.6509))
|
asymptotic safety | https://ncatlab.org/nlab/source/asymptotic+safety | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the context of [[quantum field theory]] (QFT) the term _asymptotic safety_ ([Weinberg 79](#Weinstein79)) refers to the situation where a QFT may not be [[renormalization|renormalizable]] (is not defined on all scales by the choice of a [[finite number]] of [[coupling constants]]), but has the property that its [[renormalization group flow]] has a non-trivial [[non-perturbative effect|non-perturbative]] [[fixed point]] such that the subspace of the space of [[coupling constants]] whose RG-flow converges to this fixed point is [[finite number|finite]] [[dimension|dimensional]].
> One defines the "UV-critical hypersurface" as the set of all those points in the infinite-dimensional theory space which are "pulled" into the fixed point by the inverse RG flow: trajectories lying in this surface approach the fixed point for increasing momentum scales. General arguments and known examples suggest that the UV-critical hypersurface has a finite dimensionality. This dimensionality equals the number of (infrared-) relevant couplings, i.e. couplings which get attracted to the fixed point in the UV. The important point is that once the value of these (few) couplings are known at some scale all other (irrelevant) couplings are fixed by requiring an asymptotically safe theory, that is a trajectory which lies entirely in the UV-critical hypersurface. By this means we achieve that, first, the couplings are determined by a finite number of measurements rendering the theory predictive, and, second, the UV behavior is unproblematic without any unphysical divergences. ([Nink-Reuter 12, p. 2](#NinkReuter12))
A key example of a non-renormalizable QFT is [[Einstein gravity]], and there is speculation that it might be asymptotically safe, and that this might be the solution to the construction of [[quantum gravity]] ([Reuter 96](#Reuter96)).
## Issues
{#Issues}
Issues that the program of asymptotic safety of gravity is facing include the following:
### Theoretical issues
1. All existing computations that see hints for a UV-fixed point do so by first applying a drastic truncation to the space of couplings, and then checking only whether there is a UV-fixed point for the RG-flow in the remaining small subspace. It seems unclear to which extent these approximate considerations may be extrapolated.
1. Most existing computations consider only pure Einstein gravity without matter coupling. It seems unclear to which extent these results may be extrapolated to the situation where matter is taken account of (but see [Biemans-Platania-Saueressig 17](#BiemansPlataniaSaueressig17)).
1. {#DonoghueSummarizes} [Donoghue 19](#Donoghue19) claims technical problems with the handling of [[renormalization group flow]] in Newton's constant $G_N$ as currently practiced in the asymptotic safety program:
1. The "[[running coupling constant|running]]" $G_N(E)$ comes from power-divergent corrections that vanish in [[dimensional regularization]].
1. Computing different [[Minkowski spacetime]] processes gives different power-divergent corrections: there is no generic common $G_N(E)$.
1. There is no generic [[energy]] $E$: some processes depend on $s \sim + E^2$, others on $t \sim -E^2$ and (unlike in the usual log running) this makes a difference.
1. If one assumes (as is widely, but no generally believed) that [[Bekenstein-Hawking entropy]] seen in classical gravity is to correspond to a microscopic [[entropy]] of its quantum degrees of freedom, then the scaling of this entropy with area as opposed to volume contradicts the assumption that [[quantum gravity]] is a [[local field theory]] at small scales and higher energies ([Shomer 07, section IV](#Shomer07)) and hence then it contradicts the asymptotic safety of gravity.
Similarly, if one trusts the [[AdS/CFT correspondence]] then gravity is fundamentally not a local field theory, only its boundary [[CFT]] is, in contradiction with asymptotic safety of gravity ([Shomer 07, section IV](#Shomer07)). But see [FR22](#FR22).
### Experimental issues
The near criticality of the [[Higgs field]] [[vacuum]] (see there at _[Higgs mass and vacuum (in-)stability](Higgs+field#MassAndVacuumInstability)_) implies that the [[coefficient]] $\lambda$ of the quartic part of the Higgs potential is close to zero after [[renormalization group flow]] ("RGE") to around the [[Planck scale]] of about $10^{19}$ [[GeV]] (e.g. [BDGGSSS 13, p. 17-18](#BDGGSSS13)):
<img src="https://ncatlab.org/nlab/files/HiggsQuarticCoupling.png" width="400"/>
In fact also the [[beta function]] $\beta_\lambda$ of the quartic coupling $\lambda$ (i.e. its logarithmic [[derivative]] with respect to [[scale]]) is close to zero around the [[Planck scale]] of about $10^{19}$ [[GeV]] ([BDGGSSS 13, p. 18](#BDGGSSS13)):
<img src="https://ncatlab.org/nlab/files/HiggsQuarticBetaFunctionRelative.png" width="400"/>
Earlier it has been suggested that this reflects the principle of asymptotic safety ([Shaposhnikov-Wetterich 09](#ShaposhnikovWetterich09)). But this would mean that not only $\lambda$ and its [[beta-function|RGE-derivative]] $\beta_\lambda$ vanish around the [[Planck scale]], but that in fact all higher derivatives do, too (see e.g [Niedermaier 06, equation (1.5)](#Niedermaier06)) hence that $\beta_\lambda$ asymptotes to zero. But this does not seem to be the case; in ([BDGGSSS 13, p. 17-18](#BDGGSSS13)) it says:
> As shown in fig. 2 (upper right), the corresponding Higgs quartic [[beta-function]] vanishes at a [[scale]] of about $10^{17}$-$10^{18}$ [[GeV]]. In order to quantify the degree of cancellation in the β-function, we plot in fig. 2 (lower right) $\beta_\lambda$ in units of its pure [[top quark]] contribution. The vanishing of $\beta_\lambda$ looks more like an accidental cancellation between various large contributions, rather than an asymptotic approach to zero.
<img src="https://ncatlab.org/nlab/files/HiggsQuarticBetaFunction.png" width="700"/>
## References
The idea of asymptotic safety as such and as a cure for [[quantum gravity]] is due to
* {#Weinstein79} [[Steven Weinberg]], _Ultraviolet divergences in quantum theories of gravitation_, in "General Relativity: An Einstein centenary survey", ed. S. W. Hawking and W. Israel. Cambridge University Press. pp. 790–831 (1979) ([spire](https://inspirehep.net/record/159043/))
It gained new popularity with this result:
* {#Reuter96} [[Martin Reuter]], _Nonperturbative Evolution Equation for Quantum Gravity_, Phys.Rev. D57 (1998) 971-985 ([arXiv:hep-th/9605030](https://arxiv.org/abs/hep-th/9605030))
An attempt to conceptually explain why [[gravity]] might have a UV-fixed point is in this article:
* {#NinkReuter12} Andreas Nink, [[Martin Reuter]], _On quantum gravity, Asymptotic Safety, and paramagnetic dominance_, Int. J. Mod. Phys. D22 (2013) 1330008 ([arXiv:1212.4325](https://arxiv.org/abs/1212.4325))
Observation of a special role of spacetime dimension 2:
* {#LauscherReuter06} O. Lauscher, [[Martin Reuter]], _Asymptotic Safety in Quantum Einstein Gravity: nonperturbative renormalizability and fractal spacetime structure_, In: Fauser B., Tolksdorf J., Zeidler E. (eds.) )Quantum Gravity_ Birkhäuser Basel 2006 ([arXiv:hep-th/0511260](https://arxiv.org/abs/hep-th/0511260), [doi:10.1007/978-3-7643-7978-0_15](https://doi.org/10.1007/978-3-7643-7978-0_15))
and speculation of this being related to the [[Connes-Lott model]] (see [here](2-spectral+triple#Connes06OnRelationToStringVacua)):
* {#Connes06} [[Alain Connes]], p. 8 of _Noncommutative Geometry and the standard model with neutrino mixing_, JHEP0611:081,2006 ([arXiv:hep-th/0608226](http://arxiv.org/abs/hep-th/0608226))
Review:
* {#Niedermaier06} [[Max Niedermaier]], _The Asymptotic Safety Scenario in Quantum Gravity -- An Introduction_, Class.Quant.Grav.24:R171-230,2007 ([arXiv:gr-qc/0610018](https://arxiv.org/abs/gr-qc/0610018))
* [[Max Niedermaier]], [[Martin Reuter]], _The Asymptotic Safety Scenario in Quantum Gravity_, Living Reviews in Relativity December 2006, 9:5 [doi:10.12942/lrr-2006-5](http://link.springer.com/article/10.12942/lrr-2006-5)
* Riccardo Martini, Gian Paolo Vacca, Omar Zanusso, *Perturbative approaches to non-perturbative quantum gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.13910](https://arxiv.org/abs/2210.13910)]
* Astrid Eichhorn, Marc Schiffer, *Asymptotic safety of gravity with matter*, in: [[Handbook of Quantum Gravity]], Springer (2023) [[arXiv:2212.07456](https://arxiv.org/abs/2212.07456)]
* Alexander Bednyakov, Alfiia Mukhaeva, *Perturbative Asymptotic Safety and Its Phenomenological Applications* [[arXiv:2309.08258](https://arxiv.org/abs/2309.08258)]
* Jan M. Pawlowski, Manuel Reichert, *Quantum Gravity from dynamical metric fluctuations* in *[[Handbook of Quantum Gravity]]* [[arXiv:2309.10785](https://arxiv.org/abs/2309.10785)]
Critical review:
* {#Shomer07} Assaf Shomer, _A pedagogical explanation for the non-renormalizability of gravity_ ([arXiv:0709.3555](https://arxiv.org/abs/0709.3555))
* {#Donoghue19} [[John Donoghue]], _A Critique of the Asymptotic Safety Program_, Frontiers in Physics 8, 56 (2020) ([arXiv:1911.02967](https://arxiv.org/abs/1911.02967), [doi:10.3389/fphy.2020.00056](https://www.frontiersin.org/articles/10.3389/fphy.2020.00056/full))
See also:
* [[Roberto Percacci]], _Asymptotic Safety FAQs_ ([web](http://www.percacci.it/roberto/physics/as/faq.html))
* {#DonoghuePercacci21} [[John Donoghue]] vs. [[Roberto Percacci]], *Debate on Asymptotically Safe Quantum Gravity*, Perimeter Institute, April 2021 ([video](https://pirsa.org/21040021), [slides](https://pdf.pirsa.org/files/21040021.pdf))
* {#BiemansPlataniaSaueressig17} Jorn Biemans, Alessia Platania, Frank Saueressig, _Renormalization group fixed points of foliated gravity-matter systems_ ([arXiv:1702.06539](https://arxiv.org/abs/1702.06539))
The suggestion that asymptotic safety explains the observed [[mass]] of the [[Higgs particle]] is attributed to
* {#ShaposhnikovWetterich09} [[Mikhail Shaposhnikov]], [[Christof Wetterich]], _Asymptotic safety of gravity and the Higgs boson mass_, Phys. Lett. B 683 (2010) 196 [[arXiv:0912.0208](https://arxiv.org/abs/0912.0208)]
but see p. 17 of
* {#BDGGSSS13} Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, [[Gian Giudice]], Filippo Sala, Alberto Salvio, [[Alessandro Strumia]], section 7 of _Investigating the near-criticality of the Higgs boson_ ([arXiv:1307.3536](https://arxiv.org/abs/1307.3536))
and for more see at _[Higgs field -- Asymptotic safety?](https://ncatlab.org/nlab/show/Higgs%20field#AsymtoticSafetyOrNot)_
Application to [[scattering amplitudes]]:
* Jan H. Kwapisz, Krzysztof A. Meissner, _Asymptotic safety and quantum gravity amplitudes_ ([arXiv:2005.03559](https://arxiv.org/abs/2005.03559))
Unitarity, causality and stability in Asymptotic Safety:
* Alessia Platania, [[Christof Wetterich]], _Non-perturbative unitarity and fictitious ghosts in quantum gravity_ ([arXiv:2009.06637](https://arxiv.org/abs/2009.06637))
* Alessia Platania _Causality, unitarity and stability in quantum gravity: a non-perturbative perspective_ ([arXiv:2206.04072](https://arxiv.org/abs/2206.04072))
Review on early-universe cosmology in the context of asymptotic safety:
* Alessia Platania, *From renormalization group flows to cosmology*, Front.in Phys. 8 (2020) 188 [[arXiv:2003.13656](https://arxiv.org/abs/2003.13656)]
Review on [[black holes]] in the context of asymptotic safety:
* Alessia Platania, *Black Holes in Asymptotically Safe Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.04272](https://arxiv.org/abs/2302.04272)]
See also:
* Edoardo D'Angelo, *Asymptotic Safety in Lorentzian quantum gravity* [[arXiv:2310.20603](https://arxiv.org/abs/2310.20603)]
Relation to String Theory:
* Senarath de Alwis, et al. _Asymptotic safety, string theory and the weak gravity conjecture_ ([arXiv:1907.07894](https://arxiv.org/abs/1907.07894))
* Ivano Basile, Alessia Platania, _Cosmological α′-corrections from the functional renormalization group_ ([arXiv:2101.02226](https://arxiv.org/abs/2101.02226))
* Ivano Basile, Alessia Platania, _String tension between de Sitter vacua and curvature corrections_ ([arXiv:2103.06276](https://arxiv.org/abs/2103.06276))
* Ivano Basile, Alessia Platania, _Asymptotic Safety: Swampland or Wonderland?_ ([arXiv:2107.06897](https://arxiv.org/abs/2107.06897))
* Fei Gao, Masatoshi Yamada, _Determining holographic wave functions from Wilsonian renormalization group_ ([arXiv:2202.13699](https://arxiv.org/abs/2202.13699))
* Renata Ferrero, [[Martin Reuter]], *On the possibility of a novel (A)dS/CFT relationship emerging in Asymptotic Safety* [[arXiv:2205.12030](https://arxiv.org/abs/2205.12030)]
|
asymptotically flat spacetime | https://ncatlab.org/nlab/source/asymptotically+flat+spacetime |
#Contents#
* table of contents
{:toc}
## Idea
A [[spacetime]] is _asymptotically flat_ if far enough away from any point it looks like flat [[Minkowski spacetime]].
On asymptotically flat spacetimes a total gravitational [[mass]]/[[energy]] is well defined, the _[[ADM mass]]_. See also at _[[positive energy theorem]]_.
Regarding a field configuration of [[Einstein gravity]] as a [[Cartan connection]] for the inclusion of the [[Lorentz group]] into the [[Poincaré group]] (see at _[[first-order formulation of gravity]]_), then asymptotic flatness means that the [[Cartan geometry]] asymptotivally approaches its model [[Klein geometry]]. From this perspective there are various generalizations of asymptotic flatness:
for instance for [[anti de Sitter gravity]] it means that spacetime asymptootes to [[anti de Sitter spacetime]]. For more on this see at _[[AdS-CFT correspondence]]_.
## Related concepts
* [[conformally flat manifold]]
* [[conformal compactification]]
* [[ADM mass]]
## References
### Asymptotically Minkowski spacetimes
* Wikipedia, _[Asymptotically flat spacetimes](http://en.wikipedia.org/wiki/Asymptotically_flat_spacetime)_
* Jörg Frauendiener, _Conformal Infinity_ ([living reviews](http://relativity.livingreviews.org/Articles/lrr-2000-4/))
### Asymptotically AdS spacetimes
* [[Marc Henneaux]], [[Claudio Teitelboim]], _Asymptotically Anti-de Sitter Spaces_, Comm. Math. Phys. 98, 391-424 (1985) ([pdf](http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/HenneauxTeitelboim.pdf))
[[!redirects asymptotically flat spacetimes]]
[[!redirects asymptotically anti de Sitter spacetime]]
[[!redirects asymptotically anti de Sitter spacetimes]] |
asymptotics | https://ncatlab.org/nlab/source/asymptotics |
## References
* Wikipedia, _[Asymptotic analysis](http://en.wikipedia.org/wiki/Asymptotic_analysis)_
[[!redirects asymptotic]]
[[!redirects asymptotic analysis]]
|
asynchronous automaton | https://ncatlab.org/nlab/source/asynchronous+automaton | Asynchronous automata are a generalisation of both [[transition systems]] and [[Mazurkiewicz traces]]. Their study has influences other models for concurrency such as _transition systems with independence_ (also called _asynchronous transition systems_). The idea is to decorate transition systems with an independence relation (much as in (Mazurkiewicz) [[trace alphabets]]) between actions that allow one to distinguish true concurrency from mutual exclusion (i.e. _non-determinism_). Following the paper by Goubault and Mimram, we use a slight modification called _automata with concurrency relations_:
+--{: .un-definition}
##Definition##
An _automaton with concurrency relations_ $(S,i,E,Tran,I)$ consists of
* a [[transition system]] $(S,i,E,Tran)$, such that whenever $(s,a,s')$, and $(s,a,s'')$ are in $Tran$, then $s' = s''$;
and
* $I = \{I_s\mid s\in S\}$ is a family of irreflexive, symmetric binary relations, $I_s$ on $E$ such that whenever $a_1I_s a_2$ (with $a_1,a_2 \in E$), there exist transitions $(s,a_1,s_1)$, $(s,a_2,s_2)$, $(s_1,a_2,r)$, and $(s_2,a_1,r)$ in $Tran$.
=--
##References##
* [[Eric Goubault]] and [[Samuel Mimram]], [Formal Relationships Between Geometrical and Classical Models for Concurrency](http://fr.arxiv.org/abs/1004.2818)
[[!redirects asynchronous automaton]]
[[!redirects asynchronous automata]]
|
AT category | https://ncatlab.org/nlab/source/AT+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
[Freyd (1999)](#Freyd99) gave a sharp and clever description of the commonalities and differences between [[abelian categories]] and [[toposes]] (or, in the first place, between abelian categories and [[pretoposes]]), by introducing a finitely axiomatized theory of "AT categories". Thus abelian categories and pretoposes are examples of AT categories; underscoring how much these structures have in common, Freyd's theory hews so closely to both of these two basic examples that in fact every AT category splits cleanly as a [[product]] of a pretopos part and an abelian part.
Here, the properties held in common between abelian categories and pretoposes are all exactness conditions; in the AT set-up, the decisive difference between abelian categories and pretoposes is concentrated solely in the behavior of the [[initial object]] with respect to [[cartesian product]] (being in one case the cartesian monoidal unit, i.e. an element that is universally absorbed into any other, and in the other an absorbing element, i.e. an element that absorbs any other element into itself, aka a zero element or annihilating element).
There is no research we know of on AT categories. One might say that they were invented by Freyd really for the sole theoretical purpose of demonstrating the commonalities and differences between abelian categories and pretopos categories, not because AT categories are found in nature. Indeed, a "typical" AT category is something of a mythological creature with the head of an abelian lion and the body of a pretopos dragon, and serves no purpose as such, even though pure lions and pure dragons themselves serve a purpose. Some of the axioms, particular axiom "AE" below, seem expressly designed to fuse them together in ad hoc fashion.
## Definition
Freyd introduces some baseline assumptions on the categories in question:
* [[finitely complete category|Finite completeness]],
* Presence of an [[initial object]],
* Presence of [[pushout|pushouts]] for pairs of [[morphism|arrows]] one of which is [[monomorphism|monic]],
* Presence of pushouts for [[kernel pair]]s
Each of these assumptions obviously holds in any [[abelian category]] and in any [[pretopos]].
+-- {: .num_remark}
###### Remarks
If we introduce predicates to express these four baseline assumptions, such as a predicate formula $P(p, q; f, g)$ whose meaning is "$(p, q)$ is a pushout pair for pairs of arrows $(f, g)$", then the remaining AT axioms given below all have a simple logical structure: they can be expressed as [[Horn theory|Horn clauses]] in the language generated by the baseline predicates. The first 8 axioms are in fact universal Horn clauses in these predicates.
This might look like a side comment for logicians, but it might also signal that Freyd intends to invoke, if pressed for details of proofs, one of a battery of representation theorems, such as the representation theorem for regular categories whose precise formulation is in terms of Horn theories -- see [[Categories, Allegories]] for examples.
Furthermore, by choosing a *specific* initial object, a specific pullback for each pair of arrows with common target, etc., an AT category can be cast as a model for a certain [[essentially algebraic theory]]. In other words, such "algebraic AT categories" can be internalized in any finitely complete category.
=--
The crucial difference between "abelianness" and "toposness" is concentrated in the following definition:
\begin{definition}
\label{TypeATObjects}
Let $0$ denote the [[initial object]], and for any [[object]] $X$ let
* $\pi_1 \,\colon\, 0 \times X \longrightarrow 0$,
* $\pi_2 \,\colon\, 0 \times X \longrightarrow X$
denote the two product [[projections]].
We say $X$ is
1. of **type T** if $\pi_1$ is an isomorphism,
1. of **type A** if $\pi_2$ is an isomorphism.
\end{definition}
With this:
* A pretopos will turn out to be precisely an AT category in which every object is of type T.
* An abelian category will turn out to be an AT category where every object is of type A.
Here then are the AT exactness axioms. Again, each of them is satisfied in every abelian category and in every pretopos, and according to Freyd's thesis, any exactness condition satisfied in both classes of categories is a logical consequence of this set of axioms. Some of Freyd's remarks in his original posting are included in parentheses.
+-- {: .num_defn}
###### Definition
A category meeting the baseline assumptions above is an **AT category** if the following 8 axioms plus axiom "AE" are satisfied.
1. The category is an effective [[regular category]].
("Yes this can be stated as universal Horn conditions on pullbacks and the special pushouts mentioned above.")
1. The arrow $0 \to 1$ is monic. ("Note that it follows that all maps from $0$ are monic.")
1. If $i\colon A \to C$ is [[monomorphism|mono]], then any [[pushout]] square
$$\array{
A & \stackrel{i}{\to} & C \\
\downarrow & & \downarrow \\
B & \underset{j}{\to} & D
}$$
is also a [[pullback]], and $j$ is mono.
1. The functor $0 \times -$ preserves pushouts of kernel pairs, and pushouts of pairs of arrows one of which is monic.
1. If $f\colon B \to 0 \times C$ is [[epimorphism|epi]] and
$$\array{
A & \to & B \\
\downarrow & & \downarrow f \\
0 & \to & 0 \times C
}$$
is a [[pullback]], then it's also a [[pushout]].
1. In the [[full subcategory]] of type T objects (Def. \ref{TypeATObjects}), pushouts of pairs of morphisms, one of which is monic, are universal (i.e., stable under pullback).
1. Define a functor $T$ by the pushout diagram
$$\array{
0 \times X & \stackrel{\pi_1}{\to} & 0 \\
\pi_2 \downarrow & & \downarrow \\
X & \to & T X
}$$
Then $T$ preserves pullbacks.
1. Given a morphism $f\colon X \to Y$, if $T f$ and $0 \times f$ are isomorphisms, then so is $f$.
=--
This is the basic list of exactness assumptions which permit a sharp comparison between pretoposes and abelian categories. From this list alone one can prove that every AT category embeds faithfully in a product of a pretopos and an abelian category. To show that every AT category is _equivalent_ to such a product, Freyd appends the following axiom involving an existential Horn condition:
+-- {: .num_defn #AxiomAE}
###### Definition
**axiom (AE)**
For every object $X$, there is a map $\xi\colon T X \to X$ such that
$$0 \times X \stackrel{\pi_1}{\to} X \stackrel{\xi}{\leftarrow} T X$$
is a coproduct diagram, and the canonical map $X \to T X$ is a retraction of $\xi$.
=--
To bring actual toposes into the picture, Freyd adds some more axioms, but let's first take stock and see what this gives us so far.
## Basic consequences of the AT axioms
+-- {: .num_prop}
###### Proposition
For any $X$ the unique map $0 \to X$ is monic.
=--
+-- {: .proof}
###### Proof
By axiom 2, the map $0 \to 1$ is monic. The pullback-preserving functor $- \times X$ preserves monos, so
$$0 \times X \stackrel{! \times 1_X}{\to} 1 \times X \cong X$$
(which is just the second projection $\pi_2: 0 \times X \to X$) is monic. The unique map $0 \to 0 \times X$ is monic since it has a retraction $\pi_1: 0 \times X \to0$. The result follows.
=--
+-- {: .num_prop}
###### Proposition
Binary coproducts exist.
=--
+-- {: .proof}
###### Proof
Because we can take the pushout of a pair of monos $0 \to X$, $0 \to Y$.
=--
+-- {: .num_prop}
###### Proposition
Coproducts are disjoint.
=--
+-- {: .proof}
###### Proof
By axiom 3, the two coprojections $i_X \colon X \to X + Y$, $i_Y \colon Y \to X + Y$ are monic, and their pullback is initial.
=--
## Category of type A objects is abelian
In this section we show that the full subcategory of type A objects forms an abelian category.
+-- {: .num_lemma #tozero}
###### Lemma
An object is of type A if and only if there exists a map to $0$.
=--
+-- {: .proof}
###### Proof
If $X$ is of type A, then we clearly have $X \cong 0 \times X \stackrel{\pi_1}{\to} 0$. Conversely, suppose there exists $p: X \to 0$. Note that since $0 \to 1$ is monic, there is at most one map $Y \to 0$ for any object $Y$. It follows quickly that maps $Y \to X$ are in natural bijective correspondence with maps $Y \to X \times 0$ by composition with $\pi_1: X \times 0 \to X$, so that $\pi_1$ is an isomorphism by the Yoneda lemma.
=--
+-- {: .num_prop #coreflective}
###### Proposition
The full subcategory of type A objects is coreflective.
=--
+-- {: .proof}
###### Proof
By Lemma \ref{tozero}, the subcategory of type A objects in an AT category $C$ is equivalent to the subcategory $C/0 \hookrightarrow C/1 \simeq C$, which is the category of coalgebras for the functor $A(X) = 0 \times X$. Thus the category of type A objects is comonadic.
=--
+-- {: .num_lemma}
###### Lemma
Objects of type A are closed under binary products, finite coproducts, subobjects, and quotient objects (= cokernels of kernel pairs).
=--
+-- {: .proof}
###### Proof
This follows from the fact that the category of type A objects is the slice category $C/0$.
=--
Before we prove the next lemma, recall that in a category with zero objects, a **kernel** of an arrow $f: A \to B$ is an equalizer of the pair $f, 0: A \stackrel{\to}{\to} B$. This is the same as a pullback
$$\array{
K & \to & A \\
\downarrow & & \downarrow f \\
0 & \to & B
}$$
Cokernels are defined dually, and can be formulated dually as certain pushouts. Since pushouts along monos exist, we can take the cokernel of any mono, and in particular the cokernel of any kernel.
+--{: .num_lemma}
###### Lemma
In the category of type A objects, every mono is the kernel of its cokernel, and every epi is the cokernel of its kernel.
=--
+-- {: .proof}
###### Proof
Of course, $0$ is a zero object in the category of type A objects. If $i\colon A \to B$ is a mono in the category of type A objects, then the pushout
$$\array{
A & \stackrel{i}{\to} & B \\
\downarrow & & \downarrow \\
0 & \to & coker(i)
}$$
exists and is also a pullback, by axiom 3, and hence $i$ is the kernel of its cokernel. Now suppose $f: A \to C$ is an epi in the category of type A objects. Since $0 \times C \cong C$, we have an epi $f\colon A \to 0 \times C$. Then, by axiom 5, the pullback
$$\array{
ker(f) & \to & A \\
\downarrow & & \downarrow f \\
0 & \to & 0 \times C
}$$
is also a pushout, so $f$ is the cokernel of its kernel.
=--
+-- {: .num_theorem}
###### Theorem
The category of type A objects is an abelian category.
=--
+-- {: .proof}
###### Proof
Any category with zero objects, binary products and coproducts, and in which every mono is the kernel of its cokernel and every epi is the cokernel of its kernel, is in fact an abelian category. See Freyd-Scedrov, _Categories, Allegories_, 1.598 (p. 95).
=--
## Category of type T objects is a pretopos
Now we show that the full subcategory of type T objects is a pretopos. It is clear that $0 \times 0 \cong 0$ (since $0 \to 1$ is monic), so $0$ is a type T object.
+-- {: .num_lemma #exactlyone}
###### Lemma
If $A$ is type A and $T$ is type T, then there exists exactly one map $A \to T$. Type T objects are characterized by this property.
=--
+-- {: .proof}
###### Proof
There is exactly one morphism $A \to 0$. Hence morphisms $A \to T$ are in bijection with maps $A \to 0 \times T \cong 0$, of which there is exactly one. For the second statement, suppose that $X$ has the property that there is exactly one map $A \to X$ for each type A object. Such objects $X$ are closed under products, and $0$ is such an object; therefore $0 \times X$ is such an object. On the other hand, $0 \times X$ is of type A since it projects to $0$. Hence there is at most one morphism $0 \times X \to 0 \times X$, and it follows that $0 \times X \to 0 \to 0 \times X$ is the identity, so that $0 \times X \cong 0$. Thus $X$ is of type T.
=--
+-- {: .num_cor #strict}
###### Corollary
The initial object $0$ is [[strict initial object|strict]] in the category of type T objects.
=--
+-- {: .proof}
###### Proof
Given $T$ of type T and $T \to 0$, we know $T$ is type A, and therefore by Lemma \ref{exactlyone} there is exactly one map $T \to T$. Hence $T \to 0 \to T$ is the identity, and of course so is $0 \to T \to 0$. So $T$ is initial.
=--
+-- {: .num_cor #closed}
###### Corollary
The full subcategory of objects of type T is closed under products, coproducts, subobjects, and quotient objects.
=--
+-- {: .proof}
###### Proof
Closure under products and subobjects is immediate from Lemma \ref{exactlyone}. Closure under quotients and coproducts follows from axiom 4.
=--
+-- {: .num_theorem}
###### Theorem
The full subcategory of objects of type T is a pretopos.
=--
+-- {: .proof}
###### Proof
Corollary \ref{closed} gives finite completeness, coproducts, and quotients of kernel pairs. Axiom 6 gives that in the full subcategory of $T$ objects, pushouts along monos are stable under pullback, and the initial object is stable under pullback in the category of T objects, because it is strict by Corollary \ref{strict}. It follows that coproducts are universal in the category of T objects. They are also disjoint by an earlier result, so the category of T objects is extensive. It is also effective regular by axiom 1, hence a pretopos.
=--
## Splitting into type A and type T objects
In this section we show that every AT category $C$ is the product of the abelian category $C_A$ of type A objects and the pretopos $C_T$ of type T objects.
Let us first observe that the coreflector (see Proposition \ref{coreflective})
$$0 \times {-}\colon C \to C_A$$
is left exact (as all coreflectors are), and
* Preserves the initial object (by axiom 2),
* Preserves pushouts of kernel pairs, and pushouts of pairs of arrows one of which is monic (axiom 4),
* And therefore also preserves coproducts.
A functor between AT categories which is left exact and which preserves such classes of finite colimits may be called a **morphism of AT categories**. Hence the coreflector is a morphism of AT categories.
Next let us observe that the subcategory $C_T \hookrightarrow C$ is _reflective_; the reflector is the functor $T$ defined above by means of a suitable pushout. The reflector clearly preserves any colimits that exist, and preserves pullbacks (by axiom 7), so
$$T\colon C \to C_T$$
is also a morphism of AT categories. Therefore we have a morphism of AT categories
$$F = \langle 0 \times {-}, T \rangle\colon C \to C_A \times C_T$$
and now we wish to prove, under a suitable additional axiom (also satisfied by every abelian category and every pretopos), that this is an equivalence.
+-- {: .num_lemma}
###### Lemma
The functor $F$ is faithful.
=--
+-- {: .proof}
###### Proof
The functor $F$ is left exact and therefore preserves kernels. By axiom 8, $F$ reflects isomorphisms. It follows immediately from these two facts that $F$ is faithful.
=--
+-- {: .num_remark}
###### Remark
With this result, Freyd's "first task" is complete: any AT category may be faithfully represented in a product of a pretopos and an abelian category.
=--
At this point we bring in axiom **AE**, def. \ref{AxiomAE}, for its debut appearance. It may be helpful to consider that $T X$ is by definition the [[cokernel]] appearing in the exact sequence
$$0 \times X \stackrel{\pi_2}{\to} X \to coker(\pi_2)$$
and that axiom AE asserts that this exact sequence splits (with splitting $\xi: coker(\pi_2) \to X$), in such a way that $X$ is the coproduct of the end terms.
+-- {: .num_prop}
###### Proposition
The functor $F$ is full.
=--
+-- {: .proof}
Suppose given maps $f: 0 \times X \to 0 \times Y$, $g: T X \to T Y$, and contemplate the diagram
$$\array{
0 \times X & \stackrel{\pi_2}{\to} & X & \to & T X \\
f \Big\downarrow & & & & \Big\downarrow g \\
0 \times Y & \underset{\pi_2}{\to} & Y & \stackrel{\overset{\xi}{\leftarrow}}{\to} & T Y
}$$
Since $X = (0 \times X) + T X$, the two obvious arrows $0 \times X \to Y$, $T X \to Y$ combine to give an arrow $(f, g): X \to Y$. It is straightforward to check that $0 \times (f, g) = f$ (because the functor $0 \times -$ kills the type T summands) and that $T(f, g) = g$.
=--
+-- {: .num_theorem}
###### Theorem
The functor $F$ is an equivalence.
=--
+-- {: .proof}
###### Proof
All that is left to check is that $F$ is essentially surjective, but this is clear because given a type A object $X$ and a type T object $Y$, we have $0 \times (X + Y) \cong X$ (use axiom 4 plus the fact that $0 \times Y \cong 0$), and
$$T(X + Y) \cong T X + T Y \cong 0 + T Y \cong Y$$
which completes the proof.
=--
This completes Freyd's "second task".
## Topos-theoretic considerations
In order to beef up the type T objects to a topos, Freyd imposes some extra structure on top of the AT axioms. Since it is now the topos axioms that are in the ascendant, I will christen these categories _TA categories_.
Thus, a **TA category** is an AT category $C$ together with
* Functions $P, E\colon Ob(C) \to Ob(C)$,
* Functions $l, r\colon Ob(C) \to Mor(C)$, of the form $l X\colon E X \to P X$, $r X\colon E X \to X$,
* An operation $\Lambda$ which assigns to each pair of morphisms $f\colon R \to Y$, $g\colon R \to X$ (where $R$ and $Y$ are assumed to be type T), a morphism
$\Lambda(f, g)\colon Y \to P X$.
This data is to satisfy the following axioms:
1. $\langle l X, r X \rangle: E X \to P X \times X$ is monic and $P X$, $E X$ are of type T,
2. The composite of relations
$$Y \stackrel{\Lambda(f, g)}{\to} P X \stackrel{(l X)^{op}}{\to} E X \stackrel{r X}{\to} X$$
is equal to
$$Y \stackrel{f^{op}}{\to} R \stackrel{g}{\to} X,$$
3. If
$$\array{
R & \stackrel{g}{\to} & E X \\
f \downarrow & & \downarrow l X \\
Y & \underset{h}{\to} & T X
}$$
is a pullback and $R$ is of type T, then $\Lambda(f, (r X) \circ g) = h$.
Freyd: "Note that in the full category of type-T objects, $P$ yields
power-objects with $E$, $l$, $r$ naming the universal relations. (The third axiom provides the uniqueness condition.)"
And: "In any abelian category the only type T object is the zero
object, which forces $P = E = 0$ for abelian categories."
And finally: "It's routine that both $0 \times {-}$ and $T$ preserve the new structure. We can now remove the existential from AE. Define $\xi\colon T X \to X$ as the image of $r X$. The third task is finished with:
1. For every $X$, $0 \times X \to X \stackrel{\xi}{\leftarrow} T X$ is a coproduct diagram."
_This section is likely to be rewritten and cleaned up._
## References
* {#Freyd99} [[Peter Freyd]] in reply to [[Vaughan Pratt]], *Abelian-topos (AT) categories*, mailing list comments (1999) [[catlist:1999/atcat](http://www.mta.ca/~cat-dist/catlist/1999/atcat), [[Freyd_AT-categories.txt:file]]]
[[!redirects AT category]]
[[!redirects AT categories]]
[[!redirects AT-category]]
[[!redirects AT-categories]]
[[!redirects TA category]]
[[!redirects TA categories]]
[[!redirects TA-category]]
[[!redirects TA-categories]] |
Athanasios Chatzistavrakidis | https://ncatlab.org/nlab/source/Athanasios+Chatzistavrakidis | Athanasios Chatzistavrakidis is a Greek theoretical and mathematical physicist working at the Ruđer Bošković Institute in Croatia.
* [webpage](https://www.irb.hr/eng/Divisions/Division-of-Theoretical-Physics/Quantum-Gravity-and-Mathematical-Physics-Group/Employees/Athanasios-Chatzistavrakidis)
category: people |
Atihya-Bott-Shapiro orientation > history | https://ncatlab.org/nlab/source/Atihya-Bott-Shapiro+orientation+%3E+history |
see at _[[K-orientation]]_
|
Atish Dabholkar | https://ncatlab.org/nlab/source/Atish+Dabholkar |
* [webpage](https://en.wikipedia.org/wiki/Atish_Dabholkar)
## Selected writings
On [[orientifolds]] and [[discrete torsion]]:
* [[Atish Dabholkar]], [[Jaemo Park]], _Strings on Orientifolds_, Nucl. Phys. B477 (1996) 701-714 ([arXiv:hep-th/9604178](https://arxiv.org/abs/hep-th/9604178))
On [[orientifolds]] and [[duality in string theory]]:
* [[Atish Dabholkar]], _Lectures on Orientifolds and Duality_, In *Trieste 1997, High energy physics and cosmology* 128-191 ([arXiv:hep-th/9804208](https://arxiv.org/abs/hep-th/9804208), [spire:454332](https://inspirehep.net/literature/454332))
Introducing the [[generalized Scherk-Schwarz mechanism]]:
* [[Atish Dabholkar]], [[Chris Hull]], _Duality Twists, Orbifolds, and Fluxes_, JHEP 0309:054, 2003 ([arXiv:hep-th/0210209](https://arxiv.org/abs/hep-th/0210209))
category: people |
Atiyah 2-framing | https://ncatlab.org/nlab/source/Atiyah+2-framing |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cohomology
+-- {: .hide}
[[!include cohomology - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
In the general terminology of _$n$-[[framing]]_ then a _2-framing_ of a [[manifold]] $\Sigma$ of [[dimension]] $d \leq 2$ is a trivialization of $T \Sigma \oplus \mathbb{R}^{2-d}$.
In ([Atiyah 90](#Atiyah1990)) the term "2-framing" is instead used for a trivialization of the double of the tangent bundle of a 3-manifold. So this is a different concept, but it turns out to be closely related to the _3-framing_ (in the previous sense) of surfaces.
For $X$ a [[compact space|compact]], [[connected]], [[orientation|oriented]] 3-[[dimension|dimensional]] [[manifold]], write
$$
2 T X := T X \oplus T X
$$
for the [[direct sum of vector bundles|fiberwise direct sum]] of the [[tangent bundle]] with itself. Via the [[diagonal]] embedding
$$
SO(3) \to SO(3) \times SO(3) \hookrightarrow SO(6)
$$
this naturally induces a [[special orthogonal group|SO(6)]]-[[principal bundle]].
+-- {: .num_prop}
###### Proposition
The underlying $SO(6)$-principal bundle of $2 T X$ always admits a [[lift of structure group|lift]] to a [[spin group|spin(6)]]-[[principal bundle]].
=--
+-- {: .proof}
###### Proof
By the sum-rule for [[Stiefel-Whitney classes]] (see at [SW class -- Axiomatic definition](Stiefel-Whitney+class#AxiomaticDefinition)) we have that
$$
w_2(2 T X) = 2 w_0(T X) \cup w_2(T X) + w_1(T X) w_1(T X)
\,.
$$
Since $T X$ is assumed [[orientation|oriented]], $w_1(T X) = 0$ (since this is the [[obstruction]] to having an [[orientation]]). So $w_2(2 T X) = 0 \in H^2(X,\mathbb{Z}_2)$ and since this in turn is the further [[obstruction]] to having a [[spin structure]], this does exist.
=--
Therefore the following definition makes sense
+-- {: .num_defn}
###### Definition
A **2-framing** in the sense of ([Atiyah 90](#Atiyah1990)) on a [[compact space|compact]], [[connected]], [[orientation|oriented]] 3-[[dimension|dimensional]] [[manifold]] $X$ is the [[homotopy class]] of a trivializations of the [[spin-group]]-[[principal bundle]] underlying [[Whitney sum|twice]] its [[tangent bundle]].
=--
More in detail, we may also remember the [[groupoid]] of 2-framings and the smooth structure on collections of them:
+-- {: .num_defn}
###### Definition
The [[moduli stack]] $At2\mathbf{Frame}$ is the [[homotopy pullback]] in
$$
\array{
At2\mathbf{Frame} &\to& *
\\
\downarrow && \downarrow
\\
\mathbf{B}SO(3) &\stackrel{}{\to}&
\mathbf{B} Spin(6)
}
$$
in [[Smooth∞Grpd]].
=--
In terms of this a 2-framing on $X$ with [[orientation]] $\mathbf{o} \colon X \to \mathbf{B}SO(3)$ is a lift $\hat {\mathbf{o}}$ in
$$
\array{
&& At 2 \mathbf{Frame}
\\
& {}^{\mathllap{\hat {\mathbf{o}}}}\nearrow & \downarrow
\\
X &\stackrel{\mathbf{o}}{\to}& \mathbf{B}SO(3)
}
\,.
$$
## Properties
### Relation to bounding 4-manifolds
{#RelationToBounding4Manifolds}
In ([Atiyah](#Atiyah1990)) it is shown how a framing on a compact connected oriented 3-manifold $X$ is induced by a 4-manifold $Z$ with [[boundary]] $\partial Z \simeq X$. In fact, a framing is equivalently a choice of [[cobordism]] class of bounding 4-manifolds ([Kerler](#Kerler)).
Discussion of 2-framing entirely in terms of bounding 4-manifolds is for instance in ([Sawin](#Sawin)).
### Relation to String-structures
{#RelationToStringStructures}
By ([Atiyah 2.1](#Atiyah1990)) an Atiyah 2-framing of a 3-manifold $X$ is equivalently a
$p_1$-[[twisted differential c-structures|structure]], where $p_1$ is the first [[Pontryagin class]], hence is a homotopy class of a trivialization of
$$
p_1(X) \colon X \to B SO(3) \stackrel{p_1}{\to} K(\mathbb{Z},4)
\,.
$$
This perspective on Atiyah 2-framings is made explicit in ([Bunke-Naumann, section 2.3](#BunkeNaumann)). It is mentioned for instance also in ([Freed, page 6, slide 5](#Freed08)).
## Related concepts
* [[framed manifold]]
* [[p1-structure]], [[string structure]].
## References
The notion of "2-framing" in the sense of framing of the double of the tangent bundle is due to
* {#Atiyah1990} [[Michael Atiyah]], _On framings of 3-manifolds_ , Topology, Vol. 29, No 1, pp. 1-7 (1990) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/atiyahfr.pdf))
making explicit a structure which slightly implicit in the discussion of the [[perturbation theory|perturbative]] [[path integral]] [[quantization of 3d Chern-Simons theory]] in
* [[Edward Witten]], _Quantum field theory and the Jones Polynomial_ , Comm. Math. Phys. 121 (1989)
reviewed for instance in
* {#Young} M. B. Young, section 2 of _Chern-Simons theory, knots and moduli spaces of connections_ ([pdf](http://www.math.sunysb.edu/~myoung/CS.pdf))
(see [Atiyah, page 6](#Atiyah)). For more on the role of 2-framings in [[Chern-Simons theory]] see also
* [[Daniel Freed]], Robert Gompf, _Computer calculation of Witten's 3-Manifold invariant_, Commun. Math. Phys. 141,79-117 (1991) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/freedgompf.pdf))
* [[Gregor Masbaum]], section 2 of _Spin TQFT and the Birman-Craggs Homomorphism_, Tr. J. of Mathematics 19 (1995) [pdf](http://journals.tubitak.gov.tr/math/issues/mat-95-19-2/pp-189-199.pdf)
* {#Freed08} [[Daniel Freed]], _Remarks on Chern-Simons theory_ ([arXiv:0808.2507](http://arxiv.org/abs/0808.2507), [pdf slides](http://www.ma.utexas.edu/users/dafr/MSRI_25.pdf))
and for discussion in the context of the [[M2-brane]] from p. 7 on in
* [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]] II: Twisted $String$ and $String^c$-structures_ ([arXiv:1007.5419](http://arxiv.org/abs/1007.5419)).
The relation to $p_1$-structure is made explicit in
* {#BunkeNaumann} [[Ulrich Bunke]], [[Niko Naumann]], section 2.3 of _Secondary Invariants for String Bordism and tmf_, Bull. Sci. Math. 138 (2014), no. 8, 912–970 ([arXiv:0912.4875](http://arxiv.org/abs/0912.4875))
* C. Blanchet, N. Habegger, [[Gregor Masbaum]], [[Pierre Vogel]], _Topological quantum field theories derived from the Kauffman bracket_, Topology Vol 34, No. 4, pp. 883-927 (1995) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/bhmv.pdf))
More discussion in terms of bounding 4-manifolds is in
* {#Kerler}Thomas Kerler, _Bridged links and tangle presentations of cobordism categories_. Adv. Math., 141(2):207–281, (1999) ([arXiv:math/9806114](http://arxiv.org/abs/math/9806114))
* {#Sawin} Stephen F. Sawin, _Three-dimensional 2-framed TQFTS and surgery_ (2004) ([pdf](http://digitalcommons.fairfield.edu/cgi/viewcontent.cgi?article=1020&context=mathandcomputerscience-facultypubs))
and page 9 of
* Stephen Sawin, _Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras_ ([arXiv:math/9910106](http://arxiv.org/abs/math/9910106)).
and more discussion for 3-manifolds with boundary includes
* {#KerlerLyubashenko01} Thomas Kerler, [[Volodymyr Lyubashenko]], section 1.6.1 of _Non-semisimple topological quantum field theories for 3-manifolds with corners_, Lecture notes in mathematics 2001
See also
* [[Greg Kuperberg]], _[MO comment](http://mathoverflow.net/a/4389/381)_
[[!redirects 2-framing]]
[[!redirects 2-framings]]
|
Atiyah Lie algebroid | https://ncatlab.org/nlab/source/Atiyah+Lie+algebroid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##Idea
The **Atiyah Lie algebroid** associated to a $G$-[[principal bundle]] $P$ over $X$ is a [[Lie algebroid]] structure on the [[vector bundle]] $T P/ G$, the [[quotient]] of the [[tangent bundle]] of the total space $P$ by the canonical induced $G$-[[action]].
The [[Lie groupoid]] that the Atiyah Lie algebroid [[Lie integration|integrates to]] is the _[[Atiyah Lie groupoid]]_. See there for more background and discussion.
## Definition
Let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ and let $P \to X$ be a $G$-[[principal bundle]]:
the **Atiyah Lie algebroid sequence** of $P$ is a sequence of [[Lie algebroid]]s
$$
ad(P) \to at(P) \to T X
\,,
$$
where
* $ad(P) = P \times_G \mathfrak{g}$ is the [[adjoint bundle]] of Lie algebras, associated via the [[adjoint action]] of $G$ on its Lie algebra;
* $at(P) := (T P)/G$ is the **Atiyah Lie algebroid**
* $T X$ is the [[tangent Lie algebroid]] of $X$.
The [[Lie bracket]] on the sections of $at(P)$ is that inherited from the tangent Lie algebroid of $P$.
## Relation to connections
A splitting $\nabla_{flat} : T X \to at(P)$ of the Atiyah Lie algebroid sequence in the category of [[Lie algebroid]]s is precisely a flat [[connection on a bundle|connection on]] $P$.
To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of [[vector bundle]]s. In that case one finds the curvature of the connection precisely as the [[obstruction]] to having a splitting even in Lie algebroids.
One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely $L_\infty$-[[L-infinity-algebroid|algebroids]], in terms of an [[horizontal categorification]] of [[nonabelian Lie algebra cohomology]]:
## Atiyah class
The $Ext^1$-cohomology class corresponding to the Atiyah exact sequence (usually in a version for vector bundles/coherent sheaves) is the __Atiyah class__.
## Related concepts
* [[Courant Lie 2-algebroid]]
## References
* [[Michael Atiyah]], _Complex analytic connections in fibre bundles_, Trans. Amer. Math. Soc. 85 (1957), 181--207, [doi](http://dx.doi.org/10.2307/1992969),[MR0086359](http://www.ams.org/mathscinet-getitem?mr=0086359)
* Pietro Tortella, _Representations of Atiyah algebroids and logarithmic connections_, [arxiv/1505.04763](http://arxiv.org/abs/1505.04763)
A discussion with an emphasis on the relation to [[connection on a 2-bundle|2-connections]] and [[Lie 2-algebras]] is on the first pages of
* {#Stevenson06} [[Danny Stevenson]], *Lie 2-algebras and the geometry of gerbes*, Unni Namboodiri Lectures 2006 [[pdf](http://math.ucr.edu/home/baez/namboodiri/stevenson_maclane.pdf), [[Stevenson-Lie2AlgebrasAndGerbes.pdf:file]]]
For Atiyah classes see:
* [[Luc Illusie]], _Complexe cotangent et déformations_ (vol. 1) IV.2.3
* [MO:atiyah-class-for-non-locally-free-sheaf](http://mathoverflow.net/questions/56405/atiyah-class-for-non-locally-free-sheaf)
* [[M. Kapranov]], _Rozansky–Witten invariants via Atiyah classes_, Compositio Math. 115 (1999), 71–113.
* [[Ugo Bruzzo]], Igor Mencattini, [[Vladimir Rubtsov]],
_Nonabelian holomorphic Lie algebroid extensions_, Intern. J. Math. 26, No. 05, 1550040 (2015) [doi](https://doi.org/10.1142/S0129167X15500408) [arXiv:1305.2377](https://arxiv.org/abs/1305.2377)
* Zhuo Chen, Mathieu Stiénon, Ping Xu, _From Atiyah classes to homotopy Leibniz algebras_, [arXiv/1204.1075](http://arxiv.org/abs/1204.1075); _A Hopf algebra associated to a Lie pair_, [arxiv/1409.6803](http://arxiv.org/abs/1409.6803)
* R. A. Mehta, M. Stiénon, P. Xu, _The Atiyah class of a dg-vector bundle_, [arxiv/1502.03119](http://arxiv.org/abs/1502.03119)
* [[Nikita Markarian]], _The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem_, J. Lond. Math. Soc. (2) 79 (2009), no. 1, 129--143 [doi](https://doi.org/10.1112/jlms/jdn064)
* F. Bottacin, _Atiyah classes for Lie algebroids_, [pdf](http://www.math.unipd.it/~bottacin/papers/liealgebroids.pdf)
* Ajay C. Ramadoss, _The big Chern classes and the Chern character_, Internat. J. Math. 19 (2008), no. 6, 699--746.
* [[Zhuo Chen]], [[Mathieu Stiénon]], [[Ping Xu]], _From Atiyah classes to homotopy Leibniz algebras_, Commun. Math. Phys. __341__ (2016) 309-349 [[arXiv:1204.1075](https://arxiv.org/abs/1204.1075), [doi:10.1007/s00220-015-2494-6](https://doi.org/10.1007/s00220-015-2494-6)]
* Stack Project 92.19 ([tag/09DF](https://stacks.math.columbia.edu/tag/09DF)) The Atiyah class of a sheaf of modules
[[!redirects Atiyah Lie algebroids]]
[[!redirects Atiyah algebroid]]
[[!redirects Atiyah algebroids]]
[[!redirects Atiyah class]]
[[!redirects Atiyah sequence]] |
Atiyah Lie groupoid | https://ncatlab.org/nlab/source/Atiyah+Lie+groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **Atiyah Lie groupoid** $At(P)$ of a smooth $G$-[[principal bundle]] $P \to X$ is the [[Lie groupoid]] whose [[objects]] are the [[fibers]] of the bundle, and whose [[morphisms]] are the $G$-[[equivariant maps]] between the fibers.
Schematically:
$$
At(P)
\,=\,
\left\{
P_x \stackrel{\alpha}{\to} P_y
| x,y \in X
\right\}
\,.
$$
Its [[Lie algebroid]] is the [[Atiyah Lie algebroid]] $at(P)$ of $P$.
Both the Atiyah Lie groupoid and its Lie algebroid are used to characterize and are characterized by [[connection on a bundle|connections]] on $P$.
## Definition
As generally for every [[Lie algebroid]], there are different [[Lie groupoids]] [[Lie integration|integrating]] the [[Atiyah Lie algebroid]]. We describe two of them.
The Aityah Lie algebroid $at(P)$ of the [[principal bundle]] $P \to X$ comes canonically with a morphism $at(X) \to T X$ to the [[tangent Lie algebroid]].
The simplest [[Lie integration]] of the tangent Lie algebroid is the [[pair groupoid]] $X \times X$ of $X$. On the other hand, the universal integration is the [[fundamental groupoid]] $\Pi(X)$ (both coincide precisey if $X$ is a [[simply connected space]]).
Accordingly, there is a version of the Atiyah Lie groupoid over $X \times X$, and a richer version over $\Pi(X)$.
### Over the pair groupoid
For $G$ a Lie group and $p \colon P \to X$ a $G$-principal bundle, the **Atiyah groupoid** $At(P)$ -- also called the **gauge groupoid** or **transport groupoid** -- of $P$ is the [[Lie groupoid]] with
* the [[smooth manifold]] of [[objects]] is $Obj(At(P)) \coloneqq X$;
* the [[smooth manifold]] of [[morphisms]] $Mor(At(P)) = (P \times P)/G$, where the [[quotient object|quotient]] is taken with respect to the [[diagonal action]] of $G$ on $P \times P$;
* the [[source]]/[[target]] maps are those induced by the bundle projection $p$;
notice that a point $f \colon * \to (P \times P)/G$ over $(x_1 = s(f), x_2 = t(f))$, being an [[equivalence class]] of a pair $(s_1, s_2) \in P \times P$ is canonically identified with the unique $G$-equivariant [[function]] $f \colon P_{x_1} \to P_{x_2}$ which sends $s_1$ to $s_2$;
* [[composition]] is the given by ordinary composition of these functions.
### The integrated Atiyah sequence
The Atiyah groupoid sits in a sequence of groupoids
$$
Ad(P) \to At(P) \to Pair(X)
$$
where
* $Ad(P) = P \times_G G$ is the **adjoint bundle** of groups associated via the adjoint action of $G$ on itself; regarded as a smooth union $\coprod_{x \in X} \mathbf{B} P_x \times_G G$ of one-object groupoids coming from [[group]]s;
* $Pair(X) = (X \times X \rightrightarrows X)$ is the [[pair groupoid]] of $X$
* the [[functor]] $Ad(P) \to At(P)$ is the identity on objects and on morphisms given by the canonical identification $P_x \times_G G \stackrel{\simeq}{\to} (P_x \times P_x)/G$, where again we use the diagonal action of $G$ on $P_x \times P_x$.
* the functor $At(P) \to Pair(X)$ is the unique one that is the identity on objects.
Notice that a splitting (a [[section]])
$$
Pair(X) \to At(P)
$$
of the Atiyah groupoid is a trivialization of $P$. On the other hand, locally on [[contractible space|contractible]] $U \subset X$ we have $Pair(U) \simeq \Pi_1(U)$ with $U$ the [[fundamental groupoid]] of $U$, and a splitting $Pair(U) \simeq \Pi_1(U) \to At(P)|_U$ is still a trivialization over $U$ but indicates now that one may want to interpret it as giving rise to a flat [[connection]].
### Over a path groupoid
We have the sequence of surjective and [[full functor]]s of
[[path category|path categories]]
$$
P_1(X) \to \Pi_1(X) \to Pair(X)
$$
with $\Pi_1(X)$ the [[fundamental groupoid]] and $P_1(X)$ the smooth [[path groupoid]] and may refine the Atiyah groupoid by pulling back along these.
Write therefore $At'(P) := At(P) \times_{Pair(X)} \Pi_1(X)$ for the [[pullback]]
$$
\array{
At'(P) &\to& \Pi_1(X)
\\
\downarrow && \downarrow
\\
At(P) &\to& Pair(X)
}
\,.
$$
A splitting $\Pi_1(X) \to At'(P)$ of the top row is now precisely a flat [[connection]] on $P$.
If we pull back further to $A''$
$$
\array{
At''(P) &\to& P_1(X)
\\
\downarrow && \downarrow
\\
At'(P) &\to& \Pi_1(X)
\\
\downarrow && \downarrow
\\
At(P) &\to& Pair(X)
}
\,.
$$
then splittings of $P_1(X) \to At''(X)$ are precisely (not necessarily flat) connections on $P$.
All this is more well known in terms of the [[Lie algebroid]] underlying the Atiyah Lie groupoid, i.e. the **Atiyah Lie algebroid** sequence
$$
ad(P) \to at(P) \to T X
\,,
$$
where
* $ad(P) = P \times_g Lie(G)$ is the adjoint bundle of Lie algebras, associated via the adjoint action of $G$ on its Lie algebra;
* $at(P) = (T P)/G$ is the **Atiyah Lie algebroid**
* $T X$ is the [[tangent Lie algebroid]].
Indeed, a splitting $\nabla_{flat} : T X \to at(P)$ of this sequence in the category of [[Lie algebroid]]s is precisely again a flat connection on $P$ and integrates under [[Lie integration]] to the splitting of $At'(P)$ discussed above.
To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of vector bundles. In that case one finds the curvature of the connection precisely as the obstruction to having a splitting even in Lie algebroids.
One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely $L_\infty$-[[L-infinity-algebroid|algebroids]].
## Related concepts
* [[higher Atiyah groupoid]]
* [[Atiyah Lie groupoid]], [[Atiyah Lie algebroid]]
* [[Courant Lie 2-groupoid]], [[Courant Lie 2-algebroid]]
* [[quantomorphism group]], [[quantomorphism n-group]]
* [[Schauenburg bialgebroid]] (analogue for affine noncommutative principal bundles)
[[!include higher Atiyah groupoid - table]]
[[!redirects Atiyah Lie groupoids]]
[[!redirects Atiyah Lie-groupoid]]
[[!redirects Atiyah groupoid]]
[[!redirects Atiyah groupoids]]
|
Atiyah Lie-groupoid > history | https://ncatlab.org/nlab/source/Atiyah+Lie-groupoid+%3E+history | < [[Atiyah Lie-groupoid]]
[[!redirects Atiyah Lie-groupoid -- history]] |
Atiyah-Bott fixed point formula | https://ncatlab.org/nlab/source/Atiyah-Bott+fixed+point+formula | > under construction
#Contents#
* table of contents
{:toc}
## Related entries
* [[fixed point]]
* [[Lefschetz fixed point theorem]]
* [[Lawvere's fixed point theorem]]
* [[Kleene's fixed point theorem]]
* [[Brouwer's fixed point theorem]]
## References
* [Getzler:bott.pdf](http://www.math.northwestern.edu/~getzler/Papers/bott.pdf)
* [[Loring Tu]], bio web on R. Bott, node18:[Atiyah-Bott fixed point theorem](http://www.math.harvard.edu/history/bott/bottbio/node18.html)
[[!redirects Atiyah-Bott fixed point theorem]] |
Atiyah-Bott-Shapiro isomorphism | https://ncatlab.org/nlab/source/Atiyah-Bott-Shapiro+isomorphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Atiyah-Bott-Shapiro isomorphism_ (due to [Atiyah-Bott-Shapiro 63](#AtiyahBottShapiro)) is an [[isomorphism]] between the real/complex [[topological K-theory]] ring of the point in degree $q$ and the [[quotient]] of [[Clifford modules]] of rank $q$ by those that have an extension to Clifford modules of rank $q+1$.
The generalization of this to [[Clifford module bundles]] is the content of [[Karoubi K-theory]].
## Related concepts
* [[Karoubi K-theory]]
* [[K-orientation]]
## References
The orginal reference is
* {#AtiyahBottShapiro} [[Michael Atiyah]], [[Raoul Bott]], [[Arnold Shapiro]], _Clifford modules_, Topology 3(Suppl 1):3–38 (1963) ([pdf](http://dell5.ma.utexas.edu/users/dafr/Index/ABS.pdf))
The result is reviewed as theorem I 9.27 in
* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], chapter I, section 9 of _[[Spin geometry]]_, Princeton University Press (1989)
|
Atiyah-Segal completion theorem | https://ncatlab.org/nlab/source/Atiyah-Segal+completion+theorem |
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By the general discussion at _[[equivariant K-theory]]_, given a suitable [[topological group]] $G$ with an [[action]] on a [[topological space]] $X$ there is a canonical map
$$
K_G(X) \to K_G^{Bor}(X) \simeq K_G(X \times E G) \simeq K(X \!\sslash\! G)
$$
from the [[equivariant K-theory]] of $X$ to the ordinary [[topological K-theory]] of the [[homotopy quotient]] ([[Borel construction]]).
While this map is never an [[isomorphism]] unless $G$ is the [[trivial group]], the _Atiyah-Segal completion theorem_ says that this map exhibits $K(X//G)$ as the [[formal completion]] of the ring $K_G(X)$ at the [[augmentation ideal]] of the [[representation ring]] of $G$ (hence, regarded as a [[ring of functions]], the restriction to an [[infinitesimal neighbourhood]] of the base point).
See also at _[formal completion -- Examples -- Atiyah-Segal theorem](completion+of+a+ring#AtiyahSegalTheorem)_.
The analog stable for [[stable cohomotopy]] is the _[[Segal-Carlsson completion theorem]]_:
[[!include Segal completion -- table]]
## Consequences
In the case where $X=*$ (i.e. a point), we have that $K_G(*) \simeq R(G)$ and $K(*//G)=KU^0(B G)$, thus we conclude that
$$
KU^0(BG) \simeq R(G)\hat{_I}
$$
where $I$ is the [[augmentation ideal]] of the [[representation ring]] of $G$.
## Examples and applications
{#ExamplesAndApplications}
\begin{example}\label{ComplexTopologicalKTheoryOfBS1}
**(complex topological K-theory of $B S^1$)** \linebreak
The [[complex topological K-theory]] of the [[classifying space]]
$B S^1$ of the [[circle group]] is the [[power series ring]]:
$$
KU^0
\big(
B S^1
\big)
\;\simeq\;
\mathbb{Z}
[ [ c_1^{KU} ] ]
\,,
$$
where $c_1^{K U}$ is any [[complex oriented cohomology theory|complex orientation]] of [[KU]] (see [this Prop](complex+oriented+cohomology+theory#CohomologyRingOfBU1)).
On the other hand, the [[representation ring]] of the [[circle group]] is (see [this Example](representation+ring#RepresentationRingOfUnitaryGroups))
$$
R_{\mathbb{C}}(S^1) \;\simeq\; \mathbb{Z}[x, x^{-1}]
\,.
$$
Here $x$ is the class of the 1-dimensional [[irrep]] $U(1) \hookrightarrow \mathbb{C}$, the [[augmentation ideal]] is clearly generated by $c \coloneqq x - 1$. The corresponding completion is again
$$
\begin{aligned}
\big(
R_{\mathbb{C}}(S^1)
\big)\hat{_{(c)}}
&
\;\simeq\;
\mathbb{Z}[x, x^{-1}][ [ c ] ] / (c - x + 1)
\\
&
\;\simeq\;
\mathbb{Z}[ [ c ] ][ (1 + c)^{-1} ]
\\
&
\;\simeq\;
\mathbb{Z}[ [ c ] ]
\,,
\end{aligned}
$$
where the first step is [this example](completion+of+a+ring#CompletionOfNoetherianRingByPowerSeries) and the third step observes that $1 + c$ is invertible in the $c$-power series ring (by [this Prop.](power+series#InvertibleElements)).
(e.g. [Buchholtz 08, Sec. 8.2](#Buchholtz08), also [Math.SE:a/3282578](https://math.stackexchange.com/a/3282578/58526))
\end{example}
compare also [Greenlees 1994, p. 74](https://www.sciencedirect.com/science/article/pii/002240499490006X)
## Related theorems
* [[Baum-Connes conjecture]], [[Green-Julg theorem]]
* As explained in _[[Equivariant stable homotopy theory]]_, there is a related characterization of the [[K-homology]] of $B G$ as a [[localization]].
## References
The original articles:
* [[Michael Atiyah]], _Characters and cohomology of finite groups_, Publications Mathématiques de l'IHÉS, Volume 9 (1961) , p. 23-64 ([numdam:PMIHES_1961__9__23_0]( http://www.numdam.org/item?id=PMIHES_1961__9__23_0))
* [[Michael Atiyah]], [[Friedrich Hirzebruch]], _Vector bundle and homogeneous spaces_, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 ([[AtiyahHirzebruch61.pdf:file]])
* [[Michael Atiyah]], [[Graeme Segal]], _Equivariant $K$-theory and completion_, J. Differential Geom. Volume 3, Number 1-2 (1969), 1-18. ([euclid:jdg/1214428815](http://projecteuclid.org/euclid.jdg/1214428815))
Review and survey:
* {#Buchholtz08} [[Ulrik Buchholtz]], _The Atiyah-Segal completion theorem_, Master Thesis 2008 ([pdf](http://www.math.ku.dk/~jg/students/buchholtz.ms.2008.pdf), [[Buchholtz_AtiyahSegalCompletion.pdf:file]])
* Wikipedia, _[Atiyah-Segal completion theorem](http://en.wikipedia.org/wiki/Atiyah%E2%80%93Segal_completion_theorem)_
[[!redirects Atiyah-Segal completion]]
[[!redirects Atiyah-Segal completions]]
|
Atiyah-Singer index theorem | https://ncatlab.org/nlab/source/Atiyah-Singer+index+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Statement
[[Michael Atiyah]] and [[Isadore Singer]] associated two differently defined numbers to an [[elliptic operator]] on a manifold: the **[[topological index]]** and the **[[analytical index]]**. The index theorem asserts that the two are equal.
The index theorem generalizes earlier results such as the [[Riemann-Roch theorem]].
## References
### General
The original articles:
* [[Michael Atiyah]], [[Isadore Singer]], _The index of elliptic operators I_, Ann. of Math. (2) __87__ (1968) pp. 484–530; _III_, Ann. of Math. (2) __87__ (1968) pp. 546–604; _IV_, Ann. of Math. (2) __93__ (1971) pp. 119–138; _V_, Ann. of Math. (2) __93__ (1971) pp. 139–149
* [[Michael Atiyah]], [[Graeme Segal]], _The index of elliptic operators II_, Ann. of Math. (2) __87__ (1968) pp. 531–545
Review:
* {#HirzebruchGergerJung92} [[Friedrich Hirzebruch]], Thomas Berger, Rainer Jung, chapter 6 of: *Manifolds and Modular Forms*, Aspects of Mathematics **20**, Viehweg (1992), Springer (1994) [[doi:10.1007/978-3-663-10726-2](https://doi.org/10.1007/978-3-663-10726-2), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf)]
* {#Gilkey95} [[Peter Gilkey]], Section 5.2 of: _Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem_, 1995 ([pdf](https://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf))
* Rafe Mazzeo, _The Atiyah-Singer Index theorem: what it is and why you should care_, ([slides](https://docplayer.net/52573131-The-atiyah-singer-index-theorem-what-it-is-and-why-you-should-care-rafe-mazzeo-october-10-2002.html))
* [[Daniel Freed]], _The Atiyah-Singer index theorem_, ([slides](https://web.ma.utexas.edu/users/dafr/cmsa.pdf))
See also
* Wikipedia, _[Atiyah-Singer index theorem](http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem)_
A proof of the Atiyah-Singer index theorem in terms of [[KK-theory]]/[[E-theory]] has been given by [[Nigel Higson]], an account is in
* {#HigsonRoe} [[Nigel Higson]], [[John Roe]], _Lectures on operator K-theory and the Atiyah-Singer Index Theorem_ (2004) ([pdf](http://folk.uio.no/rognes/higson/Book.pdf))
A lightning review of the proof is on the last pages of
* {#Introduction} _Introduction to KK-theory and E-theory_, Lecture notes (Lisbon 2009) ([pdf slides](http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf))
### From the point of view of physics
{#ReferencesPhysics}
The index theorem has an interpretation in terms of the [[quantum field theory]] of the [[superparticle]] on the given space. See also at [supersymmetric quantum mechanics](https://ncatlab.org/nlab/show/supersymmetric+quantum+mechanics#RelationToIndexTheory).
Traditional physics arguments along these lines include for instance
* P. Windey, _Supersymmetric quantum mechanics and the Atiyah–Singer index theorem_, Acta Physica Polonica, __B15__ (1984). ([PDF](http://th-www.if.uj.edu.pl/~acta/vol15/pdf/v15p0435.pdf))
* [[Luis Alvarez-Gaumé]], _Supersymmetry and the Atiyah-Singer index theorem_, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. ([EUCLID](http://projecteuclid.org/euclid.cmp/1103940278))
* Florian Hanisch, Matthias Ludewig, _A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold_, ([arXiv:1709.10027](https://arxiv.org/abs/1709.10027))
[[!redirects Atiyah-Singer index theorems]]
|
Atiyah-Sutcliffe conjecture | https://ncatlab.org/nlab/source/Atiyah-Sutcliffe+conjecture |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
In the course of providing a [[geometry|geometric]] [[proof]] of the [[spin-statistics theorem]], [Berry & Robbins 1997](#BerryRobbins97) asked, at each [[natural number]] $n \in \mathbb{N}$, for a [[continuous function|continuous]] and [[symmetric group|$Sym(n)$]]-[[equivariant function]]
\[
\label{TheBerryRobbinsMap}
\underset{ {}^{\{1,\cdots, n\}} }{Conf}\big(\mathbb{R}^3\big)
\xrightarrow{\phantom{AAA}}
\mathrm{U}(n)/\big(\mathrm{U}(1)\big)^n
\]
* from the [[configuration space of points|configuration space of $n$ points]] (ordered and unlabeled) in [[Euclidean space]] $\mathbb{R}^3$
* to the [[coset space]] of the [[unitary group]] $\mathrm{U}(n)$ by its [[maximal torus]], hence the complete [[flag manifold]] of flags in $\mathbb{C}^n$,
both equipped with the evident [[group action]] by the [[symmetric group]] $Sym(n)$.
For the first non-empty case $n = 2$ this readily reduces to asking for a [[continuous map]] of the form $\mathbb{R}^3 \setminus \{0\} \xrightarrow{\;\;} \mathbb{C}P^1 \simeq S^2$ which is [[equivariant function|equivariant]] with respect to passage to antipodal points. This is immediately seen to be given by the radial projection.
But this special case turns out not to be representative of the general case, as this simple construction idea does not generalize to $n \gt 2$.
That a continuous and $Sym(n)$-equivariant Berry-Robbins map (eq:TheBerryRobbinsMap) indeed exists for all $n$ was proven in [Atiyah 2000](#Atiyah00).
In this article, [[Michael Atiyah|Atiyah]] turned attention to the stronger question asking for a function (eq:TheBerryRobbinsMap) which is [[smooth function|smooth]] and $Sym(n) \times $[[SO(3)|$SO(3)$]]-[[equivariant]] and provided an elegant proof strategy for this stronger statement, which however hinges on some [[conjecture|conjectural]] positivity properties of a certain [[determinant]] (discussed in more detail and with first numerical evidence in [Atiyah 2001](#Atiyah01)), interpreted as the [[electromagnetism|electrostatic]] [[energy]] of $n$-[[particles]] in $\mathbb{R}^3$.
Extensive numerical checks of this stronger but conjectural construction was recorded, up to $n \lt 30$ , in [Atiyah & Sutcliffe 2002](#AtiyahSutcliffe02), together with a refined formulation of the [[conjecture]], whence it came to be known as the *Atiyah-Sutcliffe conjecture*.
The Atiyah-Sutcliffe conjecture has been [[proof|proven]] for $n = 3$ in [Atiyah 2000](#Atiyah00)/[01](#Atiyah01) and for $n = 4$ by [Eastwood & Norbury 01](#EastwoodNorbury01).
## References
The origin of the question in investigation of the [[spin-statistics theorem]]:
* {#BerryRobbins97} [[Michael V. Berry]], [[Jonathan M. Robbins]], *Indistinguishability for quantum particles: spin, statistics and the geometric phase*, Proceedings of the Royal Society A **453** 1963 (1997) 1771-1790 ([doi:10.1098/rspa.1997.0096](https://doi.org/10.1098/rspa.1997.0096))
First form and first checks of the conjecture:
* {#Atiyah00} [[Michael F. Atiyah]], _The geometry of classical particles_, in Surveys in Differential Geometry, Surv. Differ. Geom. __7__, Int. Press 2000, 1–15 ([doi:10.4310/SDG.2002.v7.n1.a1](https://doi.org/10.4310/SDG.2002.v7.n1.a1))
* {#Atiyah01} [[Michael F. Atiyah]], _Configurations of points_, R. Soc. Lond. Philos. Trans. Ser. A, Math. Phys. Eng. Sci. 359 (2001) 1375–1387 ([doi:10.1098/rsta.2001.0840](https://doi.org/10.1098/rsta.2001.0840))
Generalization of the [[codomain]] to [[flag manifolds]] of other [[compact Lie groups]]:
* [[Michael F. Atiyah]], [[Roger Bielawski]], _Nahm’s equations, configuration spaces and flag manifolds_, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 157–176 ([math.RT/0110112](https://arxiv.org/abs/math.RT/0110112), [doi:10.1007/s005740200007](https://doi.org/10.1007/s005740200007))
> (using [[Nahm's equation]])
Full formulation of the Atiyah-Sutcliffe conjecture:
* {#AtiyahSutcliffe02} [[Michael F. Atiyah]], [[Paul M. Sutcliffe]], _The geometry of point particles_, Proc. Roy. Soc. London Ser. A 458 (2002), 1089–1115 ([hep-th/0105179](https://arxiv.org/abs/hep-th/0105179), [doi:10.1098/rspa.2001.0913](https://doi.org/10.1098/rspa.2001.0913))
Proof for $n = 4$:
* {#EastwoodNorbury01} [[Michael Eastwood]], [[Paul Norbury]], _A proof of Atiyah’s conjecture of four points in Euclidean three-space_, Geom. Topol. 5 (2001) 885–893 ([arXiv:math/0109161](https://arxiv.org/abs/math/0109161), [doi:10.2140/gt.2001.5.88510.2140/gt.2001.5.885](https://msp.org/gt/2001/5-2/p12.xhtml))
Further discussion:
* Dragutin Svrtan, Igor Urbiha, _Atiyah-Sutcliffe conjectures for almost collinear configurations and some new conjectures for symmetric functions_, ([math.AG/0406386](https://arxiv.org/abs/math/0406386))
* Dragutin Svrtan, Igor Urbiha, _Verification and strengthening of the Atiyah--Sutcliffe conjectures for several types of configurations_ ([math.MG/0609174](https://arxiv.org/abs/math/0609174))
* Marcin Mazur, Bogdan V. Petrenko, _On the conjectures of Atiyah and Sutcliffe_, Geom Dedicata **158** (2012) 329–342
([doi:10.1007/s10711-011-9636-6](https://doi.org/10.1007/s10711-011-9636-6), [arxiv:1102.4662](https://arxiv.org/abs/1102.4662))
* [[Joseph Malkoun]], *Root Systems and the Atiyah-Sutcliffe Problem*, Journal of Mathematical Physics 60, 101702 (2019) ([arXiv:1903.00325](https://arxiv.org/abs/1903.00325))
* [[Joseph Malkoun]], _The Atiyah-Sutcliffe determinant_ ([arXiv:1903.05957](https://arxiv.org/abs/1903.05957))
|
Atiyah–Hirzebruch spectral sequence | https://ncatlab.org/nlab/source/Atiyah%E2%80%93Hirzebruch+spectral+sequence | +-- {: .rightHandSide}
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###Context###
#### Homological algebra
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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Atiyah-Hirzebruch spectral sequence_ (AHSS) is a type of [[spectral sequence]] that generalizes the [[Serre spectral sequence]] from [[ordinary cohomology]] $H^\bullet$ to any [[generalized (Eilenberg-Steenrod) cohomology]] theory $E^\bullet$.
For any ([[finite homotopy type|finite]]) [[homotopy fiber sequence]]
$$
\array{
F &\longrightarrow& P
\\
&& \downarrow
\\
&& X
}
$$
then the corresponding $E$-Atiyah-Hirzebruch spectral sequence has on its second page the [[ordinary cohomology]] of $X$ with [[coefficients]] in the $E$-[[cohomology groups]] of the [[fiber]] and converges to the proper $E$-cohomology of the total space:
$$
E_2^{p,q} =
H^p(X,E^q(F))
\Rightarrow
E^{\bullet}(P)
\,.
$$
This is of interest already for $F \simeq \ast$, as then it expresses generalized cohomology in terms of ordinary cohomology with coefficients in the base cohomology ring.
+-- {: .num_remark #NoteOnTerminology}
###### Remark
**(note on terminology)**
Often the terminology "Atiyah-Hirzebruch spectral sequence" is taken to refer to only this case with $F = \ast$, while the general case is then referred to as "Serre spectral sequence for generalized cohomology" or similar. In ([Atiyah-Hirzebruch 61,p. 17](#AtiyahHirzebruch61)) the case $F = \ast$ is labeled "Theorem", while the general case, stated right after the theorem, is labeled "2.2 Remark".
The proof of the theorem that is given is very short, it just says that since topological K-theory satisfies the [exactness axiom](generalized+%28Eilenberg-Steenrod%29+cohomology#ExactnessUnreduced) of a generalized cohomology theory, it is immediate that the conditions for a spectral sequence stated as Axioms (SP.1)-(SP.5) in ([Cartan-Eilenberg 56, section XV.7](#CartanEilenberg56)) are met. Indeed Example 2 in ([Cartan-Eilenberg 56, section XV.7](#CartanEilenberg56)) observes that the spectral sequence in question exists for $E$ "some fixed [[cohomology theory]]" because "Axioms (SP.1)-(SP.4) are consequences of usual properties of cohomology groups".
In view of this, the contribution of ([Atiyah-Hirzebruch 61](#AtiyahHirzebruch61)) would not be so much the observation of what is now called the AHSS, rather than the proof that K-theory satisfies the axioms of a generalized cohomology theory. Indeed, according to ([Adams 74, p. 127-128, 215](#Adams74)), the AHSS was earlier observed by [[George Whitehead]] and "then became a folk-theorem" which was "eventually published by Atiyah and Hirzebruch".
Maybe it should be called the "Cartan-Eilenberg-Whitehead spectral sequence".
=--
There is a generalization to [[equivariant cohomology theory]] ([Davis-Lueck 98, theorem 4.7](#DavisLueck98)).
For [[genuine G-spectrum|genuine G-equivariance]], with [[RO(G)-grading]] for [[representation spheres]] $S^V$, then for $P \to X$ an $F$-fibration of [[topological G-spaces]] and for $A$ any $G$-[[Mackey functor]], the equivariant Serre spectral sequence looks like ([Kronholm 10, theorem 3.1](#Kronholm10)):
$$
E_2^{p,q} = H^p(X, H^{V+q}(F,A)) \,\Rightarrow\, H^{V+p+q}(P,A)
\,,
$$
where on the left in the $E_2$-page we have [[ordinary cohomology]] with [[coefficients]] in the genuine equivariant cohomology groups of the fiber.
## Construction
{#Construction}
### Statement
{#Statement}
+-- {: .num_prop}
###### Proposition
Let $A^\bullet$ be a an [additive](#UnreducedAdditivity) unreduced [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology functor]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#ReducedGeneralizedCohomologyHomotopyHomotopicalFunctor)). Let $B$ be a [[CW-complex]] and let $X \stackrel{\pi}{\to} B$ be a [[Serre fibration]], such that all its [[fibers]] are [[weakly contractible topological space|weakly contractible]] or such that $B$ is [[simply connected topological space|simply connected]]. In either case all [[fibers]] are identified with a typical fiber $F$ up to [[weak homotopy equivalence]] by connectedness ([this example](Introduction+to+Stable+homotopy+theory+--+P#FibersOfSerreFibrations)), and well-defined up to unique iso in the homotopy category by simply-connectedness:
$$
\array{
F &\longrightarrow& X
\\
&&
\Big\downarrow\mathrlap{{}^{\in Fib_{cl}}}
\\
&& B
}
\,.
$$
If at least one of the following two conditions is met
* $B$ is [[finite number|finite]]-dimensional as a [[CW-complex]];
* $A^\bullet(F)$ is bounded below in degree and the sequences $\cdots \to A^p(X_{n+1}) \to A^p(X_n) \to \cdots$ satisfy the [[Mittag-Leffler condition]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#MittagLefflerCondition)) for all $p$;
then there is a cohomology [[spectral sequence]], ([def.](Introduction+to+Stable+homotopy+theory+--+S#CohomologySpectralSequence)), whose $E_2$-page is the [[ordinary cohomology]] $H^\bullet(B,A^\bullet(F))$ of $B$ with [[coefficients]] in the $A$-[[cohomology groups]] $A^\bullet(F)$ of the fiber, and which converges to the $A$-cohomology groups of the total space
$$
E_2^{p,q}
=
H^p(B, A^q(F))
\;
\Rightarrow
\;
A^\bullet(X)
$$
with respect to the filtering given by
$$
F^p A^\bullet(X)
\coloneqq
ker\left(
A^\bullet(X)
\to
A^\bullet(X_{p-1})
\right)
\,,
$$
where $X_{p} \coloneqq \pi^{-1}(B_{p})$ is the fiber over the $p$th stage of the [[CW-complex]] $B = \underset{\longleftarrow}{\lim}_n B_n$.
Generally, without assumptions on the connectivity of $B$, there is a spectral sequence of this form with ordinary cohomology with coefficients in $A^\bullet(F)$ replaced by ordinary cohomology with [[local coefficients]] $(b \mapsto A^\bullet(F_b))$.
=--
### Construction by filtering the base space
{#ConstructionByFilteringTheBaseSpace}
The [following proof](#ProofOfAHSS) is the standard and original argument due to ([Atiyah-Hirzebruch 61, p. 17](#AtiyahHirzebruch61)).
+-- {: .proof #ProofOfAHSS}
###### Proof
The [exactness axiom](#ExactnessUnreduced) for $A$ gives an [[exact couple]], ([def.](Introduction+to+Stable+homotopy+theory+--+S#ExactCoupleAndDerivedExactCouple)), of the form
$$
\array{
\underset{s,t}{\prod} A^{s+t}(X_{s})
&&
\stackrel{}{\longrightarrow}
&&
\underset{s,t}{\prod} A^{s+t}(X_{s})
\\
& \nwarrow && \swarrow
\\
&& \underset{s,t}{\prod} A^{s+t}(X_{s}, X_{s-1})
}
\;\;\;\;\;\;\;
\left(
\array{
A^{s+t}(X_s) & \longrightarrow & A^{s+t}(X_{s-1})
\\
\uparrow && \downarrow_{\mathrlap{\delta}}
\\
A^{s+t}(X_s, X_{s-1}) && A^{s+t+1}(X_{s}, X_{s-1})
}
\right)
\,,
$$
where we take $X_{\gg 1} = X$ and $X_{\lt 0} = \emptyset$.
In order to determine the $E_2$-page, we analyze the $E_1$-page: By definition
$$
E_1^{s,t}
=
A^{s+t}(X_s, X_{s-1})
$$
Let $C(s)$ be the set of $s$-dimensional cells of $B$, and notice that for $\sigma \in C(s)$ then
$$
(\pi^{-1}(\sigma), \pi^{-1}(\partial \sigma)) \simeq (D^n, S^{n-1}) \times F_\sigma
\,,
$$
where $F_\sigma$ is [[weak homotopy equivalence|weakly homotopy equivalent]] to $F$ ([exmpl.](Introduction+to+Stable+homotopy+theory+--+P#FibersOfSerreFibrations)).
This implies that
$$
\begin{aligned}
E_1^{s,t}
& \coloneqq
A^{s+t}(X_s, X_{s-1})
\\
& \simeq \tilde A^{s+t}(X_s/X_{s-1})
\\
& \simeq \tilde A^{s+t}(\underset{\sigma \in C(n)}{\vee} S^s \wedge F_+)
\\
& \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} \tilde A^{s+t}(S^s \wedge F_+)
\\
& \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} \tilde A^t(F_+)
\\
& \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} A^t(F)
\\
& \simeq C^s_{cell}(B,A^t(F))
\end{aligned}
\,,
$$
where we used the relation to [[reduced cohomology]] $\tilde A$, ([prop.](Introduction+to+Stable+homotopy+theory+--+S#ReducedToUnreducedGeneralizedCohomology)) together with ([lemma](Introduction+to+Stable+homotopy+theory+--+S#EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient)),
then the [wedge axiom](Introduction+to+Stable+homotopy+theory+--+S#WedgeAxiom) and the [suspension isomorphism](Introduction+to+Stable+homotopy+theory+--+S#SuspensionIsomorphismForReducedGeneralizedCohomology) of the latter.
The last group $C^s_{cell}(B,A^t(F))$ appearing in this sequence of isomorphisms is that of [[cellular cohomology|cellular cochains]] ([def.](Introduction+to+Stable+homotopy+theory+--+I#CellularChainComplex)) of degree $s$ on $B$ with [[coefficients]] in the group $A^t(F)$.
Since [[cellular cohomology]] of a [[CW-complex]] agrees with its [[singular cohomology]] ([thm.](Introduction+to+Stable+homotopy+theory+--+I#CelluarEquivalentToSingularFromSpectralSequence)), hence with its [[ordinary cohomology]], to conclude that the $E_2$-page is as claimed, it is now sufficient to show that the differential $d_1$ coincides with the differential in the [[cellular cochain complex]] ([def.](Introduction+to+Stable+homotopy+theory+--+I#CellularChainComplex)).
We discuss this now for $\pi = id$, hence $X = B$ and $F = \ast$. (The general case works the same, just with various factors of $F$ replacing the point.)
Consider the following diagram, which [[commuting diagram|commutes]] due to the [[natural transformation|naturality]] of the [connecting homomorphism](#ConnectingHomomorphismOfUnreducedCohomology) $\delta$ of $A^\bullet$:
$$
\array{
\partial^\ast
\colon
& C^{s-1}_{cell}(X,A^t(\ast))
& =& \underset{i \in I_{s-1}}{\prod} A^t(\ast)
&& \longrightarrow &&
\underset{i \in I_s}{\prod} A^t(\ast)
& = &
C_{cell}^{s}(X,A^t(\ast))
\\
&& & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}}
\\
&& & \underset{i \in I_{s-1}}{\prod} \tilde A^{s+t-1}(S^{s-1})
&& &&
\underset{i \in I_s}{\prod} \tilde A^{s+t}(S^{s})
\\
&& & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}}
\\
&& d_1 \colon &
A^{s+t-1}(X_{s-1}, X_{s-2})
&\overset{}{\longrightarrow}&
A^{s+t-1}(X_{s-1})
&\overset{\delta}{\longrightarrow}&
A^{s+t}(X_s, X_{s-1})
\\
&& &
\downarrow && \downarrow && \downarrow
\\
&& &
A^{s+t-1}(S^{s-1}, \emptyset)
&\overset{}{\longrightarrow}&
A^{s+t-1}(S^{s-1})
&\overset{\delta}{\longrightarrow}&
A^{s+t}(D^s , S^{s-1})
}
\,.
$$
Here the bottom vertical morphisms are those induced from any chosen cell inclusion $(D^s , S^{s-1}) \hookrightarrow (X_s, X_{s-1})$.
The differential $d_1$ in the spectral sequence is the middle horizontal composite. From this the vertical isomorphisms give the top horizontal map. But the bottom horizontal map identifies this top horizontal morphism componentwise with the restriction to the boundary of cells. Hence the top horizontal morphism is indeed the coboundary operator $\partial^\ast$ for the [[cellular cohomology]] of $X$ with coefficients in $A^\bullet(\ast)$ ([def.](Introduction+to+Stable+homotopy+theory+--+I#CellularChainComplex)). This cellular cohomology coincides with [[singular cohomology]] of the [[CW-complex]] $X$ ([thm.](https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+I#CelluarEquivalentToSingularFromSpectralSequence)), hence computes the [[ordinary cohomology]] of $X$.
Now to see the convergence. If $B$ is finite dimensional then the convergence condition as stated in [this prop.](Introduction+to+Stable+homotopy+theory+--+S#CohomologicalSpectralSequenceOfAnExactCouple) is met. Alternatively, if $A^\bullet(F)$ is bounded below in degree, then by the above analysis the $E_1$-page has a horizontal line below which it vanishes. Accordingly the same is then true for all higher pages, by each of them being the cohomology of the previous page. Since the differentials go right and down, eventually they pass beneath this vanishing line and become 0. This is again the condition needed in the proof of [this prop.](Introduction+to+Stable+homotopy+theory+--+S#CohomologicalSpectralSequenceOfAnExactCouple) to obtain convergence.
By that proposition the convergence is to the [[inverse limit]]
$$
\underset{\longleftarrow}{\lim}
\left(
\cdots \stackrel{}{\to} A^\bullet(X_{s+1}) \longrightarrow A^\bullet(X_{s})
\to \cdots
\right)
\,.
$$
If $X$ is finite dimensional or more generally if the sequences that this limit is over satisfy the [[Mittag-Leffler condition]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#MittagLefflerCondition)), then this limit is $A^\bullet(X)$, by [this prop.](Introduction+to+Stable+homotopy+theory+--+S#Lim1VanihesUnderMittagLeffler).
=--
### Construction by filtering the coefficient spectrum
{#ConstructionByFilteringTheCoefficientSpectrum}
One also gets the Atiyah-Hirzebruch spectral sequence, up to isomorphism, by instead using the filtering given by the [[Postnikov tower]] of an [[Omega-spectrum]] [[Brown representability theorem|representing]] the given generalized cohomology theory.
This is due to ([Maunder 63, theorem 3.3](#Maunder63))
This alternative construction should be the one that is discussed in ([Shulman 13](#Shulman13)) from the perspective of [[homotopy type theory]].
[[!include Lurie spectral sequences -- table]]
## Properties
### Multiplicative structure
for $E$ a [[ring spectrum]], then the AHSS is multiplicative...
([MO discussion](http://mathoverflow.net/q/225579/381))
### Kronecker pairing
+-- {: .num_prop #KroneckerPairingOnAHSS}
###### Proposition
For $E$ a [[ring spectrum]] and $X$ a [[CW complex]] of finite [[dimension]], then the [[Kronecker pairing]] $\langle -,-\rangle \colon E^\bullet(X)\otimes E_\bullet(X)\to \pi_\bullet(E)$ passes to a page-wise pairing of the corresponding Atiyah-Hirzebruch spectral sequences for $E$-cohomology/homology
$$
\langle-,-\rangle_r
\;\colon\;
\mathcal{E}_r^{n,-s}
\otimes
\mathcal{E}^r_{n,t}
\longrightarrow
\pi_{s+t}(E)
$$
such that
1. on the $\mathcal{E}_2$-page this restricts to the Kronecker pairing for [[ordinary cohomology]]/[[ordinary homology]] with [[coefficients]] in $\pi_\bullet(E)$;
1. the [[differentials]] act as [[derivations]]
$$
\langle d_r(-),-\rangle = \langle -, d^r(-)\rangle
\,,
$$
1. The pairing on the $\mathcal{E}_\infty$-page is compatible with the Kronecker pairing.
=--
([Kochman 96, prop. 4.2.10](#Kochman96))
## Examples and Applications
### To complex oriented cohomology theory
...([Adams 74](#Adams74))....
### For topological K-theory
{#ForKTheory}
For $E = $ [[KU]], hence for [[topological K-theory]], the differential $d_3$ of the Atiyah-Hirzebruch spectral sequence with $F = \ast$ is given by a [[Steenrod square]] operation
$$
d_3 = Sq^3
$$
([Atiyah-Hirzebruch 61](#AtiyahHirzebruch61)).
For [[twisted K-theory]] this picks up in addition the [[cup product]] with the 3-class $H$ of the twist:
$$
d_3 = Sq^3 + H \cup(-)
$$
([Rosenberg 82](#Rosenberg82), [Atiyah-Segal 05 (4.1)](#AtiyahSegal05)). The higher differentials $d_5$ and $d_7$ here are given by higher [[Massey products]] with the twisting class ([Atiyah-Segal 05 sections 5-7](#AtiyahSegal05)).
### D-brane charges in string theory
{#ApplicationDBraneChargesInStringTheory}
In [[string theory]] [[D-brane charges]] are classes in $E = KU$-cohomology, i.e. in [[K-theory]]. The second page of of the corresponding Atiyah-Hirzebruch spectral sequence (see [above](#ForKTheory)) for $F = \ast$ hence expresses ordinary cohomology in all even or all odd degrees, and being in the kernel of all the differentials is hence the constraint on such ordinary cohomology data to lift to genuine K-theory classes, hence to genuine D-brane charges. In this way the Atiyah-Hirzebruch spectral sequences is used in ([Maldacena-Moore-Seiberg 01](#MaldacenaMooreSeiberg01), [Evslin-Sati 06](#EvslinSati06))
## References
### General
The statement of the existence of the spectral sequence first appears in print (with $E = $ [[topological K-theory]]) in
* {#AtiyahHirzebruch61} [[M. F. Atiyah]], [[F. Hirzebruch]], _Vector bundles and homogeneous spaces_, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7–38 American Mathematical Society, Providence, R.I. ([web](http://hirzebruch.mpim-bonn.mpg.de/87/), <a href="https://doi.org/10.1142/9789814401319_0008">doi:10.1142/9789814401319_0008</a>, [MR 0139181](http://www.ams.org/mathscinet-getitem?mr=0139181))
but the proof given consists essentially in pointing to section XV.7 ("a more general setting in which the theory of spectral sequences may be developed") of
* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, Princeton Univ. Press (1956)
This is by filtering over the stages of the base space CW-complex. That one gets an isomorphic spectral sequence by instead filtering over the [[Postnikov tower]] of any [[Omega-spectrum]] [[Brown representability theorem|representing]] the given [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] theory is due to
* {#Maunder63} [[C. R. F. Maunder]], *The spectral sequence of an extraordinary cohomology theory*, Mathematical Proceedings of the Cambridge Philosophical Society **59** 3 (1963) 567- 574 $[$[doi:10.1017/S0305004100037245](https://doi.org/10.1017/S0305004100037245)$]$
Early lecture notes:
* {#Adams74} [[Frank Adams]], part III section 7 of _[[Stable homotopy and generalised homology]]_, 1974
where the idea is attributed to [[George Whitehead]]:
> the Atiyah-Hirzebruch spectral sequence, which was really invented by G. W. Whitehead but not published by him ([Adams 74, p. 127-128](#Adams74))
> These spectral sequences were probably first invented by G. W. Whitehead, but he got them just after he wrote the paper $[$Whitehead 56 'Homotopy groups of joins and unions'$]$ in which they ought to have appeared. They then became a folk-theorem and were eventually published by Atiyah and Hirzebruch ([Adams 74, p. 215](#Adams74))
A more detailed account of the proof is in
* {#Kochman96} [[Stanley Kochman]], theorem 4.2.5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996
Further discussion of the case of [[twisted K-theory]]:
* {#Rosenberg82} [[Jonathan Rosenberg]], _Homological Invariants of Extensions of $C^\ast$-algebras_, Proc. Symp. Pure Math 38 (1982) 35.
* {#AtiyahSegal05} [[Michael Atiyah]], [[Graeme Segal]], _Twisted K-theory and cohomology_, Nankai Tracts Math. 11, World Sci. Publ., Hackensack, NJ, pp. 5–43, ([arXiv:math/0510674](http://arxiv.org/abs/math/0510674))
* [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin Schottenloher]], Section 21 of: _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1))
An analogous spectral sequence for higher [[twisted de Rham cohomology]] is discussed in [Li, Liu & Wang 2009](twisted+de+Rham+cohomology#LiLiuWang09).
A discussion of the $E$-AHSS as the spectral sequence of a tower induced by forming [[mapping spectra]] $[X,-]$ into the [[Postnikov tower]] is due to
* {#Shulman13} [[Mike Shulman]], _[[homotopytypetheory:spectral sequences]]_ (2013)
### Equivariant version
{#EquivariantVersion}
Discussion of the AHSS in [[Bredon cohomology|Bredon]] [[equivariant stable homotopy theory]]/[[equivariant cohomology]] includes
* {#DavisLueck98} [[James Davis]], [[Wolfgang Lueck]], _Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory_, K-Theory 15: 201–252, 1998 ([pdf](http://www.indiana.edu/~jfdavis/papers/assembly.pdf), [K-Theory archive](http://www.math.uiuc.edu/K-theory/0131/))
Discussion in genuine equivariant cohomology, i.e. including [[RO(G)-grading]], is in
* {#Kronholm10} [[William Kronholm]], _The $RO(G)$-graded Serre spectral sequence_, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. ([pdf](http://www.swarthmore.edu/NatSci/wkronho1/serre.pdf), [Euclid](https://projecteuclid.org/euclid.hha/1296223823))
### Examples
Application for the case of [[K-theory]] to [[D-brane charges]] in [[string theory]] is discussed in
* {#MaldacenaMooreSeiberg01} [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], _D-Brane Instantons and K-Theory Charges_, JHEP 0111:062,2001 ([arXiv:hep-th/0108100](http://arxiv.org/abs/hep-th/0108100))
* {#EvslinSati06} [[Jarah Evslin]], [[Hisham Sati]], _Can D-Branes Wrap Nonrepresentable Cycles?_, JHEP0610:050,2006 ([arXiv:hep-th/0607045](http://arxiv.org/abs/hep-th/0607045))
and detailed review of this is in
* [[Fabio Ruffino]], _Topics on topology and superstring theory_ ([arXiv:0910.4524](http://arxiv.org/abs/0910.4524))
Discussion for the $\mathbb{Z}/2$-equivariant [[KR cohomology theory]] (relevant for D-branes in [[orientifolds]]) includes
* [[Daniel Dugger]], _An Atiyah-Hirzebruch spectral sequence for KR-theory_ ([arXiv:math/0304099](http://arxiv.org/abs/math/0304099)
Discussion for the case of [[Morava K-theory]] and [[Morava E-theory]] with comments on application to charges of [[M-branes]] is in
* {#SatiWesterland11} [[Hisham Sati]], [[Craig Westerland]], _Twisted Morava K-theory and E-theory_ ([arXiv:1109.3867](http://arxiv.org/abs/1109.3867))
Application to [[motivic homotopy theory|motivic]] [[cobordism cohomology theory]] is discussed in
* Nobuaki Yagita, _Applications of the Atiyah-Hirzebruch spectral sequences for motivic cobordisms_ ([pdf](http://www.math.uiuc.edu/K-theory/0627/ah.pdf))
Application to the [[K-theory classification of topological phases of matter]]:
* [[Ken Shiozaki]], [[Masatoshi Sato]], [[Kiyonori Gomi]], *Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups*, Phys. Rev. B **106** (2022) 165103 [[arXiv:1802.06694](https://arxiv.org/abs/1802.06694), [doi:10.1103/PhysRevB.106.165103](https://doi.org/10.1103/PhysRevB.106.165103)]
[[!redirects Atiyah-Hirzebruch spectral sequence]]
[[!redirects Atiyah-Hirzebruch spectral sequences]]
[[!redirects Atiyah-Hirzeburch spectral sequence]]
|
atlas | https://ncatlab.org/nlab/source/atlas | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[general topology|basic topology]] and [[differential geometry]], by an _atlas_ of/for a [[topological manifold|topological]]-, [[differentiable manifold|differentiable]]- or [[smooth manifold]] $X$ one means a collection of [[coordinate charts]] $U_i \subset X$ which form an [[open cover]] of $X$.
If one considers here the [[disjoint union]] $\mathcal{U} \coloneqq \underset{i}{\sqcup} U_i$ of all the [[coordinate charts]], then the separate chart embeddings $U_i \subset X$ give rise to a single [[map]] ([[continuous function|continuous]]/[[differentiable function]])
$$
\mathcal{U} \longrightarrow X
$$
and now the condition for an atlas is that this is a [[surjective function|surjective]] [[étale map]]/[[local diffeomorphism]].
If, next, one regards this morphism, under the [[Yoneda embedding]], inside the [[topos]] of [[formal smooth sets]], then these conditions on an atlas say that this morphism is
1. an [[effective epimorphism]];
1. a [[formally étale morphism]].
In this abstract form the concept of an atlas generalizes to any [[cohesion|cohesive]] [[higher geometry]] ([KS 17, Def. 3.3](#KhavkineSchreiber17), [Wellen 18, Def 4.13](#Wellen18), [Sati & Schreiber 2020, p. 27](#SatiSchreiber20)).
Next, for a [[geometric stack]] $\mathcal{X}$, an atlas is a [[smooth manifold]] $\mathcal{U}$ (for [[differentiable stacks]]) or [[scheme]] $\mathcal{U}$ (for [[algebraic stacks]]) or similar, equipped with a morphism
$$
\mathcal{U} \longrightarrow \mathcal{X}
$$
that is an [[effective epimorphism]] and [[formally étale morphism]]
in the corresponding [[(infinity,1)-topos|higher topos]] (for instance in that of [[formal smooth infinity-groupoids]]).
Here the terminology has a bifurcation:
1. In the general context of [[geometric stacks]] one typically drops the second condition and calls any [[effective epimorphism]] from a [[smooth manifold]] or [[scheme]] to a [[differentiable stack]] or [[algebraic stack]], respectively, an _atlas_ (e.g. [Leman 10, 4.4](#Leman10)).
1. If in addition the condition is imposed that such an effective epimorphism exists which is also [[formally étale morphism|formally étale]], then the [[geometric stack]] is called an _[[orbifold]]_ or _[[Deligne-Mumford stack]]_ (often with various further conditions imposed).
From here, the terminology generalizes to [[infinity-stacks|$\infty$-stacks]] in general [[(infinity,1)-toposes|$\infty$-toposes]], see [this Remark](groupoid+objects+in+an+infinity1-topos+are+effective#InterpretationInTermsOfInfinityStacksWithAtlases) at *[[groupoid objects in an (∞,1)-topos are effective]]*.
{#AtlasForACategory} Yet more generally, the notion generalizes to [[2-topos theory]] and higher. Over the point this yields the notion of *[[category with an atlas]]* and *[[flagged categories]]* (depending on the truncation of the atlas) and relates to the notions of *flagged higher categories* ([Ayala & Francis 2018](category+with+an+atlas#AyalaFrancis18)).
## Related concepts
* [[category with an atlas]]
* [[n-types cover]]
## References
Review of the classical concept of atlases for geometric stacks:
* {#Leman10} [[Eugene Lerman]], Section 4.4 of: _Orbifolds as stacks?_, L'Enseign. Math. (2) 56 (2010), no. 3-4, 315--363 ([arXiv:0806.4160](https://arxiv.org/abs/0806.4160))
Formalization in [[cohesive homotopy theory]] and [[cohesive homotopy type theory|cohesive]]/[[modal homotopy type theory]]:
* {#KhavkineSchreiber17} [[Igor Khavkine]], [[Urs Schreiber]], _Synthetic geometry of differential equations_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))
* {#Wellen18} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))
* {#SatiSchreiber20} [[Hisham Sati]], [[Urs Schreiber]], p. 27 of: *[[schreiber:Proper Orbifold Cohomology]]* ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))
[[!redirects atlases]]
|
Atle Selberg | https://ncatlab.org/nlab/source/Atle+Selberg |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Atle_Selberg)
## related $n$Lab entries
* [[Selberg trace formula]]
* [[Selberg zeta function]]
* [[Rankin-Selberg method]]
category: people
[[!redirects Selberg]]
|
atom | https://ncatlab.org/nlab/source/atom |
> This article is on the mathematical concept of atom as used in the theory of [[preorder|preorders]], and related mathematical notions. For small projective objects in categories see at [[atomic object]]. For still other uses, see [[atom (disambiguation)]].
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Compact objects
+-- {: .hide}
[[!include compact object - contents]]
=--
=--
=--
# Atoms
* table of contents
{: toc}
## Idea
An atom in a [[poset]] is a [[minimal element]] among those which are not actually the [[bottom element|minimum]]. Thus an atom is as small as possible without being nothing. In an atomic poset, every element may be broken down (typically not uniquely) into atoms.
A related but slightly weaker concept is that of "tiny element", which has important generalizations in the context of enriched category theory.
## Definitions
Let $S$ be a [[poset]] (or [[proset]]) with a [[bottom element]] $\bot$. Recall that an element of $S$ is __[[positive element|positive]]__ if it is not a bottom element. An element $a$ of $S$ is __atomic__ if, given any element $p \leq a$, $p$ is positive iff $a \leq p$. An __atom__ of $S$ is simply an atomic element of $S$. Note that every atom must be positive (since $a \leq a$).
The atoms are precisely the [[minimal elements]] of the set of positive elements. For a poset, $a$ is atomic iff every $p \leq a$ is positive iff $p = a$. Using [[classical logic]] too, $a$ is atomic iff every $p \leq a$ satisfies $p = \bot$ [[xor]] $p = a$.
The p(r)oset $S$ is __atomic__ (or more commonly in the literature, **atomistic**; see remarks below) if every element is a [[supremum]] of atoms. In this case, every element $x$ is a supremum of those atoms $a \leq x$. Note that $\bot$ is a supremum of no atoms, and every atom is a supremum of itself, so the condition is really about the nontrivial nonatomic elements.
In [[constructive mathematics]], we require a more complicated definition of a [[positive element]], but the other definitions above remain correct (under the stated conditions), once we have that. In [[predicative mathematics|predicative]] constructive mathematics, positivity cannot be defined at all, and $S$ must come equipped with a [[positivity predicate]] before we may consider its atoms.
### Remarks on terminology
There is some terminological variance in the literature to the notion of atomic poset as defined here. In particular, [Wikipedia](http://en.wikipedia.org/wiki/Atom_%28order_theory%29) defines an _atomic poset_ to be a poset in which every positive element has an atom below it, and refers to our stronger notion of atomic poset by the term "atomistic poset". Note well that the Wikipedia conventions seem to be the ones observed in most lattice-theoretic texts.
"Atomic" and "Atomistic" differ for the simple example of the [[divisor lattice]] for some number $n$. The atoms in this lattice are prime numbers while it may also contain [[semi-atom|semi-atoms]] which are powers of primes. This lattice is atomic because any object not the [[bottom]], $1$, is divisible by a prime in the lattice. However it is not generally atomistic, but is instead uniquely semi-atomistic (every object is the product of a unique set of semi-atoms with bottom corresponding to the empty set), which is one way of stating the [[fundamental theorem of arithmetic]], also known as the *unique factorization theorem*.
The two notions coincide in the case of complete Boolean algebras $B$. Indeed, suppose $B$ is atomic in the Wikipedia sense, and for any element $b \in B$, consider the relative complement
$$c = b \wedge \neg (\bigvee \{atoms\, a: a \leq b\})$$
To show $B$ is atomistic, it suffices to show $c = 0$. If not, then there is an atom $a'$ such that $a' \leq c$, which means both $a' \leq b$ and
$$a' = a' \wedge a' \leq a' \wedge \bigvee \{atoms\, a: a \leq b\} = 0$$
since $a' \leq \neg(\bigvee \{atoms\, a: a \leq b\})$. This is a contradiction.
Our (*pro tem*) decision to define the word "atomic" in the idiosyncratic nLab sense above is consistent with its use elsewhere in category theory; see the sections below on atomic objects and on categorification.
## Examples
+-- {: .num_example}
###### Example
In a [[power set]] the atoms are the [[singleton subsets]]. Every power set is atomic, and in fact every atomic [[complete boolean algebra]] is (up to [[isomorphism]]) a power set.
=--
+-- {: .num_example}
###### Example
In a [[lattice of subtoposes]] the atoms are the 2-valued [[Boolean toposes]]. See [this proposition](subtopos#BooleantoposesAreAtoms).
=--
## Properties
If $a$ is an atom in a [[lattice]] or more generally a [[meet]] [[semilattice]] and $b$ any other element then (using classical logic)
$$ a \wedge b \in \{a, \bot\} .$$
This is simply because $a \wedge b \leq a$, so equals either $a$ or $\bot$.
## Atoms and tiny elements
If $E$ is a poset or preorder, in other words a $\mathbf{2}$-[[enriched category]], an element $e \in E$ is _[[tiny object|tiny]]_ if the hom $E(e, -)\colon E \to \mathbf{2}$ preserves all [[supremum|sups]] that exist in $E$. It is arguable (from an [[nPOV]]) that the weaker concept of tiny element is more fundamental than the notion of atom; for example, as we will see below, replacing atoms by tiny elements permits one to generalize the characterization of power sets as complete atomic Boolean algebras.
+-- {: .num_prop}
###### Proposition
A tiny element in a Boolean algebra is precisely an atom.
=--
+-- {: .proof}
###### Proof
Let $a$ be an atom. Let $\{x_i\}$ be a collection of elements that admits a supremum such that $a \leq \bigvee_i x_i$. Then
$$a = a \wedge \bigvee_i x_i = \bigvee_i a \wedge x_i$$
(where the second equation holds since $a \wedge -$ is a left [[adjoint functor|adjoint]], because $B$ is a [[Heyting algebra]]). Since $a$ is positive, for some $i$ the element $a \wedge x_i$ is positive as well. Trivially it holds that $a \wedge x_i \leq a$; since $a$ is an atom, the inequality is an equality. Thus $a \leq x_i$ for some $i$, which is what we want.
If $a$ is not an atom, i.e., if $0 \lt b \lt a$ for some $b$, then
$$a = b \vee (a \wedge \neg b)$$
If $B(a, -)$ preserved the join on the right, then either $a \leq b$ which is evidently false, or $a \leq a \wedge \neg b$, i.e., $a \leq \neg b$, i.e., $b = a \wedge b \leq 0$, also evidently false. Thus $B(a, -)$ does not preserve suprema.
=--
Only one half of this proposition holds (an atom is a tiny element) if we replace the Boolean algebra $B$ by a general [[frame]]. (In fact, this direction even holds in impredicative constructive mathematics, if the frame is equipped with a [[positive element|positivity predicate]].) On the other hand, tiny elements need not be atoms (an easy example is the frame of down-sets of a [[poset]], where principal down-sets are atomic objects, but generally not atoms in the underlying poset of the frame).
Be this as it may, [Lawvere](http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf#page=6) has written, "In order to settle once and for all the various terminological differences, perhaps we can use a.t.o.m. as an abbreviation for 'amazing tiny object model'." This is Lawvere's 'objective' way of abbreviating "atomic object"; the word 'amazing' here is presumably chosen to evoke what Lawvere has called the "amazing right adjoint" to an exponential functor $(-)^D$, particularly in the case of [[synthetic differential geometry]] where such adjoints exist for [[infinitesimal object|infinitesimal objects]] $D$.
## Generalization and categorification
The result that an atomic complete Boolean algebra is isomorphic to a power set -- hence to a [[presheaf]] with values in the [[0-category]] $\mathbf{2} = (-1)Grpd$ of [[(-1)-groupoid|(-1)-groupoids]] -- may be generalized and categorified as follows.
Let $E$ be a $V$-category, where $V$ is a [[cosmos]] (a complete, cocomplete, symmetric monoidal closed category). We define an object $e$ of $E$ to be **tiny** or **atomic** if $E(e, -) \colon E \to V$ preserves any $V$-colimit that exists in $E$. (As usual, the appropriate notion of colimit in the enriched setting is [[weighted colimit]].)
In what follows, we suppose the full $V$-subcategory $Tiny(E)$ of atomic objects in $E$ is essentially small. The inclusion $i \colon Tiny(E) \hookrightarrow E$ induces a restricted Yoneda embedding
$$E \to V^{Tiny(E)^{op}}$$
sending an object $e$ to $E(i-, e)$. We say that $E$ is **[[atomic category|atomic]]** if $i \colon Tiny(E) \hookrightarrow E$ is $V$-[[dense functor|dense]], in other words if every object $e$ of $E$ is a canonical colimit of atomic objects below it, in the precise sense that the following enriched [[coend]] exists, and its canonical map to $e$,
$$\int^{a \in Tiny(E)} E(i a, e) \cdot i a \to e,$$
is an isomorphism.
If $E$ is a preorder, i.e., is $\mathbf{2}$-enriched where $\mathbf{2}$ is the category of $(-1)$-categories, the coend amounts to the supremum
$$\sup \{i a: i a \leq p\}$$
so that $E$ is atomic precisely if every element is the sup of the tiny elements below it.
+-- {: .num_thm}
###### Theorem
A small-cocomplete atomic preorder $E$ is equivalent to the free sup-lattice $2^{T^{op}}$ generated by the preorder $T = Tiny(E)$ of tiny elements. Conversely, every free sup-lattice $2^{T^{op}}$ is small-cocomplete and atomic, where $T$ is the poset of tiny elements.
=--
N.B. "Free sup-lattice" refers to a left adjoint of the forgetful functor $U \colon SupLat \to Preord$ from sup-lattices to preorders.
+-- {: .proof}
###### Proof
Since $E$ is cocomplete, and since $\mathbf{2}^{Tiny(E)^{op}}$ is the free sup-lattice or cocomplete preorder generated from $Tiny(E)$, the inclusion $i \colon Tiny(E) \to E$ extends uniquely to a sup-preserving map
$$L \colon \mathbf{2}^{Tiny(E)^{op}} \to E$$
which sends $X \colon Tiny(E)^{op} \to \mathbf{2}$ to
$$\int^{a \in Tiny(E)} X(a) \cdot i a = \sup \{i a: X(a) = 1\}.$$
This $L$ is left adjoint to the restricted Yoneda embedding $R \colon E \to \mathbf{2}^{Tiny(E)^{op}}$. The condition that $E$ is atomic says that for each $e \in E$, the value of the counit of $L \dashv R$ at $e$ is an isomorphism
$$\int^a E(i a, e) \cdot i a \cong e.$$
On the other hand, the value of the unit of $L \dashv R$ at an object $X$ is given by a string of isomorphisms
$$\array{
X & \stackrel{Yoneda}{\cong} & \int^a X(a) \cdot Tiny(E)(-, a) \\
& \cong & \int^a X(a) \cdot E(i-, i a) \\
& \cong & E(i-, \int^a X(a) \cdot i a)
}$$
where the last isomorphism obtains from the fact that $E(i a, -)$ preserves colimits if $a$ is tiny. Thus the unit is also an isomorphism.
For the converse: each representable object $T(-, t)$ of $\mathbf{2}^{T^{op}}$ is tiny, because the covariant functor $2^{T^{op}}(T(-, t), -)$, being the same as evaluation at $t$ by the Yoneda lemma, preserves colimits. Furthermore, every functor $X: T^{op} \to \mathbf{2}$ is a canonical colimit of representables, so that $2^{T^{op}}$ is atomic in addition to being cocomplete.
=--
+-- {: .num_cor}
###### Corollary
A complete atomic Boolean algebra $B$ is isomorphic to $2^T$, where $T$ is the discrete preorder of atoms of $B$.
=--
The argument given for the theorem above carries over without obstruction to the general enriched setting. In particular, replacing $\mathbf{2} = (-1)$-$Cat$ by its categorification $Set = 0$-$Cat$, we get the following result, first enunciated in Bunge's thesis.
+-- {: .num_thm}
###### Theorem (Bunge)
A category $E$ is equivalent to a [[presheaf topos]] (functors with values in the 1-category [[Set]] of [[0-groupoids]]) if and only if it is cocomplete and atomic as a $Set$-category. Representables $C(-, c)$ are (among the) atomic objects of $Set^{C^{op}}$, and generate the presheaf topos by closing under all small colimits.
=--
(The literal statement in Bunge's thesis is that a category is equivalent to a presheaf category $Set^{C^{op}}$ if and only if it is cocomplete, regular, and has a generating set of atomic objects, but this is trivially the same since presheaf toposes are of course [[regular category|regular]].)
## Related concepts
* [[compact object]]
* [[connected object]]
* [[Cauchy completion]]
[[!redirects atom]]
[[!redirects atoms]]
[[!redirects atomic element]]
[[!redirects atomic elements]] |
atom (disambiguation) | https://ncatlab.org/nlab/source/atom+%28disambiguation%29 |
_Atom_ may refer to one of the following:
* In mathematics, an element which is minimal among non-minimal elements (in a preorder or poset). Related notions may be found at [[atom]].
* In set theory, there are theories (for example [[ZFA]]) which admit [[urelements]], which are also called atoms.
* In [[physics]], a basic unit of [[matter]] that represents a chemical element or one of its isotopes, consisting of a central [[nucleus]] of [[protons]] and [[neutrons]] and a "cloud" of [[electrons]] surrounding it. See [[atom (physics)]].
* In [[William Lawvere]]'s work a characterization of [[infinitesimally thickened point]], an _[[atomic object]]_.
* In [[Georg Hegel]], _[[Science of Logic]]_ : [Atomism](Science+Of+Logic#Atomism) |
atom (physics) | https://ncatlab.org/nlab/source/atom+%28physics%29 |
> This entry is about the concept in [[physics]]. For other notions of the same name see [[atom (disambiguation)]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[physics]] an _atom_ is a [[bound state]] of a [[nucleus]] and a shell of [[electrons]], where the nucleus itself is a bound state of [[protons]] and [[neutrons]], which themselves are bound states of [[quarks]], which, finally, in the present [[standard model of particle physics]], are [[elementary particles]], as are the [[electrons]] in the shell of the atom.
In [[chemistry]] the different types of atoms that exist are also called _[[chemical elements]]_. Their [[bound states]] in turn are called _[[molecules]]_. The [[phenomenology]] of the [[elements]] that is captured by the [[periodic table of the elements]] is explained by the [[quantum physics]] of atoms. This transition area between [[quantum physics]] and [[chemistry]] is called _[[quantum chemistry]]_.
## Terminology
When the term _atom_ ("indivisible") for these physical objects was established, the inner structure of atoms was not known yet. Similarly what today are called [[elementary particles|elementary]] (hence again: indivisible) [[particles]] are subject to speculation that they might be compound after all (e.g. in [[technicolor]] models, or [[string theory]]).
## Examples
* [[hydrogen atom]]
## Related concepts
* [[chemistry]], [[chemical element]]
* [[molecule]]
* [[particle]]
* [[periodic table of the elements]]
* [[atomtronics]]
## References
See also
* Wikipedia, _[Atom (physics)](http://en.wikipedia.org/wiki/Atom)_
|
atom physics | https://ncatlab.org/nlab/source/atom+physics |
## Idea
The [[physics]] of [[atom (physics)|atoms]].
## References
* Wikipedia, _[Atom physics](http://en.wikipedia.org/wiki/Atomic_physics)_ |
atomic Boolean algebra | https://ncatlab.org/nlab/source/atomic+Boolean+algebra |
* table of contents
{: toc}
## Definition
Given an element $a$ of a [[Boolean algebra]] (or other [[poset]]) $A$, recall that $a$ is an **[[atom]]** in $A$ if $a$ is [[minimal element|minimal]] among non-trivial (non-[[bottom]]) elements of $A$. That is, given any $b \in A$ such that $b \leq a$, either $b = 0$ or $b = a$.
$A$ is __atomic__ if we have $b = \bigvee_I a_i$ for every $b \in A$, where $\{a_i\}_I$ is some set of atoms in $A$.
If $A$ is complete we can write it: if for every $b \in A$, we have $b = \bigvee \mathcal{A}(b)$ where $\mathcal{A}(b)$ is the set of all the atoms $a$ in $A$ such that $a \leq b$. Or: for every $b \in A$, we have $b \le \bigvee \mathcal{A}(b)$.
## Properties
* A [[complete lattice|complete]] atomic Boolean algebra is necessarily a [[power set]]; see [[CABA]].
[[!redirects atomic Boolean algebra]]
[[!redirects atomic Boolean algebras]]
|
atomic category | https://ncatlab.org/nlab/source/atomic+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An ordinary $Set$-enriched [[category]] $C$ is called **atomic** if it has a [[small set|small]] [[dense subcategory|dense]] [[full subcategory]] of [[atomic objects]], $Atom(C)$, so that every object $c$ of $C$ is a small colimit of the functor
$$Atom(C) \downarrow c \stackrel{proj}{\to} Atom(C) \stackrel{i}{\hookrightarrow} C.$$
More generally, for $V$ a [[cosmos]], a $V$-[[enriched category]] $C$ is _atomic_ if it admits a small $V$-dense full subcategory of atomic objects $Atom(C)$, such that every object $c$ is an enriched coend
$$\int^{a \in Atom(C)} C(i a, c) \cdot i a.$$
## Properties
### Relation to presheaf toposes
+-- {: .num_thm}
###### Theorem
A category $E$ is equivalent to a [[presheaf topos]] (functors with values in the 1-category [[Set]] of [[0-groupoids]]) if and only if it is cocomplete and atomic.
=--
This is due to [[Marta Bunge]], who showed it is enough to have a regular cocomplete category with a [[generating set]] of atomic objects.
## Related concepts
* [[atom]]
* [[compact object]]
* [[Cauchy completion]]
[[!redirects atomic categories]]
|
atomic geometric morphism | https://ncatlab.org/nlab/source/atomic+geometric+morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn}
###### Definition
A [[geometric morphism]] $f : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{F}$ is called **atomic** if its [[inverse image]] $f^*$ is a [[logical functor]].
=--
+-- {: .num_defn}
###### Definition
A [[sheaf topos]] $\mathcal{E}$ is called **atomic** if its [[global section]] geometric morphism is atomic, or in other words, if the [[constant sheaf|constant sheaf functor]] $\Delta\colon Set\to\mathcal{E}$ is logical.
Generally, a topos over a [[base topos]] $\Gamma : \mathcal{E} \to \mathcal{S}$ is called an [[atomic topos]] if $\Gamma$ is atomic.
=--
+-- {: .num_note}
###### Note
As shown in prop. \ref{AtomicMeansLocallyConnected} below, every atomic morphism $f : \mathcal{E} \to \mathcal{S}$ is also a [[locally connected geometric morphism]]. The connected objects $A \in \mathcal{E}$, $f_! A \simeq *$ are called the **atoms** of $\mathcal{E}$.
=--
See ([Johnstone, p. 689](#Johnstone)).
## Properties
### General
+-- {: .num_prop}
###### Proposition
Atomic morphisms are closed under composition.
=--
+-- {: .num_prop #AtomicMeansLocallyConnected}
###### Proposition
An atomic geometric morphism is also a [[locally connected geometric morphism]].
=--
+-- {: .proof}
###### Proof
By <a href="https://ncatlab.org/nlab/show/logical+functor#LeftRightAdjoint">this proposition</a> a logical morphism with a [[right adjoint]] has also a [[left adjoint]].
=--
+-- {: .num_prop #ConnectedAtomicImpliesHyperconnected}
###### Proposition
If an atomic morphism is also a [[connected geometric morphism|connected]], then it is even [[hyperconnected geometric morphism|hyperconnected]].
=--
This appears as ([Johnstone, lemma 3.5.4](#Johnstone)).
+-- {: .num_prop}
###### Proposition
A [[localic geometric morphism]] is atomic precisely if it is an [[etale geometric morphism]].
=--
This appears as ([Johnstone, lemma 3.5.4 (iii)](#Johnstone)).
+-- {: .num_prop}
###### Proposition
Every [[étale geometric morphism]] is atomic.
=--
## Related concepts
* [[atomic topos]]
* [[atomic site]]
## References
* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol. 2_ , Oxford UP 2002. (section C3.5, pp.684-695)
[[!redirects atomic geometric morphisms]]
|
atomic number | https://ncatlab.org/nlab/source/atomic+number |
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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _atomic number_ of a [[chemical element]] is the [[natural number|number]] of [[protons]] in any [[atomic nuclei]] of this element.
For instance [[carbon]] has atomic number 6.
The [[periodic table of chemical elements]] organizes them by their atomic number, together with other properties.
The corresponding [[natural number|number]] of [[neutrons]] on the nucleus tends to be close to the number of protons, hence to the atomic number of the element, but need not equal it and my differ considerably for heavy elements. These variants of neutron number for given atomic number specify the _[[isotopes]]_ of the [[chemical element]].
## References
See also
* Wikipedia, _[Atomic number](https://en.wikipedia.org/wiki/Atomic_number)_
Speculative analogy to [[characteristic classes]] of [[complex surfaces]] (different from but inspired by [[Skyrmion]] [[model (in theoretical physics)|models]] for [[atomic nuclei]]):
* {#AtiyahManton16} [[Michael Atiyah]], [[Nicholas Manton]], _Complex Geometry of Nuclei and Atoms_ ([arXiv:1609.02816](https://arxiv.org/abs/1609.02816))
[[!redirects atomic numbers]] |
atomic site | https://ncatlab.org/nlab/source/atomic+site |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
**Atomic sites** are [[sites]] $(\mathcal{C}, J_{at})$ equipped with the _atomic topology_ $J_{at}$. The corresponding [[sheaf toposes]] $Sh(\mathcal{C}, J_{at})$ are precisely the [[atomic topos|atomic Grothendieck toposes]].
## Definition
+-- {: .num_defn}
###### Definition
A [[site]] $(\mathcal{C}, J_{at})$ is called **atomic** if the [[covering sieve|covering]] [[sieves]] $S$ of $J_{at}$ are exactly the [[inhabited set|inhabited]] sieves $S\neq\emptyset$. A [[Grothendieck topology]] $J_{at}$ of this form is called _atomic_.
=--
## Example
Let $FinSet^{op}_{mono}$ be the opposite of the category $FinSet_{mono}$ with objects finite sets and monomorphisms. Then $(FinSet^{op}_{mono}, J_{at})$ is an atomic site and the corresponding sheaf topos $Sh(FinSet^{op}_{mono}, J_{at})$ is the [[Schanuel topos]].
That $J_{at}$ is indeed a [[Grothendieck topology]] is ensured by prop. \ref{atomic_ore}.
## Properties
+-- {: .num_prop #atomic_ore}
###### Proposition
Let $\mathcal{C}$ be a [[category]]. Then $\mathcal{C}$ can be made into an atomic site if and only if for any diagram
$$
\array{
& & A \\
& & \downarrow\\
B & \to & C
}
$$
there is an object $D$ and arrows $D \to A, B$ such that the following diagram commutes:
$$
\array{
D & \to & A \\
\downarrow & & \downarrow\\
B & \to & C
}
$$
=--
+-- {: .proof}
###### Proof
This is exactly what is needed for the pullback stability axiom to hold, and the other axioms are immediate.
=--
The condition occurring in the proposition is called the (right) [[Ore condition]]. It is a result by [[Peter Johnstone|P. T. Johnstone]] (1979) that $Set^{\mathcal{C}^{op}}$ is a [[De Morgan topos]] precisely if $\mathcal{C}$ satisfies the Ore condition. Whence we see that every [[atomic topos|atomic Grothendieck toposes]] is a ([[Boolean topos|Boolean]]) subtopos of a De Morgan [[presheaf topos]].
Recall that the [[dense topology]] $J_d$ on a category $\mathcal{C}$ consists of all sieves $S\in J_d(C)$ with the property that given $f:D\to C$ there exists $g:E\to D$ such that $f\cdot g\in S$. The atomic topology is a special case of this:
+-- {: .num_prop #atomic_dense}
###### Proposition
Let $\mathcal{C}$ be a [[category]] satisfying the [[Ore condition]]. Then the atomic topology $J_{at}$ coincides with the dense topology $J_d$.
=--
+-- {: .proof}
###### Proof
For $\mathcal{C}=\emptyset$ the claim is trivial. So let $C\in\mathcal{C}$ be an object and $S$ a sieve on $C$.
Assume $S\in J_d(C)$, then for $id\colon C\to C$ there exists $g:E\to C$ with $id\cdot g\in S$ whence $S\in J_{at}(C)$.
Conversely, assume $S\in J_{at}(C)$ and let $f:D\to C$ be a morphism. Then there exists $g\in S$ by assumption and the diagram $D\overset{f}{\rightarrow} C \overset{g}{\leftarrow} E$ can be completed to a commutative square $f\cdot i = g\cdot h$ but $g\cdot h\in S$ since $g\in S$ and $S$ is a sieve. Whence $f \cdot i\in S$ and, accordingly, $S\in J_d(C)$.
=--
In other words, the atomic topology is just the [[dense topology]] on categories satisfying the Ore condition. Since the corresponding sheaf toposes of the dense topology are just the double negation subtoposes of the corresponding presheaf topos we finally get:
+-- {: .num_prop}
###### Proposition
Atomic Grothendieck toposes i.e. toposes (equivalent to) $Sh(\mathcal{C}, J_{at})$ for $(\mathcal{C}, J_{at})$ an atomic site are precisely (the toposes equivalent to) the [[double negation|double negation subtoposes]] $Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ for a De Morgan presheaf topos $Set^{\mathcal{C}^{op}}$. $\qed$
=--
The sheaves of atomic sheaf toposes $Sh(\mathcal{C}, J_{at})$ are easy to describe:
+-- {: .num_prop #atomic_sheaf}
###### Proposition
Let $(\mathcal{C}, J_{at})$ be an atomic site. A presheaf $P\in Set^{\mathcal{C}^{op}}$ is a sheaf for $J_{at}$ iff for any morphism $f:D\to C$ and any $y\in P(D)$ , if $P(g)(y)=P(h)(y)$ for all diagrams
$$
E\overset{g}{\underset{h}{\rightrightarrows}} D\overset{f}{\to} C
$$
with $f\cdot g=f\cdot h$ , then $y=P(f)(x)$ for a unique $x\in P(C)$.
=--
For the proof see Mac Lane-Moerdijk ([1994](#MM94), pp.126f).
## The more general definition
Thus far we have presented the classical approach as presented in Mac Lane-Moerdijk ([1994](#MM94)) going back to Barr-Diaconescu ([1980](#BD80)) but it was observed by [[Olivia Caramello|O. Caramello]] ([2012](#Ca12)) that the atomic topology can in fact be defined on arbitrary categories not only on those satisfying the [[Ore condition]].
+-- {: .num_defn}
###### Definition'
Let $\mathcal{C}$ be a category. The _atomic topology_ $J_{at}$ on $\mathcal{C}$ is the smallest [[Grothendieck topology]] containing all the nonempty sieves. A site of the form $(\mathcal{C}, J_{at})$ is called _atomic_.
=--
Note that $J_{at}$ is well defined as the intersection of all Grothendieck topologies with the property that all nonempty sieves cover. The following proposition justifies the terminology:
+-- {: .num_prop}
###### Proposition
Let $(\mathcal{C}, J_{at})$ be an atomic site. Then $Sh(\mathcal{C}, J_{at})$ is an [[atomic topos|atomic Grothendieck topos]].
=--
+-- {: .proof}
###### Proof
The main idea is to consider the full subcategory $\mathcal{C}'$ on those objects $U$ with $\emptyset\notin J_{at}(U)$ together with the [[dense sub-site|induced topology]] $J'_{at}=J_{at}|_{\mathcal{C}'}$. Then one shows that $\mathcal{C}'$ satisfies the Ore condition and concludes by the [[comparison lemma]] that $Sh(\mathcal{C}', J'_{at})\simeq Sh(\mathcal{C}, J_{at})$. For the details see Caramello ([2012](#Ca12), prop.1.4).
=--
+-- {: .num_example}
###### Example
Consider the category $\mathcal{C}$ on the 'walking co-span' $A\overset{f}{\rightarrow} C\overset{g}{\leftarrow} B$. $\mathcal{C}$ does not satisfy the [[Ore condition]]. The atomic topology $J_{at}$ is given by
$$
J_{at}(A)=\{\{id_A\},\emptyset\} \qquad J_{at}(B)=\{\{id_B\},\emptyset\}
$$
$$
J_{at}(C)=\{\{ id_C, f,g\},\{f \}, \{ g \},\{f,g\},\emptyset\} \quad .
$$
Here $\emptyset\in J_{at}(A)$ , respectively $\emptyset\in J_{at}(B)$, due to the stability axiom applied to $f^\ast(\{g\})=\emptyset$ , respectively to $g^\ast(\{f\})=\emptyset$ . Whereas $\emptyset\in J_{at}({C})$ by the transitivity axiom applied to $\{f\}\in J_{at}(C)$ and the sieve $\emptyset$ since $f^{\ast}(\emptyset)=\emptyset\in J_{at}(A)$.
Accordingly the subcategory $\mathcal{C}'$ is empty and $Sh(\mathcal{C},J_{at})\simeq 1$ is degenerate. In particular, $Sh(\mathcal{C},J_{at})$ is not equivalent to $Sh(\mathcal{C},J_d)\simeq Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})\simeq Set\times Set$. So we see that the atomic topology on $\mathcal{C}$ is distinct from the [[dense topology]]. For completeness we describe the latter:
$$
J_{d}(A)=\{\{id_A\}\} \qquad J_{d}(B)=\{\{id_B\}\}
$$
$$
J_{d}(C)=\{\{id_C, f,g\},\{f,g\}\} \quad .
$$
Further details on $Set^{\mathcal{C}^{op}}$, the topos of hypergraphs, may be found at [[hypergraph]].
=--
In the example, we observed that dense and the atomic topology need not coincide for categories not satisfying the [[Ore condition]].
In fact more can be said here:
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be a [[category]]. Then $J_{d}\subseteq J_{at}$ in general, but $J_d=J_{at}$ precisely if $\mathcal{C}$ satisfies the [[Ore condition]].
=--
+-- {: .proof}
###### Proof
The proof of prop. \ref{atomic_dense} already showed that the sieves of the dense topology $J_d$ are never empty regardless of the Ore condition. From prop. \ref{atomic_ore} follows that the atomic topology $J_{at}$ will additionally contain empty sieves precisely if $\mathcal{C}$ does not satisfy the Ore condition.
=--
In particular, $Sh(\mathcal{C},J_{at})\subseteq Sh(\mathcal{C},J_{d})\simeq Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ .
## Related entries
* [[dense topology]]
* [[atomic topos]]
* [[atomic geometric morphism]]
* [[Ore condition]]
* [[De Morgan topos]]
## Reference
* {#BD80}[[Michael Barr]], [[Radu Diaconescu]], _Atomic Toposes_ , JPAA **17** (1980) pp.1-24. ([pdf](http://www.math.mcgill.ca/barr/papers/atom.top.pdf))
* {#Ca12}[[Olivia Caramello]], _Atomic toposes and countable categoricity_ , Appl. Cat. Struc. **20** no. 4 (2012) pp.379-391. ([arXiv:0811.3547](http://arxiv.org/abs/0811.3547))
* {#MM94} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (pp.115, 126)
[[!redirects atomic topology]] |
atomic spectrum | https://ncatlab.org/nlab/source/atomic+spectrum |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
> This entry is about the conept in [[physics]]. For other uses of the term see at _[[spectrum - disambiguation]]_.
## References
* Wikipedia, _[Spectroscopy](https://en.wikipedia.org/wiki/Spectroscopy#Atoms)_
[[!redirects atomic spectra]]
[[!redirects spectroscopy]] |
atomic topos | https://ncatlab.org/nlab/source/atomic+topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An atomic topos is a topos $\mathcal{E}$ where the global sections functor $\Gamma: \mathcal{E} \to Set$ is [[atomic geometric morphism|atomic]]. In the case where $\mathcal{E}$ is a Grothendieck topos, there are some (arguably more explicit) alternative characterizations of atomic toposes, as in Theorem \ref{EquivChar}, which make it clearer why they are called "atomic".
## Definition
Recall that a [[geometric morphism]] $f$ is called [[atomic geometric morphism|atomic]] if its [[inverse image functor]] $f^*$ is [[logical functor|logical]].
+-- {: .num_defn}
###### Definition
A [[topos]] over a [[base topos]] $\Gamma : \mathcal{E} \to \mathcal{S}$ is called an **atomic topos** if $\Gamma$ is [[atomic geometric morphism|atomic]]. Unless otherwise specified, the base topos will be taken to be $Set$.
=--
+-- {: .num_defn}
###### Definition
A non-zero object $A$ of a topos $\mathcal{E}$ is an [[atom]] if its only subobjects are $A$ and $0$.
=--
+-- {: .num_theorem #EquivChar}
###### Theorem
Let $\mathcal{E}$ be a [[Grothendieck topos]]. Then the following are equivalent:
1. $\mathcal{E}$ is an atomic topos.
1. $\mathcal{E}$ is the [[category of sheaves]] on an [[atomic site]].
1. The [[subobject lattice]] of every object of $\mathcal{E}$ is a [[complete lattice|complete]] [[atomic Boolean algebra]].
1. $\mathcal{E}$ has a [[small set|small]] [[generating set]] of atoms.
1. Every object of $\mathcal{E}$ can be written as a disjoint union of atoms.
=--
+-- {: .proof}
###### Proof
See [Johnstone, C3.5.8](#Johnstone) and [Barr-Diaconescu, Theorem A](#Barr-Diaconescu80).
=--
## Properties
+-- {: .num_prop}
###### Proposition
Let $\mathcal{E}$ be an atomic topos. Then $\mathcal{E}$ is Boolean.
=--
This appears as one direction of ([Johnstone, cor. C3.5.2](#Johnstone)).
+-- {: .proof}
###### Proof
If $\Gamma^*$ is logical then it preserves the isomorphism $* \coprod * \simeq \Omega$ characterizing a [[Boolean topos]].
=--
+-- {: .num_prop}
###### Proposition
Let $\mathcal{E}$ be a [[Boolean topos|Boolean]] [[Grothendieck topos]] with [[point of a topos|enough points]]. Then $\mathcal{E}$ is an atomic topos.
=--
+-- {: .proof}
###### Proof
See ([Johnstone, cor. C3.5.2](#Johnstone))
=--
+-- {: .num_prop}
###### Proposition
Atomic Grothendieck toposes $Sh(\mathcal{C}, J_{at})$ for $(\mathcal{C}, J_{at})$ an [[atomic site]] are precisely the [[double negation|double negation subtoposes]] $Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ for a [[De Morgan topos|De Morgan]] [[presheaf topos]] $Set^{\mathcal{C}^{op}}$.
=--
+-- {: .proof}
###### Proof
For the argument see at [[atomic site]].
=--
### Decomposition of atomic toposes
Atomic toposes decompose as [[disjoint unions]] of [[connected topos|connected]] atomic toposes. Connected atomic toposes with a [[point of a topos|point]] are the [[classifying toposes]] of [[localic groups]].
An example of a connected atomic topos without a [[point of a topos|point]] is given in ([Johnstone, example D3.4.14](#Johnstone})).
## Examples
* A [[category of presheaves]] $Set^{\mathcal{C}^{op}}$ is atomic precisely iff $\mathcal{C}$ is a groupoid (cf. [Barr-Diaconescu (1980)](#Barr-Diaconescu80)).
* Another example of an atomic Grothendieck topos is the [[Schanuel topos]]. More generally, any [[category of G-sets]] is an atomic Grothendieck topos.
## Related concepts
* [[atomic geometric morphism]]
* [[atomic site]]
* [[connected topos]]
## References
* {#Barr-Diaconescu80}[[Michael Barr]], [[Radu Diaconescu]], *Atomic Toposes*, J. Pure Appl. Algebra **17** (1980) 1-24 [<a href="https://doi.org/10.1016/0022-4049(80)90020-1">doi:10.1016/0022-4049(80)90020-1</a>, [pdf](http://www.math.mcgill.ca/barr/papers/atom.top.pdf), [[BarrDiaconescu-Atomictoposes.pdf:file]]]
* [[Olivia Caramello]], _Atomic toposes and countable categoricity_ , Appl. Cat. Struc. **20** no. 4 (2012) pp.379-391. ([arXiv:0811.3547](http://arxiv.org/abs/0811.3547))
* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol. 2_ , Oxford UP 2002. (section C3.5, pp.684-695)
[[!redirects Atomic topos]]
[[!redirects atomic toposes]]
[[!redirects atomic topoi]] |
atomtronics | https://ncatlab.org/nlab/source/atomtronics |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
(...)
## Related concepts
* [[atom (physics)]]
* [[quantum circuit]]
* [[quantum information theory]], [[quantum computation]]
## References
* [[Luigi Amico]], [[Malcolm G. Boshier]], *Atomtronics* [[arXiv:1511.07215](https://arxiv.org/abs/1511.07215)]
* [[Luigi Amico]], [[Malcolm G. Boshier]], [[Juan Polo Gomez]] et al.: *Roadmap on Atomtronics: State of the art and perspective*, AVS Quantum Sci. **3** 039201 (2021) [[arXiv:2008.04439](https://arxiv.org/abs/2008.04439), [doi:10.1116/5.0026178](https://doi.org/10.1116/5.0026178)]
* [[Luigi Amico]], Dana Anderson, [[Malcolm Boshier]], Jean-Philippe Brantut, Leong-Chuan Kwek, Anna Minguzzi, Wolf von Klitzing: *Colloquium: Atomtronic circuits: From many-body physics to quantum technologies*, Rev. Mod. Phys. **94** (2022) 041001 [[doi:10.1103/RevModPhys.94.041001](https://doi.org/10.1103/RevModPhys.94.041001)]
See also:
* Wikipedia, *[Atomtronics](https://en.wikipedia.org/wiki/Atomtronics)*
|
atoroidal 3-manifold | https://ncatlab.org/nlab/source/atoroidal+3-manifold |
#Contents#
* table of contents
{:toc}
## Idea
A [[3-manifold]] that does not contain an essential [[torus]].
## Related concepts
* [[virtually fibered conjecture]]
## References
* Wikipedia, _[Atoroidal 3-manifold](http://en.wikipedia.org/wiki/Atoroidal)_
[[!redirects atoroidal 3-manifold]]
|
Atsushi Hosaka | https://ncatlab.org/nlab/source/Atsushi+Hosaka |
* [webpage](https://www.phys.sci.osaka-u.ac.jp/en/research_groups/group/12-0_quark/hosaka/index.html)
## Selected writings
Equivalence between [[hidden local symmetry]]- and [[massive Yang-Mills theory]]-description of [[Skyrmion]] [[quantum hadrodynamics]]:
* [[Atsushi Hosaka]], H. Toki, [[Wolfram Weise]], _Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 ([spire:24079](http://inspirehep.net/record/24079))
Combination of the [[omega-meson]]-stabilized [[Skyrme model]] with the [[bag model for quark confinement|bag model]] for [[nucleons]]:
* [[Atsushi Hosaka]], _Omega stabilized chiral bag model with a surface $\omega q q$ coupling_, Nuclear Physics A Volume 546, Issue 3, 31 (1992) Pages 493-508 (<a href="https://doi.org/10.1016/0375-9474(92)90544-T">doi:10.1016/0375-9474(92)90544-T</a>)
On [[kaon]]-[[K*-meson]]-[[photon]] [[interaction]] via the [[WZW term]]:
* S. Ozaki, H. Nagahiro, [[Atsushi Hosaka]], Equations (3) and (9) in: _Magnetic interaction induced by the anomaly in kaon-photoproductions_, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 ([arXiv:0710.5581](https://arxiv.org/abs/0710.5581), [doi:10.1016/j.physletb.2008.06.020](https://doi.org/10.1016/j.physletb.2008.06.020))
On [[heavy baryons]] in [[holographic QCD]] coming from higher [[KK-compactification|KK-modes]]:
* Daisuke Fujii, [[Atsushi Hosaka]], _Heavy baryons in holographic QCD with higher dimensional degrees of freedom_ ([arXiv:2003.13415](https://arxiv.org/abs/2003.13415))
category: people |
Atsushi Matsuo | https://ncatlab.org/nlab/source/Atsushi+Matsuo |
* [personal page](https://www.ms.u-tokyo.ac.jp/~matsuo/)
* [institute page](https://www.s.u-tokyo.ac.jp/en/people/matsuo_atsushi/)
## Selected writings
On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:
* [[Etsuro Date]], [[Michio Jimbo]], [[Atsushi Matsuo]], [[Tetsuji Miwa]], *Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation*, International Journal of Modern Physics B **04** 05 (1990) 1049-1057 $[$[doi:10.1142/S0217979290000528](https://doi.org/10.1142/S0217979290000528)$]$
* [[Atsushi Matsuo]], *An application of Aomoto-Gelfand hypergeometric functions to the $SU(n)$ Knizhnik-Zamolodchikov equation*, Communications in Mathematical Physics **134** (1990) 65–77 $[$[doi:10.1007/BF02102089](https://doi.org/10.1007/BF02102089)$]$
category: people
|
Atsushi Tomoda | https://ncatlab.org/nlab/source/Atsushi+Tomoda |
* [webpage](http://ton.prosou.nu/official/)
## Related $n$Lab entries
* [[splitting principle]]
* [[twisted vector bundle]], [[gerbe module]]
category: people |
Attempt to Introduce the Concept of Negative Quantities into Philosophy | https://ncatlab.org/nlab/source/Attempt+to+Introduce+the+Concept+of+Negative+Quantities+into+Philosophy | [[!redirects empty 154]]
+-- {: .rightHandSide}
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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
**"Versuch, den Begriff der negativen Grössen in die Weltweisheit einzuführen""** (1763) is a short article of [[Immanuel Kant]] concerning the philosophical foundations of the [[integer|integers]], in particular the _negative_ integers.
## Overview
> Diese kleine Schrift ist eine der tiefsinnigsten und lichtvollsten, die nicht bloß Kant geschrieben, sondern welche die philosophische Literatur überhaupt aufzuweisen hat. Man tut Kant nicht unrecht, wenn man behauptet, daß sie ihm wie ein Meteor entschlüpft und selbst nicht wieder zu Gesicht gekommen ist.[^trans] Karl Rosenkranz ([1840, p.118](#Rosenkranz40))
[^trans]: _'This small text is one of the deepest and most lightfull not only in the writings of Kant but in the whole philosophical literature. One does no injustice to Kant if one says that it snapped out of him like a meteor and was never seen again by him.'_
Starting with a discussion of the meaning of the minus sign in negative numbers and as a designator of the subtraction operation, Kant sets out to distinguish between (predicate) negation in logic and substraction in arithmetic. He is thereby led to argue for a sharp distinction between _logical_ and _real opposition_: whereas the former leads to logical contradictions, the latter does not.
Since he conceives logical contradictions as the simultaneous predication of a predicate $p$ and its negative $\neg p$ to the same subject $S$, a contradiction destroys the possibility of the thing represented by the subject.
In contrast, in a real opposition, e.g. two forces acting in opposite directions on the same point mass, merely the effects of the opposites are cancelled without affecting their reality, i.e. the cause is still effective but its effect are covered by the effects of its opposite.
As result, a gulf opens between the realm of logic with its analytic modes of reasoning and the realm of reality throwing into crisis the rationalist thought of the Leibniz-Wolffian school that maintained that in principle all truth was analytical and that denied any difference in principle between empirical and logical truths.
Hence the distinction between logical and real oppositions gave Kant strong incentive to reconceptualize the relation between logical and empirical propositions finally leading to the distinction between the formal and transcendental logic in his [[Critique of Pure Reason]] of 1781.
On more general grounds, Kant also recalibrates in the article the relation between philosophy and mathematics, urging the former to modesty e.g. when it comes to deny the existence of [[infinitesimal|infinitesimals]] on metaphysical grounds.
## Repercussions
The concept of real opposition preserving (the reality of) the opposites positively in its result proved to be decisive not only for Kantian but also for the post Kantian philosophy since, as already pointed out by [[Karl Rosenkranz|K. Rosenkranz]] surrounding the discussions around the [[Science of Logic|Hegelian logic]] in the 1840-50s, it can be viewed as a germ of Hegel's concept of contradiction and [[Aufhebung|sublation]] in the _'Wissenschaft der Logik'_. In fact, the section on 'contradiction' in the second volume even has a discussion of arithmetic and negative numbers (cf. [Wolff 2010](#Wolff10)).
## Related entries
* [[Immanuel Kant]]
* [[Karl Rosenkranz]]
* [[Hermann Grassmann]]
* [[transcendental ideal]]
* [[infinite judgement]]
* [[Aufhebung]]
* [[negation]]
* [[Ausdehnungslehre]]
* [[construction in philosophy]]
## References
* {#Giovanelli15} M. Giovanelli, *Trendelenburg and the Concept of Negation in Post-Kantian Philosophy*, to appear in Munk (ed.), *Proceedings of the Amsterdam 2010 Colloquium: Natur des Denkens und das Denken der Natur: Spinoza, Trendelenburg und H. Cohen*. ([draft](https://www.dropbox.com/s/mxm4db8id7je924/Giovanelli%2C%20Marco%20-%20Trendenleburg%20and%20the%20Concept%20of%20Negation%20in%20Post-Kantian%20Philosophy%20Proceedings%20of%20the%20Connference.pdf?dl=0))
* {#Kaestner58} Abraham Gotthelf Kästner, _Anfangsgründe der Arithmetik, Algebra, Geometrie, ebenen und sphärischen Trigonometrie, und Perspectiv_, Göttingen 1758. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/toc/?PID=PPN812429885))
* Srećko Kovač, _In what sense is Kantian principle of contradiction non-classical?_, Logic and Logical Philosophy **17** (2008) pp.251-274. ([link](http://dx.doi.org/10.12775/LLP.2008.013))
* {#Rosenkranz40} [[Karl Rosenkranz]], _Geschichte der Kant'schen Schule_, Akademie-Verlag Berlin 1987[1840].
* {#Wolff10} Michael Wolff, _Der Begriff des Widerspruchs - Eine Studie zur Dialektik Kants und Hegels_, Frankfurt UP ²2010.
[[!redirects Attempt to introduce the concept of negative quantities into philosophy]]
[[!redirects concept of negative quantities]]
[[!redirects Concept of Negative Quantities]] |
Attila Szabo | https://ncatlab.org/nlab/source/Attila+Szabo |
* <a href="https://en.wikipedia.org/wiki/Attila_Szabo_(scientist)">Wikipedia entry</a>
## Selected writings
On [[quantum chemistry]]:
* [[Attila Szabo]], [[Neil S. Ostlund]], *Modern Quantum Chemistry -- Introduction to Advanced Electronic Structure Theory*, Macmillan (1982), McGraw-Hill (1989), Dover (1996) $[$[pdf](https://chemistlibrary.files.wordpress.com/2015/02/modern-quantum-chemistry.pdf)$]$
## Related $n$Lab entries
* [[Slater determinant]]
category: people
|
Atul Sharma | https://ncatlab.org/nlab/source/Atul+Sharma |
* [inspire page](https://inspirehep.net/authors/1737875)
## Selected writings
On [[celestial holography]] via [[topological strings]] on [[twistor space]]:
* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Top-down holography in an asymptotically flat spacetime* [[arXiv:2208.14233](https://arxiv.org/abs/2208.14233)]
* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Burns space and holography* [[arXiv:2306.00940](https://arxiv.org/abs/2306.00940)]
category: people
|
AUC | https://ncatlab.org/nlab/source/AUC |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
\tableofcontents
## Idea
AUC (ACU in the French language) is a method of naming [[magmoidal categories]]:
* A stands for associative
* U stands for unital
* C stands for commutative
Thus an AU category would be a [[monoidal category]]. The AUC terminology is commonly used in [[rewriting]] theory.
## References
For the AUC terminology in [[category theory]], see:
* [[Hoàng Xuân Sính]], _Gr-catégories_, Ph.D. thesis (1973) [[web](http://w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/Kuenzer/sinh.html)]
* [[Neantro Saavedra-Rivano]], *[[Catégories Tannakiennes]]*, Bulletin de la Société Mathématique de France 100 (1972) 417-430
* [[John Baez]], *Who Invented Monoidal Categories?*, nCafé ([web](https://golem.ph.utexas.edu/category/2023/07/who_invented_monoidal_categori.html))
> "John Baez wrote:
> > Right, it’s an efficient way to name all the possibilities. What makes it look baroque is that nowadays we’ve realized that 99% of the time, we only need two of the possibilities.
> It turns out that in the area of CS called rewriting theory, all the possibilities show up. So they’ve stuck to the AUC naming scheme."
> - [[Jacques Carette]]
[[!redirects ACU]] |
Aufhebung | https://ncatlab.org/nlab/source/Aufhebung |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
[[!redirects aufhebung]]
[[!redirects Aufhebungs relation]]
[[!redirects jump operator]]
[[!redirects sublation]]
## Idea
> I am not a "Hegelian". F. W. Lawvere [^LQ]
**Aufhebung** (sublation) is a central concept[^ref] in the [[dialectic|dialectical]] logic of the German philosopher [[Georg Hegel|G. W. F. Hegel]]. The German expression has several meanings for which _tollere, elevare, conservare_ would be Latin equivalents.[^fine]
[^ref]: A pertinent passage is e.g. [SoL §209](Science+of+Logic#209).
[^fine]: As this polysemy is important for the concept and difficult to preserve in translation we prefer to use the German term in the following.
In his quest to [[axiom|axiomatize]] the concepts of [[space]] and [[cohesion]], [[F. W. Lawvere]], inspired by [[homotopy theory]] proposed a mathematical rendering of the _Aufhebungs_ relation within [[topos theory]] or [[category theory]] more generally. It is the mathematical concept that will constitute the primary subject in the following.
##_Aufhebung_ in Hegel's 'Wissenschaft der Logik'
Although the two volumes of _'[[Science of Logic|Wissenschaft der Logik]]'_ (1st ed. 1812-1816) can be considered as one of the main texts of [[Hegel]]'s philosophy they fell into disfavour in the second half of the 19th century and most of the 20th century, and accordingly received much less attention than the 'Phänomenologie des Geistes' or the 'Rechtsphilosophie'. They shared this fate with Hegelian philosophy as a whole which apart from the philological interest it generated, was continued only through the political wing of Lefthegelianism which in either its existentialist interpretation by A. Kojève or its Marxist interpretation by G. Lukács openly rejected the concept of objective dialectics in nature thereby cutting the social thought from its broad foundation in [[ontology]] and [[logic]], whereas the [[natural philosophy|natural philosophical]] tradition in the vein of F. Engels petrified to the doctrines of dialectical materialism.
The 'Wissenschaft der Logik' has to be viewed against the background of philosophy in the early 19th century: **Kant** had embarked on a project of 'refoundation', or rather demolition, of [[metaphysics]] from an epistemological perspective and this project had been pushed further by his followers especially Fichte in his _Wissensschaftslehre_. However critical these [[idealism|idealist]] systems had been to the claims of traditional metaphysics and epistemology they all left the traditional logic untouched and in this respect fell behind Leibniz. It is at this point where Hegel starts: he sets out to extend the critical examination of the foundations of knowledge to logic itself.
Heavily influenced by the transcendental deductions and the chapters on dialectical paralogisms in Kant's 'Kritik der reinen Vernunft' he intends to start from indeterminate, immediate [[being]] and justify the autonomous development of the system of categories. Here dialectics and Aufhebung enter the picture as Hegel conceives the categories not only as not given apriorily but as actually [[becoming]]: _logic ceases to be an inventory of categories but becomes a system of transformations of categories!_ (Had Eilenberg and MacLane in 1945 intended their terminological loans from philosophy as a kind of joke, Lawvere would 25 years later take this terminological proximity at face value.)
Hegel parts with the traditional conception mainly in two points: the foundations of his logic coalesce with [[ontology]] into an **objective logic** as the first part is titled (a 'logic of things' as [[Charles Sanders Peirce|C.S. Peirce]] would later put it), i.e. he rejects the subject as a possible ground for logic, and he reassesses the status of **negativity** or conflict-contradiction in logic. The cornerstone of the edifice is the anti-eleatic unity of [[being]] and [[nothing|nothingness]] in the idea of [[becoming]]. It is precisely this 'positively being negative' that finds its expression in the concept of 'Aufhebung'.
A key passage on Aufhebung in 'Wissenschaft der Logik' comes at the end of the first chapter ([I.1.1Cc, p.113](#WdL)): after a deduction of the categories of '[[being]]', '[[nothing|nothingness]]', and their unity in '[[becoming]]' Hegel determines **[[dialectics]]** as 'the higher rational movement... in which the precondition of the separatedness (of the seemingly separated) is lifted ( _sich aufhebt_ )' ([p.111](#WdL)).
He goes on ([p.113](#WdL)) to explicate _Aufheben_ as one of the most important concepts in all of philosophy that constantly recurs everywhere. The sublated - _das Aufgehobene_ is not nothing which is an _unmediated_, but is a mediated - _ein Vermitteltes_; it is nothing - _das Nichtseiende_, but as a result that originated from a being and therefore still carries with it the determination from which it derives. This is inspired by Spinoza's 'omnis determinatio est negatio': **Aufhebung** is the mode of this coexistence of negation-affirmation.
Hegel draws his logic from a rich tradition of dialectic going back to Plato in general and to its renewal in Fichte and Schelling's attempts to transcendental philosophy in the 1790s for which in turn Kant's attitude was of capital importance: the latter had established in a short paper _'[[Attempt to Introduce the Concept of Negative Quantities into Philosophy|Versuch den Begriff der negativen Größen in die Weltweisheit einzuführen]]'_ (1763) a distinction between contradictions and real oppositions[^trendel] anticipating the later analytic-synthetic division - here originates the term 'Aufhebung' and gets tied to synthesis of oppositions which is terminologically present in Schelling's use of the terms 'thesis', 'anti-thesis' and 'synthesis' for the dialectical triad.[^triad]
Secondly, Kant had arranged the table of categories in triadic fashion with the third terms _grosso modo_ being the synthesis of the preceding positive respectively negative terms and claimed to have demonstrated the completeness of the resulting table though his deduction was generally considered inconclusive. Starting with Reinhold the deduction of the categories soon became a main concern of transcendental philosophy. In particular, Fichte rejected the primacy of the judgement forms in Kant's approach i.e. the primacy of propositions in logic and gave direct derivations of the categories from the transcendental ego by dialectical steps. Schelling and Hegel followed Fichte in this shift.[^schluss]
Thirdly, Kant's view that reason gets necessarily entangled in the contradictions of _transcendental dialectic_ by its own nature was interpreted by Hegel as an indication of the positive role of contradictions as the driving force of thought.
[^triad]: Hegel apparently didn't use these terms though, probably through their use by F. Engels who was a student of Schelling, the terms stick to Hegelian dialectic today.
[^trendel]: On the avatars of this synthetic negation from Kant through von Trendelenburg to Neo-Kantianism see Giovanelli ([2015](#Giovanelli15)). The connection between Kant's approach to negative quantities and Hegel's concept of contradiction against the background of the mathematics of their time is developed in the groundbreaking study Wolff ([2010](#Wolff10)) first published in 1981. The importance of this link has been stressed already in the 19th century by [[Karl Rosenkranz]].
[^schluss]: Eventually both distanced themselves from the Fichtean ego as the starting point though. Whereas Hegel gives primacy to 'deduction' in logic (cf. [[absolute conclusion]] - _'Alles Vernünftige ist ein Schluss'_) the philosophy of the mature Schelling with its emphasis on the [[transcendental ideal]] can be partly be seen as a return to the classical Pre-Kantian primacy of 'concept' in logic.
Clearly, these remarks can not do justice to the richness and subtlety of Hegel's logic and should only serve as canvas against which to get a better grasp of Lawvere's conceptual translation. The points to keep in mind from this view are:
* Aufhebung is a _pervasive_ concept: although Lawvere proposes only mathematical definitions for two terms of Hegel's logic, namely _[[unity of opposites]]_ and _Aufhebung_, these are in fact the key terms and go already far in a reconstruction of the whole edifice!
* Aufhebung unites _determinateness_ with annihilation of being. These recur at the mathematical level as the correspondance between being-being a sheaf, annihilation -being the adjoint opposite of sheaf (anti-sheaf), and the determination of Aufhebung as being the least (=the Leibnizian best) level of being simultaneously a sheaf and an anti-sheaf.
This is the original text on _Aufhebung_ from [Hegel 1812, book 1, section 1, chapter 1, C, 3. ](Science+of+Logic#AufhebenDesWerdens):
> Aufheben und das Aufgehobene (das Ideelle) ist einer der wichtigsten Begriffe der Philosophie, eine Grundbestimmung, die schlechthin allenthalben wiederkehrt, deren Sinn bestimmt aufzufassen und besonders vom Nichts zu unterscheiden ist.—Was sich aufhebt, wird dadurch nicht zu Nichts. Nichts ist das Unmittelbare; ein Aufgehobenes dagegen ist ein Vermitteltes, es ist das Nichtseyende, aber als Resultat, das von einem Seyn ausgegangen ist; es hat daher die Bestimmtheit aus der es herkommt, noch an sich.
> Aufheben hat in der Sprache den gedoppelten Sinn, daß es so viel als aufbewahren, erhalten bedeutet, und zugleich so viel als aufhören lassen, ein Ende machen. Das Aufbewahren selbst schließt schon das Negative in sich, daß etwas seiner Unmittelbarkeit und damit einem den äußerlichen Einwirkungen offenen Daseyn entnommen wird, um es zu erhalten.—So ist das Aufgehobene ein zugleich Aufbewahrtes, das nur seine Unmittelbarkeit verloren hat, aber darum nicht vernichtet ist. —Die angegebenen zwei Bestimmungen des Aufhebens können lexikalisch als zwei Bedeutungen dieses Wortes aufgeführt werden. Auffallend müßte es aber dabei seyn, daß eine Sprache dazu gekommen ist, ein und dasselbe Wort für zwei entgegengesetzte Bestimmungen zu gebrauchen. Für das spekulative Denken ist es erfreulich, in der Sprache Wörter zu finden welche eine spekulative Bedeutung an ihnen selbst haben; die deutsche Sprache hat mehrere dergleichen. Der Doppelsinn des lateinischen: tollere (der durch den ciceronianischen Witz tollendum esse Octavium, berühmt geworden) geht nicht so weit, die affirmative Bestimmung geht nur bis zum Emporheben. Etwas ist nur insofern aufgehoben, als es in die Einheit mit seinem Entgegengesetzten getreten ist; in dieser nähern Bestimmung als ein reflektirtes kann es passend Moment genannt werden. Gewicht und Entfernung von einem Punkt heißen beim Hebel, dessen mechanische Momente, um der Dieselbigkeit ihrer Wirkung willen bei aller sonstigen Verschiedenheit eines Reellen, wie das ein Gewicht ist, und eines Ideellen, der bloßen räumlichen Bestimmung, der Linie; s. Encykl. der philos. Wissenschaft 3te Ausg. _ 261. Anm.—Noch öfter wird die Bemerkung sich aufdringen, daß die philosophische Kunstsprache für reflektirte Bestimmungen lateinische Ausdrücke gebraucht, entweder weil die Muttersprache keine Ausdrücke dafür hat, oder wenn sie deren hat, wie hier, weil ihr Ausdruck mehr an das Unmittelbare, die fremde Sprache aber mehr an das Reflektirte erinnert.
## Lawvere's path to _Aufhebung_
> In early 1985, while I was studying the foundations of homotopy theory, it occurred to me that the explicit use of a certain simple categorical structure might serve as a link between mathematics and philosophy. ([Lawvere 1996](#Law96), p.167)
## Extracting the rational kernel
So now let's get down to business and do some mathematics!
### The mathematics of Yin and Yang
In ([Lawvere 2000](#Law00)) a particularly simple example of the [[adjoint cylinder]] was suggested that we use here as a warm up. Note that the categories involved are not toposes and even lack a terminal object!
Let $N$ be the [[natural numbers]] $\{0, 1,\dots\}$ viewed as a [[category]] via their usual [[poset|ordering]]. Let $L,R:N\to N$ be the two parallel [[functors]] '_even_' and '_odd_' defined by $L(n) \coloneqq 2n$ and $R(n) \coloneqq 2n+1$.
Both are [[fully faithful functor|full and faithful]], which means that they correspond to two subcategory inclusions and, accordingly, to two subcategories $N_{even}$ and $N_{odd}$. We are now in situation where we have two subcategories that 'oppose' each other in that $N_{even}\neq N_{odd}$ but are nevertheless 'identical' in that there is a bijection $N_{even}\overset{\simeq}{\to} N_{odd}$. Furthermore, both are 'united' as different parts in the encompassing $N$ whose overall structure they represent in that $N_{even}\simeq N\simeq N_{odd}$ - that is somewhat unusual for what is to follow below where the opposing parts are seldom equivalent to the whole but they will always be a pair consisting of a reflective and a coreflective subcategory.
Now it was Lawvere's observation that a third functor $N\to N$ which with a clin d'oeil to [[Charles Sanders Peirce|C. S. Peirce's]] concept of _thirdness_ we call $T$ , can encapsulate this bunch of relations in one sweep when it forms an [[adjoint triple]] $L\dashv T\dashv R$ with $L$ and $R$:
1. The triple expresses the _unity_ by the idempotency of $(R\circ T)^2=R\circ T$ and $(L\circ T)^2=L\circ T$ typical for (co)reflective subcategories,
1. it expresses the _opposition_ between $L$ and $R$ by an entailed adjunction $L\circ T\dashv R\circ T$,
3. it expresses the _identity_ between $L$ and $R$ by the entailed equivalence $T\circ L\simeq T\circ R$ .
In other words, *$T$ unites, opposes and identifies $L$ and $R$ at the same time*!
For our simple [[poset]] example the [[adjunctions]] $L\dashv T$ and $T\dashv R $ amount to $L(n)\leq m$ iff $n\leq T(m)$ and $n\leq R(m)$ iff $T(n)\leq m$. When $T$ exists it must satisfy $T\circ L \cong id\cong T\circ R$ which in our case just gives $T\circ L = id = T\circ R$. Spelled out this says $T(2n)=n$ and $T(2n+1)=n$ which indicates as definition for $T$:
$$
T(k)=\Bigg\{ \array{\frac{k}{2}\quad k\in N_{even} \\
\frac{k-1}{2} \quad k\in N_{odd}}
$$
Whereas $T\circ L$ and $T\circ R$ are each the identity, the reverse compositions $L\circ T$ and $R\circ T$ yield an [[idempotent comonad]] $sk:N\to N$ and an [[idempotent monad]] $cosk:N\to N$, respectively, where $sk(2n)=2n$ and $sk(2n+1)=2n$ and $cosk(2n)=2n+1$ and $cosk(2n+1)=2n+1$: in new guises $L$ and $R$ resurface again but this time within an 'opposition' $sk\dashv cosk$ which expresses formally the 'conflict' between $N_{even}$ and $N_{odd}$, even and odd, as well as their essential identity and unity.
### The mathematics of _Aufhebung_
{#TheMathematicsOfAufhebung}
For convenience let us briefly recall the following
+-- {: .num_defn #EssentialLocalization}
###### Definition
A [[localization of a category]] $\mathcal{B}$ with [[finite limits]] is a [[reflective subcategory]] $\mathcal{A}$ whose reflection preserves finite limits. The localization is called _[[essential geometric morphism|essential]]_ when the reflection has furthermore a [[left adjoint]].
=--
If $l\dashv r\dashv i$ is an essential localization then $l$ is also [[fully faithful functor|full and faithful]]. If $\mathcal{B}$ is a [[topos]], $\mathcal{A}$ is called an _[[essential geometric morphism|essential]] [[subtopos]]_ and we write $i_!\dashv i^*\dashv i_*$ in this case and call $i_!$ the _essentiality_.
It is a result in ([Kelly-Lawvere 89](#KL89)) that the [[essential geometric morphism|essential]] [[subtoposes]] of a topos form a [[complete lattice]]. Therefore we say:
+-- {: .num_defn #Level}
###### Definition
An [[essential geometric morphism|essential]] [[subtopos]] of $\mathcal{B}$ is referred to as a _[[level of a topos|level]]_ of $\mathcal{B}$ and levels are denoted by small letters $i,j,\dots$ .
=--
An [[adjoint triple]] $i_!\dashv i^*\dashv i_*$ yields two [[adjoint modalities]] $\Box _i\dashv\bigcirc _i$ on $\mathcal{B}$, namely $\Box _i \coloneqq i_!i^*$ and $\bigcirc _i \coloneqq i_*i^*$.
The [[modalities]] yield notions of _[[modal types]]_, which may be called
* the _i-sheaves_ : $X\in\mathcal{B}$ with $\bigcirc _i X\simeq X$ (following the terminology at _[[Lawvere-Tierney operator]]_);
* the _i-skeleta_ : $X\in\mathcal{B}$ with $\Box _i X\simeq X$ (following the example of [[simplicial skeleta]] discussed [below](#SimplicialAndCubicalSets)).
+-- {: .num_defn #Aufhebung}
###### Definition
([Lawvere 1989b](#Law89b))
Let $i,j$ be [[level of a topos|levels]], def. \ref{Level}, of a topos $\mathcal{A}$ we say that the level $i$ is _lower_ than level $j$, written
$$
\array{
\Box_i &\prec & \Box_j
\\
\bot && \bot
\\
\bigcirc_i &\prec & \bigcirc_j
}
$$
(or $i\prec j$ for short) when every i-sheaf ($\bigcirc_i$-[[modal type]]) is also a j-sheaf and every i-skeleton ($\Box_i$-[[modal type]]) is a j-skeleton. This is equivalent to say that both $\bigcirc_j \bigcirc_i =\bigcirc_i$ and $\Box_j \Box_i =\Box_i$.
Let $i\prec j$, we say that the level $j$ _resolves the opposite_ of level $i$, written
$$
\array{
\Box_i &\ll& \Box_j
\\
\bot && \bot
\\
\bigcirc_i &\ll& \bigcirc_j
}
$$
(or just $i\ll j$ for short[^wayb]) if $\bigcirc _j\Box_i=\Box _i$.
Finally a [[level of a topos|level]] $\bar{i}$ is called the _Aufhebung_ of level $i$
$$
\array{
\Box_i &\ll& \Box_{\bar i}
\\
\bot &\searrow& \bot
\\
\bigcirc_i &\ll& \bigcirc_{\bar i}
}
$$
iff it is a minimal level which resolves the opposites of level $i$, i.e. iff $i\ll\bar{i}$ and for any $k$ with $i\ll k$ then it holds that $\bar{i}\leq k$ in the order relation (by subtopos inclusion) between levels.
=--
[^wayb]: $\ll$ is called the _way below_ relation in ([KRRZ11](#KRRZ11)).
+-- {: .num_remark}
###### Remark
The condition $\bigcirc_j \Box_i=\Box_i$ amounts to saying that every $i$-skeleton is a $j$-sheaf:
Suppose the condition holds and $X$ is an $i$-skeleton ($\Box_i X= X$) then $\bigcirc_j X =\bigcirc_j \Box_i X =\Box_i X =X$ i.e. $X$ is a $j$-sheaf. Conversely, if every $i$-skeleton is a $j$-sheaf then, since by the idempotency of $\Box_i$ $i$-skeleta are precisely the objects of form $\Box_i X$ for some $X$, $\Box_i X$ is by assumption a $j$-sheaf and that's precisely what $\bigcirc_j \Box_i X=\Box_i X$ asserts.
The resolution condition $\bigcirc_j \Box_i=\Box_i$ ensures that $i$-skeleta are in the intersection of the $j$-skeleta and $j$-sheaves at the resolving level.
Thinking of $\bigcirc_i\Box_i=\bigcirc_i$ as an expression of the negation of $\Box_i$ by $\bigcirc_i$ one could think of $\bigcirc_j \Box_i=\Box_i$ as dialogical refinement of the opposition through a _negation of the negation_: 'Proponent' $\bigcirc_i$ gets updated - _sublated_ to $\bigcirc_j$ in order to absorb the 'opponent' $\Box_i$.
Note that $\bigcirc_j \Box_i=\Box_i$ does not imply $\Box_j \bigcirc_i =\bigcirc_i$ , e.g. in the [[Sierpinski topos]] $Set^\to$ the level $\emptyset\dashv\ast$ is resolved by $\nabla\dashv ʃ$ but $(\nabla\circ\ast)\neq\ast$ (see [below](#SierpExample)).
We say that $j$ _co-resolves_ $i$ if $\Box_j \bigcirc_i =\bigcirc_i$. If $j$ resolves and co-resolves $i$ we say that $j$ bi-resolves $i$. In the latter case, all $i$-sheaves and $i$-skeleta are simultaneously $j$-sheaves and $j$-skeleta at the higher level $j$.
=--
+-- {: .num_remark}
###### Remark
The Aufhebung of a level is the smallest level that resolves its opposites or contradictions. Such a level need not exist in general for every level but in certain cases like [[presheaf toposes]] over [[graphic category|graphic categories]] or, more generally, over [[von Neumann regular categories]] ([Lawvere 2002](#Law02)), it does. The Aufhebungs relation is also called the _jump operator_ in [Lawvere (2009)](#Law09).
=--
+-- {: .num_remark}
###### Remark
Comparing with [[Science of Logic|WdL]] under Lawvere's translation and identifing the [[level of a topos|levels]] with logical categories of thinking in the ordinary sense (thought determinations - _Gedankenbestimmungen_), one sees that $\emptyset\dashv\ast$ corresponds to Hegel's logical category of _indeterminate being_ whereas the higher levels correspond to logical categories of _determinate being_ - _Bestimmtheit_.
Furthermore one sees that the subtoposes corresponding to the levels trace out as _mathematical_ categories the _logical_ categories of thought as envisioned by Hegel thereby corroborating the terminological choices of Eilenberg and Mac Lane made in their 1945 paper!
=--
+-- {: .num_remark}
###### Remark
We can use the definition to try to shed some light on the apparently rather odd contention by Hegel that the method of logic is analytic and synthetic at the same time:
> Der Gang oder die Methode des absoluten Wissens ist ebensosehr analytisch als synthetisch. Die Entwicklung dessen, was im Begriff enthalten ist, die Analysis, ist das Hervorgehen verschiedener Bestimmungen, die im Begriff enthalten sind, somit zugleich synthetisch. ([Begriffslehre für die Mittelklasse (1809/10)](#Begriff), p.161)
We can think of the inclusion of the sheaf category of a lower level into the higher sheaf category as an _analytic_ relation between the concepts involved: when viewed as a relation between the [[geometric theories]] classified by the respective subtoposes an inclusion relation corresponds indeed to an unpacking of the richer theory of the smaller subtopos e.g. the subtopos corresponding to the theory of [[local ring|local rings]] is included in the topos corresponding to the theory of rings which on the conceptual side is spelled out as _a local ring is a ring_, or, the concept 'local ring' implies the concept 'ring'. So the passage from subtopos to including supratopos corresponds to an unfolding of the concepts implied in the subtopos concept.
This analytic procedure seems close to the 'analytical reading' of Hegel's dialectic as a refinement of meaning postulates proposed by F. Berto e.g. in ([Berto 2007](#Berto07)).
Whereas on the _synthetic_ side, by demanding _essentiality_ of the subtoposes we get at each level skeletal 'determinations' corresponding to features _not_ contained in the concept on the sheaf side which by the resolution condition nevertheless get synthesized into the sheaf side on the higher levels.
=--
## Examples
### Aufhebung of Becoming -- Determinate being
{#AufhebungOfBecoming}
#### From Faust's study
In the context of a **[[category of being]]**, aka a (sufficiently) [[cohesive topos]], which has a connected [[subobject classifier]] $\Omega$ and product preserving components functor $\Pi _0$, there is an opposition
$$
\empty\dashv \ast
$$
between _[[nothing|non being]]_ (the [[idempotent comonad]] constant on the [[initial object]]) and _[[being|pure being]]_ (the idempotent comonad constant in the [[terminal object]]) whose Aufhebung is (at least in suitable cases, see [below](#ExamplesBecomingFormalization)) the [[unity of opposites|opposition]] of [[becoming]]
$$
\flat\dashv \sharp
$$
given by [[flat modality]] $\dashv$ [[sharp modality]], between _non-becoming_ vs. _pure becoming_ (cf. Lawvere 1989a, 1989b, [1991a](#Law91a))[^Pure]. This is what in [SoL § 191](Science+of+Logic#DaseinUberhaupt) is called _[[determinate being]]_ as it corresponds to localization at $\neg\neg$ and the double negation creates the determinateness of the 'Etwasse'.
[^Pure]: Lawvere suggests more generally to read $\Box_ i\dashv\bigcirc_i$ as an opposition _non-F_ vs. _pure-F_ where $F$ is a property descriptively appropriate for the level.
In terms of [[topos theory]] the Aufhebungs-condition $\sharp \emptyset \simeq \emptyset$ says equivalently that the [[subtopos]] of $\sharp$-[[modal objects]] is a [[dense subtopos]].
This lowest essential subtopos arises more generally for categories $\mathcal{A}$ with [[initial object|initial]] and [[terminal objects]], via the adjoints to $\mathcal{A}\to \{*\}$ that map $*$ to $0$ and $1$. Especially, the imposition of conditions that ensure the existence of $\flat\dashv \sharp$ can be viewed as intended to provide a specific resolution of the 'identity' $0=1$, the indeterminate confluence of truth and falsity at the lowest level which [[syntax|syntactically]] corresponds to the inconsistent [[geometric theory]].
Following Lawvere's suggestive terminology and identifying a level with its sheaf part, we could somewhat more loosely say that [[becoming]] is the Aufhebung of the opposition between [[nothing]] and [[being]], or more shortly, that _becoming is the Aufhebung of being_.
The Aufhebungs relation expresses precisely that the (positive) sheaf part of the higher level $j$ subsumes (the opposition between) the skeleton and the sheaf part of the lower level in a universal way - it is the smallest context in which negative and positive poles of the lower level can positively coexist. To elaborate this intuition somewhat, it is the minimal way to turn the negative part into a positive part yet retaining the positivity of its positive opposite.
For more on the relevant _metaphysical_ modalities see at [[adjoint modality]].
#### Over cohesive sites
{#ExamplesBecomingFormalization}
We discuss Aufhebung of [[becoming]] in the above sense in [[cohesive toposes]] ([[cohesive (∞,1)-toposes]]) with a [[cohesive site]] ([[∞-cohesive site]]) of definition.
(More general discussion is now also in [Lawvere-Menni 15, lemma 4.1, 4.2](#LawvereMenni15), [Shulman 15, section 3](#Shulman15), see at [pieces-to-points-transform -- Relation to Aufhebung](points-to-pieces+transform#RelationToAufhebung)).
+-- {: .num_prop #AufhebungOfBecomingMeansOnlyInitialObjectHasNoGlobalPoints}
###### Proposition
Given a [[topos]] equipped with a [[level of a topos]] given by an [[adjoint modality]] $(\Box\dashv \bigcirc) \coloneqq (\flat \dashv \sharp)$, then the condition $\sharp \emptyset \simeq \emptyset$ is equivalent to $(\flat X \simeq \emptyset) \Leftrightarrow (X \simeq \emptyset)$.
=--
+-- {: .proof}
###### Proof
In a topos the [[initial object]] $\emptyset$ is a [[strict initial object]], and hence $(X \simeq \emptyset) \simeq (X \to \emptyset)$.
In one direction, assuming $\sharp \emptyset \simeq \emptyset$ then
$$
\begin{aligned}
(X \simeq \emptyset)
& \simeq
(X \to \emptyset)
\\
& \simeq (X \to \sharp \emptyset)
\\
& \simeq (\flat X \to \emptyset)
\\
& \simeq (\flat X \simeq \emptyset)
\end{aligned}
\,.
$$
Conversely, assume that $(\flat X \simeq \emptyset) \Leftrightarrow (X \simeq \emptyset)$. Then for all $X$
$$
\begin{aligned}
(X\to \emptyset)
& \simeq
(X\simeq \emptyset)
\\
& \simeq (\flat X \simeq \emptyset)
\\
& \simeq (\flat X \to \emptyset)
\\
& \simeq (X\to \sharp \emptyset)
\end{aligned}
$$
and hence by the [[Yoneda lemma]] $\emptyset \simeq \sharp \emptyset$.
=--
+-- {: .num_example}
###### Example
In the [[Sierpinski topos]] $Set^{\to}$ with objects maps $X\to Y$ between sets $X,Y$, the initial object is $\emptyset\to\emptyset$ and the respective adjoint modalities are given by $\sharp(X\to Y)=X\to 1$ and $\flat(X\to Y)=X\overset{id}{\to} X$. Since not only $\flat(\emptyset\to\emptyset)=\emptyset\to\emptyset$ but also $\flat(\emptyset\to Y)=\emptyset\to\emptyset$, we find that $\flat\dashv \sharp$ does not resolve $\emptyset\dashv\ast$ (we expand on this example [below](#SierpExample)).
=--
+-- {: .num_cor}
###### Corollary
An [[(∞,1)-topos]] with Aufhebung $(\flat \dashv \sharp)$ of [[being]] has [[homotopy dimension]] $\leq 0$ with respect to the $\flat$-[[modal types|modal]] [[base (∞,1)-topos]].
=--
+-- {: .num_prop #OverCohesiveSiteBecomingIsResolved}
###### Proposition
Let $\mathcal{S}$ be a [[cohesive site]] (or [[∞-cohesive site]]) and $\mathbf{H} = Sh(\mathcal{S})$ its [[cohesive topos|cohesive]] [[sheaf topos]] (or $\mathbf{H} = Sh_\infty(S)$ its [[cohesive (∞,1)-topos]] ).
Then in $\mathbf{H}$ we have $\sharp \emptyset \simeq \emptyset$, hence that $(\flat \dashv \sharp)$ resolves, def. \ref{Aufhebung}, the [[unity of opposites]] $(\emptyset \dashv \ast)$ which is [[becoming]].
=--
+-- {: .proof}
###### Proof
The [[flat modality]] $\flat$ in this case is given by forming [[global sections]] and re-embedding the resulting [[set]] as a [[constant sheaf]].
Since by assumption $\mathcal{S}$ has a [[terminal object]] $\ast$, it follows that for $X\in \mathbf{H}$ any sheaf $X \colon \mathcal{S}^{op}\to Set$ then
$$
\flat X \simeq X(\ast)
$$
(where we may leave the constant re-embedding implicit, due to it being [[fully faithful functor|fully faithful]]).
Moreover by assumption, for every object $U\in \mathcal{S}$ there exists a morphism $i \colon \ast \to U$ hence for every $X\in \mathbf{H}$ and every $U$ there exists a morphism $i^\ast \colon X(U)\to \flat X$. This means that if $\flat X \simeq \emptyset$ then $X(U) \simeq \emptyset$ for all $U \in \mathcal{S}$ and hence $X\simeq \emptyset$. From this the claim follows with prop. \ref{AufhebungOfBecomingMeansOnlyInitialObjectHasNoGlobalPoints}.
=--
+-- {: .num_prop #OverCohesiveSiteBecomingIsAufgehoben}
###### Proposition
Let $\mathcal{S}$ be a [[cohesive site]] (or [[∞-cohesive site]]) and $\mathbf{H} = Sh(\mathcal{S})$ its [[cohesive topos|cohesive]] [[sheaf topos]] with values in [[Set]] (or $\mathbf{H} = Sh_\infty(S)$ its [[cohesive (∞,1)-topos]] ).
Then in $\mathbf{H}$ we have Aufhebung, def. \ref{Aufhebung}, of the [[duality of opposites]] of [[becoming]] $\emptyset \dashv \ast$ ("[[Dasein]]").
$$
\array{
\flat &\dashv& \sharp
\\
\vee &\nearrow& \vee
\\
\emptyset &\dashv& \ast
}
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{OverCohesiveSiteBecomingIsResolved} we have that $(\flat\dashv \sharp)$ resolves $(\emptyset \dashv \ast)$ and so it remains to see that it is the minimal [[level of a topos|level]] with this property. But the [[subtopos]] of [[sharp modality|sharp]]-[[modal types]] is $\simeq$ [[Set]] which is clearly a [[two-valued topos|two-valued]] [[Boolean topos]]. By [this proposition](subtopos#BooleantoposesAreAtoms) these are the [[atoms]] in the [[subtopos lattice]] hence are minimal as non-trivial subtoposes and hence also as non-trivial [[level of a topos|levels]].
=--
+-- {: .num_remark #OnDoubleNegation}
###### Remark
As mentioned above, the Aufhebung of $\emptyset\dashv \ast$ is necessarily given by a _[[dense subtopos|dense]]_ subtopos $\mathcal{E}_j$. Since the [[double negation topology]] $\neg\neg$ is the unique largest dense topology it follows in general that $\mathcal{E}_{\neg\neg}\subseteq\mathcal{E}_j$ , in particular in the case that $\mathcal{E}_{\neg\neg}$ happens to be [[essential geometric morphism|essential]] and hence happens to be a [[level of a topos|level]], the minimality condition on the Aufhebung of the initial opposition means that $\mathcal{E}_j = \mathcal{E}_{\neg\neg}$ is, in particular, a [[Boolean topos]].
=--
A special case of this are toposes $\mathcal{E}$ such that $\mathcal{E}_{\neg\neg}$ is [[open subtopos|open]] whence essential in particular; these are called $\bot$-scattered toposes. For the record we state:
**Proposition.**
Let $\mathcal{E}$ be a $\bot$-[[scattered topos]]. The Aufhebung of $\emptyset\dashv\ast$ is given by $\mathcal{E}_{\neg\neg}$. $\qed$
Another consequence is that the Aufhebung $\mathcal{E}_j$ of $\emptyset\dashv\ast$ is [[Boolean topos|Boolean]] precisely when $\mathcal{E}_{\neg\neg}$ is essential e.g. for Boolean $\mathcal{E}$ this happens trivially and accordingly the Aufhebung of $\emptyset\dashv \ast$ is $\mathcal{E}$ in this case.
It also follows from the [above proposition](#OverCohesiveSiteBecomingIsAufgehoben) that for [[cohesive sites]] over [[Set]]: $\mathbf{H}_{\neg\neg}=Set$ i.e. the double negation topos coincides with the base.
+-- {: .num_example}
###### Example
Examples of $\infty$-toposes satisfying the assumptions of prop. \ref{OverCohesiveSiteBecomingIsAufgehoben} and hence exhibiting Aufhebung of becoming include
* [[∞-groupoids]]
* [[Euclidean-topological ∞-groupoids]]
* [[smooth ∞-groupoids]]
* [[formal smooth ∞-groupoids]]
* [[super ∞-groupoids]]
* [[smooth super ∞-groupoids]]
Here [[formal smooth ∞-groupoid]] has its [[cohesion]] further refined to [[differential cohesion]], yielding
$$
\array{
id & \dashv & id
\\
\vee && \vee
\\
\Re &\dashv& ʃ_{inf} &\dashv& \flat_{inf}
\\
&& \vee && \vee
\\
&& ʃ &\dashv& \flat &\dashv& \sharp
\\
&& && \vee &\nearrow& \vee
\\
&& && \emptyset &\dashv& \ast
}
$$
=--
### Absolute identity and _Selbstaufhebung_
Given a topos $\mathcal{E}$ the highest level is always given by $id_\mathcal{E}$ yielding the trivial [[adjoint string]] $id_\mathcal{E}\dashv id_\mathcal{E}\dashv id_\mathcal{E}$ with no less trivial adjoint modalities $id_\mathcal{E}\dashv id_\mathcal{E}$. Obviously, the sheaves and skeleta for this level coincide. Since the corresponding subtopos is simply $\mathcal{E}$, this level _resolves every other level_ and suggests to view the ascension from $\emptyset\dashv\ast$ to $id_\mathcal{E}\dashv id_\mathcal{E}$ as a process of increasing stepwise the number of objects that are sheaves as well as skeleta at a given level. The definition of resolution ensures that a level inherits the objects in the intersection from lower levels but also that all non-sheaves from the lower levels will be henceforth in the intersection.
Note that though $id_\mathcal{E}$ resolves every level it need not be the Aufhebung of any strictly lower level. This situation occurs e.g. in the examples from combinatorial topology discussed [below](#SimplicialAndCubicalSets): Here the levels correspond to the geometrical dimension of the 'triangulated spaces' involved plus the highest level 'at infinity' and the Aufhebung is a simple numerical relation between finite levels e.g. in the topos of ball complexes [[Roy (1997)](#Roy97)] the Aufhebung of level $n$ is $n+1$. Of course, $id_\mathcal{E}$ is always its own Aufhebung and we see, incidentally, that a level might be the Aufhebung of more than one level, namely e.g. itself and perhaps several other levels - this might happen with $id_\mathcal{E}$ when the lattice of levels is finite e.g. in the [[Sierpinski topos]] $Set^{\to}$ (see [below](#SierpExample)).
Though $id_\mathcal{E}$ is trivial from a mathematical point of view, paradoxically it nevertheless captures on the philosophical side one of the most enigmatic concepts of the early _identity philosophy_ of Hegel and Schelling, namely the **absolute** as the identity of identity and non-identity:
>Das Absolute selbst aber ist darum die Identität der Identität und der Nichtidentität; Entgegensetzen und Einssein ist zugleich in ihm. (Hegel [1801](#Diff), p.96)
We might call a topos $\mathcal{E}$ with the property that $id_\mathcal{E}$ is the Aufhebung of no other level than itself _absolute_ and $id_\mathcal{E}$ the absolute level. Note that these toposes will occasionally have infinitely many levels and are then from the perspective of the mature Hegel vulnerable to the charge of being a _'bad infinity'_.
The discussion so far might suggest that being self-sublating is a property peculiar to $id_\mathcal{E}$ but this is not the case. In fact $id_\mathcal{E}$ is only an instance of a whole class of essential localizations enjoying this property:
+-- {: .num_defn #quintessentialloc}
###### Definition
Let $\mathcal{C}$ be a finitely complete category. An [[level|essential localization]] $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ is called **quintessential** if $l$ is naturally isomorphic to $i$.
=--
To say that $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ is a quintessential localization amounts to say in Lawvere's terminology ([2007](#07)) that $i:\mathcal{L}\to\mathcal{C}$ exhibits $\mathcal{C}$ as a [[quality type]] over $\mathcal{L}$ with $r$ providing the right adjoint to $i\simeq l$ (provided $\mathcal{L}$, $\mathcal{C}$ are [[extensive category|extensive]]).
The following is immediate:
+-- {: .num_prop #quintAuf}
###### Proposition
Let $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ be a quintessential localization. Then the corresponding [[adjoint modality]] $l\cdot r\dashv i\cdot r$ coincides up to natural isomorphism and provides its own Aufhebung. $\qed$
=--
+-- {: .num_example}
###### Example
A simple _example_ of a non-trivial quintessential localization is given by the category $\mathcal{C}$ with objects pairs $(X, e)$ where $X$ is a set and $e=e^2$ an idempotent map $X\to X$. A morphism $f:(X_1, e_1)\to (X_2, e_2)$ is a function $f:X_1\to X_2$ with $f\cdot e_1=e_2\cdot f$. These equivariant morphisms are bound to preserve fixpoints: when $e_1(x)=x$ then $f(e_1(x))=f(x)=e_2(f(x))$. Then the fixpoint set functor $r:\mathcal{C}\to Set$ with $r(X, e)=\{x\in X | e(x)=x \}$ is left as well as right adjoint to $i(X)=(X, id_X)$ since an equivariant morphism $f:(X,e)\to (Y,id_Y)$ is uniquely determined by its restriction to the fixpoints of $e$ and its values are given by $f(e(x))$. The [[adjoint modality]] $i\cdot r\dashv i\cdot r:\mathcal{C}\to\mathcal{C}$ corresponding to $i\dashv r\dashv i:Set\to\mathcal{C}$ maps $(X,e)$ to $(r(X),id_{r(X)})$. The corresponding level sublates $\emptyset\dashv\ast$ as well as itself whereas $id_\mathcal{E}$ only sublates itself.
=--
For further properties of quintessential localizations see at [[quality type]].
### The example of the Sierpinski topos{#SierpExample}
(...)
### Simplicial and cubical sets
{#SimplicialAndCubicalSets}
(...) [[simplicial set]] (...)
[[simplicial skeleton]] $\dashv$ [[simplicial coskeleton]]
(...) [[cubical set]] (...)
([Kennett-Riehl-Roy-Zaks (2011)](#KRRZ11))
### An open problem: the presheaf topos over non-empty finite sets
...
## Related pages
* [[Hegelian taco]]
* [[Science of Logic]]
* [[adjoint modality]]
* [[adjoint logic]]
* [[adjoint triple]]
* [[level of a topos]]
* [[negative moment]]
* [[quality type]]
* [[graphic category]]
* [[Attempt to Introduce the Concept of Negative Quantities into Philosophy|Concept of Negative Quantities]]
* [[construction in philosophy]]
* [[infinite judgement]]
* [[co-Heyting boundary]]
* [[homotopy dimension]]
## Links
* [Online Journal of the Brazilian Hegel Society](http://revista.hegelbrasil.org/english-version-introduction/)
## A guide to the literature
On the philosophical side, the lectures notes [Koch (2009)](#Koch09) that suggest the use of [[pure set|non-wellfounded set theory]] as interpretative tool may serve as a good general introduction to Hegel's ideas on logic and metaphysics. [Krahn (2014)](#Krahn14) considers Hegel's concept of Aufhebung in the context of postmodern thought. For lucid accounts of Hegel's concept of dialectics in general consult Brauer ([1982](#Brauer82)) and Wolff ([2010](#Wolff10)) the latter highlighting Hegel as 'a philosopher of mathematics' in this context. Wegerhoff ([2008](#Wegerhoff08)) offers an interesting structural account of the dialectics inspired by Dedekind's theory of natural numbers.
On the mathematical side, the book by [La Palme-Reyes-Zolfaghari (2004)](#RRZ04) provides a good general entry to the 'mathematics of Lawvere' from an elementary point of view and contains even a page on the adjoint cylinder. Goldblatt's book on [[topos theory]] ([Goldblatt 1984](#Goldblatt84)) covers a good deal of ground yet stays accessible and is available online.
Lawvere introduced the Hegelian concepts in Lawvere ([1989b](#Law89b)). They get some attention in Lawvere ([1991a](#Law91a),[1992](#Law92),[1994a](#Law94a)) with the second containing his 'philosophical program'. By all means have a look at [Lawvere (1996)](#Law96), this together with Lawvere ([1989a](#Law89a),[1999](#Law99)) exposes his ideas on homotopy theory. The work on graphic toposes ([1989b](#Law89b),[1991b](#Law91b),[2002](#Law02)) concerns the _Aufhebungs_ relation with the latter containing a discussion of the relevant concepts. [Kelly-Lawvere (1989)](#KL89) provides the technical prerequisites on essential localizations for _Aufhebung_.
The known mathematical results on the Aufhebungs relation are contained in the paper by [Kennett-Riehl-Roy-Zaks (2011)](#KRRZ11) which is based on older phd-works by some of the authors (e.g. [Roy (1997)](#Roy97)).
Further results on essential localizations can be found in the papers by [Borceux-Korotenski (1991)](#BK91), [Johnstone (1996)](#JS96), [Vitale (2001)](#VT01) and [Lucyshyn-Wright (2011)](#LW11) or in [SGA 4](#SGA4).
***
<img src="http://ncatlab.org/nlab/files/AufhebungDetail.jpg" width="900">
([full pdf](http://www.in-situ-art-society.com/docs/2015-09-18+19_aufhebung_flyer.pdf) by [In Situ Art Society](http://www.in-situ-art-society.com/aufhebung.html))
***
##References
* {#SGA4} [[Michael Artin|M.Artin]], [[Alexander Grothendieck|A.Grothendieck]], [[J. L. Verdier]] (eds.), *Théorie des Topos et Cohomologie Etale des Schémas - SGA 4*, LNM **269** Springer Heidelberg 1972. (sec. IV 7.6., pp.414-416)
* [[John Baez|J. C. Baez]], [[Mike Shulman|M. Shulman]], *Lectures on n-categories and cohomology*, pp.1-68 in J. C. Baez, P. May (eds.), *Towards Higher Categories*, Springer Heidelberg 2010. ([preprint](http://math.ucr.edu/home/baez/cohomology.pdf))
‎
* {#Berto07} F. Berto, *Hegel's Dialectic as a Semantic Theory: An Analytical Reading*, European Journal of Philosophy **15** no.1 (2007) pp.19-39. ([philpapers](http://philpapers.org/rec/BERHDA))
* {#BK91}[[Francis Borceux|F. Borceux]], M. Korostenski, *Open Localizations*, JPAA **74** (1991) pp.229-238.
* {#Brandom04} Robert Brandom, _Selbstbewusstsein und Selbstkonstitution_ , pp.46-77 in Halbig, Quante, Siep (eds.), _Hegels Erbe_ , Suhrkamp Frankfurt am Main 2004.
* {#Brauer82} Oscar Daniel Brauer, _Dialektik der Zeit - Untersuchungen zu Hegels Metaphysik der Weltgeschichte_ , frommann-holzboog Stuttgart 1982.
* J. Climent Vidal, J. Soliveres Tur, *Functors of Lindenbaum-Tarski, Schematic Interpretations, and Adjoint Cylinders between Sentential Logics*, Notre Dame Journal of Formal Logic **49** no.2 (2008) pp.185-202. ([pdf](https://projecteuclid.org/download/pdfview_1/euclid.ndjfl/1210859927))
‎
* H. F. Fulda, *Aufheben*, pp.318-320 in Ritter (ed.), *Historisches Wörterbuch der Philosophie*, Schwabe Basel 1971. ([Heidi](http://archiv.ub.uni-heidelberg.de/volltextserver/16307/))
* {#Giovanelli15} M. Giovanelli, *Trendelenburg and the Concept of Negation in Post-Kantian Philosophy*, to appear in Munk (ed.), *Proceedings of the Amsterdam 2010 Colloquium: Natur des Denkens und das Denken der Natur: Spinoza, Trendelenburg und H. Cohen*. ([draft](https://www.dropbox.com/s/mxm4db8id7je924/Giovanelli%2C%20Marco%20-%20Trendenleburg%20and%20the%20Concept%20of%20Negation%20in%20Post-Kantian%20Philosophy%20Proceedings%20of%20the%20Connference.pdf?dl=0))
* {#Goldblatt84} R. Goldblatt, *Topoi - The Categorical Analysis of Logic*, 2nd ed. North-Holland Amsterdam 1984. (Dover reprint New York 2006; [project euclid](http://projecteuclid.org/euclid.bia/1403013939))
* {#Diff} [[G. W. F. Hegel]], *Differenz des Fichte'schen und Schelling'schen Systems der Philosophie*, pp.7-138 in Moldenhauer, Michel (eds.), *Werke 2*, Suhrkamp Frankfurt 1986[1801].
* {#Begriff} [[G. W. F. Hegel]], *Begriffslehre für die Mittelklasse (1809/10)*, pp.139-162 in Moldenhauer, Michel (eds.), *Werke 4*, Suhrkamp Frankfurt 1986.
* {#WdL} [[G. W. F. Hegel]], *[[Science of Logic|Wissenschaft der Logik]] I*, Suhrkamp Frankfurt 1986[1812/13; revised 1831].
* {#JS96}[[Peter Johnstone|P. Johnstone]], *Remarks on Quintessential and Persistent Localizations*, TAC **2** no.8 (1996) pp.90-99. ([pdf](http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf))
* {#KL89} [[G. M. Kelly]], [[F. W. Lawvere]], *On the Complete Lattice of Essential Localizations*, Bull. Soc. Math. de Belgique **XLI** (1989) 289-319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]
* {#KRRZ11} C. Kennett, [[Emily Riehl]], [[Michael Roy]], M. Zaks, *Levels in the toposes of simplicial sets and cubical sets*, JPAA **215** 5 (2011) 949-961 [[arXiv:1003.5944](http://arxiv.org/abs/1003.5944), [doi:j.jpaa.2010.07.002](https://doi.org/10.1016/j.jpaa.2010.07.002)]
* {#Koch09} A. F. Koch, *Hegel's Science of Logic*, Lectures Emory University 2009. ([pdf](http://www.philosophie.uni-hd.de/md/philsem/personal/koch_hegel_science_of_logic_winter_2016.pdf))
* {#Krahn14} R. Krahn, *The Sublations of Dialectics: Hegel and the Logic of Aufhebung*, PhD Guelf University Ontario 2014. ([link](https://atrium.lib.uoguelph.ca/xmlui/handle/10214/8222))
* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, *Generic Figures and their Glueings*, Polimetrica Milano 2004.
* {#Law89a} [[F. W. Lawvere]], *Qualitative Distinctions between some Toposes of Generalized Graphs*, Cont. Math. **92** (1989) pp.261-299.
* {#Law89b} [[F. W. Lawvere]], *Display of graphics and their applications, as exemplified by 2-categories and the Hegelian "taco"*, Proceedings of the first international conference on algebraic methodology and software technology University of Iowa, May 22-24 1989, Iowa City, pp.51-74.
* {#Law91a} [[F. W. Lawvere]], *[[Some Thoughts on the Future of Category Theory]]*, pp.1-13 in LNM **1488** Springer Heidelberg 1991.
* {#Law91b} [[F. W. Lawvere]], *More on Graphic Toposes*, Cah. Top. Géom. Diff. Cat. **XXXII** no.1 (1991) pp.5-10. ([pdf](archive.numdam.org/article/CTGDC_1991__32_1_5_0.pdf))
‎
* {#Law92} [[F. W. Lawvere]], _Categories of Space and Quantity_, pp.14-30 in: J. Echeverria et al (eds.), *The Space of mathematics*, de Gruyter Berlin 1992.
* {#Law94a} [[F. W. Lawvere]], *Cohesive Toposes and Cantor's 'lauter Einsen'*, Phil. Math. **2** no.3 (1994) pp.5-15.
* {#Law94b} [[F. W. Lawvere]], _Tools for the Advancement of Objective Logic: Closed Categories and Toposes_, pp.43-56 in: J. Macnamara, G. E. Reyes (eds.), *The Logical Foundations of Cognition*, Oxford UP 1994.
* {#Law96} [[F. W. Lawvere]], *[[Unity and Identity of Opposites in Calculus and Physics]]*, App. Cat. Struc **4** (1996) pp.167-174.
* {#Law99} [[F. W. Lawvere]], *Kinship and Mathematical Categories*, pp.411-425 in: R. Jackendoff, P. Bloom, K. Wynn (eds), _Language, Logic, and Concepts - Essays in Memory of John Macnamara_, MIT Press 1999.
* {#Law00} [[F. W. Lawvere]], *Adjoint Cylinders*, message to catlist November 2000. ([link](http://permalink.gmane.org/gmane.science.mathematics.categories/1683))
* {#Law02} [[F. W. Lawvere]], *Linearization of graphic toposes via Coxeter groups*, JPAA **168** (2002) pp.425-436.
* {#Law07} [[F. W. Lawvere]], *Axiomatic cohesion*, TAC **19** no.3 (2007) pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))
* {#Law09} [[F. W. Lawvere]], *Open Problems in Topos Theory*, ms. (2009). ([pdf](http://cheng.staff.shef.ac.uk/pssl88/lawvere.pdf)) {#Law09}
* {#Law13} [[F. W. Lawvere]], *Combinatorial Topology*, message to catlist November 2013. ([link](http://comments.gmane.org/gmane.science.mathematics.categories/7920))
* [[F. William Lawvere]], _Toposes generated by codiscrete objects in combinatorial topology and functional analysis_, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, [tac](http://tac.mta.ca/tac/reprints/articles/27/tr27abs.html).
* {#LawvereMenni15} [[F. W. Lawvere]], [[Matías Menni|M. Menni]], *Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness*, TAC **30** no. 26 (2015) pp.909-932. ([tac:30-26](http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html))
* {#LW11}[[Rory Lucyshyn-Wright|R. Lucyshyn-Wright]], *Totally Distributive Toposes*, arXiv.1108.4032 (2011). ([pdf](http://arxiv.org/pdf/1108.4032v3))
* {#MM19}[[Francisco Marmolejo|F. Marmolejo]], [[Matías Menni|M. Menni]], *Level $\epsilon$* , arXiv:1909.12757 (2019). ([abstract](https://arxiv.org/abs/1909.12757))
* {#Menni09} [[Matías Menni|M. Menni]], *Algebraic Categories whose Projectives are Explicitly Free*, TAC **22** no.29 (2009) pp.509-541. ([pdf](http://www.tac.mta.ca/tac/volumes/22/20/22-20.pdf))
* {#Menni12} [[Matías Menni|M. Menni]], *Bimonadicity and the Explicit Base Property*, TAC **26** no.22 (2012) pp.554-581. ([pdf](http://www.tac.mta.ca/tac/volumes/26/22/26-22.pdf))
* {#Menni19} [[Matías Menni|M. Menni]], *Monic skeleta, Boundaries, Aufhebung, and the meaning of 'one-dimensionality'*, TAC **34** no.25 (2019) pp.714-735. ([tac](http://www.tac.mta.ca/tac/volumes/34/25/34-25abs.html))
* {#Menni22} [[Matías Menni|M. Menni]], *Maps with Discrete Fibers and the Origin of Basepoints*, Appl. Categor. Struct. **30** (2022) 991–1015 [[doi:10.1007/s10485-022-09680-2](https://doi.org/10.1007/s10485-022-09680-2)]
* {#Menni23} [[Matías Menni|M. Menni]], *The successive dimension, without elegance* [[arXiv:2308.04584](https://arxiv.org/abs/2308.04584)]
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* J. P. Pertille, *Aufhebung - Meta-categoria da Lógica Hegeliana*, Revista Eletrônica Estudos Hegelianos **8** no.15 (2011) pp.58-66. ([pdf](http://www.hegelbrasil.org/reh_2011_2_art4.pdf))
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[^LQ]: ([Lawvere 1989](#Law89b), p.74). |
augmentation | https://ncatlab.org/nlab/source/augmentation |
#Contents#
* table of contents
{:toc}
## Idea
An **augmentation** of a [[simplicial set]] or generally a [[simplicial object]] $S_\bullet$ is a [[homomorphism]] of simplicial objects to a simplicial object constant ([[discrete object|discrete]]) on an object $A$:
$$
\epsilon \colon S_\bullet \to A
\,.
$$
Equivalently this is an _[[augmented simplicial set|augmented simplicial object]]_, namely a [[diagram]] of the form
$$
\array{
\cdots S_2 \stackrel{\to}{\stackrel{\to}{\to}} S_1 \stackrel{\to}{\to} S_0 \stackrel{\epsilon_0}{\to} A
}
$$
(showing here only the face maps).
Under the [[Dold-Kan correspondence]] this yields:
The **augmentation of a chain complex** $V_\bullet$ (in non-negative degree) is a [[chain map]]
$$
\epsilon \colon V_\bullet \to A
\,.
$$
If $V_\bullet$ and $A$ are equipped with [[algebra]]-[[structure]] ($V$ might be an [[augmented algebra]] over $A$), then the [[kernel]] of the augmentation map is called the **[[augmentation ideal]]**.
## Related concepts
* [[reduced homology]]
* [[augmented algebra]], [[augmented A-infinity algebra]]
* [[augmentation ideal]]
[[!redirects augmentations]].
|
augmentation ideal | https://ncatlab.org/nlab/source/augmentation+ideal |
+-- {: .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
For $R \hookrightarrow A$ an [[associative algebra]] over a [[ring]] $R$ equipped with the structure of an [[augmented algebra]] $\epsilon \colon A \to R$, the _augmentation ideal_ is the [[kernel]] of $\epsilon$.
Specifically for $G$ a [[group]], and $R[G]$ its [[group algebra]] over a [[ring]] $R$, the _augmentation ideal_ is the [[ideal]] in $R[G]$ which consists of those [[formal linear combinations]] over $R$ of elements in $G$ whose sum of [[coefficients]] vanishes in $R$.
## Examples
### For group algebras
Let $G$ be a [[discrete group]] and $R$ a [[ring]]. Write $R[G]$ for the [[group algebra]] of $G$ over $R$.
+-- {: .num_defn }
###### Definition
Write
$$
\epsilon \colon \mathbb{Z}[G] \to \mathbb{Z}
$$
for the [[homomorphism]] of [[abelian groups]] which forms the sum of $R$-[[coefficients]] of the [[formal linear combinations]] that constitute the group ring
$$
\epsilon \colon r \mapsto \sum_{g \in G} r_g
\,.
$$
This is called the [[augmentation]] map. Its [[kernel]]
$$
ker(\epsilon) \hookrightarrow \mathbb{Z}[G]
$$
is the **augmentation ideal** of $\mathbb{Z}[G]$. (It is often denoted by $I(G)$.
=--
## Properties
### General
+-- {: .num_prop }
###### Proposition
The augmentation ideal is indeed a left and right [[ideal]] in $R[G]$.
=--
### For group algebras
+-- {: .num_prop }
###### Proposition
The $R$-[[module]] underlying the augmentation ideal of a [[group algebra]] is a [[free module]], free on the set of elements
$$
\{
g - e | g \in G,\; g \neq e
\}
$$
in $R[G]$.
=--
+-- {: .num_prop }
###### Proposition
(For the case $R= \mathbb{Z}$)
As a $\mathbb{Z}[G]$-module, considered with the same generators, the relations are generated by those of the form
$$g_1(g_2-e)= (g_1g_2-e)-(g_1-e).$$
=--
##Related entries
* [[derived module]]
[[!redirects augmentation ideals]]
|
augmented A-infinity algebra | https://ncatlab.org/nlab/source/augmented+A-infinity+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _augmented $A_\infty$-algebra_ is the analogue in [[higher algebra]] of the notion of _[[augmented algebra]]_ in ordinary algebra: an [[A-∞ algebra]] euipped with a [[homomorphism]] to the base [[E-∞ ring]] (which might be a plain [[commutative ring]]).
## Definition
{#Definition}
Let $R$ be an [[E-∞ ring]] and $A$ an [[A-∞ algebra]] over $R$.
+-- {: .num_defn}
###### Definition
An **augmentation** of $A$ is an $R$-[[A-∞ algebra]] [[homomorphism]]
$$
\epsilon \colon A \to R
\,.
$$
=--
+-- {: .num_remark}
###### Remark
In as far as one considers [[A-∞ algebras]] are presented by [[simplicial objects]] or similar, there might also be a (less intrinsic) notion of [[augmentation]] as in _[[augmented simplicial sets]]_. This is _not_ what the above defines.
=--
Fully generally, a definition of augmentation of [[∞-algebras over an (∞,1)-operad]] is in ([Lurie, def. 5.2.3.14](#Lurie)).
## Examples
+-- {: .num_example}
###### Example
An augmentation of an [[E-∞ ring]] $R$, being an [[E-∞ algebra]] over the [[sphere spectrum]] $\mathbb{S}$, is a homomorphism
$$
\epsilon \colon R \to \mathbb{S}
$$
to the [[sphere spectrum]], regarded as an [[E-∞ ring]].
Forming [[augmentation ideals]] constitutes an [[equivalence of (∞,1)-categories]]
$$
E_\infty Ring_{/\mathbb{S}}
\stackrel{\simeq}{\longrightarrow}
E_\infty Ring^{nu}
$$
of $\mathbb{S}$-augmented $E_\infty$-rings and [[nonunital E-∞ rings]] ([Lurie, prop. 5.2.3.15](#Lurie)).
=--
+-- {: .num_example}
###### Example
A [[bipermutative category]] $\mathcal{C}$ induces (as discussed there) an [[E-∞ ring]] $\vert \mathcal{C}\vert$. If $\mathcal{C}$ is equipped with a bi-[[monoidal functor]] $\mathcal{C} \to \mathcal{Z}$ then this induces an augmentation of $\vert \mathcal{C}\vert$ over $H \mathbb{Z}$, the [[Eilenberg-MacLane spectrum]] of the [[integers]].
=--
See for instance ([Arone-Lesh](#AroneLesh))
## Related concepts
* [[augmented algebra]], [[augmentation ideal]]
* [[augmentation]], [[augmented simplicial set]]
* [[augmented ∞-group]]
## References
For $A_\infty$-algebras in [[characteristic]] 0 (in chain complexes) augmentation appears for instance as def. 2.3.2.2 on p. 81 in
* Kenji Lefèvre-Hasegawa, _Sur les A-infini catégories_ ([arXiv:math/0310337](http://arxiv.org/abs/math/0310337))
augmentation of $\mathbb{F}_p$ [[E-∞ algebras]] is considered in definition 7.1 of
* [[Michael Mandell]], _$E_\infty$-Algebras and $p$-adic homotopy theory_ ([pdf](hopf.math.purdue.edu/Mandell/einffinal.pdf))
The following articles discuss (just) [[augmented ∞-groups]].
Augmentation (of [[∞-groups of units]] of [[E-∞ rings]]) over the [[sphere spectrum]] appears in
* Steffen Sagave, _Spectra of units for periodic ring spectra_ ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))
Augmentation over the [[Eilenberg-MacLane spectrum]] $H\mathbb{Z}$ appears in
* Gregory Arone, Kathryn Lesh, _Augmented $\Gamma$-spaces, the stable rank filtration, and a $b u$-analogue of the Whitehead conjecture_ ([pdf](http://www.math.union.edu/~leshk/papers/FilteredGammaSpaces-revision.pdf))
{#AroneLesh}
See also
* {#Fresse06} [[Benoit Fresse]], _The Bar Complex of an E-infinity Algebra_, Adv. Math. 223 (2010), pages 2049-2096 ([arXiv:math/0601085](http://arxiv.org/abs/math/0601085))
and
* {#Schwede01} [[Stefan Schwede]], section 7.8 of _Stable Homotopy of Algebraic Theories_, 2001 ([pdf](http://www.math.uni-bonn.de/people/schwede/stable.pdf))
with comments on the relation to [[nonunital algebras]].
Fully general discussion in [[higher algebra]] is in
* {#Lurie} [[Jacob Lurie]], section 5.2.3 in _[[Higher Algebra]]_
[[!redirects augmented A-infinity-algebras]]
[[!redirects augmented A-∞ algebra]]
[[!redirects augmented A-∞ algebras]]
[[!redirects augmented E-∞ algebra]]
[[!redirects augmented E-∞ algebras]]
[[!redirects augmented E-∞ ring]]
[[!redirects augmented E-∞ rings]]
|
augmented algebra | https://ncatlab.org/nlab/source/augmented+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $R$ a [[ring]], an [[associative algebra]] over $R$ is a [[ring]] $A$ equipped with a ring inclusion $R \hookrightarrow A$.
+-- {: .num_defn}
###### Definition
If the $R$-algebra $A$ is equipped with an $R$-algebra [[homomorphism]] the other way around,
$$
\epsilon \colon A \to R
\,,
$$
then it is called an _augmented $R$-algebra_.
=--
+-- {: .num_remark}
###### Remark
In [Cartan-Eilenberg](#CartanEilenberg) this is called a _supplemented algebra_.
=--
+-- {: .num_defn}
###### Definition
The [[kernel]] of $\epsilon$ is called the corresponding [[augmentation ideal]] in $A$.
=--
## Examples
+-- {: .num_example}
###### Example
An augmentation of a bare [[ring]] itself, being an [[associative algebra]] over the ring of [[integers]] $\mathbb{Z}$, is a ring homomorphism to the integers
$$
\epsilon \colon R \to \mathbb{Z}
$$
=--
+-- {: .num_example}
###### Example
Every [[group algebra]] $R[G]$ is canonically augmented, the augmentation map being the operation that forms the sum of [[coefficients]] of the canonical basis elements.
=--
+-- {: .num_example}
###### Example
If $X$ is a variety over an algebraically closed field $k$ and $x\in X(k)$ is a closed point, then the local ring $\mathcal{O}_{X,x}$ naturally has the structure of an augmented $k$-algebra. The augmentation map $\mathcal{O}_{X,x}\rightarrow k$ is the evaluation map, and the augmentation ideal is the maximal ideal of $\mathcal{O}_{X,x}$.
=--
## Related concepts
* [[augmentation]]
* [[augmented A-∞ algebra]]
## References
* [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_
{#CartanEilenberg}
[[!redirects augmented algebras]]
[[!redirects augmentation map]]
[[!redirects augmentation maps]]
|
augmented infinity-group | https://ncatlab.org/nlab/source/augmented+infinity-group |
#Contents#
* table of contents
{:toc}
## Idea
An [[∞-group]] equipped with a map to a fixed base [[∞-group]], such as that underlying the [[sphere spectrum]] or an [[Eilenberg-MacLane spectrum]].
## Related concepts
* [[augmentation]]
* [[augmented algebra]], [[augmentation ideal]]
* [[augmented A-infinity algebra]]
## References
Discussion of [[∞-groups of units]] of [[E-∞ rings]] as [[abelian ∞-groups]] augmented over the [[sphere spectrum]] is in
* Steffen Sagave, _Spectra of units for periodic ring spectra_ ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))
Augmentation over the [[Eilenberg-MacLane spectrum]] $H\mathbb{Z}$ appears in
* Gregory Arone, Kathryn Lesh, _Augmented $\Gamma$-spaces, the stable rank filtration, and a $b u$-analogue of the Whitehead conjecture_ ([pdf](http://www.math.union.edu/~leshk/papers/FilteredGammaSpaces-revision.pdf))
{#AroneLesh}
[[!redirects augmented infinity-groups]]
[[!redirects augmented ∞-group]]
[[!redirects augmented ∞-groups]]
|
augmented simplicial set | https://ncatlab.org/nlab/source/augmented+simplicial+set |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
# Augmented simplicial sets
* table of contents
{: toc}
## Idea
Where an ordinary [[simplicial set]] $X$ may be thought of a [[space]] made up of $n$-[[simplices]] $\sigma \in X_n$ for all $n \in \{0,1,2,3, \cdots \}$, an _augmented simplicial set_ in addition has a set $X_{-1}$ of "$(-1)$-simplices" such that each 0-simplex has a single $(-1)$-dimensional face and such that the $(-1)$-dimensional faces of the two faces of any $1$-simplex coincide.
Equivalently this may be thought of as the data that encodes a [[morphism]] $X \to const X_{-1}$ in [[sSet]] between ordinary simplicial sets from an _underlying_ simplicial set to a _constant_ simplicial set with $X_{-1}$ as its set of $k$-simplices for all $k$.
It is in this latter form that augmented simplicial sets maybe mostly arise in practice, whereas the former incarnation offers a more succinct way of thinking about them.
For instance a major source of augmented [[simplicial object]]s are given by [[colimit]]s or rather [[homotopy colimit]]s over simplicial diagrams in a [[model category]]: for $X\colon \Delta^{op} \to C$ a simplicial object in a category with colimits, its colimit [[cocone]] may be thought of as a morphism
$$
X_\bullet \to const \lim_{\to} X_\bullet
$$
into the constant simplicial object.
## Definitions
Denote by $\Delta_+$ (also denoted $\Delta_a$) the [[augmented simplex category]], which may be defined as the [[full subcategory]] of [[Cat]] on [[free categories]] over finite and _possibly empty_ linear [[directed graphs]], which are
* $[-1] := \emptyset$;
* $[0] := (0)$;
* $[1] := (0 \to 1)$;
* $[2] := (0 \to 1 \to 2)$;
* and so on.
An **augmented simplicial set** $X$ is a [[presheaf]] on $\Delta_+$ and the [[category]] of augmented simplicial sets, [[ASSet|$sSet_+$]], is the presheaf category
$$
[\Delta_+^{op}, Set]
\,.
$$
There is a canonical inclusion $\Delta \hookrightarrow \Delta_+$ of the ordinary [[simplex category]] and that the [[restriction]] of an augmented simplicial set along this inclusion is a simplicial set. This gives a [[forgetful functor]]
$$
U : sSet_+ \to sSet
\,.
$$
A cartoon of an augmented simplicial set, showing just the face maps, looks like
$$
X_\bullet =
\left(
\cdots
X_2 \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{d^0_1}{\to}}{\underset{d^0_0}{\to}} X_0 \stackrel{d^{-1}}{\to} X_{-1}
\right)
$$
where the only new [[simplicial identity]] satisfied by the new face map in degree $-1$ is that it coequalizes the degree-$0$ face maps in that
$$
d^{-1}\circ d^0_1 = d^{-1}\circ d^0_0
\,.
$$
It follows that $d^{-1}$ coequalizes in fact every pair of composites of face maps $X_n \to X_0$, so that equivalently an augmented simplicial set $X_\bullet$ is a morphism of ordinary simplicial sets
$$
U(X_\bullet) \to const X_{-1}
\,.
$$
We say that that $U(X_\bullet)$ is _augmented_ over $X_{-1}$.
More explicitly, an __augmented simplicial set__ consists of
* for each [[integer]] $n \geq -1$, a [[set]] $X_n$ (so an [[infinite sequence]] $(X_{-1},X_0,X_1,X_2,\ldots)$ of sets),
* for each integer $n \geq -1$ and each [[natural number]] $m \leq n + 1$, a __face map__ $d^n_m\colon X_{n+1} \to X_n$,
* for each integer $n \geq -1$ and each natural number $m \leq n$, a __degeneracy map__ $s^n_m\colon X_n \to X_{n+1}$,
* satisfying the [[simplicial identities]].
Above, we use the traditional system of numbering for a [[simplicial set]]. However, part of the motivation behind augmented simplicial sets is that this is a more sensible numbering system:
* for each [[natural number]] $n$, a [[set]] $X_n$ (so an [[infinite sequence]] $(X_0,X_1,X_2,\ldots)$ of sets),
* for each natural number $n$ and each natural number $m \leq n$, a __face map__ $d^n_m\colon X_{n+1} \to X_n$,
* for each natural number $n$ and each natural number $m \lt n$, a __degeneracy map__ $s^n_m\colon X_n \to X_{n+1}$,
* satisfying the [[simplicial identities]].
While this numbering is very nice for augmented simplicial sets, it is not standard and is can be easily misunderstood, so we don\'t use it in this article.
## Properties
Anything that applies to [[simplicial sets]] should also apply to augmented simplicial sets, if one properly takes care of the [[negative thinking]] necessary to deal with $X_{-1}$.
Every augmented simplicial set has an underlying unaugmented simplicial set found by forgetting $X_{-1}$ (and $d^{-1}_0$). Conversely, every unaugmented simplicial set gives rise to a [[free object|free]] augmented simplicial set by augmentation over $\pi_0(X)$ (the set of [[connected component]]s of $X$) and a [[cofree object|cofree]] augmented simplicial set by augmentation over the [[point]] (the [[singleton set]]). This defines [[adjunctions]]:
$$
\vdash\mathclap{\underoverset{\textsize{\operatorname{AugSimpSet}}}{\textsize{\operatorname{SimpSet}}}{\begin{matrix}\begin{svg}
<svg width="47" height="66" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="12691">
<g>
<title>Layer 1</title>
<path fill="none" stroke="#000000" d="m15.14844,3c-21,22.035261 -18,43.469551 0,62.5" id="svg_12691_1" marker-start="url(#se_marker_start_svg_12691_1)"/>
<path fill="none" stroke="#000000" d="m31.500429,65.25c21.000002,-21.947319 18.000002,-43.295465 0,-62.25" id="svg_12691_2" marker-end="url(#se_marker_end_svg_12691_2)"/>
<line fill="none" stroke="#000000" x1="23.148436" y1="0.500004" x2="23.148436" y2="63.504595" id="svg_12691_3" marker-end="url(#se_marker_end_svg_12691_3)"/>
</g>
<defs>
<marker id="se_marker_start_svg_12691_1" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50">
<path id="svg_12691_4" d="m0,50l100,40l-30,-40l30,-40l-100,40z" fill="#000000" stroke="#000000" stroke-width="10"/>
</marker>
<marker id="se_marker_end_svg_12691_2" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50">
<path id="svg_12691_5" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#000000" stroke="#000000" stroke-width="10"/>
</marker>
<marker id="se_marker_end_svg_12691_3" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50">
<path id="svg_12691_6" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#000000" stroke="#000000" stroke-width="10"/>
</marker>
</defs>
</svg>
\end{svg}\includegraphics[width=35]{vertarrows}\end{matrix}}}\vdash
$$
## Examples
* For $C$ a [[site]], $\{U_i \to X\}$ a [[covering]] family we have the [[Cech nerve]] [[simplicial presheaf]] $C(U) \in [C^{op}, sSet]$. This comes canonically equipped with a morphism of simplicial presheaves to the one represented by $X\colon C(U) \to X$. This is an augmented simplicial object in the category of presheaves $[C^{op}, Set]$. Morover, over each object $V \in C$ its component
$$
C(U)(V) \to X(V)
$$
is an augmented simplicial set.
More generally this applies to [[hypercover]]s.
## Related concepts
* [[augmentation]], [[augmentation ideal]]
* [[augmented algebra]]
[[!redirects augmented simplicial set]]
[[!redirects augmented simplicial sets]]
[[!redirects augmented simplicial object]]
[[!redirects augmented simplicial objects]]
|
augmented Teichmüller space | https://ncatlab.org/nlab/source/augmented+Teichm%C3%BCller+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _augmented Teichmüller space_ ([Bers 73](#Bers73)) is [[Teichmüller space]] with also the points for the degenerate [[nodal curve]] admitted.
Accordingly, the [[orbifold quotient]] of the augmented Teichmüller space by the [[mapping class group]] is equivalently the [[Deligne-Mumford compactification]] of the [[moduli space of curves]] over the complex numbers ([Hubbard-Koch 13, main theorem](#HubbardKoch13)).
## References
The concept was introduced in
* {#Bers73} [[Lipman Bers]], _Spaces of degenerating Riemann surfaces. Discontinuous groups and Riemann surfaces_, Ann. Math. Stud. 79, 43–55 (1973).
Discussion of [[orbifold]] structure is in
* {#HinichVaintrob07} [[Vladimir Hinich]], [[Arkady Vaintrob]], _Augmented Teichmuller Spaces and Orbifolds_ ([arXiv:0705.2859](http://arxiv.org/abs/0705.2859))
Discussion of the relation to the [[Deligne-Mumford compactification]] of the [[moduli stack of curves]] is in
* {#HubbardKoch13} [[John Hubbard]], [[Sarah Koch]], _An analytic construction of the Deligne-Mumford compactification of the moduli space of curves_ ([arXiv:1301.0062](http://arxiv.org/abs/1301.0062))
[[!redirects augmented Teichmuller space]] |
augmented virtual double category | https://ncatlab.org/nlab/source/augmented+virtual+double+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Augmented virtual double categories
* table of contents
{: toc}
## Idea
An **augmented virtual double category** is a [[virtual double category]] enhanced with additional [[2-cells]] whose vertical [[target]] has length 0 (i.e. is a single object rather than a horizontal arrow).
In particular, there is always a "vertical 2-category" consisting of the objects, vertical arrows, and 2-cells whose vertical source and target are both length 0. A virtual double category, by contrast, does not have a vertical 2-category unless it has all units.
## Examples
[[large category|Large]] (not necessarily [[locally small category|locally small]]) [[categories]], [[functors]], small-set-valued [[profunctors]], and transformations form an augmented virtual double category. Note that only locally small categories have units therein, so the underlying virtual double category does not have enough data to reconstruct the 2-category of large categories, functors, and natural transformations; but the augmented virtual double category does.
## Applications
Augmented Virtual double categories are a natural context in which to compare [[proarrow equipments]] and [[Yoneda structures]], which are two different approaches to [[formal category theory]]. In both cases there is a notion of "[[profunctor]]" which are generally considered to be small-set-valued, but [[Yoneda structures]] require non-locally-small categories (the presheaf categories of non-small categories). In this context the Yoneda embedding can be given a universal property relative to the horizontal arrows. See [(Koudenburg)](#Koudenburg19).
## Related pages
* [[virtual double category]]
* [[virtual equipment]]
* [[Yoneda structure]]
## Reference
* {#Koudenburg15} [[Seerp Roald Koudenburg]], *A double-dimensional approach to formal category theory* ([arXiv:1511.04070](http://arxiv.org/abs/1511.04070)) 2015.
> (motivated by [[formal category theory]])
* {#Koudenburg19} [[Seerp Roald Koudenburg]], *Augmented virtual double categories*, Theory and Applications of Categories, Vol. 35, 2020, No. 10, pp 261-325 ([arXiv:1910.11189](https://arxiv.org/abs/1910.11189), [tac:35-10](http://www.tac.mta.ca/tac/volumes/35/10/35-10abs.html))
> (streamlined and expanded version of Sec. 1-3 of [Koudenburg 15](#Koudenburg15) )
[[!redirects hypervirtual double category]]
[[!redirects hypervirtual double categories]]
[[!redirects augmented virtual double categories]]
|
August Ferdinand Möbius | https://ncatlab.org/nlab/source/August+Ferdinand+M%C3%B6bius |
* [Wikipedia entry](https://en.wikipedia.org/wiki/August_Ferdinand_M%C3%B6bius)
## related $n$Lab entries
* [[projective space]]
* [[Möbius strip]]
* [[Möbius transformation]]
category: people
[[!redirects August Möbius]]
[[!redirects Möbius]]
[[!redirects August Ferdinand Moebius]]
[[!redirects August Moebius]]
[[!redirects Moebius]]
[[!redirects August Ferdinand Mobius]]
[[!redirects August Mobius]]
[[!redirects Mobius]] |
Auguste Bravais | https://ncatlab.org/nlab/source/Auguste+Bravais |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Auguste_Bravais)
## Selected writings
On [[crystallography]] and introducing the notion now known as *[[Bravais lattice]]*:
* {#Bravais1850} [[Auguste Bravais]], *Mémoire sur les Systèmes Formés par les Points Distribués Régulièrement sur un Plan ou dans L'espace*, J. Ecole Polytech. 19 (1850) 1 $[$[ark:12148/bpt6k96124j](https://gallica.bnf.fr/ark:/12148/bpt6k96124j)$]$
category: people
|
Auguste Sagnotti > history | https://ncatlab.org/nlab/source/Auguste+Sagnotti+%3E+history |
see _[[Augusto Sagnotti]]_ |
Augustin-Louis Cauchy | https://ncatlab.org/nlab/source/Augustin-Louis+Cauchy | [[!redirects Augustin Cauchy]]
__Augustin-Louis Cauchy__ was a pioneer in [[analysis]] and [[group theory]]. He wrote an influential 1821 textbook, _[[Cours d'Analyse]]_.
Cauchy described the basic concepts of [[differential calculus|differential]] and [[integral calculus|integral]] [[infinitesimal calculus|calculus]] in terms of [[convergence|limits]]. His conception of limits was based on [[infinitesimal]] [[variables]], which do not appear as such in modern [[mathematics]], although they have been variously identified with [[sequences]] (that converge to zero), [[ultrafilters]] (that converge to zero), hyperpoints (in the [[infinitesimal neighborhood|infinitesimal neighbourhood]] of zero) in the sense of [[nonstandard analysis]], etc. (His infinitesimals were *not* [[nilpotent infinitesimal|nilpotent]].)
Cauchy\'s student [[Karl Weierstrass]] defined [[limit of a sequence|limits]] in terms of [[Richard Dedekind]]\'s static conceptions of *[[real numbers]]* and *[[functions]]*, thereby creating modern [[analysis]].
Cauchy is associated with:
* [[Cauchy sequences]]
* [[Cauchy sequences]] (and thus [[Cauchy nets]], [[Cauchy filters]], and [[Cauchy spaces]], although Cauchy himself knew none of these)
* [[Cauchy–Riemann equations]]
* [[Cauchy-Schwarz inequality]]
* [[Cauchy integral]]
* [[Cauchy surface]]
* [[Cauchy real number]]
* [[Cauchy principal value]]
* [[Cours d'Analyse]], a textbook on [[infinitesimal analysis]]/[[epsilontic analysis]]
* [[Cauchy's theorems]]
* [[Cauchy integral theorem]] and [[Cauchy integral formula]] (for [[contour integrals]] in [[complex analysis]])
* [[Cauchy sum theorem]] ("[[Cauchy's mistake]]") from his 1821 textbook [[Cours d'Analyse]]
* [[Cauchy group theorem]]
category: people
[[!redirects Cauchy]]
[[!redirects Augustin Cauchy]]
[[!redirects Augustin-Louis Cauchy]]
[[!redirects Augustin Louis Cauchy]]
|
Augusto Sagnotti | https://ncatlab.org/nlab/source/Augusto+Sagnotti |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Augusto_Sagnotti)
## Selected writings
Review of the [[Green-Schwarz mechanism]] with an eye towards [[KK-compactification]] to 6d (see also at [[D=6 N=(1,0) SCFT]]):
* [[Augusto Sagnotti]], _A Note on the Green - Schwarz Mechanism in Open - String Theories_, Phys. Lett. B294:196-203, 1992 ([arXiv:hep-th/9210127](https://arxiv.org/abs/hep-th/9210127))
On [[D=6 supergravity]]:
* [[Sergio Ferrara]], [[Fabio Riccioni]], [[Augusto Sagnotti]], _Tensor and Vector Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. B519 (1998) 115-140 ([arXiv:hep-th/9711059](https://arxiv.org/abs/hep-th/9711059))
* [[Fabio Riccioni]], [[Augusto Sagnotti]], _Some Properties of Tensor Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. Proc. Suppl. 67 (1998) 68-73 ([arXiv:hep-th/9711077](https://arxiv.org/abs/hep-th/9711077))
On [[orientifolds]] and [[RR-field tadpole cancellation]]:
* {#Sagnotti88} [[Augusto Sagnotti]], _Open strings and their symmetry groups_ in G. Mack et. al. (eds.) Cargese ’87, "Non-perturbative Quantum Field Theory," (Pergamon Press, 1988) p. 521 ([arXiv:hep-th/0208020](https://arxiv.org/abs/hep-th/0208020))
* Carlo Angelantonj, [[Augusto Sagnotti]], _Open Strings_, Phys. Rept. 371:1-150,2002; Erratum ibid. 376:339-405, 2003 ([arXiv:hep-th/0204089](https://arxiv.org/abs/hep-th/0204089))
On [[higher spin gauge theory]] as [[string field theory]] for vanishing [[string tension]]:
* {#SagnottiTsulaia03} [[Augusto Sagnotti]], M. Tsulaia, _On higher spins and the tensionless limit of String Theory_, Nucl. Phys. B682:83-116, 2004 ([arXiv:hep-th/0311257](http://arxiv.org/abs/hep-th/0311257))
* [[Auguste Sagnotti]], M. Taronna, _String Lessons for Higher-Spin Interactions_, Nucl. Phys. B842:299-361,2011 ([arXiv:1006.5242](http://arxiv.org/abs/1006.5242))
* {#Sagnotti11} [[Augusto Sagnotti]], _Notes on Strings and Higher Spins_ ([arXiv:1112.4285](http://arxiv.org/abs/1112.4285))
Discussion of the [[Velo-Zwanziger problem]] for the higher [[string]] excitations that appear in [[string theory]]:
* [[Massimo Porrati]], Rakibur Rahman, [[Augusto Sagnotti]], _String Theory and The Velo-Zwanziger Problem_, Nucl. Phys. B846:250-282, 2011 ([arXiv:1011.6411](http://arxiv.org/abs/1011.6411))
## Related $n$Lab entries
* [[orientifold]]
* [[RR-field tadpole cancellation]]
* [[5d supergravity]]
category: people |
Augustus De Morgan | https://ncatlab.org/nlab/source/Augustus+De+Morgan |
Augustus De Morgan is a 19th-century mathematician and logician most famous for the [[De Morgan laws]], even though he did not formulate them. He did, however, contribute (along with [[George Boole]], [[Charles Peirce]], etc) to the work of making [[logic]] into a mathematical discipline.
## Quotations
'Imagine a person with a gift of ridicule. \[He might say\] First that a negative quantity has no logarithm; secondly that a negative quantity has no square root; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter.' (That is, $\ln(-1)$ and $\sqrt{-1}$ are both imaginary, but their quotient is $\pi$.)
'I end with a word on the new symbols which I have employed. Most writers on logic strongly object to all symbols. \[...\] I should advise the reader not to make up his mind on this point until he has well weighed two facts which nobody disputes, both separately and in connexion. First, logic is the only science which has made no progress since the revival of letters; secondly, logic is the only science which has produced no growth of symbols.'
## Related entries
* [[De Morgan laws]]
* [[De Morgan duality]]
* [[De Morgan Heyting algebra]]
* [[De Morgan Heyting category]]
* [[De Morgan algebra]]
* [[De Morgan topos]]
category: people
[[!redirects Augustus De Morgan]]
[[!redirects Augustus de Morgan]]
|
Auke Booij | https://ncatlab.org/nlab/source/Auke+Booij | [[!redirects Auke Bart Booij]]
[[!redirects Auke B. Booij]]
## Selected writings
On [[parametricity]], [[type universe]]-[[automorphisms]] and [[excluded middle]]
* [[Auke Bart Booij]], [[Martín Hötzel Escardó]], [[Peter LeFanu Lumsdaine]], [[Michael Shulman]], *Parametricity, automorphisms of the universe, and excluded middle*, in 22nd International Conference on Types for Proofs and Programs (TYPES 2016) $[$[arxiv:1701.05617](https://arxiv.org/abs/1701.05617), [drops:2018/9855](https://drops.dagstuhl.de/opus/volltexte/2018/9855), [blog entry](https://homotopytypetheory.org/2017/01/26/parametricity-automorphisms-of-the-universe-and-excluded-middle/)$]$
On [[constructive analysis]] with [[real numbers]] in [[univalent foundations]] ([[homotopy type theory]] with the [[univalence axiom]]):
* {#Booij18} [[Auke Booij]], _Extensional constructive real analysis via locators_, Mathematical Structures in Computer Science **31** 1 (2021) 64-88 [[arXiv:1805.06781](https://arxiv.org/abs/1805.06781), [doi:10.1017/S0960129520000171](https://doi.org/10.1017/S0960129520000171)]
* {#Booij20} [[Auke Booij]], *Analysis in Univalent Type Theory* (2020) [[etheses:10411](http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf), [[Booij-AnalysisInUF.pdf:file]]]
category: people |
Aurelio Carboni | https://ncatlab.org/nlab/source/Aurelio+Carboni | Aurelio Carboni was an Italian category theorist, who was based in Como. He died on 11 December 2012.
* [webpage](http://www.uninsubria.eu/research/physmath/cv_Carboni.htm)
## Selected writings
On [[extensive categories]] and [[distributive categories]]:
* {#CarboniLackWalters93} [[Aurelio Carboni]], [[Stephen Lack]], [[Bob Walters|R. F. C. Walters]], _Introduction to extensive and distributive categories_, JPAA **84** (1993) pp. 145-158 (<a href="https://doi.org/10.1016/0022-4049(93)90035-R">doi:10.1016/0022-4049(93)90035-R</a>)
On [[reflective factorization systems]]:
* {#CJKP} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], _On localization and stabilization for factorization systems_, Appl. Categ. Structures 5 (1997), 1--58 ([doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444))
On [[regular and exact completions]] (and on [[free coproduct completions]], etc.):
* [[Aurelio Carboni]], [[Enrico Vitale]], *Regular and exact completions*, Journal of Pure and Applied Algebra **125** 1–3 (1998) 79-116 (<a href="https://doi.org/10.1016/S0022-4049(96)00115-6">doi:10.1016/S0022-4049(96)00115-6</a>)
On [[Cartesian bicategories]]:
* [[Aurelio Carboni]], [[Bob Walters]], _Cartesian Bicategories, I_, Journal of Pure and Applied Algebra, Volume 49, Issues 1–2, November 1987, (<a href="https://doi.org/10.1016/0022-4049(87)90121-6">doi:10.1016/0022-4049(87)90121-6</a>)
* [[Aurelio Carboni]], [[Max Kelly]], [[Bob Walters]], [[Richard Wood]], _Cartesian Bicategories II_, Theory and Applications of Categories, Vol. 19, 2008, No. 6, pp 93-124. ([arXiv:0708.1921](https://arxiv.org/abs/0708.1921), [tac:19-06](http://www.tac.mta.ca/tac/volumes/19/6/19-06abs.html))
category:people
[[!redirects A. Carboni]] |
Ausdehnungslehre | https://ncatlab.org/nlab/source/Ausdehnungslehre | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
This page collects material related to the book
* {#Grassmann44} [[Hermann Grassmann]],
_Die Wissenschaft der extensiven Grössen oder die Ausdehnungslehre_
_Erster Teil, die lineale Ausdehnungslehre_,
1844
([pdf scan of original](http://www.uni-potsdam.de/u/philosophie/grassmann/Werke/Hermann/Ausdehnungslehre_1844.pdf), [Internet Archive copy](https://archive.org/details/dielinealeausde00grasgoog/page/n11))
which introduced for the first time basic concepts of what today is known as [[linear algebra]] (including [[affine spaces]] as [[torsors]] over [[vector spaces]]) and introduced in addition an _exterior product_ (§37, §55) on [[vectors]], forming what today is known as _[[exterior algebra]]_ or _[[Grassmann algebra]]_, hence in fact _[[superalgebra]]_ (see [below](#SupercommutativeSuperalgebra)).
Grassmann advertizes his work (p. xxv) as being the theory of _[[extensive quantity]]_. The modern way of speaking about this is that the elements of the [[exterior algebra]] he considered are [[differential forms]] on [[Euclidean space]].
Prominent followups are:
* [[Giuseppe Peano]], *Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva*, Fratelli Bocca Editori, Torino, 1888, pp. XI, 171 [[Mathematica Italiana opere 138](http://mathematica.sns.it/opere/138), [pdf](http://mathematica.sns.it/media/volumi/138/Calcolo%20geometrico%20secondo%20l'Ausdehnungslhere%20di%20H.%20Grassmann_bw.pdf)]
which introduces the modern concept of *[[vector spaces]]*.
and
* [[William Clifford]], *Applications of Grassmann's extensive algebra*, American Journal of Mathematics **1** 4 (1878) 350-358 [[doi:10.2307/2369379](https://doi.org/10.2307/2369379), [jstor:2369379](https://www.jstor.org/stable/2369379)]
which introduces the notion of [[Clifford algebra]].
#Contents#
* table of contents
{:toc}
## Content
### Supercommutative superalgebra
{#SupercommutativeSuperalgebra}
{#GrassmannDefinesSuperalgebra} Here is Grassmann introducing the [[signs in supergeometry|sign rule]] of [[supercommutative superalgebra]]:
<center>
<img src="https://ncatlab.org/nlab/files/GrassmannGradedCommutativityI.jpg" width="460">
</center>
> from [Grassmann 1844, p. 61](#Grassmann44)
<center>
<img src="https://ncatlab.org/nlab/files/GrassmannGradedCommutativityII.jpg" width="460">
</center>
> from [Grassmann 1844, p. 84](#Grassmann44)
## Reception
{#Reception}
The appreciation of Grassmann's ideas took a long time:
From [Drew Armstrong](https://plus.google.com/103061162497127117651/posts/f4HHK1XhUL1):
> Perhaps owing to the abstraction of his work, he was not recognized by the mathematical community and was more famous during his lifetime for his work in philology (including a translation of the Rig Veda). Despite his failure to achieve mathematical success, Grassmann never doubted that his work was significant. Here's a striking quote from the preface of the 1862 second edition of the _[[Ausdehnungslehre]]_:
>> {#CompletelyConvinced} I remain completely confident that the labour I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development of science, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit. I know that if I also fail to gather around me (as I have until now desired in vain) a circle of scholars, whom I could fructify with these ideas, and whom I could stimulate to develop and enrich them further, yet there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into a living communication with contemporary developments. For truth is eternal and divine.
> (Translation from Michael Crowe, _A History of Vector Analysis_)
{#WhyNoOneHasTranslated} Here is R. W. Genese in 1893 ([Nature volume 48, page 517 ](https://www.nature.com/articles/048517a0)) lamenting the delay in its translation:
> Sir Robert Ball asks why no one has translated the "[[Ausdehnungslehre]]" into English. The answer is as regretable as simple—it would not pay. The number of mathematicians who, after the severe courses of the universities, desire to extend their reading is very small. It is something that a respectable few seek to apply what they have already learnt. The first duty of those who direct the studies of the universities is to provide that students may leave in possession of all the best means of future investigation. That fifty years after publication the principles of the "[[Ausdehnungslehre]]" should find no place in English mathematical education is indeed astonishing. Half the time given to such a wearisome subject as Lunar Theory would place a student in possession of many of the delightful surprises of Grassmann's work, and set him thinking for himself. The "[[Ausdehnungslehre]]" has won the admiration of too many distinguished mathematicians to remain longer ignored. Clifford said of it: "I may, perhaps, be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science.” Useful or not, the work is “a thing of beauty," and no mathematician of taste should pass it by. It is possible, nay, even likely, that its principles may be taught more simply; but the work should be preserved as a classic.
Grassmann's ideas were finally studied in
* Hermann Hankel, _Vorlesungen über die Complexen Zahlen und ihre Functionen_, Leipzig: Leopold Voss, 1867.
His ideas were developed by [[William Kingdon Clifford]] who turned Grassmann algebras into _[[Clifford algebras]]_ in
* [[William Clifford]], _Applications of Grassmann's extensive algebra_, American Journal of Mathematics 1 (4): 350–358. (1878). doi:[10.2307/2369379](https://doi.org/10.2307/2369379).
Today [[Grassmann algebra]] is ubiquituous in [[differential geometry]] ([[de Rham algebra]], [[supergeometry]]) and [[superalgebra]]. In fact, Grassmann had invented the [[signs in supergeometry|sign rule]] of [[supercommutative superalgebra]], which was fully appreciated only in the second half of the 20th century.
This way Grassmann's prediction on the eventual impact of his work did become true. From [[Freeman Dyson]], _Missed opportunities_ ([doi:10.1090/S0002-9904-1972-12971-9](https://doi.org/10.1090/S0002-9904-1972-12971-9)):
> In the year 1844 two remarkable events occurred, the publication by Hamilton of his discovery of [[quaternions]], and the publication by Grassmann of his "[[Ausdehnungslehre]]." With the advantage of hindsight we can see that Grassmann's was the greater contribution to mathematics, containing the germ of many of the concepts of modern algebra, and including vector analysis as a special case. However, Grassmann was an obscure high-school teacher in Stettin, while Hamilton was the world-famous mathematician whose official titles occupy six lines of print after his name at the beginning of his 1844 paper. So it is regrettable, but not surprising, that quaternions were hailed as a great discovery, while Grassmann had to wait 23 years before his work received any recognition at all from professional mathematicians. When Grassmann's work finally became known, mathematicians were divided into quaternionists and antiquaternionists, and were spending more energy in polemical arguments for and against quaternions than in trying to understand how Grassmann and Hamilton might be fitted together into a larger scheme of things.
Grassmann also had a profound influence on the thought of [[Gottlob Frege]]. See Sec. 2.1.1 of
* Paolo Mancosu, _Abstraction and infinity_, Oxford University Press, 2017.
## Discussion
Discussion of the book includes
* {#Lawvere96} [[William Lawvere]], _Grassmann's Dialectics and Category Theory_, in _Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar_, Boston Studies in the Philosophy of Science Volume 187, 1996, pp 255-264, doi:[10.1007/978-94-015-8753-2_21](https://doi.org/10.1007/978-94-015-8753-2_21)
and the similar text
* {#Lawvere95} [[William Lawvere]], _A new branch of mathematics, "The Ausdehnungslehre of 1844," and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis_, _Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106_, doi:[10.1016/j.hm.2004.07.004](https://doi.org/10.1016/j.hm.2004.07.004)
which says at one point that full appreciation of the _Ausdehnungslehre_ requires concepts of [[category theory]]
> The modern conceptual apparatus, involving levels of structure, [[categories]] of [[morphisms]] preserving given [[structure]], [[forgetful functor|forgetful reduct functors]] between categories, the [[adjoints]] to such functors, etc., seems to be necessary for ordinary mortals to be able to find their way through the riches of Grassmann's geometry.
The first part of the introduction of the _Ausdehnungslehre_ is concerned with [[philosophy]], about which
> Grassmann insists that his reason for including it is an attempt to provide an orientation to help the student form for himself the proper estimation of the relation between general and particular at every stage of the learning process ([Lawvere 95](#Lawvere95)).
The second part of the introduction, titled _Survey of the general theory of forms_ considers key concepts of [[algebra]]. For instance it considers the [[associativity law]] and states its [[coherence law]] (§3).
Grassmann writes that he uses the term "form" in place of "quantity" (German: "Grösse") (Introduction A.3, §2). It is "forms" that his algebraic operations are defined on, and which are produced by these.
> The last half of that introduction is essentially one of the first expositions of the rudimentary principles of what today might be called [[universal algebra]]. The content of the first half, after considerable study of the compact formulations, appears to be a simple and clear natural scientist's version of the basic principles of dialectical materialism, as applied to the formal sciences. ([Lawvere 95](#Lawvere95))
Curiously, while Grassmann complains (on p. xv) about the "unclarity and arbitrariness" of [[Hegel]]'s school of philosophy ([[German idealism]], predominant in Germany at Grassmann's time), the introduction of the _Ausdehnungslehre_ has much the same sound as Hegel, notably it discusses "[[category (philosophy)|categories]]" such as _[[being]]_, _[[becoming]]_ (p. xxii), _[[concrete particular|particulars]]_ (p.xx) and the [[dialectic]] of [[unities of opposites|opposites]] such as _[[flat modality|discrete]] $\dashv$ [[sharp modality|continuous]]_ (p.xxii) and, notably, of _[[intensive and extensive quantity]]_ (p. xxiv-xxv), which Grassmann advertizes as the very topic of his mathematical theory. That of course is the difference to [[Hegel]], that unambiguous mathematical formalization of these otherwise vague concepts is provided (according to [Lawvere 95](#Lawvere95) Grassmannn's formalization of the pair _[[being]]_ and _[[becoming]]_ is via points and [[vectors]] in an [[affine space]]), and in this sense Grassmann is clearly a forerunner of Lawvere's various proposals for formalizing Hegel's [[objective logic]] in [[categorical logic]]/[[topos theory]] (as discussed at _[[Science of Logic]]_).
## Related entries
* [[Grassmann algebra]]
* [[supercommutative superalgebra]]
* [[Berezin integral]], [[integration over supermanifolds]]
## References
* Wikipedia, _[Grassmann -- Mathematician](http://en.wikipedia.org/wiki/Hermann_Grassmann#Mathematician)_
* {#Dyson72} [[Freeman Dyson]], _Missed opportunities_, Bulletin of the AMS, Volume 78, Number 5 (1972) pp 635-652, doi:[10.1090/S0002-9904-1972-12971-9](https://doi.org/10.1090/S0002-9904-1972-12971-9)
> In the year 1844 two remarkable events occurred, the publication by Hamilton of his discovery of [[quaternions]], and the publication by Grassmann of his "Ausdehnungslehre." With the advantage of hindsight we can see that Grassmann's was the greater contribution to mathematics, containing the germ of many of the concepts of modern algebra, and including vector analysis as a special case. However, Grassmann was an obscure high-school teacher in Stettin, while Hamilton was the world-famous mathematician whose official titles occupy six lines of print after his name at the beginning of his 1844 paper. So it is regrettable, but not surprising, that quaternions were hailed as a great discovery, while Grassmann had to wait 23 years before his work received any recognition at all from professional mathematicians. When Grassmann's work finally became known, mathematicians were divided into quaternionists and antiquaternionists, and were spending more energy in polemical arguments for and against quaternions than in trying to understand how Grassmann and Hamilton might be fitted together into a larger scheme of things.
category: reference |
Auslander-Buchsbaum formula | https://ncatlab.org/nlab/source/Auslander-Buchsbaum+formula | Let $R$ be a commutative Noetherian local ring. Then for any nonzero finitely generated $R$-module $M$ of finite [[projective dimension]] $pd_R(M)$ the following formula holds
$$
pd_R(M) + depth(M) = depth(R)
$$
As a corollary a commutative ring is regular iff it has a finite global dimension. This is also proved by Serre.
This in turn is the essence of the fact that for any [[regular scheme]], hence in particular for any [[smooth scheme]] over a field, the bounded [[derived category of coherent sheaves]] coincides with the [[full triangulated subcategory]] of [[perfect complex]]es. In other words, the [[triangulated category of singularities]] of a smooth scheme is trivial.
One should be warned that another result on regular rings is usually known as [[Auslander-Buchsbaum theorem]] (from a later article in 1959), namely that every regular local ring is a [[unique factorization domain]].
## Literature
* Maurice Auslander, [[David Buchsbaum|David A. Buchsbaum]], _Homological dimension in local rings_, Trans. Amer. Math. Soc. __85__ (2): 390--405 (1957) [doi](https://doi.org/10.2307/1992937) MR0086822
* stacks project 10.111 Auslander-Buchsbaum tag:[090U](https://stacks.math.columbia.edu/tag/090U)
* wikipedia:[Auslander-Buchsbaum_formula](https://en.wikipedia.org/wiki/Auslander%E2%80%93Buchsbaum_formula)
[[!redirects Auslander–Buchsbaum formula]] |
Australian Category Theory Seminar | https://ncatlab.org/nlab/source/Australian+Category+Theory+Seminar | The **Australian Category Theory Seminar** (also know as **AusCat**), previously called the **Sydney Category Theory Seminar**, is a weekly meeting on ([[higher category theory|higher]]) [[category theory]] held by the Centre of Australian Category Theory that has been held since 1970. The first seminar was probably held on the 23rd September, 1970.
> During 1970 Bob Walters, who had been a student of Bernhard Neumann at Canberra, and who had developed an interest in category theory, took a Lectureship at Sydney University. I went off with my family to spend the academic year 70/71 at Chicago, and Brian Day too had a post at Chicago for that year. While I was away, Ross and Bob began weekly meetings, along with my research student Geoffrey Lewis, to discuss category theory. This was the beginning of the Australian Category Seminar, which has met ever since for a half day each week. Using Fulbright funds at first to support visitors, we learnt that the Australian Research Council was willing to receive proposals from mathematicians. So we prepared such a proposal, and were successful, and have ever since had such support.
-[Kelly, 2007](#Kelly)
## Links
- [Seminar webpage](http://web.science.mq.edu.au/groups/coact/seminar/)
- [Blog post](https://rfcwalters.blogspot.com/2010/08/category-seminar-in-sydney.html) by [[RFC Walters]] regarding the first seminar
- [Seminar abstracts 1986](https://web.archive.org/web/20170210231156/https://dl.dropboxusercontent.com/u/92056191/Archive/seminars/sydney-category-seminar-abstracts/sydney_category_seminar_1986.pdf)
- [Seminar abstracts 1987](https://web.archive.org/web/20170210231156/https://dl.dropboxusercontent.com/u/92056191/Archive/seminars/sydney-category-seminar-abstracts/sydney_category_seminar_1987.pdf)
- [Seminar abstracts 1988](https://web.archive.org/web/20170210231156/https://dl.dropboxusercontent.com/u/92056191/Archive/seminars/sydney-category-seminar-abstracts/sydney_category_seminar_1988.pdf)
[[!redirects Sydney Category Theory Seminar]]
## References
* {#Kelly} [[G. Maxwell Kelly]], _The beginnings of category theory in Australia_, in: *Categories in Algebra, Geometry and Mathematical Physics*, Contemporary Mathematics **431** (2007) 1-6 [[doi:10.1090/conm/431](https://doi.org/10.1090/conm/431)]
category: reference
|
autoequivalence type | https://ncatlab.org/nlab/source/autoequivalence+type |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--
\tableofcontents
## Definition
In [[dependent type theory]], given a [[type]] $A$, the **autoequivalence type** of $A$ is the [[equivalence type]] between $A$ and $A$ itself, $\mathrm{Aut}(A) \coloneqq (A \simeq A)$. The elements ([[terms]]) of $\mathrm{Aut}(A)$ are called **autoequivalences** or **self-equivalences**.
## Properties
{#Properties}
* $\mathrm{Aut}(A)$ is an [[infinity-group|$\infty$-group]].
* Given a [[univalent universe|univalent]] [[Russell universe]] $U$ and an element $A:U$, there is an equivalence between $\mathrm{Aut}(A)$ and the [[loop space type]] $\Omega(U, A)$, $\mathrm{ua}(A, A):\Omega(U, A) \simeq \mathrm{Aut}(A)$. $\Omega(U, A)$ is called the *[[automorphism infinity-group|automorphism $\infty$-group]]* in $U$ at $A$.
* The autoequivalence type on an [[n-truncation modality|$n$-truncated type]] $A$ is an [[n-group|$(n + 1)$-group]].
* If $A$ is a [[set]], then $\mathrm{Aut}(A)$ is the **[[symmetric group]]** on $A$ and is also written as $S_A$, $\Sigma_A$, or $\mathrm{Sym}(A)$, and the elements of $\mathrm{Aut}(A)$ are called **[[permutations]]**. Every autoequivalence type is a symmetric group in a [[dependent type theory]] with [[axiom K]] or [[uniqueness of identity proofs]].
* If $A$ is a [[mere proposition]], then $\mathrm{Aut}(A)$ is a [[contractible type]].
* Given types $A$ and $B$, there is a [[function]] $\mathrm{ae}_\mathrm{Aut}:(A \simeq B) \to (\mathrm{Aut}(A) \simeq \mathrm{Aut}(B))$.
### Deloopings
Given an [[infinity-group]] $A$, the [[delooping]] of $A$ is defined to be a [[higher inductive type]] $B(A)$ generated by equivalences $\delta_A:\mathrm{Aut}(B(A)) \simeq A$ and $\epsilon_A:B(\mathrm{Aut}(A)) \simeq A$, and identity
$$\kappa_A:\mathrm{ae}_\mathrm{Aut}(\epsilon_A) =_{\mathrm{Aut}(B(\mathrm{Aut}(A))) \simeq \mathrm{Aut}(A)} \delta_{\mathrm{Aut}(A)}$$
## See also
* [[equivalence type]]
* [[permutation]], [[symmetric group]]
* [[automorphism group]], [[automorphism infinity-group]]
* [[loop space type]] (for the equivalent for [[identity types]])
[[!redirects autoequivalence]]
[[!redirects autoequivalences]]
[[!redirects self-equivalence]]
[[!redirects self-equivalences]]
[[!redirects autoequivalence type]]
[[!redirects autoequivalence types]]
[[!redirects type of autoequivalences]]
[[!redirects types of autoequivalences]] |
AUTOMATH | https://ncatlab.org/nlab/source/AUTOMATH | { ----------------------------- pseudoterms ------------------------------1-- }
pterm : TYPE := PRIM
{ ----------------------------- judgements -------------------------------4-- }
* [A:pterm] type : TYPE := PRIM
* [A:pterm][B:pterm] eq_type : TYPE := PRIM
* [a:pterm][A:pterm] in : TYPE := PRIM
* [a:pterm][b:pterm][A:pterm] eq_in : TYPE := PRIM
{ ----------------------------- equality rules ---------------------------8-- }
* [A:pterm][a:pterm][b:pterm][c:pterm]
A * [B:pterm][C:pterm]
A * [_1:type(A)] refl_type : eq_type(A,A) := PRIM
B * [_1:eq_type(A,B)] sym_type : eq_type(B,A) := PRIM
C * [_1:eq_type(A,B)][_2:eq_type(B,C)] trans_type : eq_type(A,C) := PRIM
a * [_1:in(a,A)] refl : eq_in(a,a,A) := PRIM
b * [_1:eq_in(a,b,A)] sym : eq_in(b,a,A) := PRIM
c * [_1:eq_in(a,b,A)][_2:eq_in(b,c,A)] trans : eq_in(a,c,A) := PRIM
B * [a:pterm][_1:in(a,A)][_2:eq_type(A,B)] conv_in : in(a,B) := PRIM
a * [b:pterm][_1:eq_in(a,b,A)][_2:eq_type(A,B)]
conv_eq_in : eq_in(a,b,B) := PRIM
{ ----------------------------- congruence rules -------------------------2-- }
* [A:pterm][a:pterm][b:pterm][C:[x,pterm]pterm]
[_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>C)][_3:eq_in(a,b,A)]
cong_type : eq_type(<a>C,<b>C) := PRIM
C * [c:[x,pterm]pterm]
[_1:type(A)][_2:[x,pterm][_,in(x,A)]in(<x>c,<x>C)][_3:eq_in(a,b,A)]
cong : eq_in(<a>c,<b>c,<a>C) := PRIM
{ ----------------------------- the empty type ---------------------------4-- }
* '0' : pterm := PRIM
* [a:pterm] R_0 : pterm := PRIM
* '0'_form : type('0') := PRIM
a * [C:[x,pterm]pterm]
[_1:[x,pterm][_,in(x,'0')]type(<x>C)][_2:in(a,'0')]
'0'_elim : in(R_0(a),<a>C) := PRIM
{ ----------------------------- the unit type ----------------------------7-- }
* '1' : pterm := PRIM
* star : pterm := PRIM
* [c:pterm][a:pterm] R_1 : pterm := PRIM
* '1'_form : type('1') := PRIM
* '1'_intro : in(star,'1') := PRIM
a * [C:[x,pterm]pterm]
[_1:[x,pterm][_,in(x,'1')]type(<x>C)][_2:in(a,'1')][_3:in(c,<star>C)]
'1'_elim : in(R_1(c,a),<a>C) := PRIM
c * [C:[x,pterm]pterm]
[_1:[x,pterm][_,in(x,'1')]type(<x>C)][_2:in(c,<star>C)]
'1'_eq : eq_in(R_1(c,star),c,<star>C) := PRIM
{ ----------------------------- the Booleans ----------------------------13-- }
* '2' : pterm := PRIM
* 1 : pterm := PRIM
* 2 : pterm := PRIM
* [c:pterm][d:pterm][a:pterm] R_2 : pterm := PRIM
* '2'_form : type('2') := PRIM
* '2'_intro_1 : in(1,'2') := PRIM
* '2'_intro_2 : in(2,'2') := PRIM
a * [C:[x,pterm]pterm]
[_1:[x,pterm][_,in(x,'2')]type(<x>C)][_2:in(a,'2')]
[_3:in(c,<1>C)][_4:in(d,<2>C)]
'2'_elim : in(R_2(c,d,a),<a>C) := PRIM
d * [C:[x,pterm]pterm]
[_1:[x,pterm][_,in(x,'2')]type(<x>C)][_2:in(c,<1>C)][_3:in(d,<2>C)]
'2'_eq_1 : eq_in(R_2(c,d,1),c,<1>C) := PRIM
'2'_eq_2 : eq_in(R_2(c,d,2),d,<2>C) := PRIM
* [A:pterm][B:pterm][c:pterm][_1:type(A)][_2:type(B)][_3:in(c,'2')]
'2'_elim_type : type(R_2(A,B,c)) := PRIM
B * [_1:type(A)][_2:type(B)]
'2'_eq_type_1 : eq_type(R_2(A,B,1),A) := PRIM
'2'_eq_type_2 : eq_type(R_2(A,B,2),B) := PRIM
{ ----------------------------- product types ----------------------------9-- }
* [A:pterm][B:[x,pterm]pterm] Pi : pterm := PRIM
A * [B':pterm] arrow : pterm := Pi(A,[x,pterm]B')
A * [b:[x,pterm]pterm] lambda : pterm := PRIM
* [f:pterm][a:pterm] app : pterm := PRIM
B * [C:[x,pterm]pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]eq_type(<x>B,<x>C)]
Pi_cong : eq_type(Pi(A,B),Pi(A,C)) := PRIM
B * [b:[x,pterm]pterm][c:[x,pterm]pterm]
[_1:type(A)][_2:[x,pterm][_,in(x,A)]eq_in(<x>b,<x>c,<x>B)]
lambda_cong : eq_in(lambda(A,b),lambda(A,c),Pi(A,B)) := PRIM
B * [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
Pi_form : type(Pi(A,B)) := PRIM
B * [b:[x,pterm]pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]in(<x>b,<x>B)]
Pi_intro : in(lambda(A,b),Pi(A,B)) := PRIM
B * [a:pterm][f:pterm]
[_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)][_3:in(f,Pi(A,B))][_4:in(a,A)]
Pi_elim : in(app(f,a),<a>B) := PRIM
a * [b:[x,pterm]pterm]
[_1:type(A)][_2:[x,pterm][_,in(x,A)]in(<x>b,<x>B)][_3:in(a,A)]
Pi_eq {beta} : eq_in(app(lambda(A,b),a),<a>b,<a>B) := PRIM
{ ----------------------------- sum types -------------------------------11-- }
* [A:pterm][B:[x,pterm]pterm] Sigma : pterm := PRIM
* [a:pterm][b:pterm] pair : pterm := PRIM
a * pi_1 : pterm := PRIM
a * pi_2 : pterm := PRIM
B * [C:[x,pterm]pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]eq_type(<x>B,<x>C)]
Sigma_cong : eq_type(Sigma(A,B),Sigma(A,C)) := PRIM
B * [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
Sigma_form : type(Sigma(A,B)) := PRIM
B * [a:pterm][b:pterm]
[_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)][_3:in(a,A)][_4:in(b,<a>B)]
Sigma_intro : in(pair(a,b),Sigma(A,B)) := PRIM
B * [c:pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
[_3:in(c,Sigma(A,B))]
Sigma_elim_1 : in(pi_1(c),A) := PRIM
Sigma_elim_2 : in(pi_2(c),<pi_1(c)>B) := PRIM
b * [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)][_3:in(a,A)][_4:in(b,<a>B)]
Sigma_eq_1 : eq_in(pi_1(pair(a,b)),a,A) := PRIM
Sigma_eq_2 : eq_in(pi_2(pair(a,b)),b,<a>B) := PRIM
{ ----------------------------- W types ----------------------------------8-- }
* [A:pterm][B:[x,pterm]pterm] W : pterm := PRIM
* [a:pterm][f:pterm] sup : pterm := PRIM
* [b:pterm][e:pterm] rec : pterm := PRIM
B * [C:[x,pterm]pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]eq_type(<x>B,<x>C)]
W_cong : eq_type(W(A,B),W(A,C)) := PRIM
B * [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
W_form : type(W(A,B)) := PRIM
B * [a,pterm][f:pterm]
[_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)][_3:in(a,A)]
[_4:in(f,arrow(<a>B,W(A,B)))]
W_intro : in(sup(a,f),W(A,B)) := PRIM
B * [C:[z,pterm]pterm]
[x:pterm][u:pterm]
D : pterm := arrow(Pi(<x>B,[y,pterm]<app(u,y)>C),<sup(x,u)>C)
C * [b:pterm][e:pterm]
[_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
[_3:[z,pterm][_,in(z,W)]type(<z>C)]
[_4:in(b,Pi(A,[x,pterm]Pi(arrow(<x>B,W),[u,pterm]D(x,u))))][_5:in(e,W)]
W_elim : in(rec(b,e),<e>C) := PRIM
b * [a:pterm][f:pterm]
g : pterm := lambda(<a>B,[y,pterm]rec(b,app(f,y)))
[_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
[_3:[z,pterm][_,in(z,W)]type(<z>C)]
[_4:in(b,Pi(A,[x,pterm]Pi(arrow(<x>B,W),[u,pterm]D(x,u))))]
[_5:in(a,A)][_6:in(f,arrow(<a>B,W))]
W_eq : eq_in(rec(b,sup(a,f)),app(app(app(b,a),f),g),<sup(a,f)>C) := PRIM
{ ----------------------------- extensionality --------------------------10-- }
* [A:pterm][a:pterm][b:pterm] Eq : pterm := PRIM
b * [_1:type(A)][_2:in(a,A)][_3:in(b,A)] Eq_form : type(Eq(A,a,b)) := PRIM
b * [_1:eq_in(a,b,A)] Eq_intro : in(star,Eq(A,a,b)) := PRIM
b * [c:pterm][_1:in(c,Eq(A,a,b))] Eq_elim : eq_in(a,b,A) := PRIM
Eq_eq : eq_in(c,star,Eq(A,a,b)) := PRIM
A * [B:pterm] EQ : pterm := PRIM
B * [_1:type(A)][_2:type(B)] EQ_form : type(EQ(A,B)) := PRIM
B * [_1:eq_type(A,B)] EQ_intro : in(star,EQ(A,B)) := PRIM
B * [c:pterm][_1:in(c,EQ(A,B))] EQ_elim : eq_type(A,B) := PRIM
EQ_eq : eq_in(c,star,EQ(A,B)) := PRIM
|
Automath | https://ncatlab.org/nlab/source/Automath |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--
\tableofcontents
## Idea
_Automath_ was historically the first [[logical framework]].
The goal of the Automath project, initiated by [[Nicolaas de Bruijn]], was to provide a tool for the
formalization of [[mathematics]] without [[foundations|foundational]] prejudice.
## Related concepts
* [[proof assistant]]
* [[dependent type theory]]
* [[Coq]], [[Agda]]
## References
* [The Automath Archive](http://www.win.tue.nl/automath/)
* [Automath Restaurant](https://cs.ru.nl/~freek/zfc-etc/) has examples of different foundations of mathematics encoded in Automath.
* [`aut`](http://www.cs.ru.nl/~freek/aut/) - An implementation in ANSI C.
On [[mathematical structures]] in Automath via [[type telescopes]]:
* {#Zucker1975} [[Jeffery Zucker]], *Formalization of Classical Mathematics in Automath*, Colloques Internationaux du Centre National de la Recherche Scientifique **249** (1975) 135-145 [[web](https://www.win.tue.nl/automath/archive/webversion/aut042/aut042.html), [pdf](https://www.win.tue.nl/automath/archive/pdf/aut042.pdf)]
also in: Studies in Logic and the Foundations of Mathematics **133** (1994) 127-139 [<a href="https://doi.org/10.1016/S0049-237X(08)70202-7">doi:10.1016/S0049-237X(08)70202-7</a>]
> [[Zucker (1975, §10.2)](#Zucker1975):] Now a general framework in which to view linear orders, or other algebraic structures, has been proposed by de Bruijn. It uses the notion of “telescope”. [...] A telescope therefore functions like a “generalized $\sum$”.
* [[Nicolaas de Bruijn]], *Telescopic mappings in typed lambda calculus*, Information and Computation **91** 2 (1991) 189-204 [<a href="https://doi.org/10.1016/0890-5401(91)90066-B">doi:10.1016/0890-5401(91)90066-B</a>]
category: software |
automatic differentiation | https://ncatlab.org/nlab/source/automatic+differentiation | ## Contents
* table of contents
{: toc}
## Idea
_Automatic differentiation_ (AD) is a technique for computing ([[transpose matrix|transposed]]) [[derivative|derivatives]] of functions implemented by computer programs, essentially by applying the chain-rule across program code.
It is typically the method of choice for computing derivatives in machine learning and scientific computing because of its efficiency and numerical stability.
## Forward and reverse automatic differentiation
AD works by calculating the (transposed) derivative of a composite program in terms of the (transposed) derivatives of the parts, by using the chain-rule. The distinction between derivatives and transposed derivatives leads to the main distinction in automatic differentiation modes:
* _forward mode AD_: this implements the [[tangent bundle]] functor $T$ and calculates derivatives by a forward pass to calculate function values (primals), followed by another forward pass to calculate derivative values (tangents); these two forward passes can be interleaved into a single forward pass for efficiency.
* _reverse mode AD_: this implements the [[cotangent bundle]] functor $T^*$ and calculates transposed derivatives by a forward pass to calculate function values (primals), followed by a _reverse_ pass (inverting the control flow of the original program) to calculate transposed derivative values (cotangents); these two passes cannot easily be interleaved due to their differing direction.
When calculating a derivative of a program that implements a function $f:R^n\to R^m$, reverse mode tends to be the more efficient algorithm if $n \gg m$ and forward mode tends to be more efficient if $n\ll m$.
Seeing that many tasks in machine learning and statistics require the calculation of derivatives (for use in gradient-based optimization or Monte-Carlo sampling) of functions $f:R^n\to R$ (e.g. probability density functions) for $n$ very large, reverse mode AD tends to be the most popular algorithm.
## Combinatory Homomorphic Automatic Differentiation (CHAD) - a categorical take on AD
Let us fix some class of categories with [[stuff]] $S$. We will call its members $S$-categories. For example, for our purposes, $S$ might include [[property|properties]] like
* closure under finite [[products]] (aka tuples, in programming-speak);
* [[cartesian closed category|Cartesian closure]] (aka higher-order functions);
* closure under finite [[coproducts]] (aka sum or variant types);
and [[structure]] like
* designated objects like $R$ and $Z$ (which we think of as real numbers and integers)
* designated morphisms like $sin:R\to R$, $cos: R\to R$, $(+): R\times R\to R$, $(*):R\times R\to R$ (which we think of as the corresponding mathematical operations).
The idea behind CHAD will be to view forward and reverse AD as the unique structure (stuff) preserving functor (homomorphism of $S$-categories) from the initial $S$-category $Syn$ to two suitably chosen $S$-categories $\Sigma_{CSyn}LSyn$ and $\Sigma_{CSyn}LSyn^{op}$.
### The source language
Consider the initial $S$-category $Syn$ (put differently, the S-properties category that is freely generated from $S$-structure).
We can think of this category as a programming language: its objects are types and its morphisms are programs modulo [[beta-equivalence]] and [[eta-equivalence]].
In fact, for a wide class of programming languages, we can find a suitable choice of stuff $S$, such that the programming language arises as the initial $S$-category.
We will refer to this category as the _source language_ of our AD transformation: its morphisms are the programs we want to differentiate.
For example, if we choose the property part of $S$ to consist of Cartesian closure and the structure part of $S$ consists of designated objects $R$ and $Z$ and morphisms $sin$, $cos$, $(+)$, and $(*)$, then $Syn$ is the simply typed [[lambda-calculus]] with base types $R$ and $Z$ and the primitive operations $sin$, $cos$, $(+)$, and $(*)$.
### The target language and deriving AD
Given a strictly [[indexed category]] $L:C^{op}\to Cat$, we can form its [[Grothendieck construction]] $\Sigma_C L$, which is a [[split fibration]] over $C$.
Similarly, we can take the [[opposite category]] $L^{op}:C^{op}\to Cat$ of $L$ and form $\Sigma_C L^{op}$ from that.
For most natural choices of stuff $S$, we can find elegant sufficient conditions on $C$ and $L$ that guarantee that $\Sigma_C L$ and $\Sigma_C L^{op}$ are both $S$-categories.
Let us call a strictly indexed category $L:C^{op}\to Cat$ satisfying these conditions a $S'$-category (where we think of $S'$ again as [[stuff]]).
We give the corresponding $S'$ for some examples for different choices of $S$:
* for the property of closure under finite products, we demand that $C$ has finite products and $L$ has finite ([[indexed]]) [[biproducts]] (or, equivalently, that $L$ has finite products and is enriched over [[commutative monoids]]); (this can be extended with conditions to guarantee Cartesian closure; see the CHAD paper below;)
* for the property of closure under finite coproducts, we demand that $C$ has finite coproducts and that $L$ is [[extensive]], in the sense that the canonical functors $L(0)\to 1$ and $L(A+B)\to L(A)\times L(B)$ are isomorphisms of categories;
* for structure like distinguished objects $R$, we need to have chosen distinguished objects $R'$ of $C$, $TR'$ (which we think of as the tangent bundle of $R'$) of $L(R')$ and $T^* R'$ (the cotangent bundle of $R'$) of $L(R')$;
* for structure like distinguished morphisms $(*):R\times R\to R$, we need to have chosen distinguished morphisms $(*)'$ in $C(R'\times R', R')$ (the primal computation of $(*)$), $T(*)'$ in $L(R'\times R')(TR'\times TR', TR')$ (the derivative of $(*)$) and $T^*(*)'$ in $L(R'\times R')(TR', TR'\times TR')$ (the transposed derivative of $(*)$).
Let us write $LSyn:CSyn^{op}\to Cat$ for the initial $S'$-category.
We think of this category as the _target language_ of automatic differentiation, in the sense that the forward/reverse derivatives of programs (morphisms) in the source language $Syn$ with consist of an associated primal program that is a morphism in $CSyn$ and an associated tangent/cotangent program that is a morphism in $LSyn$/$LSyn^{op}$.
As a programming language, $LSyn$ is a [[linear dependent type theory]] over the Cartesian type theory $CSyn$.
Indeed, as for any $S'$-category $L:C^{op}\to Cat$, $\Sigma_C L$ and $\Sigma_C L^{op}$ are $S$-categories, it follows that $\Sigma_{CSyn} LSyn$ and $\Sigma_{CSyn} LSyn^{op}$ are, in particular, $S$-categories.
Seeing that $Syn$ is the initial $S$-category, we obtain unique morphisms of $S$-categories:
* forward AD: $Df:Syn\to \Sigma_{CSyn} LSyn$;
* reverse AD: $Dr:Syn\to \Sigma_{CSyn} LSyn^{op}$.
### Semantics of the source and target languages
The category $Set$ of [[sets]] and functions gives another example of an $S$-category, for a lot of choices of $S$ (if we choose the sets $[[R]]$ and functions $[[op]]$ that we would like to denote with the types $R$ and operations $op$ in the structure part of $S$).
Moreover, the strictly indexed category $Fam(CMon):Set^{op}\to Cat$ of [[families]] of [[commutative monoids]] tends to give an example of an $S'$-category.
By initiality of $Syn$ and $LSyn:CSyn^{op}\to Cat$, we obtain
* the semantics of the source language: the unique $S$-functor $[[-]]:Syn\to Set$;
* the semantics of the target language: the unique $S'$-functor $[[-]]:(CSyn,LSyn)\to (Set,Fam(CMon))$ (with a slight abuse of notation).
In particular, we see that source language programs $t\in Syn(A, B)$ get interpreted as a function $[[t]]\in Set([[A]], [[B]])$.
Similarly, programs $t\in CSyn(A, B)$ in the target language with _Cartesian_ type get interpreted as a function $[[t]]\in Set([[A]], [[B]])$.
Finally, programs $t\in LSyn(A)(B,C)$ in the target language with _linear_ type get interpreted as a function $[[t]] \in Fam(CMon)([[A]])([[B]],[[C]])$:
families of monoid homomorphisms.
### Correctness of CHAD
We say that CHAD calculates the correct derivative (resp. transposed derivative) of a program $s$ if
the [[semantics]] $[[Df(s)]]$ of the program $Df(s)$ (resp. $Dr(s)$) equals the pair of the semantics $[[s]]$ of $s$ and the derivative $T[[s]]$ (resp. transposed derivative $T^*[[s]]$) of the semantics $[[s]]$ of $s$.
CHAD is correct in the sense that it calculates the correct (transposed) derivative of any composite (possibly higher-order) program between first-order types (meaning: types built using only [[positive type]] formers), provided that it calculates the correct (transposed) derivatives of all primitive operations like $(*)$ that we used to generate the source language.
That is, CHAD is a valid way for compositionally calculating (transposed) derivatives of composite computer programs, as long as we correctly implement the derivatives for all primitive operations (basic mathematical functions like multiplication, addition, sine, cosine) in the language.
We can prove this by a standard [[logical relations]] argument, relating smooth curves to their primal and (co)tangent curves.
Viewed more abstractly, the proof follows automatically because the [[Artin gluing]] along a representable functor (like the hom out of the real numbers) of an $S$-category is itself again an $S$-category, for common most choices of $S$.
## Related concepts
* [[differential programming]]
* [[backpropagation]]
## References
Forward mode automatic differentiation was introduced by Robert Edwin Wengert in
* [[Robert Edwin Wengert]], A simple automatic derivative evaluation program. Communications of the ACM 7.8. 1964.
An early description of reverse mode automatic differentiation can be found in
* [[Bert Speelpenning]], Compiling fast partial derivatives of functions given by algorithms. Illinois Univ., Urbana (USA). Dept. of Computer Science. Technical report. 1980.
but it was already described earlier by others such as Seppo Linnainmaa:
* [[Seppo Linnainmaa]], The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors. Master's Thesis (in Finnish), Univ. Helsinki. 1970.
A categorical analysis of (non-interleaved) forward and reverse AD is given by
* [[Matthijs Vákár]], CHAD: Combinatory Homomorphic Automatic Differentiation. 2021. [arXiv preprint 2103.15776](https://arxiv.org/abs/2103.15776) [Haskell implementation](https://github.com/VMatthijs/CHAD; funded by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 895827)
* [[Fernando Lucatelli Nunes]], [[Matthijs Vákár]], CHAD for Expressive Total Languages. 2021. [arXiv preprint 2110.00446] (https://arxiv.org/abs/2110.00446; funded by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 895827)
A categorical analysis of (interleaved) forward mode AD for calculating higher order derivatives is given by
* [[Mathieu Huot]], [[Sam Staton]], [[Matthijs Vákár]], Higher Order AD of Higher Order Functions. 2021. [arXiv preprint 2101.06757](https://arxiv.org/abs/2101.06757; funded by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 895827) |
automaton | https://ncatlab.org/nlab/source/automaton |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computing
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--
# Automata
* table of contents
{: toc}
## Idea
An *automaton* is an abstract concept of machine, modelled as a collection of states and transitions between states, together with an assignment of some external behavior (typically input and/or output) to these transitions. A quintessential example of an automaton is a vending machine, which can be in any number of different states (e.g., "READY", "INSERT 2 TOKENS", "OUT OF SERVICE", etc.), and will transition between these states in response to user input (e.g., from "READY" to "INSERT 2 TOKENS" after the user makes a selection, and from "INSERT 2 TOKENS" to "INSERT 1 TOKEN" after the user inserts a token).
Many different notions of automaton exist in the literature. For now this article only considers one fairly basic notion taken from [[Joy of Cats]], although it is by no means the simplest nor the most general.
## Definition
A __deterministic, sequential, Moore automaton__ is formally definable (as in [[Joy of Cats]]) as a sextuple ($Q$, $\Sigma$, $Y$, $\delta$, $q_{0}$, $y$), where $Q$ is the set of states, $\Sigma$ and $Y$ are the sets of input symbols and output symbols, respectively, $\delta$: $\Sigma$ $\times$ $Q$ $\to$ $Q$ is the transition map, $q_{0}$ $\epsilon$ $Q$ is the initial state, and $y$: $Q$ $\to$ $Y$ is the output map. [[morphism|Morphisms]] from an automaton ($Q$, $\Sigma$, $Y$, $\delta$, $q_{0}$, $y$) to an automaton ($Q$′, $\Sigma$′, $Y$′, $\delta$′, $q_{0}$′, $y$′) are triples ($f_{Q}$, $f_{\Sigma}$, $f_{Y}$) of [[function|functions]] $f_{Q}: Q \to Q\prime$, $f_{\Sigma}: \Sigma \to \Sigma\prime$, and $f_{Y}: Y \to Y\prime$ satisfying the following conditions:
(i) preservation of transition:
$\delta\prime$($f_{\Sigma}$($\sigma$), $f_{Q}$($q$)) = $f_{Q}$($\delta$($\sigma$, $q$)),
(ii) preservation of outputs: $f_{Y}$($y$($q$)) = $y$$\prime$($f_{Q}$($q$)),
(iii) preservation of initial state:
$f_{Q}$($q_{0}$) = $q_{0}\prime$.
Note that in such an automaton, the outputs are determined by the current state alone (and do not depend directly on the input).
A morphism $f$ : ($Q$, $\delta$, $q_{0}$, $F$) $\to$ ($Q\prime$, $\delta\prime$, $q_{0}\prime$, $F\prime$) (called a __simulation__) is a function $f : Q \to Q\prime$ that preserves:
(i) the transitions, i.e., $\delta\prime$($\sigma$, $f$($q$)) = $f$($\delta$($\sigma$, $q$)),
(ii) the initial state, i.e., $f$($q_{0}$) = $q_{0}\prime$, and
(iii) the final states, i.e., $f[F] \subseteq F\prime$.
## The category of automata
There is a [[category]] $Aut$ whose [[object|objects]] are deterministic sequential Moore automata and whose [[morphism|morphisms]] are simulations.
## Examples
* [[Mealy machine]]
* [[Moore machine]]
* [[Conway's game of life]]
## Variants
There are several variant forms of automaton. The above just gives a basic one. Others are treated in the entries:
* [[asynchronous automaton]];
* [[deterministic automaton]];
* [[nondeterministic automaton]].
There are tentative definitions of
*[[higher dimensional automaton]]
which take a more nPOV of automata theory.
## References
* [[Jiri Adamek]], [[Horst Herrlich]], [[George Strecker]], *[[Abstract and Concrete Categories]]*, John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories **17** (2006) 1-507 [[tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [book webpage](http://katmat.math.uni-bremen.de/acc/)]
* [[Mark V. Lawson]], _[Finite automata](http://www.ma.hw.ac.uk/~markl/books.html)_, CRC Press, see also [here](http://www.ma.hw.ac.uk/~markl/teaching/AUTOMATA/kleene.pdf) for a shorter version in the form of Course Notes.
* [[Jiří Adámek]], *Free algebras and automata realizations in the language of categories*, Commentationes Mathematicae Universitatis Carolinae **15**.4 (1974) 589-602 [[eudml:16649](https://eudml.org/doc/16649)]
* [[Jiří Adámek]], Věra Trnková, *Automata and algebras in categories* **37** Springer (1990) [[ISBN:9780792300106](https://link.springer.com/book/9780792300106)]
* Liang-Ting Chen, Henning Urbat, _A fibrational approach to automata theory_, [arxiv/1504.02692](http://arxiv.org/abs/1504.02692)
An early discussion of automata via [[string diagrams]] in the [[Cartesian monoidal category]] of [[finite sets]]:
* {#Hotz65} [[Günter Hotz]], _Eine Algebraisierung des Syntheseproblems von Schaltkreisen_, EIK, Bd. 1, (185-205), Bd, 2, (209-231) 1965 ([part I](https://www.magentacloud.de/lnk/LiPMlYfh), [part II](https://www.magentacloud.de/lnk/YivslUWJ), [[HotzSchaltkreise.pdf:file]])
Discussion of [[non-deterministic automata]] as 1-dimensional [[defect QFT|defect]] [[TQFTs]]:
* Paul Gustafson, [[Mee Seong Im]], Remy Kaldawy, [[Mikhail Khovanov]], Zachary Lihn, *Automata and one-dimensional TQFTs with defects* [[arXiv:2301.00700](https://arxiv.org/abs/2301.00700)]
Review:
* [[Mee Seong Im]], *Correspondence between automata and one-dimensional Boolean topological theories and TQFTs*, 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#ImMar2023), video:[YT](https://www.youtube.com/watch?v=q5kynYxwTmk)]
category:computer science
[[!redirects automata]]
[[!redirects automatons]]
[[!redirects Aut]]
[[!redirects finite state automaton]]
[[!redirects finite state automatons]]
[[!redirects finite state automata]]
[[!redirects cellular automaton]]
|
automorphic form | https://ncatlab.org/nlab/source/automorphic+form | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
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## Idea
{#Idea}
Quite generally, _automorphic forms_ are suitably well-behaved [[functions]] on a [[quotient]] space $K\backslash X$ where $K$ is typically a [[discrete group]], hence suitable functions on $X$ which are [[invariant]] under the [[action]] of a discrete group. The precise definition has evolved a good bit through time.
[[Henri Poincaré]] considered [[analytic functions]] invariant under a discrete infinite group of [[fractional linear transformations]] and called them _[[Fuchsian functions]]_ (after his advisor [[Lazarus Fuchs]]).
More generally, automorphic forms in the modern sense are suitable functions on a [[coset space]] $K \backslash G$, hence functions on [[groups]] $G$ which are [[invariant]] with respect to the [[action]] of the [[subgroup]] $K \hookrightarrow G$. The archetypical example here are [[modular forms]] regarded as functions on $K\backslash PSL(2,\mathbb{R})$ where $K$ is a [[congruence subgroup]], and for some time the terms "modular form" and "automorphic form" were used essentially synonymously, see [below](#ModularForms). Based on the fact that a [[modular form]] is a [[section]] of some [[line bundle]] on the [[moduli stack of elliptic curves]], [[Pierre Deligne]] defined an automorphic form to be a section of a line bundle on a [[Shimura variety]].
By pullback of functions the linear space of such functions hence constitutes a [[representation]] of $G$ and such representations are then called _automorphic representations_ (e.g. [Martin 13, p. 9](#Martin13)) , specifically so if $G = GL_n(\mathbb{A}_K)$ is the [[general linear group]] with [[coefficients]] in a [[ring of adeles]] of some [[global field]] and $K = GL_n(K)$. This is the subject of the _[[Langlands program]]_. There one also considers [[unramified]] such representations, which are constituted by functions that in addition are invariant under the action of $GL_n$ with coefficients in the [[integral adeles]], see [below](#InNumberTheory).
### Modular forms as classical automorphic forms on $PSL(2,\mathbb{R})$
{#ModularForms}
By a standard definition, a _[[modular form]]_ is a [[holomorphic function]] on the [[upper half plane]] $\mathfrak{H}$ satisfying a specified transformation property under the [[action]] of a given [[congruence subgroup]] $\Gamma$ of the [[modular group]] $G = PSL(2,\mathbb{Z})$ (e.g. [Martin 13, definition 1](#Martin13), [Litt, def. 1](#Litt)).
But the [[upper half plane]] is itself the [[coset]] of the [[projective linear group]] $G = PSL(\mathbb{R})$ by the subgroup $K = Stab_G(\{i\}) \simeq SO(2)/\{\pm I\}$
$$
f\colon \mathfrak{H} \simeq PSL(2,\mathbb{R})/K
\,.
$$
In view of this, one finds that every modular function $f \colon \mathfrak{H} \to \mathbb{C}$ lifts to a function
$$
\tilde f \colon \Gamma\backslash PSL(2,\mathbb{R})
\longrightarrow \mathbb{C}
\,,
$$
hence to a function on $G$ which is actually _invariant_ with respect to the $\Gamma$-action ("automorphy"), but which instead now satisfies some transformation property with respect to the action of $K$, as well as some well-behavedness property
This $\tilde f$ is the incarnation as an _automorphic function_ of the modular function $f$ (e.g. [Martin 13, around def. 3](#Martin13), [Litt, section 2](#Litt)). For emphasis these automorphic forms on $PSL(2,\mathbb{R})$ equivalent to modular forms are called _classical modular forms_.
This is where the concept of automorphic forms originates (for more on the history see e.g. [this MO comment](http://mathoverflow.net/a/124785/381) for the history of terminology) [and this one](http://mathoverflow.net/a/21556/381).
### Modular forms as adelic automorphic forms on $GL(2,\mathbb{A})$
{#ModularFormsAdAdelicAutomorphicForms}
Where by the [above](#ModularForms) an ordinary [[modular form]] is equivalently a suitably periodic function on $SL(2,\mathbb{R})$, one may observe that the [[real numbers]] $\mathbb{R}$ appearing as [[coefficients]] in the latter are but one of many [[p-adic number]] completions of the [[rational numbers]]. Hence it is natural to consider suitably periodic functions on $SL(2,\mathbb{Q}_p)$ of all these completions at once. This means to consider functions on $SL(2,\mathbb{A})$, for $\mathbb{A}$ the [[ring of adeles]]. These are the _adelic automorphic forms_. They may be thought of as subsuming ordinary modular forms for all [[level structures]].
(e.g. [Martin 13, p. 8](#Martin13), also [Goldfeld-Hundley 11, lemma 5.5.10](#GoldfeldHundley11), [Bump, section 3.6](#Bump), [Gelbhart 84, p. 22](#Gelbhart84)): we have
$$
\Gamma \backslash PSL(2,\mathbb{R})
\simeq
Z(\mathbb{A}) GL_2(\mathbb{Q})\backslash GL_2(\mathbb{A})/ GL_2(\mathbb{A}_{\mathbb{Z}})
\,,
$$
where $\mathbb{A}_{\mathbb{Z}}$ are the [[integral adeles]]. (The [[double coset]] on the right is analogous to that which appears in the [[Weil uniformization theorem]], see the discussion there and at _[[geometric Langlands correspondence]]_ for more on this analogy.)
This leads to the more general concept of _adelic automorphic forms_ [below](#InNumberTheory).
### General adelic automorphic forms
{#InNumberTheory}
More generally, for the [[general linear group]] $G = GL_n(\mathbb{A}_F)$, for any $n$ and with [[coefficients]] in a [[ring of adeles]] $\mathbb{A}_F$ of some [[number field]] $F$, and for the subgroup $GL_n(F)$, then sufficiently well-behaved functions on $GL_n(F)\backslash GL_n(\mathbb{A}_F)$ form [[representations]] of $GL_n(\mathbb{A}_{F})$ which are called _[[automorphic representations]]_. Here "well-behaved" typically means
1. **finiteness** -- the functions [[invariant]] under the [[action]] of the [[maximal compact subgroup]] [[span]] a [[finite number|finite]] [[dimension|dimensional]] [[vector space]];
1. **central character** -- the action by the [[center of a group|center]] is is controled by (...something...);
1. **growth** -- the functions are [[bounded functions]];
1. **cuspidality** -- (...)
(e.g. [Frenkel 05, section 1.6](#Frenkel05), [Loeffler 11, page 4](#Loeffler11), [Martin 13, definition 4](#Martin13), [Litt, def.4](#Litt)).
(These conditions are not entirely set in stone, they are being varied according to application (see e.g. [this MO comment](http://mathoverflow.net/a/66598/381))).
In particular one considers subspaces of "[[unramified]]" such functions, namely those which are in addition trivial on the subgroup of $GL_n$ of the [[integral adeles]] $\mathcal{O}_F$ ([Goldfeld-Hundley 11, def. 2.1.12](#GoldfeldHundley11)). This means that that unramified automorphic representations are spaces of functions on a [[double
coset]] of the form
$$
GL_n(F)\backslash GL_n(\mathbb{A}_F) / GL_n(\mathcal{O}_F)
\,.
$$
See at _[[Langlands correspondence]]_ for more on this. Such double cosets are [[analogy|analogous]] to those appearing in the [[Weil uniformization theorem]] in [[complex analytic geometry]], an analogy which leads to the conjecture of the [[geometric Langlands correspondence]].
### Dirichlet characters
For the special case of $n = 1$ in the discussion of adelic automorphic forms [above](#InNumberTheory), the group
$$
GL_1(\mathbb{A}_F) = (\mathbb{A}_F)^\times = \mathbb{I}_F
$$
is the [[group of ideles]] and the quotient
$$
GL_1(F) \backslash GL_1(\mathbb{A}_F)
=
F^\times \backslash (\mathbb{A}_F)^\times
$$
is the [[idele class group]]. Automorphic forms in this case are effectively [[Dirichlet characters]] in disguise... ([Goldfeld-Hundley 11, theorem 2.1.9](#GoldfeldHundley11)).
## Properties
### Function field analogy
[[!include function field analogy -- table]]
### Application in string theory
In [[string theory]] [[partition functions]] tend to be automorphic forms for [[U-duality]] groups. See the [references below](#ReferencesInStringTheory)
## Related entries
* [[modular form]], [[topological modular form]], [[topological automorphic form]]
* [[automorphic L-function]]
* [[Langlands duality]]
## References
### General
Introductions and surveys include
* {#Deligne} [[Pierre Deligne]], _Fromed Modulaires et representations de $GL(2)$_ ([](http://publications.ias.edu/sites/default/files/Number21.pdf))
* {#Gelbhart84} [[Stephen Gelbart]], starting on p. 20 (196) of _An elementary introduction to the Langlands program_, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([web](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))
* Nolan Wallach, _Introductory lectures on automorphic forms_ ([pdf](http://math.ucsd.edu/~nwallach/luminy-port2.pdf))
[[!redirects automorphic forms]]
* E. Kowalski, section 3 of _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))
* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))
* {#Bump} Daniel Bump, _Automorphic forms and representations_
* {#Loeffler11} David Loeffler, _Computing with algebraic automorphic forms_, 2011 ([[LoefflerAutomorphic.pdf:file]])
* {#Martin13} [[Kimball Martin]], _A brief overview of modular and automorphic forms_,2013 [pdf](http://www2.math.ou.edu/~kmartin/papers/mfs.pdf)
* {#Litt} [[Daniel Litt]], _Automorphic forms notes, part I_ ([pdf](http://math.stanford.edu/~dlitt/Talks/automorphicformspt1.pdf))
* Denis Trotabas, Modular Forms and Automorphic Representations (2010) ([pdf](http://math.stanford.edu/~conrad/modseminar/pdf/L10.pdf))
* Werner Muller, Spectral theory of automorphic forms (2010) ([pdf](http://www.math.uni-bonn.de/people/mueller/skripte/specauto.pdf))
* {#Miyake76} Toshitsune Miyake's _Modular Forms_ 1976 (English version 1989) ([review pdf](projecteuclid.org/euclid.bams/1183556263))
Review in the context of the [[geometric Langlands correspondence]] is in
* {#Frenkel05} [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_, in _Frontiers in number theory, physics, and geometry II_, Springer Berlin Heidelberg, 2007. 387-533. ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))
The generalization of theta functions to [[automorphic forms]] is due to
* [[André Weil]], _Sur certaines groups d'operateur unitaires_, Acta. Math. 111 (1964), 143-211
see [Gelbhart 84, page 35 (211)](Langlands+program#Gelbhart84) for review.
Further developments here include
* {#Kudla77} [[Stephen Kudla]], _Relations between automorphic forms produced by theta-functions_, in _Modular Functions of One Variable VI_, Lecture Notes in Math. 627, Springer, 1977, 277–285.
* {#Kudla78} [[Stephen Kudla]], _Theta functions and Hilbert modular forms_,Nagoya Math. J. 69 (1978) 97-106
* {#Stopple95} [[Jeffrey Stopple]], _Theta and $L$-function splittings_, Acta Arithmetica LXXII.2 (1995) ([pdf](http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7221.pdf))
### In string theory
{#ReferencesInStringTheory}
The relation between [[string theory]] on [[Riemann surfaces]] and automorphic forms was first highlighted in
* {#Witten88} [[Edward Witten]], _Quantum field theory, Grassmannians, and algebraic curves_, Comm. Math. Phys. Volume 113, Number 4 (1988), 529-700 ([Euclid](http://projecteuclid.org/euclid.cmp/1104160350))
See also
* {#GRV10} [[Michael Green]], Jorge G. Russo, Pierre Vanhove, _Automorphic properties of low energy string amplitudes in various dimensions_ ([arXiv:1001.2535](http://arxiv.org/abs/1001.2535))
[[!redirects automorphic forms]]
[[!redirects automorphic representation]]
[[!redirects automorphic representations]]
[[!redirects Fuchsian function]]
[[!redirects Fuchsian functions]] |
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## Idea
{#Idea}
An _automorphic L-function_ $L_\pi$ is an [[L-function]] built from an [[automorphic representation]] $\pi$, generalizing
1. how the [[Mellin transform]] takes an [[automorphic form]] to a [[zeta function]];
1. (nonabelian generalization) how a [[Dirichlet L-function]] $L_\chi$ is associated to a [[Dirichlet character]] $\chi$ (which is an automorphic form on the (abelian) [[idele group]]).
In analogy to how [[Artin reciprocity]] implies that to every 1-dimensional [[Galois representation]] $\sigma$ there is a [[Dirichlet character]] $\chi$ such that the [[Artin L-function]] $L_\sigma$ equals the [[Dirichlet L-function]] $L_\chi$, so the [[conjecture|conjectured]] [[Langlands correspondence]] says that to every $n$-dimensional [[Galois representation]] $\sigma$ there is an [[automorphic representation]] $\pi$ such that the automorphic L-function $L_\pi$ equals the [[Artin L-function]] $L_\sigma$.
## Related concepts
* [[Rankin-Selberg method]]
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]
## References
* Wikipedia, _[Automorphic L-function](http://en.wikipedia.org/wiki/Automorphic_L-function)_
* {#Gelbhart84} [[Stephen Gelbart]], starting on p. 26 (202) of _An elementary introduction to the Langlands program_, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([web](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))
* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))
* F. Shahidi, _Automorphic $L$-functions: A Survey_ in Laurent Clozel, [[James Milne]] (eds.) _Automorphic forms, Shimura varieties and $L$-functions_, volume I ([pdf](www.jmilne.org/math/articles/1990aT.pdf))
[[!redirects automorphic L-functions]] |
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## Definition
An __automorphism__ of an [[object]] $x$ in a [[category]] $C$ is an [[isomorphism]] $f : x \to x$. In other words, an automorphism is an [[endomorphism]] that is an [[isomorphism]].
## Automorphism group
Given an object $x$, the automorphisms of $x$ form a [[group]] under [[composition]], the __automorphism group__ of $x$, which is a submonoid of the [[endomorphism monoid]] of $x$:
$$
Aut_C(x) = End_C(x) \cap Iso(C) = Iso_C(x,x)
,$$
which may be written $Aut(x)$ if the category $C$ is understood. Up to equivalence, every group is an automorphism group; see [[delooping]].
## Examples
(...)
* [[permutation|Permutations]] are automorphisms in [[FinSet]].
* [[automorphism of a vertex operator algebra]]
## Related concepts
* **automorphism group**
* [[inner automorphism group]]
* [[outer automorphism group]]
* [[automorphism Lie group]]
* [[automorphism 2-group]]
* [[automorphism ∞-group]]
## References
Discussion of automorphism groups [[internalization|internal]] to [[sheaf toposes]] ("automorphism sheaves"):
* [[Simon Henry]], [MO:a/262687](https://mathoverflow.net/a/262687/381)
* Robert Friedman, John W. Morgan, §2.1 in: *Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections* [[arXiv:math/0209053](https://arxiv.org/abs/math/0209053)]
[[!redirects automorphisms]]
[[!redirects automorphism group]]
[[!redirects automorphism groups]]
[[!redirects group of automorphisms]]
[[!redirects groups of automorphisms]]
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## Automorphism 2-groups
For $C$ any [[2-category]] and $c \in C$ any object of it, the category $Aut_C(c) \subset Hom_C(c,c)$ of auto-equivalences of $c$ and invertible 2-morphisms between these is naturally a [[2-group]], whose group product comes from the horizontal composition in $C$.
If $C$ is a [[strict 2-category]] there is the notion of strict [[automorphism 2-group]]. See there for more details on that case.
For instance if $C = Grp_2 \subset Grpd$ is the 2-category of [[group]] obtained by regarding groups as one-object [[groupoid]]s, then for $H \in Grp$ a group, its automorphism 2-group obtained this way is the strict 2-group
$$
AUT(H) := Aut_{Grp_2}(H)
$$
corresponding to the [[crossed module]] $(H \stackrel{Ad}{\to} Aut(H))$, where $Aut(H)$ is the ordinary [[automorphism group]] of $H$.
## Inner automorphism 2-groups
See [[inner automorphism 2-group]].
## Related concepts
* [[group]], [[∞-group]],
* [[automorphism group]], [[automorphism ∞-group]],
* [[center]], [[center of an ∞-group]]
* [[outer automorphism group]], [[outer automorphism ∞-group]]
[[!redirects automorphism 2-groups]] |
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## Definition
### Externally
Let $\mathcal{C}$ be an [[(∞,1)-category]]. Let $X \in \mathcal{C}$ be an [[object]].
As a [[discrete ∞-groupoid|discrete]] [[∞-group]] the **automorphism $\infty$-group** of $X$ is the [[monomorphism in an (∞,1)-category|sub-]][[∞-groupoid]]
$$
Aut(X) \hookrightarrow \mathcal{C}(X,X)
$$
of the [[derived hom space]] of [[morphisms]] in $\mathcal{C}$ from $X$ to itself, on those that are [[equivalence in an (∞,1)-category|equivalences]].
This is an [[∞-group]] in [[∞Grpd]],
$$
Aut(X)\in Grp(\infty Grpd)
\,.
$$
### Internally
Let $\mathcal{C}$ be a [[cartesian closed (∞,1)-category]] (for instance an [[(∞,1)-topos]]). Write
$$
[-,-] : \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}
$$
for the [[internal hom]]. Then for $X \in \mathcal{C}$ an object, the _internal automorphism $\infty$-group_ is the [[subobject]]
$$
\mathbf{Aut}(X) \hookrightarrow [X,X]
$$
of the [[internal hom]] on those morphism that are [[equivalence in an (infinity,1)-category|equivalences]].
In the special case that $\mathcal{C}$ is an [[∞-topos]], the [[delooping]] $\mathbf{B}\mathbf{Aut}(X)$ of the internal automorphism $\infty$-group is equivalently the [[∞-image]]
$$
* \to \mathbf{B}\mathbf{Aut}(X) \hookrightarrow Obj
$$
of the morphism
$$
* \stackrel{\vdash X}{\to} Obj
$$
to [[generalized the|the]] [[object classifier]], that modulates $X$ (the "name" of $X$).
### In homotopy type theory
{#InHomotopyTypeTheory}
Let $\mathcal{C}$ be an [[(∞,1)-topos]]. Then its [[internal language]] is [[homotopy type theory]]. In terms of this the object $X \in \mathcal{C}$ is a [[type]] ([[homotopy type]]). In the type theory syntax the internal automorphism $\infty$-group $\mathbf{Aut}(X)$ then is (as a type, without yet the group structure)
$$
\vdash (X \stackrel{\simeq}{\to} X) : Type
\,,
$$
the [[subtype]] of the [[function type]] on the [[equivalence in homotopy type theory|equivalences]]. Its [[delooping]] $\mathbf{B}\mathbf{Aut}(X)$ is
$$
\vdash \; \left(\sum_{Y : Type} [Y = X]\right) \colon Type
\,,
$$
where on the right we have the [[dependent sum]] over one argument of the [[bracket type]]/[[inhabited type|(-1)-truncation]] $[X = Y] = isInhab(X = Y)$ of the [[identity type]] $(X = Y)$.
The equivalence of this definition to the previous one is essentially equivalent to the [[univalence axiom]].
## Examples
### In a 1-category
If $\mathcal{C}$ happens to be a [[1-category]] then the external automorphism $\infty$-group of an object is the ordinary [[automorphism group]] of that object.
If $\mathcal{C}$ happens to be a 1-[[topos]], then the internal automorphism $\infty$-group is the traditional automorphism [[group object]] in the topos. Etc.
### Of $\infty$-groups
For $G \in \infty Grp(\mathcal{X})$ an [[∞-group]] there is the direct automorphism $\infty$-group $Aut(G)$. But there is also the [[delooping]] $\mathbf{B}G \in \mathcal{X}$ and _its_ automorphism $\infty$-group.
Sometimes (for instance in the discussion of [[∞-gerbes]]) one considers
$$
AUT(G) := Aut(\mathbf{B}G)
$$
and calls this the automorphism $\infty$-group of $G$.
For instance when $G$ is an ordinary [[group]], $AUT(G)$ is the [[2-group]] discussed at [[automorphism 2-group]].
## Related concepts
* There may be the [[stuff, structure, property|structure]] of an [[∞-Lie group]] on $Aut(F)$. The corresponding [[∞-Lie algebra]] is an [[automorphism ∞-Lie algebra]].
* [[group]], [[∞-group]],
* [[automorphism group]], [[automorphism 2-group]]
* [[autoequivalence type]]
* [[center]], [[center of an ∞-group]]
* [[outer automorphism group]], [[automorphism 2-group]], [[outer automorphism ∞-group]]
* [[rigidification of a stack]]
[[!redirects automorphism ∞-group]]
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[[!redirects autoequivalence ∞-group]]
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[[!redirects ∞-automorphism]]
[[!redirects ∞-automorphisms]]
[[!redirects automorphism infinity-groups]]
[[!redirects automorphism n-group]]
[[!redirects automorphism n-groups]]
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## Idea
The _automorphism $\infty$-Lie algebra_ $aut(\mathfrak{g})$ of an [[∞-Lie algebra]] $\mathfrak{g}$ -- or dually $aut(CE(\mathfrak{g}))$ of the corresponding [[Chevalley-Eilenberg algebra]] -- has in degree $k$ the [[derivations]] on $CE(\mathfrak{g})$ of degree $-k$. The [[higher Lie algebra]] version of the [[automorphism Lie algebra]] of an ordinary [[Lie algebra]].
In terms of [[rational homotopy theory]] $aut(\mathfrak{g})$ is a model for the rationalization of the group of [[automorphisms]]s of the [[rational space]] $\exp(\mathfrak{g})$ corresponding to $CE(\mathfrak{g})$ under the [[Sullivan construction]].
## Definition
Let $A := (\wedge^\bullet \mathfrak{a}^*, d_A)$ be a [[semifree dga|semifree]] [[dg-algebra]] of [[finite type]].
Notice that for $\phi : A \to A$ a [[derivation]] of degree $-k$ and $\lambda : A\to A$ another derivation of degree $-l$ the [[commutator]]
$$
[\phi,\lambda] := \phi \circ \lambda - \lambda \circ \phi : A \to A
$$
is itself a derivation, of degree $-(k+l)$. In particular, since the [[differential]] $d_A : A \to A $ is itself a derivation of degree +1, we have that
$$
d_A \phi := [d_A, \phi] : A \to A
$$
is a derivation of degree $-(k+1)$.
+-- {: .un_defn}
###### Definition
**(automorphism $\infty$-Lie algebra)**
The [[∞-Lie algebra]] $aut(A)$ is the [[dg-Lie algebra]] which
* in degree $-k$ for $k \gt 0$ has the derivations $\phi : A \to A$ of degree $-k$;
* in degree $0$ the derivations that commute with the differential $d_A$
* whose differential $\delta_{aut(A)} := [d_A,-]$ is given by the commutator with the differential of $A$;
* whose Lie bracket is the commutator $[\phi,\lambda] = \phi \circ \lambda - \lambda \circ \phi$.
=--
## Properties
### Automorphism group {#AutomorphismGroup}
For stating the fundamental theorem about $aut(\mathfrak{g})$ below we need some facts about the ordinary [[automorphism group]] of a dg-algebra $A$.
(...)
(See chapter 6 of [Sullivan](#Sullivan)).
### Classifying space for $Aut(X)$-principal bundles
{#ClassifyingSpaces}
Let $X$ be a [[rational space]] whose [[Sullivan model]] is $\mathfrak{g}$, $X \simeq \exp(\mathfrak{g})$. Let $aut'(\mathfrak{g}) \subset aut(\mathfrak{g})$ be the sub dg-algebra of the automorphism $\infty$-Lie algebra on the maximal nilpotent ideal in degree 0. Let $G(X)$ be the maximal reductive group of genuine automorphisms of $CE(\mathfrak{g})$ (see [above](#AutomorphismGroup)).
Then the [[rational space]]
$$
\exp(aut'(\mathfrak{g}))/G(X)
\simeq
B Aut (X)
$$
is the [[classifying space]] for $Aut(X)$-[[principal bundle]]s, i.e. for [[bundle]]s with typical fiber $X$.
## Examples
* The [[inner derivation Lie 2-algebra]] $inn(\mathfrak{g})$ is the full subalgebra of the automorphism $\infty$-Lie algebra on the _inner_ derivations of the [[Chevalley-Eilenberg algebra]] of a [[Lie algebra]] $\mathfrak{g}$.
## Related concepts
* [[automorphism Lie algebra]]
* [[deformation theory]]
* [[tangent complex]]
* [[cotangent complex]]
## References
The general definition of $aut(\mathfrak{g})$ is the topic of p. 313 (45 of 63) and following in
* [[Dennis Sullivan]], _Infinitesimal computations in topology_ Publications Mathématiques de l'IHÉS, 47 (1977) ([numdam](http://www.numdam.org/item?id=PMIHES_1977__47__269_0)) {#Sullivan}
The automorphism group $Aut(A)$ of a dg-algebra is discussed in paragraph 6 there. Few details on proofs are given there. Only recently in
* [[Andrey Lazarev]], [[Jonathan Block]], ...
a detailed proof is given.
Concrete computations of $aut(\mathfrak{g})$ for some classes of [[rational space]]s $X = \exp(\mathfrak{g})$ can be found for instance in
* Samual Bruce Smith, _The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces_ , Journal of Pure and Applied Algebra
Volume 160, Issues 2-3, 25 June 2001, Pages 333-343
[[!redirects automorphism ∞-Lie algebra]]
[[!redirects automorphism dg-Lie algebra]]
[[!redirects automorphism infinity-Lie algebras]]
[[!redirects automorphism ∞-Lie algebras]]
[[!redirects automorphism dg-Lie algebras]]
[[!redirects automorphism L-infinity algebra]]
[[!redirects automorphism L-∞ algebra]]
[[!redirects automorphism L-infinity algebras]]
[[!redirects automorphism L-∞ algebras]]
|
automorphism Lie group | https://ncatlab.org/nlab/source/automorphism+Lie+group |
The [[automorphism group]] $Aut(G)$ of a [[Lie group]] $G$ itself canonically inherits the structure of a Lie group. Its [[Lie algebra]] is the [[derivation Lie algebra]] of the Lie algebra $\mathfrak{g}$ of $G$: $Lie(Aut(G)) \simeq \mathfrak{der}(\mathfrak{g})$.
[[!redirects automorphism Lie groups]]
|
automorphism of a vertex operator algebra | https://ncatlab.org/nlab/source/automorphism+of+a+vertex+operator+algebra |
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given a [[vertex operator algebra]] $\mathcal{V}$ ([[VOA]]) or [[super vertex operator algebra]] ([[sVOA]]), or more generally a full [[2d CFT]] (of which the [[VOA]] is the local and chiral data) or [[2d SCFT]] (hence a "[[2-spectral triple]]") one may ask (as for any [[object]] in any [[category]]) for its [[automorphisms]], hence the [[homomorphisms]]
$$
\mathcal{V}
\overset{\simeq}{\longrightarrow}
\mathcal{V}
$$
in the corresponding [[category]] of sVOA-s/[[2d SCFTs]], from $\mathcal{V}$ to itself, which are [[invertible morphism|invertible]] and hence constitute a _[[symmetry]]_ of $\mathcal{V}$.
## In perturbative string vacua and Connes-Lott models
If $\mathcal{V}$ is a [[direct sum|direct summand]] of a [[2d CFT]]/[[2-spectral triple]] encoding a [[perturbative string theory vacuum]], such as, typically, a [[rational 2d CFT]] such as a [[Gepner model]] encoding a "[[non-geometric string vacuum|non-geometric]]" [[KK-compactification]], then the automorphisms of $\mathcal{V}$ are the [[formal dual|formal duals]] to [[symmetries]] of that [[KK-compactification]]-[[fiber]] space (see the references [below](#AsSymmetriesOfStringCompactifications))
In the point-particle limit where the [[2d SCFT]]/[[2-spectral triple]] becomes an ordinary [[spectral triple]] (see [there](2-spectral+triple#References)) this hence reduces to the automorphisms of internal algebras as discussed in [[Connes-Lott-Chamseddine-Barrett models]].
(...)
## Examples
### Moonshine
[[moonshine]]-examples:
* The [[Conway group]] $CO_{0}$ is the [[automorphism group|group of]] [[automorphisms of a super VOA]] of the unique chiral [[number of supersymmetries|N=1]] [[super vertex operator algebra]] of [[central charge]] $c = 12$ without fields of [[conformal weight]] $1/2$
([Duncan 05](#Duncan05), see also [Paquette-Persson-Volpato 17, p. 9](#PaquettePerssonVolpato17))
* similarly, there is a super VOA, the _[[Monster vertex operator algebra]]_, whose [[automorphism group|group of]] of [[automorphisms of a VOA]] is the [[monster group]]
([Frenkel-Lepowski-Meurman 89](#FrenkelLepowskiMeurman89), [Griess-Lam 11](#GriessLam11))
## References
### General
(...)
### As symmetries of non-geometric string compactifications
{#AsSymmetriesOfStringCompactifications}
Automorphisms of [[vertex operator algebras]] regarded as [[symmetries]] of non-geometric [[perturbative string theory vacua]] (e.g. [[Gepner models]]):
* {#HullIsraelSarti17} [[Chris Hull]], Dan Israel, Alessandra Sarti, _Non-geometric Calabi-Yau Backgrounds and K3 automorphisms_, JHEP11(2017)084 ([arXiv:1710.00853](https://arxiv.org/abs/1710.00853))
### Moonshine automorphism groups
* {#FrenkelLepowskiMeurman89} [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, _Vertex operator algebras and the monster_, Pure and Applied Mathematics __134__, Academic Press, New York 1998. liv+508 pp. [MR0996026](http://www.ams.org/mathscinet-getitem?mr=996026)
* {#Duncan05} John F. Duncan, _Super-moonshine for Conway's largest sporadic group_ ([arXiv:math/0502267](https://arxiv.org/abs/math/0502267))
* {#GriessLam11} [[Robert Griess]] Jr., Ching Hung Lam, _A new existence proof of the Monster by VOA theory_ ([arXiv:1103.1414](https://arxiv.org/abs/1103.1414))
* {#KachruPaquetteVolpato16} [[Shamit Kachru]], [[Natalie Paquette]], [[Roberto Volpato]], _3D String Theory and Umbral Moonshine_ ([arXiv:1603.07330](http://arxiv.org/abs/1603.07330))
* {#PaquettePerssonVolpato16} [[Natalie Paquette]], Daniel Persson, [[Roberto Volpato]], _Monstrous BPS-Algebras and the Superstring Origin of Moonshine_ ([arXiv:1601.05412](http://arxiv.org/abs/1601.05412))
* [[Miranda Cheng]], Sarah M. Harrison, [[Roberto Volpato]], Max Zimet, _K3 String Theory, Lattices and Moonshine_ ([arXiv:1612.04404](https://arxiv.org/abs/1612.04404))
* {#PaquettePerssonVolpato17} [[Natalie Paquette]], Daniel Persson, Roberto Volpato, _BPS Algebras, Genus Zero, and the Heterotic Monster_ ([arXiv:1701.05169](https://arxiv.org/abs/1701.05169))
* [[Shamit Kachru]], Arnav Tripathy, _The hidden symmetry of the heterotic string_ ([arXiv:1702.02572](https://arxiv.org/abs/1702.02572))
[[!redirects automorphisms of a vertex operator algebra]]
[[!redirects automorphisms of vertex operator algebras]]
[[!redirects automorphism of a VOA]]
[[!redirects automorphisms of a VOA]]
[[!redirects automorphisms of VOAs]]
[[!redirects automorphism of a super vertex operator algebra]]
[[!redirects automorphisms of a super vertex operator algebra]]
[[!redirects automorphisms of super vertex operator algebras]]
[[!redirects automorphism of a super VOA]]
[[!redirects automorphisms of a super VOA]]
[[!redirects automorphisms of super VOAs]]
[[!redirects automorphism of an sVOA]]
[[!redirects automorphisms of an sVOA]]
[[!redirects automorphisms of sVOAs]]
[[!redirects automorphism of a 2d conformal field theory]]
[[!redirects automorphisms of a 2d conformal field theory]]
[[!redirects automorphisms of 2d conformal field theories]]
[[!redirects automorphism of a 2d CFT]]
[[!redirects automorphisms of a 2d CFT]]
[[!redirects automorphisms of 2d CFT]]
[[!redirects automorphism of a 2d super-conformal field theory]]
[[!redirects automorphisms of a 2d super-conformal field theory]]
[[!redirects automorphisms of 2d super-conformal field theories]]
[[!redirects automorphism of a 2d sCFT]]
[[!redirects automorphisms of a 2d sCFT]]
[[!redirects automorphisms of 2d sCFT]]
[[!redirects automorphism of a 2-spectral triple]]
[[!redirects automorphisms of a 2-spectral triple]]
[[!redirects automorphisms of 2-spectral triples]]
|
automorphism of free group | https://ncatlab.org/nlab/source/automorphism+of+free+group | It appears that the group $Aut(F_n)$ of automorphisms of the free group $F_n$ on $n$ letters and especially the group $Out(F_n)$ of outer automorphisms of the free group on $n$ letters appear often at sometimes unexpected places in geometry, theory of operads, deformation theory, study of moduli spaces etc. Some of the occurences are expressed in terms of Kontsevich's [[graph homology]], ribbon diagrams and alike.
Some of the authors who wrote extensively on related issues include Karen Vogtmann ([homepage](http://www.math.cornell.edu/~vogtmann/papers/papers.html)). Cf. also articles by [[Leila Schneps]] etc. |
automorphism of the complex numbers | https://ncatlab.org/nlab/source/automorphism+of+the+complex+numbers |
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###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An [[automorphism]] of the [[field]] of [[complex numbers]].
Beyond the two evident ones, the [[identity]] and [[complex conjugation]], there is an infinitude of "wild" automorphisms, which, however, are not [[continuous functions|continuous]] and not [[constructive mathematics|constructive]] (their existence requires [[Zorn's lemma]]).
## References
* H. Kestelman, _Automorphisms of the Field of Complex Numbers_, Proceedings of the London Mathematical Society, Volume s2-53, Issue 1, 1951 ([doi:10.1112/plms/s2-53.1.1](https://doi.org/10.1112/plms/s2-53.1.1))
* Paul B. Yale, _Automorphisms of the Complex Numbers_, Mathematics Magazine
Mathematics Magazine Vol. 39, No. 3 (May, 1966), pp. 135-141 ([doi:10.2307/2689301](https://doi.org/10.2307/2689301))
* MathOverflow, _[Wild automorphisms of the complex numbers](https://math.stackexchange.com/q/412010/58526)_
[[!redirects automorphisms of the complex numbers]] |
autopoiesis | https://ncatlab.org/nlab/source/autopoiesis |
## References
* Wikipedia, _[Autopoiesis](http://en.wikipedia.org/wiki/Autopoiesis)_
* [[Erich Jantsch]], _[[The Self-Organizing Universe]]_
|
auxiliary field | https://ncatlab.org/nlab/source/auxiliary+field |
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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[quantum field theory]] an "auxiliary field" is a [[field (physics)|field]] [[type]] that is introduced on top of the [[field (physics)|fields]] in the [[field theory]] genuinely of interest, in order to bring that field theory of actual interest into another form -- usually while not actually changing it up to [[equivalence]] -- which lends itself better to certain purposes.
## Examples
For example under suitable conditions a field theory with [[constraints]] may be formulated equivalently as another field theory without explicit constraints, but with an auxiliary [[Lagrange multiplier]] field which does induce that constraint after all, but indirectly so via its [[equation of motion]].
In [[BV-BRST formalism]], a powerful generalization of this idea of [[Lagrange multiplier]] is the usage of auxiliary fields that allow to define "[[gauge fixing fermions]]" to implement [[gauge fixing]] via their [[equations of motion]].
For instance in the [[quantization of Yang-Mills theory]] the _[[Nakanishi-Lautrup field]]_ (and its "[[antighost field]]") are auxiliary fields that are introduced in order to indirectly induce [[gauge fixing]] to [[Lorentz gauge]]. The [[BV-BRST formalism]] also makes precise how exactly this introduction of auxiliary fields does not change the field theory, up to [[equivalence]]: the auxiliary fields change the [[BV-BRST complex]] (which effectivley defines the field theory) only up to a [[quasi-isomorphism]], i.e. the relevant notion of [[equivalence]] in this context.
In fact even before it comes to [[gauge fixing]], [[BV-BRST formalism]] introduces a variety of auxiliary fields: the [[ghosts]], the [[ghosts-of-ghosts]], etc., and the [[antifields]].
[[!redirects auxiliary fields]]
[[!redirects auxiliary field bundle]]
[[!redirects auxiliary field bundles]]
|
Avogadro constant | https://ncatlab.org/nlab/source/Avogadro+constant |
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
In macroscopic [[thermodynamics|thermodynamic]] systems one deals with a large [[number]] of microscopic particles. The number of particles is, for 'human sized' systems (e.g. engines, rooms), roughly on the order of the _Avogadro constant_ (or 'Avogadro\'s number' in historical usage). This constant is used as a scaling factor between macroscopic and microscopic observations.
More precisely, **Avogadro's number** is defined as the number of [[atoms]] in $0.012$ [[kg]] of the [[carbon]] [[isotope]] $\mathrm{C}^12$. This is roughly the [[real number]]
$$
6.02214179(30) \times 10^{23} \;\in\; \mathbb{R}
\,.
$$
An amount of any chemical substance which contains precisely this number of [[molecule]]s is called a **mole** of substance. So in particular, by definition, a mole of $\mathrm{C}^{12}$ is precisely of weight $12 \,\mathrm{g}$.
Attempts to define the SI unit of mass, i.e. the kilogram, in terms of invariant and basic physical concepts generally involve trying to measure Avogadro's number at increasing levels of accuracy. As with the speed of light, once the measurement of this derived quantity (taking the kilogram as the mass of certain reference kilogram cylinders) reaches a certain threshold of accuracy, it will make sense to specify Avogadro's number exactly and then derive the units of mass.
Of course the choice of $12 \,\mathrm{g}$ of $\mathrm{C}^{12}$ is entirely arbitrary and just serves as some example of a typical laboratory-scale macroscopic quantity.
Accordingly, later it was found useful to have different such macroscopic reference quantities, for instance the _pound mole_ in addition to the (standard) gram mole. For exchanging between these it was found useful to introduce [[unit]]s and treat 'amount of substance' as a [[dimensional analysis|dimension]].
Then the **Avogadro constant** is taken to be
$$
N_A = 6.02214179(30) \times 10^{23} \,mol^{-1}
\,.
$$
(Strictly speaking this is $N_A = 1 \in \mathbb{R}$ in [[natural unit]]s.)
[[!redirects Avogadro constant]]
[[!redirects Avogadro's constant]]
[[!redirects Avogadro number]]
[[!redirects Avogadro's number]]
[[!redirects mole]]
[[!redirects moles]]
|
Awais Shaukat | https://ncatlab.org/nlab/source/Awais+Shaukat |
## Selected writings
On the [[ordinary homology]] of [[configuration spaces of points]] in a [[surface]] as a [[group representation]] of the [[mapping class group]]:
* [[Christian Blanchet]], [[Martin Palmer]], [[Awais Shaukat]], *Heisenberg homology on surface configurations* [[arXiv:2109.00515](https://arxiv.org/abs/2109.00515)]
* [[Awais Shaukat]], [[Christian Blanchet]], *Weakly framed surface configurations, Heisenberg homology and Mapping Class Group action*, Archiv der Mathematik **120** (2023) 99–109 [[arXiv:2206.11475](https://arxiv.org/abs/2206.11475), [doi:10.1007/s00013-022-01793-3](https://doi.org/10.1007/s00013-022-01793-3)]
category: people |
Awodey's proposal | https://ncatlab.org/nlab/source/Awodey%27s+proposal |
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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
### Statement
In [Awodey 09](#Awodey09), [Awodey 10](#Awodey10) was first expressed
the idea that [[dependent type theory]] with intensional [[identity types]] ([[Martin-Löf dependent type theory]]), viewed as [[homotopy type theory]], is in similar relation to the concept of [[(∞,1)-toposes]] as [[extensional type theory]] is to the ordinary concept of [[toposes]] (as discussed at [[relation between type theory and category theory]]).
From [Awodey 09, p. 13](#Awodey09), [Awodey 10, p. 15](#Awodey10):
> The homotopy interpretation of Martin-Löf type theory into Quillen model categories, and the related results on type-theoretic constructions of higher groupoids, are analogous to basic results interpreting extensional type theory and higher-order logic in (1-)toposes, and clearly indicate that the logic of higher toposes, and therewith of higher homotopy theory, is a form of intensional type theory.
{#ConciseReStatement} A concise re-statement would be that:
1. the [[internal logic of an (∞,1)-topos|internal logic of (∞,1)-toposes]] is [[univalence axiom|univalent]] [[homotopy type theory]]
(though there is fine print involved, e.g. the [[initiality conjecture]]);
1. there is a [[model of type theory in an (infinity,1)-topos|model of (univalent) homotopy type theory in any $(\infty,1)$-topos]]
(this version has a proof, see [below](#Proof));
1. [[homotopy type theory]] is *[[synthetic homotopy theory]]*
(this may be read as a suggestive colloquial version of the previous statement, remaining vague on whether [[univalence axiom|univalence]] is considered or not).
Following this suggestion, the weaker form of this idea, ignoring the [[univalence|univalent]] [[type universe]] and relating to the broader class of [[locally Cartesian closed (∞,1)-categories]], was stated more concretely as a conjecture in [Joyal 11](#Joyal11). For more precision see [Kapulkin-Lumsdaine 16, p. 9](#KapulkinLumsdaine16).
Roughly, this is about the following table of correspondences (for more see at *[[relation between type theory and category theory]]*):
| [[internal logic]]/[[type theory]] | [[higher category theory|higher]] [[category theory]] |
|-----|--------|
| [[type theory]] | [[locally Cartesian closed categories]] |
| [[homotopy type theory]] | [[locally Cartesian closed (∞,1)-categories]] |
| [[homotopy type theory]] with [[univalence|univalent]] [[type universes]] | [[elementary (∞,1)-toposes]] |
Fore more precision see [Kapulkin-Lumsdaine 16, p. 9](#KapulkinLumsdaine16).
### Proof
{#Proof}
A [[proof]] of the weaker version of the conjecture, in form of the statement that every [[locally presentable (∞,1)-category|locally presentable]] [[locally Cartesian closed (∞,1)-category]] is presented by a suitable [[type theoretic model category]] which provides [[categorical semantics]] for [[homotopy type theory]], was proven in [Shulman 12, Example 2.16](#Shulman12), following [Cisinski 12](#Cisinski12).
Generalizing this to a proof of the full conjecture required finding "strict" models for the [object classifier](object+classifier#DetailsObjClassf) by strict [[type universes]]. A series of article ([Shulman 12](#Shulman12), [Shulman 13](#Shulman13)) showed that this is possible in an increasing class of special cases.
A proof of the general case was finally announced in [Shulman 19](#Shulman19).
For more see at _[[model of type theory in an (infinity,1)-topos]]_.
## References
### Statement
The idea is due to
* {#Awodey09} [[Steve Awodey]], _Homotopy and Type Theory_, grant proposal project description ([pdf](https://ncatlab.org/homotopytypetheory/files/proposal2009.pdf))
* {#Awodey10} [[Steve Awodey]], *Type theory and homotopy*, in: Dybjer P., Lindström S., Palmgren E., Sundholm G. (eds.), *Epistemology versus Ontology*, Springer (2012) 183-201 $[$[arXiv:1010.1810](http://arxiv.org/abs/1010.1810), [doi:10.1007/978-94-007-4435-6_9](https://doi.org/10.1007/978-94-007-4435-6_9)$]$
A pronounced statement of the weaker version was highlighted in
* {#Joyal11} [[André Joyal]], _Remarks on homotopical logic_ Oberwolfach (2011) ([pdf](http://hottheory.files.wordpress.com/2011/06/report-11_2011.pdf#page=19))
and stated more precisely in
* {#KapulkinLumsdaine16} [[Chris Kapulkin]], [[Peter Lumsdaine]], *The homotopy theory of type theories*, Advances in Mathematics **337** (2018) 1-38 $[$[arXiv:1610.00037](https://arxiv.org/abs/1610.00037), [doi:10.1016/j.aim.2018.08.003](https://doi.org/10.1016/j.aim.2018.08.003)$]$
### Proof
{#ReferencesProof}
The proof of the weaker version of Awodey's conjecture (that every [[locally Cartesian closed (∞,1)-category]] has a presentation by a suitable [[type-theoretic model category]] which provides [[categorical semantics]] for [[homotopy type theory]]) is due, independently, to
* {#Shulman12} [[Michael Shulman]], Example 2.16 of *Univalence for inverse diagrams and homotopy canonicity*, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( _From type theory and homotopy theory to Univalent Foundations of Mathematics_ ) June 2015 ([arXiv:1203.3253](https://arxiv.org/abs/1203.3253), [doi:/10.1017/S0960129514000565](https://doi.org/10.1017/S0960129514000565))
following
* {#Cisinski12} [[Denis-Charles Cisinski]], [blog comment](http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html#c041306) (2012)
The proof of the stronger version (including [[univalence|univalent]] [[type universes]] modelling [object classifier](object+classifier#DetailsObjClassf) of [[(∞,1)-toposes]]) was found for the special case of [[(∞,1)-presheaf (∞,1)-toposes]] over [[elegant Reedy categories]] in
* [Shulman 12](#Shulman12)
* {#Shulman13} [[Michael Shulman]], _The univalence axiom for elegant Reedy presheaves_, Homology, Homotopy and Applications Volume 17 (2015) Number 2 ([arXiv:1307.6248](https://arxiv.org/abs/1307.6248), [doi:10.4310/HHA.2015.v17.n2.a6](http://dx.doi.org/10.4310/HHA.2015.v17.n2.a6))
A general proof was announced in
* {#Shulman19} [[Michael Shulman]], slides 5 to 10 of: _Semantics of higher modalities_, talk at _[Geometry in Modal HoTT (2019)](https://felix-cherubini.de/modal-workshop.html)_ ([pdf slides](http://home.sandiego.edu/~shulman/papers/cmu2019b.pdf), [video recording](https://www.youtube.com/watch?v=Wcpi1vVMrCs))
and appeared in
* {#Shulman19} [[Michael Shulman]], _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004))
It is reviewed in:
* {#Riehl22} [[Emily Riehl]], *On the $\infty$-topos semantics of homotopy type theory*, lecture at *[Logic and higher structures](https://conferences.cirm-math.fr/2689.html)* CIRM (Feb. 2022) $[$[pdf](https://emilyriehl.github.io/files/semantics.pdf), [[Riehl-InfinityToposSemantics.pdf:file]]$]$
[[!redirects Awodey proposal]]
[[!redirects Awodey conjecture]]
[[!redirects Awodey's conjecture]]
[[!redirects Awodey's conjecture]]
[[!redirects Awodey's conjecture]]
|
Axel Kleinschmidt | https://ncatlab.org/nlab/source/Axel+Kleinschmidt |
* [webpage](http://www.aei.mpg.de/~axkl/)
## Selected writings
On [[U-duality]] and [[diffeomorphisms]] in [[exceptional generalized geometry]]:
* [[David Berman]], [[Martin Cederwall]], [[Axel Kleinschmidt]], [[Daniel C. Thompson]], _The gauge structure of generalised diffeomorphisms_, High Energ. Phys. 2013, 64 (2013) [[arXiv:1208.5884](http://arxiv.org/abs/1208.5884)]
On [[E9]] [[exceptional field theory]]:
* Guillaume Bossard, Franz Ciceri, Gianluca Inverso, [[Axel Kleinschmidt]], [[Henning Samtleben]], _$E_9$ exceptional field theory II. The complete dynamics_ ([arXiv:2103.12118](https://arxiv.org/abs/2103.12118))
On [[E11]]-[[exceptional field theory]]:
* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _On supersymmetric E11 exceptional field theory_ ([arXiv:1907.02080](https://arxiv.org/abs/1907.02080))
* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _A master exceptional field theory_ ([arXiv:2103.13411](https://arxiv.org/abs/2103.13411))
On maximal [[gauged supergravity|gauged]] [[D=2 supergravity]] via [[KK-reduction]] of [[D=11 supergravity]]:
* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Maximal $D=2$ supergravities from higher dimensions* [[arXiv:2309.07232](https://arxiv.org/abs/2309.07232)]
* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Consistent truncation of eleven-dimensional supergravity on $S^8 \times S^1$* [[arXiv:2309.07233](https://arxiv.org/abs/2309.07233)]
## Related entries
* [[U-duality]], [[Kac-Moody group]]
category: people
|
Axel Ljungström | https://ncatlab.org/nlab/source/Axel+Ljungstr%C3%B6m |
* [institute page](https://www.su.se/english/profiles/axlj4439-1.450268)
## Selected writings and talks
On [[ordinary cohomology in homotopy type theory]], specifically on [[integral cohomology]]:
* [[Guillaume Brunerie]], [[Axel Ljungström]], [[Anders Mörtberg]], *Synthetic Integral Cohomology in Cubical Agda*, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) **216** (2022) $[$[doi:10.4230/LIPIcs.CSL.2022.11](https://doi.org/10.4230/LIPIcs.CSL.2022.11)$]$
> (in [[cubical type theory|cubical]] [[Agda]])
On [[homotopy groups of spheres in homotopy type theory]], specifically on the [[first stable homotopy group of spheres]]:
* {#Ljungstroem22} [[Axel Ljungström]], *The Brunerie Number Is -2* (June 2022) [[blog entry](https://homotopytypetheory.org/2022/06/09/the-brunerie-number-is-2/)]
* [[Axel Ljungström]], *Calculating a Brunerie Number*, [[Homotopy Type Theory Electronic Seminar Talks]], 20 October 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Ljungstrom-2022-10-20-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=MIMeQ88YMKI))
Implementation of [[ordinary cohomology|ordinary]]$\;$[[cohomology rings]] in [[cubical agda]]:
* [[Thomas Lamiaux]], [[Axel Ljungström]], [[Anders Mörtberg]], _Computing Cohomology Rings in Cubical Agda_, CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs (2023) [[arxiv:2212.04182] (https://arxiv.org/abs/2212.04182), [doi:10.1145/3573105.3575677](https://doi.org/10.1145/3573105.3575677)]
category: people
[[!redirects Axel Ljungstrom]]
[[!redirects Axel Ljungstroem]]
|
Axel Osmond | https://ncatlab.org/nlab/source/Axel+Osmond |
* [institute page](https://www.irif.fr/users/osmond/index)
## Selected writings
On the notion of [[spectrum (geometry)|spectrum in geometry]] known as the [[Diers spectrum]]:
* [[Axel Osmond]], _On Diers theory of Spectrum I: Stable functors and right multi-adjoints_, ([arXiv:2012.00853](https://arxiv.org/abs/2012.00853))
> (on [[multi-adjoints]])
* [[Axel Osmond]], _On Diers theory of Spectrum II: Geometries and dualities_, ([arXiv:2012.02167](https://arxiv.org/abs/2012.02167))
* [[Axel Osmond]], _The general construction of Spectra_, [arXiv:2102.01259](https://arxiv.org/abs/2102.01259).
category: people
|
axio-dilaton | https://ncatlab.org/nlab/source/axio-dilaton |
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[type IIB supergravity]] the [[dilaton]]/[[string coupling constant]] $g_s$ and the degree-1 [[RR-field]] $C_0$ of the [[D(-1)-brane]] combine into a [[complex number|complex]] function
$$
\tau \coloneqq C_0 + i/ g_s
$$
which transforms under the [[modular group]] $SL(2,\mathbb{Z})$ [[S-duality]] $\subset$ [[U-duality]] as
$$
\left(
\array{
a & b
\\
c & d
}
\right)
:
\tau
\mapsto
\frac{a \tau + b}{c \tau + d}
\,.
$$
This combined field is also called the **axio-dilaton**.
## Properties
* In [[F-theory]] the variation of the axio-dilaton determines an [[elliptic fibration]] which encodes non-perturbative [[vacua]] of type IIB string theory.
|
axiom | https://ncatlab.org/nlab/source/axiom |
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### Context
#### Type theory
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[[!include type theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
An _axiom_ is a [[proposition]] in [[logic]] that a given [[theory]] requires to be [[true]]: every [[model]] of the theory is required to make the axiom hold true. The sense however is that an axiom is a _basic_ true proposition, used to prove other true propositions (the [[theorems]]) in the theory.
## Definition
Given a [[language]] $L$ (perhaps specified by a [[signature (in logic)|signature]]: a collection of [[types]], [[function symbols]] and [[relation symbols]]), a [[theory]] is the collection of assertions which are derivable (using the rules of [[deduction]] of the ambient [[logic]] or [[deductive system]]) from a given set of assertions, called **axioms** of the theory. In other words, a theory is generated from a set of axioms, by starting with those axioms and applying [[rule of inference|rules of deduction]], much as terms in an algebraic system may be generated from a set of basic terms by applying operations. Axioms should therefore be considered as _presenting_ a theory; different axiom sets may well give the same theory.
In terms of a [[deductive system]], axioms can be regarded as "rules with zero hypotheses". The form of such axioms depends on the details of the deductive system used: it could be [[natural deduction]], [[sequent calculus]], a [[Hilbert system]], etc. If we take sequent calculus, for instance, then any collection of [[sequents]] written in the given language $L$
$$
\vec{\phi} \vdash_{\vec x} \vec{\psi}
$$
(asserting that "If every [[proposition]] $\phi_i$ is [[true]] in [[context]] $\vec{x}$ then also some $\psi_i$ is/has to be true") can be taken as a collection of axioms for some theory. [[model|Models]] of the theory will then be those [[structure in model theory|structures]] of the language in which the axioms are interpreted as true statements. For example, a model of [[group theory]] is a structure in the language of groups for which the group theory axioms hold, which is (of course) a [[group]].
Assuming the deductive system is [[soundness theorem|sound]], every sequent which is the conclusion of a valid sequent deduction, starting from the axioms, will also be true in every model. And if the deductive system is also [[completeness theorem|complete]], then every sequent of the language which is true in every model will in fact be provable from the axioms.
## Examples
{#Examples}
[[!include foundational axiom - contents]]
(...)
* [[pushout-product axiom]]
(...)
* [[Hilbert's sixth problem]] asks for an axiomatization of [[physics]].
## Related concepts
* [[axiom schema]]
[[!include mathematical statements --- contents]]
## References
For instance def. D1.1.6 in
* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
[[!redirects axiom]]
[[!redirects axioms]]
[[!redirects axioms]]
[[!redirects axiomatization]]
[[!redirects axiomatizations]]
|
axiom K | https://ncatlab.org/nlab/source/axiom+K | Axiom K may refer to
* [[axiom K (type theory)]]
* [[axiom K (modal logic)]]
[[!redirects K axiom]]
[[!redirects Axiom K]] |
axiom K (modal logic) | https://ncatlab.org/nlab/source/axiom+K+%28modal+logic%29 |
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### Context
#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
=--
=--
=--
*Disambiguation: For axiom K as a principle of [[type theory]], see [[axiom K (type theory)]]*
# Contents
* table of contents
{: toc}
## Idea
In [[modal logic]], axiom K, named after [[Saul Kripke]], is a basic principle which almost all versions of propositional modal logic satisfy
$$K \colon \Box(p \to q) \to \Box p \to \Box q.$$
## Developments
It is possible to consider a variant in [[dependent type theory]] ([Spitters](#Spitters))
$$
Dependent\; K \colon \Box \Pi_{y:A} B(y) \to \Pi_{x: \Box A} \Box B [open\; x/y].
$$
## Related concepts
* [[K modal logic]]
## References
* {#Spitters} [[Bas Spitters]] et al., _Modal Dependent Type Theory and Dependent Right Adjoints_, ([slides](http://www.cs.au.dk/~spitters/floc.pdf))
Axiom K is discussed in Example 6.1.7 of:
* [[Daniel Gratzer]], *Syntax and semantics of modal type theory* ([pdf](https://jozefg.github.io/papers/phd-thesis.pdf)) |
axiom K (type theory) | https://ncatlab.org/nlab/source/axiom+K+%28type+theory%29 | +-- {: .rightHandSide}
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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--
> This entry is about the axiom K in [[type theory]]. For axiom K of [[modal logic]], see *[[axiom K (modal logic)]].
# Contents
* table of contents
{: toc}
## Idea
In [[type theory]], the _axiom K_ is an [[axiom]] that when added to [[intensional type theory]] turns it into [[extensional type theory]] --- or more precisely, what is called [[extensional type theory|here]] "propositionally extensional type theory". In the language of [[homotopy type theory]], this means that all types are [[h-sets]], accordingly axiom K is incompatible with the [[univalence axiom]].
Heuristically, the axiom asserts that each [[term]] of each [[identity type]] $Id_A(x,x)$ (of [[equivalences]] of a [[term]] $x \colon A$) is [[propositional equality|propositionally equal]] to the canonical [[reflexive relation|reflexivity]] equality proof $refl_x \colon Id_A(x,x)$.
See also at _[[set-level type theory]]_.
Axiom K can also be called **loop induction** or **self-identification induction** in parallel to [[path induction]].
## Statement
$$
K
\colon
\underset{A \colon Type}{\prod}
\underset{x \colon A}{\prod}
\underset{P \colon Id_A(x,x) \to Type}{\prod}
\left(
P(refl_A x)
\to
\underset{h \colon Id_A(x,x)}{\prod} P(h)
\right)
$$
If one doesn't have [[type universes]] in the type theory, then axiom K has to be expressed as an [[inference rule]], and thus is called the **K-rule**:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, p:\mathrm{Id}_A(x,x) \vdash P(x, p) \; \mathrm{type}}{\Gamma \vdash K_A:\prod_{x:A} P(x, \mathrm{refl}_A(x)) \to \prod_{h:\mathrm{Id}_A(x,x)} P(x, h)}$$
### Using the circle type
One can, instead of using elements $x:A$ and self identifications $p:\mathrm{Id}_A(x,x)$, use a function $p:S^1 \to A$ from the [[circle type]] $S^1$ to express axiom K:
$$
K^{\prime}
\colon
\underset{A \colon Type}{\prod}
\underset{x \colon A}{\prod}
\underset{P \colon (S^1 \to A) \to Type}{\prod}
\left(
P(\lambda i:S^1.x)
\to
\underset{p:S^1 \to A}{\prod} P(p)
\right)
$$
This states that the function type $S^1 \to A$ is a [[positive copy]] of $A$, and is equivalent to the other formulation of axiom K through the recursion principle of the [[circle type]].
If one doesn't have [[type universes]] in the type theory, then axiom K has to be expressed as an [[inference rule]]:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, p:S^1 \to A \vdash P(p) \; \mathrm{type}}{\Gamma \vdash K_A^{\prime}:\prod_{x:A} P(\lambda i:S^1.x) \to \prod_{p:S^1 \to A} P(p)}$$
## Properties
### Relation to $S^1$-localization
The negative analogue of axiom K is the [[axiom of S1-localization|axiom of $S^1$-localization]], which states that
$$\mathrm{const}_{A, S^1} \equiv \lambda x:A.\lambda i:S^1.x:A \to (S^1 \to A)$$
is an [[equivalence of types]] or a definitional [[isomorphism]].
\begin{theorem}
Suppose that every type $A$ is definitionally $S^1$-local.
Then $S^1 \to A$ is a [[positive copy]] of $A$: given any type $A$, and any type family $C(p)$ indexed by loops $p:S^1 \to A$ in $A$, and given any dependent function $t:\prod_{x:A} C(\lambda i:S^1.x)$ which says that for all elements $x:A$, there is an element of the type defined by substituting the constant loop of $x:A$ into $C$, $C(\lambda i:S^1.x)$, one can construct a dependent function $K_A(t):\prod_{z:S^1 \to A} C(z)$ such that for all $x:A$, $K_A(t, \lambda i:S^1.x) \equiv t(x):C(\lambda i:S^1.x)$.
\end{theorem}
\begin{proof}
$K_A(t)$ is defined to be
$$K_A(t) \equiv \lambda p:S^1 \to A.t(\mathrm{const}_{A, S^1}^{-1}(p))$$
and by the computation rules of loop types as negative copies, one has that for all $x:A$,
$$\mathrm{const}_{A, S^1}^{-1}(\lambda i:\mathbb{I}.x) \equiv x$$
and so by definition of $\mathrm{ind}_{S^1 \to A}(t)$ and the judgmental congruence rules for substitution, one has
$$K_A(t, \lambda i:S^1.x) \equiv t(\mathrm{const}_{A, S^1}^{-1}(\lambda i:S^1.x)) \equiv t(x)$$
\end{proof}
\begin{theorem}
Suppose that for all types $A$, $S^1 \to A$ is a [[positive copy]] of $A$ through the function
$$\mathrm{const}_{A, S^1} \equiv \lambda x:A.\lambda i:S^1.x:A \to (S^1 \to A)$$
Then Streicher's axiom K is true: given a type $A$ and given a type family $C(x, P)$ indexed by $x:A$ and $p:x =_A X$, and a dependent function $t:\prod_{x:A} C(x, \mathrm{refl}_A(x))$, one can construct a dependent function
$$K_A(t):\prod_{x:A} \prod_{p:x =_A X} C(x, p)$$
such that for all $x:A$,
$$K_A(t, x, \mathrm{refl}_A(x)) \equiv t(x)$$
\end{theorem}
\begin{proof}
By the induction principle of [[positive copies]] on the type family $C(f(\mathrm{base}), \mathrm{ap}_f(\mathrm{loop}))$ indexed by $f:S^1 \to A$, we can construct a dependent function
$$K_A^{\prime}(t):\prod_{f:\mathbb{I} \to A} C(f(\mathrm{base}), \mathrm{ap}_f(\mathrm{loop}))$$
such that for all $x:A$,
$$K_A^{\prime}(t, \lambda i:S^1.x) \equiv t(x):C(x, \mathrm{refl}_A(x))$$
since by definition of constant function and reflexivity, one has
$$(\lambda i:\mathbb{I}.x)(\mathrm{base}) \equiv x \quad \mathrm{ap}_{\lambda i:\mathbb{I}.x}(\mathrm{loop}) \equiv \mathrm{refl}_A(x)$$
We define
$$K_A(t, x, p) \equiv K_A^{\prime}(t, \mathrm{rec}_{S^1}^A(x, p))$$
since by circle recursion one has a path $\mathrm{rec}_{S^1}^A(x, p):S^1 \to A$ such that
$$\mathrm{rec}_{S^1}^{A}(x, p)(\mathrm{base}) \equiv x \quad \mathrm{ap}_{\mathrm{rec}_{S^1}^{A}(x, p)}(\mathrm{base}, \mathrm{base}, \mathrm{loop}) \equiv p$$
\end{proof}
One has the following analogies between localization at a specific type and the type theoretic letter rule that it proves:
| localization rule | type theoretic letter rule |
|-------------------|----------------------------|
| [[I-localization|$\mathbb{I}$-localization]] | [[J-rule]] |
| [[S1-localization|$S^1$-localization]] | [[K-rule]] |
### Computational behavior
Unlike its logical equivalent [[axiom UIP]], axiom K can be endowed with computational behavior: $K(A,x,P,d,refl_A x)$ computes to $d$. This gives a way to specify a computational propositionally extensional type theory.
This sort of computational axiom K can also be implemented with, and is sufficient to imply, a general scheme of function definition by pattern-matching. This is implemented in the proof assistant [[Agda]]. (The flag `--without-K` alters Agda's pattern-matching scheme to a weaker version appropriate for [[intensional type theory]], including [[homotopy type theory]].)
## Related concepts
* [[axiom UIP]]
* [[axiom of S1-localization|axiom of $S^1$-localization]]
* [[J-rule]], [[interval type localization]]
## References
The axiom K was introduced in
* [[Thomas Streicher]], _Investigations into intensional type theory_ Habilitation thesis (1993) ([pdf](http://www.mathematik.tu-darmstadt.de/~streicher/HabilStreicher.pdf))
For a review and discussion of the implementation in [[Coq]], see
* Pierre Corbineau, _The K axiom in Coq (almost) for free_ ([pdf](http://coq.inria.fr/files/adt-2fev10-corbineau.pdf))
Discussion in the context of [[homotopy type theory]] is in
* [[Univalent Foundations Project]], _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_
around theorem 7.2.1
[[!redirects Streicher axiom K]]
[[!redirects Streicher's axiom K]]
[[!redirects Streicher\'s axiom K]]
[[!redirects Axiom K (type theory)]]
[[!redirects K rule]]
[[!redirects Streicher's K rule]]
[[!redirects Streicher\'s K rule]]
[[!redirects K rule (type theory)]]
[[!redirects K-rule]]
[[!redirects Streicher's K-rule]]
[[!redirects Streicher\'s K-rule]]
[[!redirects K-rule (type theory)]]
[[!redirects self-identity induction]]
[[!redirects loop induction]]
[[!redirects self-identification induction]]
[[!redirects self-equality induction]]
[[!redirects self-identity elimination]]
[[!redirects loop elimination]]
[[!redirects self-identification elimination]]
[[!redirects self-equality elimination]]
[[!redirects self-identity computation]]
[[!redirects loop computation]]
[[!redirects self-identification computation]]
[[!redirects self-equality computation]] |
axiom of choice | https://ncatlab.org/nlab/source/axiom+of+choice |
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### Context
#### Foundations
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[[!include foundations - contents]]
=--
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=--
# The axiom of choice
* table of contents
{: toc}
## Statement
The __axiom of choice__ is the following statement:
* _Every [[surjection]] in the category [[Set]] of [[sets]] [[split epimorphism|splits]]._
This means: for every [[surjection]] $f\colon A \to B$ of sets, there is a function $\sigma\colon B \to A$ (a [[section]]), such that
$$
(B \stackrel{\sigma}{\to} A \stackrel{f}{\to} B)
=
(B \stackrel{Id_B}{\to} B)
\,.
$$
Note that a surjection $A \to B$ of sets can be regarded as a $B$-indexed [[family]] of [[inhabited sets]], while the existence of a section is equivalent to a choice of one element in each set of this family. This reproduces the more classical form of the axiom of choice.
When the full axiom of choice fails, it may still be valid for some restricted class of objects $A$ and/or $B$. An object $B$ such that any [[epimorphism]] $A \to B$ [[split epimorphism|splits]] is called [[projective object|projective]]; this means that one can make choices 'indexed by' $B$. Dually, an object $A$ such that one can make choices 'with values in' $A$ is called a [[choice object]] (this is not quite equivalent to every epimorphism $A \to B$ splitting).
## In other categories
More generally, we may consider analogous statements in [[categories]] $C$ other than $Set$.
### External form
{#ExternalForm}
We say that $C$ satisfies the **external axiom of choice** if every [[epimorphism]] in $C$ [[split epimorphism|splits]].
In this form, the axiom of choice may look less mysterious than in its original formulation. For instance, it is clear that it fails in contexts such as $C =$ [[Top]] and $C = $[[Diff]], due to the existence of nontrivial topological and smooth [[fiber bundles]].
If $C$ is not [[balanced category|balanced]], such as a [[regular category|regular]] or [[coherent category]] which is not a [[pretopos]], it may be more appropriate to replace in this statement "epimorphism" by [[regular epimorphism]] (or [[extremal epimorphism]], [[effective epimorphism]], etc.) In $Set$ (and in any [[topos]]), all of these notions of epimorphism are the same.
More generally still, if $C$ is a [[site]], then the axiom of choice for $C$ may be taken to say that any [[cover]] $U\to X$ admits a section. Obviously this refers only to singleton covers, but if $C$ is [[superextensive site|superextensive]] then any covering family $(p_i\colon U_i \to X)_i$ can be replaced by a singleton cover $\coprod_u U_i \to X$.
### Internal form
However, when working in a category that has an [[internal logic]], we may want to "internalize" the axiom of choice by asserting, not that every epimorphism has a section, but that the statement "every epimorphism has a section" is true in the internal logic (or more precisely the [[stack semantics]]). An equivalent statement is that every object is [[internally projective object|internally projective]]. We call this the **internal axiom of choice**.
This is generally a weaker statement: a [[topos]] satisfies the external AC if and only if it satisfies the internal AC and also (the external form of) [[supports split]]. Often, however, this is the more relevant notion to consider.
The following characterization can be found in Freyd-Scedrov ([1990](#FreydScedrov90), p.181)
+-- {: .num_theorem}
###### Theorem
A [[Grothendieck topos]] satisfies the internal axiom of choice iff it is a [[Boolean topos|Boolean]] [[étendue]].
=--
In particular, satisfaction of IAC entails Booleanness.
Classical examples of Boolean étendues are provided by toposes of group actions hence these all satisfy IAC but satisfy AC only in case the group is trivial (cf. Johnstone [1977](#JT77), p.144 or Freyd-Scedrov [1990](#FreydScedrov90), p.84, 179)).
If a topos $C$ satisfies IAC, then so do all of its [[slice categories]], although this may not be obvious. See [this answer](http://mathoverflow.net/questions/139874/pullback-stability-of-internally-projective-objects/140262#140262).
### In toposes
+-- {: .num_theorem}
###### Theorem
A [[Grothendieck topos]] satisfies the external axiom of choice iff it is equivalent to the [[category of sheaves]] on a [[complete Boolean algebra]].
=--
## Equivalents
The following statements are all equivalent to the axiom of choice in $Set$ (although sometimes the proof in one direction requires [[excluded middle]]). This is a *very* short list; much longer lists can be found elsewhere, such as at [Wikipedia](http://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents). Some of the statements on this list, though, may be of interest to nLabbers but are not commonly mentioned as equivalents of choice.
* That any [[cartesian product]] of any [[family]] of [[inhabited sets]] is [[inhabited]].
* That given a [[surjection]] $f$ and an element $b$ in the [[codomain]] of $f$, the [[preimage]] $f^*(b)$ of $f$ at $b$ has a (local) [[choice operator]], a function $\epsilon:[f^*(b)] \to f^*(b)$ from the [[support of a set|support]] of $f^*(b)$ to $f^*(b)$ itself.
* The [[well-ordering theorem]] (that any set can be [[well-order|well-ordered]]),
* [[Zorn's lemma]],
* That ($L =$ [[monomorphism | monomorphisms]], $R =$ [[epimorphism | epimorphisms]]) is a [[weak factorization system on Set]].
* That [[Set]] is equivalent to its own [[free exact completion]].
* That there exists a [[group]] structure on every [[inhabited set]] (see [[Hartogs number]], or [this MO answer](http://mathoverflow.net/questions/12973/does-every-non-empty-set-admit-a-group-structure-in-zf/12988#12988)).
* {#EsoFunctorSplits} That every [[essentially surjective functor]] between [[small categories|small]] [[strict categories]] is [[split essentially surjective]].
* {#EsoFFFunctorEquivalence} That every [[fully faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective]] functor between [[small categories|small]] [[strict categories]] is a strong [[equivalence of categories]].
* That the [[nonabelian cohomology]] $H^1(X;G)$ is trivial for every set $X$ and every [[group]] $G$ (see [this post](http://golem.ph.utexas.edu/category/2013/07/cohomology_detects_failures_of.html)).
* If $X$ and $Y$ are infinite sets and the two [[free groups]] $F(X)$ and $F(Y)$ have equal [[cardinality]], then $X$ and $Y$ also have equal cardinality (see [Kleppmann14](#Kleppmann14)).
* That every [[vector space]] is [[free object|free]].
* That every [[subset]] $A \subseteq B$ in a universe $\mathcal{U}$ comes with a choice of [[injection]] $i:A \hookrightarrow B$. Constructive mathematicians usually use subsets equipped with the structure of an injection, as those are usually more useful than general subsets with the mere property of being a subset.
## Variants
{#Variants}
There are a number of weaker axioms which are implied by the full axiom of choice. Some of these are valid or accepted more generally than the full AC, and/or suffice for some of the usual applications of choice. In particular, the full axiom of choice is generally rejected in [[constructive mathematics]], whereas some of these weaker forms of choice may be accepted, such as (in order of increasing strength) [[countable choice]], [[dependent choice]], and [[COSHEP]].
* Many applications of choice in [[logic]], [[topology]], and [[algebra]] require only the [[ultrafilter principle]] (UF), or equivalently the *Boolean prime ideal theorem*.
* From the perspective of [[constructive mathematics]], the principle of [[excluded middle]] (EM) may be seen as a weak form of the axiom of choice; EM is equivalent to the statement that every [[Kuratowski-finite set]] is projective. See at _[[Diaconescu-Goodman-Myhill theorem]]_.
* A very weak form of choice (which follows from EM) is the statement that [[supports split]] in $Set$.
* The axioms of [[countable choice]] (CC) and [[dependent choice]] (DC) suffice for many of the usual applications of choice in the analysis of [[separable space | separable spaces]]. CC states that the set $\mathbb{N}$ of [[natural number | natural numbers]] is projective. DC strenghtens CC by allowing the set of possible choices for $n+1$ to depend on the choice made for $n$.
* The axiom [[COSHEP]], also called the "presentation axiom," says that any set admits a surjection from a projective one (whereas full AC says that all sets are projective). This implies CC and DC, and is moreover sufficient for the existence of [[projective resolution | projective resolutions]] and cofibrant replacements, as well as the usual theorems in algebra that (for example) [[Mod]] has enough projectives. For example, see the [[canonical model structure on Cat]].
* The [[axiom of small violations of choice]] (SVC) asserts there is a set $S$ such that every set is a [[subquotient]] of $C\times S$ for some [[choice object|choice set]] $C$. Intuitively, this says that the failure of AC is parametrized by a single set. It can be regarded as a "dual" of COSHEP, since it deals with choice sets rather than projective ones, it implies the existence of (at least some) [[injective resolution | injective resolutions]], and together with COSHEP and EM it implies full AC.
* The [[axiom of multiple choice]] is a different way of saying that choice is violated in only a small way, which is more "local" than SVC. It apparently follows from SVC, at least in [[ZF]].
* The [[small cardinality selection axiom]] is another similar axiom. It asserts that there is a class function selecting for every set an isomorphic set (its "cardinality") such that among each isomorphism class of sets, the collection of all "cardinalities" forms only a set.
* A still weaker axiom along the lines of "AC fails in only a small way," which is implied by AMC, is [[WISC]], i.e. that for any set $X$, the full subcategory of $Set/X$ consisting of the surjections has a [[weakly initial set]] (under COSHEP it has a single weakly initial object, namely a projective cover of $X$). Two similar assertions are that the [[free exact completion]] $Set_{ex/lex}$ of $Set$ is a [[topos]] (i.e. that $Set$ has a [[generic proof]]), and that $Set_{ex/lex}$ is [[well-powered category|well-powered]]; both of these imply WISC.
* In [[constructive mathematics]], there is also [[decidable choice]], where every [[set]] with [[decidable equality]] is projective.
The axiom of choice can also be strengthened in a few ways.
* While the ordinary axiom of choice says that any surjection of sets is split, the *axiom of global choice* says that this is also true for any surjection of [[proper classes]]. (Making this precise requires a bit of work.) It is equivalent to the existence of a well-ordering of the class of all sets.
* One can also postulate a [[choice operator]], which gives a *specified* way to choose an element from any nonempty set. This implies global choice, and conversely a choice operator can be defined from any well-ordering of the class of all sets.
Finally, one can instead adopt the *negation* of the axiom of choice, or a strengthened version of this negation:
* The assumption that every subset of the [[real line]] has the [[Baire property]] (BP) is consistent with DC but not AC; the same holds for the assumption that every subset of the real line is [[Lebesgue measure|measurable]] (LM) if at least one [[Grothendieck universe]] exists. These assumptions leads to a very nice setting for analysis called [[dream mathematics]].
* The [[axiom of determinacy]] is a natural statement in [[game theory]] that is consistent with DC; in fact, it implies DC in certain models, such as in the constructible (in the sense of Goedel) closure of the set of reals. However, determinacy contradicts full AC (for example, it implies LM and BP, as in the previous entry).
* Any of the varieties of [[constructive mathematics]] that contradict excluded middle necessarily contradict choice, but they are usually (if not always) consistent with DC (and even COSHEP).
* The existence of a [[elementary embedding#Inconsistency|Reinhardt cardinal]] contradicts AC.
### In higher category theory
To formulate a version of the axiom of choice in a higher category, one has to make an appropriate choice of the meaning of "epimorphism". In most cases, it is best to choose [[effective epimorphisms|effective epimorphism in an (infinity,1)-category]] or a related notion such as [[eso morphisms]].
Less obviously, we usually want to also impose [[truncated object|truncation]] requirements on at least some of the objects involved in the axiom of choice. It seems usually necessary to require the codomain to be 0-truncated (axioms of choice without this requirement tend to be inconsistent); as for the domain we can choose to or not.
* An $(\infty,1)$-category satisfies the **axiom of $n$-choice**, or $AC_n$, if every $n$-[[truncated morphism]] with 0-truncated codomain has a section. We write $AC_\infty$ for the **axiom of infinity-choice**: the statement that *every* morphism with 0-truncated codomain has a section.
These are stronger axioms as $n$ increases. The "difference" between $AC_0$ and $AC_\infty$ is roughly the axiom that [[sets cover]].
For $(n,k)$-categories with $k\gt 1$ it is unclear whether it is sensible to allow the domain to be non-groupoidal.
* In [[2-category theory]], the _[[michaelshulman:axiom of 2-choice]]_ has been proposed to mean that every [[eso morphism]] with groupoidal domain and 0-truncated codomain has a section.
There are also "internal" versions of these axioms.
* In [[homotopy type theory]] (the internal logic of an $(\infty,1)$-topos), the internal version of $AC_n$ is "every surjection onto a set with $n$-type fibers has a section", or equivalently
$$ \prod_{(X:Set)} \prod_{(Y:X\to n Type)} \Big( \prod_{(x:X)} \Vert Y(x)\Vert \to \Big\Vert \prod_{(x:X)} Y(x) \Big\Vert \Big) $$
* More generally, we can replace the $(-1)$-truncation by the $k$-truncation to obtain a family of axioms $AC_{k,n}$.
* We can also replace the $(-1)$-truncation by the assertion of $k$-connectedness, obtaining the **axiom of $k$-connected choice**.
### In type theory
{#InTypeTheory}
In [[dependent type theory]], there are two interpretations of [[predicate logic]] in the type theory; [[propositions as some types]] and [[propositions as types]], resulting in two notions of the axiom of choice:
* [[propositional axiom of choice]]
* [[type theoretic axiom of choice]]
The former corresponds to the [[internal axiom of choice]] in [[categorical semantics]] and [[set theory]], while the latter is provable in any [[locally cartesian closed category]] or the [[set theory]] which interprets such categories.
### In spatial type theory
In [[spatial type theory]], which is [[dependent type theory]] with the [[sharp modality]] and the [[flat modality]], there exists a version of the axiom of choice called the **sharp axiom of choice**, which is given by the following [[inference rule]]:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ac}^{\sharp}_{A, B}:\left(\prod_{x:A} \sharp [B(x)]\right) \to \sharp\forall x:A.\sharp B(x)}$$
where $\sharp A$ is the sharp modality of type $A$, $[A]$ is the [[propositional truncation]] of $A$, and the [[universal quantifier]] $\forall x:A.B(x)$ is the propositional truncation of the [[dependent product type]] $\prod_{x:A} B(x)$.
## Consequences
* The axiom of choice implies the [[principle of excluded middle]] (due to Diaconesu 1975) see [McLarty, theorem 17.9](#McLarty), and see at _[excluded middle -- Relation to axiom of choice](excluded+middle#RelationToTheAxiomOfChoice)_
See also the [choice consequences data base](#ConsequencesDatabase)
## Related concepts
* [[propositional axiom of choice]]
* [[well-ordering]]
* [[anafunctor]]
* [[foundations]]
* [[Boolean topos]]
* [[finite choice]]
## References
### General
* [[HAF|Eric Schechter's analysis book]] surveys several variants of AC and its negation with a view to applications of [[analysis]], including this nice picture:
\begin{imagefromfile}
"file_name": "ac-chart-7dd95d8c-7395-40f1-8c6d-9d3cbe59d957.gif"
\end{imagefromfile}
(Here, UF, DC, CC, BP, and LM are as defined above.)
Discussion in [[toposes]] is in
* {#McLarty} [[Colin McLarty]], section 17.6 of _Elementary Categories, Elementary Toposes_
* {#ConsequencesDatabase} The [Consequences of the Axiom of Choice Project](http://consequences.emich.edu/conseq.htm) provides an interactive data base that can be used to search for implications between various (weakened) forms of the Axiom of Choice. [Choiceless grapher](http://cgraph.inters.co/) builds on this data and provides a graphical presentation.
* Gonçalo Gutierres da Conceição, _The Axiom of Countable Choice in Topology_, [pdf](http://www.mat.uc.pt/~ggutc/teses/teseingles.pdf)
Despite the title, this covers more than countable choice, but the focus is on sequential aspects (metric spaces, first- and second-countable spaces, etc).
A classical reference for AC in toposes is section 5.2 (pp.140ff) in
* {#JT77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Reprinted by Dover Mineola 2014)
Relation to [[cohomology]] is discussed in
* [[Andreas Blass]], _Cohomology detects failures of the axiom of choice_, Trans. Amer. Math. Soc. 279 (1983), 257-269 ([web](http://www.ams.org/journals/tran/1983-279-01/S0002-9947-1983-0704615-7/))
* [[Mike Shulman]], _Cohomology_ on the [[homotopy type theory]] blog [here](http://homotopytypetheory.org/2013/07/24/cohomology/)
and to [[free groups]] in
* {#Kleppmann14} Philipp Kleppmann, *Generating sets of free groups and the axiom of choice*, Mathematical Logic Quarterly, 2014, [doi](https://doi.org/10.1002/malq.201300088)
An alternative form of AC in toposes using injectivity is explored in
* Toby Kenney, _Injective Power Objects and the Axiom of Choice_ , JPAA **215** (2011) pp.131–144.
The result on IAC in toposes stems from
* {#FreydScedrov90}[[Peter Freyd|P. J. Freyd]], A. Scedrov, _[[Categories, Allegories]]_ , North-Holland Amsterdam 1990.
### In homotopy type theory
{#ReferencesInHomotopy}
Discussion in [[homotopy type theory]]:
* {#HoTTBook} [[Univalent Foundations Project]], section 3.8 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_
* [[Egbert Rijke]], section 17.4 of *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))
### In spatial type theory
For the sharp axiom of choice in spatial type theory, see:
* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))
category: foundational axiom
[[!redirects axioms of choice]]
[[!redirects Choice]]
[[!redirects AC]]
[[!redirects full choice]]
[[!redirects full AC]]
[[!redirects axiom of global choice]]
[[!redirects global axiom of choice]]
[[!redirects axiom of infinity-choice]]
[[!redirects axiom of n-choice]]
[[!redirects axiom of k-connected choice]]
[[!redirects set-theoretic axiom of choice]]
[[!redirects set theoretic axiom of choice]]
[[!redirects external axiom of choice]]
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[[!redirects sharp axiom of choice]] |
axiom of circle type localization | https://ncatlab.org/nlab/source/axiom+of+circle+type+localization | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundational axiom
+-- {: .hide}
[[!include foundational axiom - contents]]
=--
=--
=--
\tableofcontents
## Idea
In [[dependent type theory]], the [[0-truncation]] [[modality]] of a type $A$ can be defined by [[localization of a type]] $A$ at the [[circle type]] $S^1$. This means that a type $A$ is 0-truncated, i.e. a set, if it is $S^1$-local, which means that, in addition to [[axiom K]] and [[uniqueness of identity proofs]], there is another way to make the types of a universe into sets: by stipulating that every type is $S^1$-local, or that the canonical [[function]]
$$\mathrm{const}_{A, S^1} \equiv \lambda x:A.\lambda i:S^1.x:A \to (S^1 \to A)$$
which takes elements of $A$ to [[constant functions]] in $S^1 \to A$, is an [[equivalence of types]]. This is (tentatively) called the **axiom of circle type localization** or the axiom of $S^1$-localization.
Assuming that one has the function $\mathrm{const}_{A, S^1}:A \to (S^1 \to A)$ defined in the [[dependent type theory]], the syntactic rules for the axiom of $S^1$-localization in a universe $\mathcal{U}$ is given by:
$$\frac{\Gamma \vdash A:\mathcal{U} \quad \Gamma \vdash f:S^1 \to A}{\Gamma \vdash \mathrm{circlocal}_{A}:\mathrm{isEquiv}(\mathrm{const}_{A, S^1})}$$
There is also a definitional version of $S^1$-localization which says that $\mathrm{const}_{A, S^1}$ is a definitional [[isomorphism]]:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:S^1 \to A \vdash \mathrm{const}_{A, S^1}^{-1}(p):A}$$
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A \vdash \mathrm{const}_{A, S^1}^{-1}(\lambda i:S^1.x) \equiv x:A}$$
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:S^1 \to A \vdash \lambda i:S^1.\mathrm{const}_{A, S^1}^{-1}(p) \equiv p:S^1 \to A}$$
### Relation to axiom K
One can prove axiom K (positive copy induction rules) from the axiom of circle type localization (negative copy inference rules):
\begin{theorem}
Suppose that every type $A$ is definitionally $S^1$-local.
Then definitional [[axiom K (type theory)|axiom K]] holds: given any type $A$, and any type family $C(p)$ indexed by loops $p:S^1 \to A$ in $A$, and given any dependent function $t:\prod_{x:A} C(\lambda i:S^1.x)$ which says that for all elements $x:A$, there is an element of the type defined by substituting the constant loop of $x:A$ into $C$, $C(\lambda i:S^1.x)$, one can construct a dependent function $K_A(t):\prod_{z:S^1 \to A} C(z)$ such that for all $x:A$, $K_A(t, \lambda i:S^1.x) \equiv t(x):C(\lambda i:S^1.x)$.
\end{theorem}
\begin{proof}
$K_A(t)$ is defined to be
$$K_A(t) \equiv \lambda p:S^1 \to A.t(\mathrm{const}_{A, S^1}^{-1}(p))$$
and by the computation rules of loop types as negative copies, one has that for all $x:A$,
$$\mathrm{const}_{A, S^1}^{-1}(\lambda i:\mathbb{I}.x) \equiv x$$
and so by definition of $\mathrm{ind}_{S^1 \to A}(t)$ and the judgmental congruence rules for substitution, one has
$$K_A(t, \lambda i:S^1.x) \equiv t(\mathrm{const}_{A, S^1}^{-1}(\lambda i:S^1.x)) \equiv t(x)$$
\end{proof}
One has the following analogies between localization at a specific type and the type theoretic letter rule that it proves:
| localization rule | type theoretic letter rule |
|-------------------|----------------------------|
| [[I-localization|$\mathbb{I}$-localization]] | [[J-rule]] |
| [[S1-localization|$S^1$-localization]] | [[K-rule]] |
## Consequences
Since the [[boolean domain]] $\mathbb{2}$ is $S^1$-local, the axiom of $S^1$-localization implies that $S^1$ is [[compact connected]]:
\begin{theorem}
Assuming [[propositional truncation]], where $\Omega$ is the type of all $\mathcal{U}$-small propositions with type reflector $T$, and the axiom of $S^1$-localization, if the function $\mathrm{const}_{2, S^1}$ is an equivalence of types, then for all functions $P:S^1 \to \Omega$, if for all $x:S^1$, $T(P(x)) \vee \neg T(P(x))$ is contractible, then either for all $x:S^1$, $T(P(x))$ is contractible, or for all $x:S^1$, $\neg T(P(x))$ is contractible.
\end{theorem}
\begin{proof}
If $P:S^1 \to \Omega$ is such that for all $x:S^1$, $T(P(x)) \vee \neg T(P(x))$ is contractible, then there is a function $P':S^1 \to \mathbb{2}$ into the [[booleans type]] $\mathbb{2}$ with $\delta_{P'}^{1_2}(x):(P'(x) = 1_2)) \simeq T(P(x))$ and $\delta_{P'}^{0_2}(x):(P'(x) = 0_2)) \simeq \neg T(P(x))$. Since $\mathbb{2}$ is $S^1$-local, then, by the axiom of $S^1$-localization, $P'$ is constant, which implies that either for all $x:S^1$, $T(P(x))$ is contractible, or for all $x:S^1$, $\neg T(P(x))$ is contractible. Thus, $S^1$ is compact connected
\end{proof}
## In spatial type theory
In [[spatial type theory]], a crisp type $\Xi \vert () \vdash A$ is discrete if the function $(-)_\flat:\flat A \to A$ is an [[equivalence of types]]. There is a variant of the above axiom called the **axiom of circle type cohesion** or **axiom $S^1 \flat$**, which states that given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $S^1$-local, or if $\mathrm{const}_{A, S^1}$ is an [[equivalence of types]].
$$\frac{\Xi \vert () \vdash A \; \mathrm{type}}{\Xi \vert () \vdash S^1 \flat\mathrm{ax}_A:\mathrm{isEquiv}(\lambda x:\flat A.x_\flat) \simeq \mathrm{isEquiv}(\mathrm{const}_{A, S^1})}$$
This allows us to define discreteness for non-crisp types: a type $A$ is **discrete** if $A$ is $S^1$-local, resulting in a flavor of [[cohesive homotopy type theory]] where the [[shape modality]] is the [[0-truncation]] [[modality]]. This rule is equivalent to the axiom of circle type localization if every type is discrete.
## See also
* [[set-level type theory]]
* [[axiom of set truncation]]
* [[axiom K]]
* [[uniqueness of identity proofs]]
* [[interval type localization]]
* [[axiom of cohesion]]
[[!redirects axiom of circle type localization]]
[[!redirects axiom of S1 localization]]
[[!redirects axiom of S1-localization]]
[[!redirects circle type localization]]
[[!redirects S1 localization]]
[[!redirects S1-localization]]
[[!redirects axiom of circle type cohesion]]
[[!redirects axiom of S1 cohesion]]
[[!redirects axiom of S1-cohesion]]
[[!redirects axiom circle type flat]]
[[!redirects axiom S1 flat]]
[[!redirects axiom S1-flat]]
[[!redirects circle type cohesion]]
[[!redirects S1 cohesion]]
[[!redirects S1-cohesion]]
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[[!redirects S1 flat]] |
axiom of cohesion | https://ncatlab.org/nlab/source/axiom+of+cohesion |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
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=--
\tableofcontents
## Idea
{#Idea}
Axioms of [[cohesion]] are certain [[axioms]] added to any [[spatial type theory]] in order to define the [[shape modality]] for [[cohesive homotopy type theory]]. In particular, the *axiom of real cohesion* plays a role in defining [[real-cohesive homotopy type theory]] (the setting for [[classical homotopy theory]] and [[algebraic topology]]), and the *axiom of affine cohesion* plays a role in defining [[A1-cohesive homotopy type theory|$\mathbb{A}^1$-cohesive homotopy type theory]] (the setting for [[A1-homotopy theory|$\mathbb{A}^1$-homotopy theory]]), where the [[affine line]] $\mathbb{A}^1$ plays the role that $\mathbb{R}$ does in real-cohesive homotopy type theory. Affine cohesion could hypothetically also be used in defining a cohesive version of [[Mitchell Riley]]'s [[bunched logic|bunched]] [[linear homotopy type theory]] (the setting for [[stable homotopy theory]]) for the purposes of doing [[motivic homotopy theory]].
## Definition
We assume the presentation of [[spatial type theory]] using crisp term judgments $a::A$ in addition to the usual (cohesive) type and term judgments $A \; \mathrm{type}$ and $a:A$, as well as context judgments $\Xi \vert \Gamma \; \mathrm{ctx}$ where $\Xi$ is a list of crisp term judgments, and $\Gamma$ is a list of cohesive term judgments. A crisp type is a type in the context $\Xi \vert ()$, where $()$ is the empty list of cohesive term judgments. We also assume [[identity types]], the [[sharp modality]], and the [[flat modality]].
Given a type $A$, let us define $\mathrm{const}_{A, R}:A \to (R \to A)$ to be the type of all constant functions in $R$:
$$\delta_{\mathrm{const}_{A, R}}(a, r):\mathrm{const}_{A, R}(a)(r) =_A a$$
There is an equivalence $\mathrm{const}_{A, 1}:A \simeq (1 \to A)$ between the type $A$ and the type of functions from the [[unit type]] $1$ to $A$.
Given types $B$ and $C$ and a function $F:(B \to A) \to (C \to A)$, type $A$ is **$F$-[[localization of a type at a family of functions|local]]** if the function $F:(B \to A) \to (C \to A)$ is an [[equivalence of types]].
A crisp type $\Xi \vert () \vdash A$ is **discrete** if the function $(-)_\flat:\flat A \to A$ is an [[equivalence of types]].
The **axiom of cohesion** for type $R$ states that there is a crisp type $\Xi \vert () \vdash R \; \mathrm{type}$ such that given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, R})$-local, or equivalently, if $\mathrm{const}_{A, R}$ is an [[equivalence of types]].
$$\frac{\Xi \vert () \vdash A \; \mathrm{type}}{\Xi \vert () \vdash R \flat\mathrm{ax}_A:\mathrm{isEquiv}(\mathrm{const}_{A, R}) \simeq \mathrm{isEquiv}(\lambda x:\flat A.x_\flat)}$$
This allows us to define discreteness for non-crisp types: a type $A$ is **discrete** if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, R})$-local, or equivalently, if $\mathrm{const}_{A, R}$ is an [[equivalence of types]].
Another consequence is that the [[shape]] of $R$ is [[contractible]].
\begin{theorem}
Assuming a type $R$ and the axiom of $R$-cohesion, the shape of $R$ is contractible.
\end{theorem}
\begin{proof}
The type $R$ is inhabited by $\kappa_R(\sigma_R)$, so it remains to show that for all $x:\esh R$, $x =_{\esh R} \kappa_R(\sigma_R)$. Since $\esh R$ is discrete, so is the identity type $x =_{\esh R} \kappa_R(\sigma_R)$, which means by $\esh$-induction, it suffices to prove $\sigma_R(x) =_{\esh R} \kappa_R(\sigma_R)$ for all $x:\esh R$. But this is true from the third introduction rule for $\esh R$.
\end{proof}
In addition, if the [[type of booleans]] $\mathbb{2}$ is discrete, then $R$ is [[compact connected]]:
\begin{theorem}
Assuming a type $R$, a [[type of all propositions]] $\Omega$ with type reflector $T$, and the axiom of $R$-cohesion, if the function $\mathrm{const}_{2, R}$ is an equivalence of types, then for all functions $P:R \to \Omega$, if for all $x:R$, $T(P(x)) \vee \neg T(P(x))$ is contractible, then either for all $x:R$, $T(P(x))$ is contractible, or for all $x:R$, $\neg T(P(x))$ is contractible.
\end{theorem}
\begin{proof}
If $P:R \to \Omega$ is such that for all $x:R$, $T(P(x)) \vee \neg T(P(x))$ is contractible, then there is a function $P':R \to \mathbb{2}$ into the [[booleans type]] $\mathbb{2}$ with $\delta_{P'}^{1_2}(x):(P'(x) = 1_2)) \simeq T(P(x))$ and $\delta_{P'}^{0_2}(x):(P'(x) = 0_2)) \simeq \neg T(P(x))$. But if $\mathbb{2}$ is discrete, then by $R$-cohesion $P'$ is constant, which implies that either for all $x:R$, $T(P(x))$ is contractible, or for all $x:R$, $\neg T(P(x))$ is contractible. Thus, $R$ is compact connected if $\mathbb{2}$ is discrete.
\end{proof}
## Examples
There are a number of axioms which in general could be called an axiom of cohesion for type $R$. The most general such axiom of cohesion is called **stable local connectedness** or **axiom C0**, which imposes no other restrictions on $R$. If we additionally assume that the type $R$ is pointed with point $0:R$, then the axiom becomes **punctual local connectedness** or **axiom C1**, and if we additionally assume that the type $R$ is a non-trivial bi-pointed set, with points $0:R$, $1:R$, and witnesses $\tau_0:\mathrm{isSet}(R)$ and $\mathrm{nontriv}:(0 =_{R} 1) \to \emptyset$, then the axiom becomes **contractible codiscreteness** or **axiom C2**. If we additionally assume that the type $R$ is a [[Dedekind complete]] [[Archimedean field|Archimedean]] [[ordered field|ordered]] [[lattice]] [[field]] (and usually written as $\mathbb{R}$), then the axiom becomes **real cohesion** or **axiom $\mathbb{R}$-flat**.
| number/symbol | name | associated [[shape modality]] | additional requirements |
|--------|------|------|-----------|
| $C_0$ | cohesion/stable [[locally ∞-connected (∞,1)-topos|local connectedness]] | [[localization of a type|localization]] at a [[type]] $R$ | |
| $C_1$ | punctual cohesion/punctual [[locally ∞-connected (∞,1)-topos|local connectedness]] | [[localization of a type|localization]] at a [[pointed type]] $R$ | |
| $C_2$ | contractible codiscreteness | [[localization of a type|localization]] at a non-trivial [[bi-pointed type|bi-pointed]] [[h-set]] $R$ | |
| | discrete cohesion | [[localization of a type|localization]] at a [[contractible type]] $R$ | |
| | affine cohesion/algebraic cohesion/$\mathbb{A}^1$-cohesion | [[localization of a type|localization]] at an [[affine line]] $\mathbb{A}^1$ | |
| | [[compact connectedness]] | [[localization of a type|localization]] at a [[type]] $R$ | the [[type of booleans]] $\mathbb{2}$ is discrete |
| | [[continuum]] cohesion | [[localization of a type|localization]] at a [[Hausdorff space]] $R$ | $\mathbb{2}$ is discrete. |
| | [[metric continuum]] cohesion | [[localization of a type|localization]] at a [[metric space]] $R$ | $\mathbb{2}$ is discrete. |
| $\mathbb{R} \flat$ | [[real number|real]] cohesion | [[localization of a type|localization]] at the [[real numbers]] $\mathbb{R}$ | A higher [[coinductive type]] representing the [[homotopy]] [[terminal]] [[Archimedean ordered field]] $\mathbb{R}$ |
| $\mathbb{R} \flat$ | [[Dedekind real number|real]] cohesion ([[impredicative mathematics|impredicative]]) | [[localization of a type|localization]] at the impredicative [[Dedekind real numbers]] or [[generalized Cauchy real numbers]] $\mathbb{R}$ | A [[type of all propositions]] $\Omega$ |
| $\mathbb{R}_{U} \flat$ | locally $U$-small real cohesion | [[localization of a type|localization]] at the $U$-[[Dedekind real numbers]] or $U$-[[generalized Cauchy real numbers]] $\mathbb{R}_U$ | A [[Tarski universe]] $(U, T)$ |
| | [[unit interval]] cohesion | [[localization of a type|localization]] at the [[unit interval]] $[0, 1]$ | a higher [[coinductive type]] representing the [[homotopy]] [[terminal]] [[dyadic interval coalgebra]]. |
| | $U$-small unit interval cohesion | [[localization of a type|localization]] at the $U$-small [[unit interval]] $[0, 1]_\mathbb{U}$ | A [[Tarski universe]] $(U, T)$. |
| | smooth cohesion | [[localization of a type|localization]] at a [[local Artin algebra|local Artin $\mathbb{R}$-algebra]] | A [[local Artin algebra|local Artin $\mathbb{R}$-algebra]] $R$ |
| $S^1 \flat$ | [[circle type cohesion]] | [[localization of a type|localization]] at the circle type | The [[circle type]] $S^1$ |
## See also
* [[shape modality]]
* [[localization of a type at a family of functions]]
## References
* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))
* [[Mike Shulman]], *Homotopy type theory: the logic of space*, New Spaces in Mathematics: Formal and Conceptual Reflections, ed. Gabriel Catren and Mathieu Anel, Cambridge University Press, 2021 ([arXiv:1703.03007](https://arxiv.org/abs/1703.03007), [doi:10.1017/9781108854429](https://doi.org/10.1017/9781108854429))
[[!redirects axiom of cohesion]]
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axiom of constructibility | https://ncatlab.org/nlab/source/axiom+of+constructibility |
An [[axiom]] in [[set theory]].
## References
* Wikipedia, _[Axiom of constructibility](http://en.wikipedia.org/wiki/Axiom_of_constructibility)_
|