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!-modality | https://ncatlab.org/nlab/source/%21-modality | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
In full [[linear logic]] [[Girard 1987](#Girard1987)] and more generally in [[linear type theory]] there is assumed a (comonadic) [[modality]] traditionally denoted "!" whose role is to model the "underlying" classical [[intuitionistic type theory|(intuitionistic) type]] of a given [[linear type]].
This is alternatively called the "exponential modality" (for good reasons discussed [below](#TheExponentialCondition)), or "storage modality" (as it allows to duplicate and hence "store" otherwise linear data) and sometimes pronounced "of course" (intended alongside "necessarily" and "possibly" as used in [[modal logic]]) or even "bang" (just in reference to the symbol "!").
In classical linear logic (meaning with involutive [[de Morgan duality]]), the de Morgan dual of "!" is denoted "?" and then sometimes pronounced "why not".
Today it is understood (cf. [Melliès 2009](#Melliès09), [p. 36](/nlab/files/Mellies-CategoricalSemanticsLinear.pdf#page=36))
that the exponential modality is best and generally to be thought as as, first of all a *[[comonad]]* on [[linear types]] ([Seely 1989 Β§2](#Seely89), [dePaiva 1989](#dePaiva89), [Benton, Bierman, de Paiva and Hyland 1992](#BBPH92)) which, secondly, is [induced](monad#RelationBetweenAdjunctionsAndMonads) by an adjunction to classical (meaning here: non-linear [[intuitionistic type theory|intuitionistic]]) types with special [[monoidal functor|monoidal]] properties ([Seely 1989 Β§2](#Seely89), [Bierman 1994 pp. 157](#Bierman94), [Benton 1995](#Benton95)).
In a most elementary and illuminating example (which may not have received the attention it deserves, cf. the history at *[[quantum logic]]*):
* classical intuitionistic types form (are interpreted in) the category of [[Sets]]
$ClaTypes \,\coloneqq\, Set$
* linear types form (are interpreted in) the category of [[vector spaces]] (cf. [Murfet 2014](linear+logic#Murfet14))
$LinTypes \,\coloneqq\, Vect_{\mathbb{K}}$
and the [[adjoint functors]] in question are
* forming the [[linear span]] of a set
$$
\array{
Set &\overset{\mathrm{Q}}{\longrightarrow}& Vect
\\
W &\mapsto& \oplus_W \mathbb{1}
}
$$
* forming the [[underlying set]] of a [[vector space]]:
$$
\array{
Vect &\overset{\mathrm{C}}{\longrightarrow}& Set
\\
\mathscr{V} &\mapsto& Hom(\mathbb{1}, \, \mathscr{V})
}
$$
in which case the exponential modality acts by sending a vector space to the linear span of its underlying set
\[
\array{
Vect &\overset{\;\; ! \;\;}{\longrightarrow}& Vect
\\
\mathscr{V} &\mapsto& \underset{\mathscr{V}}{\bigoplus} \mathbb{1}
\,.
}
\]
One sees that this construction takes ([[direct sums|direct]]) sums to ([[tensor product of vector spaces|tensor]]) products
$$
!(\mathscr{V} \oplus \mathscr{V}')
\;\simeq\;
(! \mathscr{V}) \otimes (! \mathscr{V}')
$$
and zero ([[zero object|objects]]) to unit ([[unit object|objects]])
$$
! 0
\;\simeq\;
\mathbb{1}
$$
as expected of any [[exponential map]], whence the name "exponential modality".
This example is the 0-sector of the more famous *[[stabilization]]* ([[(infinity,1)-adjoint functor|$\infty$-]])adjunction $(\Sigma_+^\infty \dashv \Omega_+^\infty)$ between (the [[homotopy theory]] of) [[topological spaces]] and (the [[stable homotopy theory]] of) [[module spectra|module]] [[spectra]] equipped with the [[symmetric smash product of spectra]] (given by forming [[suspension spectra]] $X \mapsto \Sigma_+^\infty X$), which points to a deeper origin of the "exponential modality", see [below](#RealizationInLinearHomotopyTypeTheory) and at *[[linear homotopy type theory]]* for more on this.
<center>
<img src="https://ncatlab.org/nlab/files/QuantClassExponential-230821.jpg" width="600"/>
</center>
## Categorical semantics
Following [Seely 1989](#Seely89), [dePaiva 1989](#dePaiva89), [Benton, Bierman, de Paiva and Hyland 1992](#BBPH92) it is common convention that "!" should be a [[comonad]] on the type system (and ? should be a [[monad]], see also at *[[monads in computer science]]*), but there is some room in which further axioms to impose.
The goal is to capture the syntactic rules allowing assumptions of the form $!A$ to be duplicated and discarded.
### Intuitionistic case
The definition from [Seely](#Seely89), adapted to the intuitionistic case and modernized, is:
+-- {: .num_defn}
###### Definition
Let $LinTypes$ be
* a [[cartesian monoidal category]], with [[Cartesian product]] "$\times$" and [[tensor unit]] the [[terminal object]] $\ast$.
which *in addition* carries the [[structure]] of
* a [[symmetric monoidal category]] with respect to a [[tensor product]] $\otimes$ with [[tensor unit]] $\mathbb{1}$.
A **Seely comonad** on $LinTypes$ is a [[comonad]] that is a [[strong monoidal functor|strong monoidal]] as a functor from cartesian monoidal structure $\times$ to the other monoidal structure $\otimes$:
$$
(LinTypes, \times)
\xrightarrow{ \; ! \; }
(LinTypes, \otimes)
$$
meaning that there are [[natural transformations]] of the form
\[
\label{StrongMonoidalPropertyOfExponential}
A,\, B \colon LinTypes
\;\;\;\;\;\;\;
\vdash
\;\;\;\;\;\;\;
!(A \times B)
\;\cong\;
(!A) \otimes (!B)
\]
and
\[
! \ast \;\;\simeq\;\; \mathbb{1}
\,.
\]
=--
(There is also an additional [[coherence]] [[axiom]] that should be imposed; see [Melliès 2009, section 7.3](#Melliès09).)
\begin{remark}\label{TheExponentialCondition}
**(the exponential condition)**
\linebreak
In [[linear logic]], the cartesian monoidal structure on linear types is often denoted "$\&$" ("[[additive conjunction]]"), in which case the condition (eq:StrongMonoidalPropertyOfExponential) reads
\[
!(A \& B)
\;\cong\;
(!A) \otimes (!B)
\,.
\]
But in key examples of categories of (the multiplicative fragment of) linear logic (such as [[Vect]], cf. [Murfet 2014](linear+logic#Murfet14)), the [[cartesian product]] is actually a [[biproduct]], hence a [[direct sum]], in which case the condition on the exponential modality is that
*it takes (direct) sums to (tensor) products*
\[
!(A \oplus B)
\;\cong\;
(!A) \otimes (!B)
\]
*it takes zero (objects) to unit (objects)*
\[
! 0
\;\cong\;
\mathbb{1}
\]
as befits any [[exponential map]] and may explain the choice of terminology here.
\end{remark}
This condition implies that the [[Kleisli category of a comonad|Kleisli category]] of $!$ (i.e. the category of cofree !-coalgebras) is cartesian monoidal. If $LinTypes$ is *[[closed monoidal category|closed]]* symmetric monoidal then the Kleisli category of a [[cartesian closed category]], which is a categorical version of the translation of intuitionistic logic into linear logic.
Of course, the above definition depends on the existence of the cartesian product. A different definition that doesn't require the existence of $\times$ was given by [Benton, Bierman, de Paiva, and Hyland](#BBPH92):
+-- {: .num_defn}
###### Definition
Let $LinTypes$ be a [[symmetric monoidal category]]; a **linear exponential comonad** on $LinTypes$ is a [[lax monoidal functor|lax monoidal]] comonad such that every cofree !-coalgebra naturally carries the structure of a [[comonoid object]] in the category of coalgebras (i.e. the cofree-coalgebra functor lifts to the category of comonoids in the category of coalgebras).
=--
It follows automatically that all !-coalgebras are comonoids, and therefore that the category of all !-coalgebras (not just the cofree ones) is cartesian monoidal. Note that for a comonad on a [[poset]], every coalgebra is free; thus the world of pure propositional "logic" doesn't tell us whether to consider the Kleisli category or the Eilenberg-Moore category for the translation.
A more even-handed approach is the following (see [Benton (1995)](#Benton95) and [Melliès (2009)](#Melliès09)), based on the observation that both Kleisli and Eilenberg-Moore categories are instances of adjunctions.
\begin{definition}
\label{LinearNonlinearAdjunction}
A **linear-nonlinear adjunction** is a [[monoidal adjunction]] $F \colon ClaTypes \rightleftarrows LinTypes \colon G$ in which $LinTypes$ is symmetric monoidal and $ClaTypes$ is cartesian monoidal. The induced !-modality is the induced [[comonad]] $F G$ on $LinTypes$.
\end{definition}
This includes both of the previous definitions where $M$ is taken respectively to be the Kleisli category or the Eilenberg-Moore category of !. Conversely, in any linear-nonlinear adjunction the induced comonad $F G$ can be shown to be a linear exponential comonad. Moreover, if $!$ is a linear exponential comonad on a symmetric monoidal category $LinTypes$ with finite products, then the cofree !-coalgebra functor is a right adjoint and hence preserves cartesian products; but the cartesian products of coalgebras are the tensor products in $C$, so we have $!(A\times B) \cong !A \otimes !B$, the Seely condition.
### Classical case
For "classical" linear logic, we want $LinTypes$ to be not just (closed) symmetric monoidal but $\ast$-[[star-autonomous category|autonomous]]. If an $\ast$-autonomous category has a linear exponential comonad $!$ one can derive a ? from the ! by [[de Morgan duality]], $?A = \big(!(A^*)\big)^*$. The resulting relationship between ! and ? was axiomatized in a way not requiring the de Morgan duality by [Blute, Cockett & Seely (1996)](#BluteCockettSeely96):
+-- {: .num_defn}
###### Definition
Let $LinTypes$ be a [[linearly distributive category]] with tensor product $\otimes$ and cotensor product $\parr$. A (!,?)-modality on $LinTypes$ consists of:
1. a $\otimes$-monoidal comonad ! and a $\parr$-comonoidal monad ?
1. ? is a !-strong monad, and ! is a ?-strong comonad
1. all free !-coalgebras are naturally commutative $\otimes$-comonoids, and all free ?-algebras are naturally commutative $\parr$-monoids.
=--
Here a functor $F$ is [[strong functor|strong]] with respect to a [[lax monoidal functor]] $G$ if there is a [[natural transformation]] of the form $F A \otimes G B \to F(A\otimes G B)$ satisfying some natural axioms, and we similarly require compatibility of the monad and comonad structure transformations. [BCS96](#BluteCockettSeely96) showed that if $LinTypes$ is in fact $\ast$-autonomous, it follows from the above definition that $?A = \big(!(A^*)\big)^*$, as expected.
## Examples
### Relation to Chu construction
\begin{theorem}
Suppose $F \colon M \rightleftarrows C : G$ is a linear-nonlinear adjunction (Def. \ref{LinearNonlinearAdjunction}), where $C$ is closed symmetric monoidal with finite limits and colimits, and $\bot\in C$ is an object. Then there is an induced linear-nonlinear adjunction $M \rightleftarrows Chu(C,\bot)$ where $Chu(C,\bot)$ is the [[Chu construction]] $Chu(C,\bot)$, which is $\ast$-autonomous with finite limits and colimits. Hence $Chu(C,\bot)$ admits a !-modality.
\end{theorem}
\begin{proof}
The embedding of $C$ in $Chu(C,\bot)$ as $A \mapsto (A, [A,\bot], ev)$ is coreflective: the coreflection of $(B^+, B^-, e_B)$ is $(B^+, [B^+,\bot], ev)$. Moreover, this subcategory is closed under the tensor product of $Chu(C,\bot)$, i.e. the embedding $C\hookrightarrow Chu(C,\bot)$ is strong monoidal, hence the adjunction is a monoidal adjunction. Therefore, the composite adjunction $M \rightleftarrows C \rightleftarrows Chu(C,\bot)$ is again a linear-nonlinear-adjunction.
\end{proof}
Note that this theorem is a direct consequence of the work of [[Yves Lafont]] and [[Thomas Streicher]] on game semantics using the Chu construction, together with Benton's reformulation of linear logic in terms of monoidal adjunctions. Since a Chu construction is $\ast$-autonomous, this !-modality implies a dual ?-modality.
\begin{corollary}
If $C$ is a cartesian closed category with finite limits and colimits and $\bot\in C$ is an object, then there is a linear-nonlinear adjunction $C \rightleftarrows Chu(C,\bot)$, and hence $Chu(C,\bot)$ admits a !-modality.
\end{corollary}
\begin{proof}
Apply the previous theorem to the identity adjunction $C\rightleftarrows C$.
\end{proof}
Note that the !-modality obtained from the corollary is [[idempotent comonad|idempotent]], while that obtained from the theorem is idempotent if and only if the original one was. Other ways of constructing !-modalities, such as by cofree coalgebras, may produce examples that are not idempotent.
### Realization in linear homotopy type theory
{#RealizationInLinearHomotopyTypeTheory}
In [[dependent linear homotopy type theory]] the "linear-nonlinear adjunction" is naturally identified ([Ponto & Shulman (2012), Ex. 4.2](#PontoShulman12), see also [Schreiber (2014), Sec. 4.2](#Schreiber14)) with the [[stabilization]] [[adjoint functor|adjunction]] between [[homotopy types]] and [[stable homotopy types]] ([Riley (2022), Prop. 2.1.31](#Riley22Thesis)), whose [[left adjoint]] (forming [[suspension spectra]] $\Sigma^\infty_+$) is equivalently given (cf. [Riley (2022), Rem. 2.4.13](#Riley22Thesis)) by sending $B \,\colon\, Type$ to the linear [[dependent sum]] $\star_B \mathbb{1} \;\coloneqq\; (p_B)_! (p_B)^\ast \mathbb{1}$ over the monoidal unit in the context $B$. In terms of [quantum modal logic](necessity+and+possibility#ModalQuantumLogic) this is forming the "linear randomization" of the given classical homotopy type (its "[[motive]]"):
\[
\label{MotivizationInLHott}
\]
\begin{imagefromfile}
"file_name": "MotivizationInLHoTT-230825b.jpg",
"width": "460",
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
Curiously, this homotopy-theoretic realization of the exponential modality (beware the little subtlety of whether or not to reflect onto [[pointed homotopy types]])
$$
\Sigma^\infty \circ \Omega^\infty
\;\colon\;
Spectra \to Spectra
$$
has independently been argued to have properties of an [[exponential map]] in the context of [[Goodwillie calculus]], see [there](Goodwillie+calculus#GoodwillieDerivativeOfExponentialModality).
{#ComonadicityOfClassicalOverLinearHomotopyTypes} Moreover $\Sigma^\infty \dashv \Omega^\infty$ is a [[comonadic adjunction]] on [[simply connected homotopy type|simply connected]] [[homotopy types]] [[Blomquist & Harper 2016 Thm. 1.8](#BlomquistHarper16); [Hess & Kedziorek 2017 Thm. 3.11](#HessKedziorek17)], meaning that simply-connected classical (but [[pointed homotopy types|pointed]]) homotopy types are identified with the $\Sigma^\infty \Omega^\infty$-[[modales]] among [[stable homotopy types]].
## Modal term calculi
Girard's original presentation of linear logic involved rules that explicitly assumed the presence of $!$ on hypotheses or on entire contexts, such as [[dereliction rule| dereliction]], [[weakening rule|weakening]] and [[contraction rule|contraction]]:
$$\frac{\Gamma, A \vdash B}{\Gamma, !A \vdash B} \qquad \frac{\Gamma \vdash B}{\Gamma, !A\vdash B} \qquad \frac{\Gamma,!A,!A \vdash B}{\Gamma, !A\vdash B}$$
and "promotion":
$$ \frac{!\Gamma \vdash A}{!\Gamma\vdash !A} $$
If this is translated into a [[natural deduction]] style term calculus, the resulting rules are more complicated than those of most type formers. This can be avoided using [[adjoint type theory]] with two context zones, one "nonlinear" one where contraction and weakening are permitted (and [[admissible rule|admissible]]) and one "linear" one where they are not, with $!$ as a modality relating the two zones.
Such a "modal" presentation of linear logic was first introduced by [Girard 1993](#Girard93) and then developed by [Plotkin 1993](#Plotkin93), [Wadler 1993](#Wadler93), [Benton 1995](#Benton95), [Barber 1996](#Barber96).
This presentation also generalizes naturally to [[dependent linear type theory]], with the nonlinear type theory being dependent, and the linear types depending on the nonlinear ones but nothing depending on linear types. In this context, the $!$-modality decomposes into "context extension" and a "dependent sum".
## Related concepts
* [[exponential map]]
[[!include logic symbols -- table]]
## References
### General
The notion of the exponential modality originates in [[linear logic]] with
* {#Girard1987} [[Jean-Yves Girard]], _Linear logic_, Theoretical Computer Science **50** 1 (1987) [<a href="https://doi.org/10.1016/0304-3975(87)90045-4">doi:10.1016/0304-3975(87)90045-4</a>, [pdf](http://iml.univ-mrs.fr/~girard/linear.pdf)]
A streamlined statement of the [[inference rules]] and the observation that these make the exponential a [[comonad]] is due to:
* {#Seely89} [[R. A. G. Seely]], Β§2 of: *Linear logic, $\ast$-autonomous categories and cofree coalgebras*, in *Categories in Computer Science and Logic*, Contemporary Mathematics **92** (1989) [[[SeelyLinearLogic.pdf:file]], [ps.gz](http://www.math.mcgill.ca/rags/nets/llsac.ps.gz), [ISBN:978-0-8218-5100-5](https://bookstore.ams.org/conm-92)]
* {#dePaiva89} [[ Valeria de Paiva]], Β§2 of: *The Dialectica Categories*, in *Categories in Computer Science and Logic*, Contemporary Mathematics **92** (1989) [[ISBN:978-0-8218-5100-5](https://bookstore.ams.org/conm-92), [doi:10.1090/conm/092](https://doi.org/10.1090/conm/092)]
Review:
* [[Anne Sjerp Troelstra]], Β§12 in: *Lectures on Linear Logic* (1992) [[ISBN:0937073776](https://web.stanford.edu/group/cslipublications/cslipublications/site/0937073776.shtml)]
Brief survey in a context of [[computer science]]/[[linear type theory]]:
* {#MihΓ‘lyiNovitzkΓ‘13} Daniel MihΓ‘lyi, Valerie NovitzkΓ‘, Section 2.2 of: *What about Linear Logic in Computer Science?*, Acta Polytechnica Hungarica **10** 4 (2013) 147-160 [[pdf](http://acta.uni-obuda.hu/Mihalyi_Novitzka_42.pdf), [[MihalyiNovitzka-LinearLogic.pdf:file]]]
Further on the semantics of exponential conjunction as a [[comonad]]:
* {#deP1988} [[Valeria de Paiva]], *The Dialectica Categories*, PhD thesis, technical report 213, Computer Laboratory, University of Cambridge (1991) [[pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf), [[dePaivaDialectica.pdf:file]]]
* {#BBP92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], Β§8 of: *Term assignment for intuitionistic linear logic*, Technical report 262, Computer Laboratory, University of Cambridge (August 1992) [[pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-262.pdf), [[BentonBiermanDePaiva-TermAssignment.pdf:file]]]
> (published abridged as [BBPH92](#BBPH92))
* {#BBPH92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], [[Martin Hyland]], *Linear $\lambda$-Calculus and Categorical Models Revisited*, in *Computer Science Logic. CSL 1992*, Lecture Notes in Computer Science **702**, Springer (1993) [[doi:10.1007/3-540-56992-8_6](https://doi.org/10.1007/3-540-56992-8_6)]
> (abridged version of [BBP92](#BBP92))
* {#BluteCockettSeely96} [[R. F. Blute]] , [[J. R. B. Cockett]], [[R. A. G. Seely]], *! and ? -- Storage as tensorial strength*, Mathematical Structures in Computer Science **6** 4 (1996) 313-351 [[doi:10.1017/S0960129500001055](https://doi.org/10.1017/S0960129500001055)]
* {#HylandSchalk01} [[Martin Hyland]] and Andreas Schalk, _Glueing and orthogonality for models of linear logic_, [pdf](http://www.cs.man.ac.uk/~schalk/publ/gomll.pdf)
and its [resolution](monad#CategoryOfAdjunctionResolutionsOfAMonad) by a [[monoidal adjunction]] between the linear and a classical (intuitionistic) type system:
* {#Bierman94} [[Gavin Bierman]], *On Intuitionistic Linear Logic*, Cambridge (1994) [[[Bierman-LinearLogic.pdf:file]], [pdf](https://www.dropbox.com/s/hdxgubjljb96rmf/Biermanthesis.pdf?dl=0)]
* {#Benton95} [[Nick Benton]], *A mixed linear and non-linear logic: Proofs, terms and models*, in *Computer Science Logic. CSL 1994*, Lecture Notes in Computer Science **933** (1995) 121-135 [[doi:10.1007/BFb0022251](https://doi.org/10.1007/BFb0022251), [[BentonLinearLogic.pdf:file]]]
Review:
* {#Melliès02} [[Paul-André Melliès]], *Categorical models of linear logic revisited* (2002) [[hal:00154229](https://hal.science/hal-00154229)]
* {#Melliès09} [[Paul-André Melliès]], *Categorical semantics of linear logic*, in *[Interactive models of computation and program behaviour](https://smf.emath.fr/publications/modeles-interactifs-de-calcul-et-de-comportement-de-programme)*, Panoramas et synthèses **27** (2009) 1-196 [[web](https://smf.emath.fr/publications/semantique-categorielle-de-la-logique-lineaire), [pdf](https://www.irif.fr/~mellies/papers/panorama.pdf), [[Mellies-CategoricalSemanticsLinear.pdf:file]]]
Construction of such comonads based on cofree comonoids:
* Mellies and Tabareau and Tasson, *An explicit formula for the free exponential modality of linear logic*. Mathematical Structures in Computer Science, 28(7), 1253-1286. doi:[10.1017/S0960129516000426](https://doi.org/10.1017/S0960129516000426)
* [[Sergey Slavnov]], *On Banach spaces of sequences and free linear logic exponential modality*, Math. Struct. Comp. Sci. 29 (2019) 215-242, [arXiv:1509.03853](https://arxiv.org/abs/1509.03853)
The relation to [[Fock space]] is discussed in:
* {#BlutePanangadenSeely94} [[Richard Blute]], [[Prakash Panangaden]], [[R. A. G. Seely]], _Fock Space: A Model of Linear Exponential Types_ (1994) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.6825), [[BPSLinear.pdf:file]])
* {#Fiore07} [[Marcelo Fiore]], _Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic_, Lecture Notes in Computer Science Volume 4583, 2007, pp 163-177 ([pdf](http://www.cl.cam.ac.uk/~mpf23/papers/Types/diff.pdf))
* {#Vicary07} [[Jamie Vicary]], _A categorical framework for the quantum harmonic oscillator_, International Journal of Theoretical Physics December 2008, Volume 47, Issue 12, pp 3408-3447 ([arXiv:0706.0711](http://arxiv.org/abs/0706.0711))
> (in the context of [[quantum information theory in terms of dagger-compact categories]])
The interpretation of $\Omega^\infty \Sigma^\infty_+$ as an exponential in the context of [[Goodwillie calculus]] is due to
* {#AroneKankaanrinta95} [[Gregory Arone]], Marja Kankaanrinta, _The Goodwillie tower of the identity is a logarithm_, 1995 ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.8306))
based on
* {#AroneMahowald98} [[Gregory Arone]], [[Mark Mahowald]], _The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres_, 1998 ([pdf](http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf))
On the [[modal type theory]]-approach to a term calculus for the $!$-modality:
* {#Girard93} [[Jean-Yves Girard]]. *On the unity of logic* Annals of Pure and Applied Logic **59** (1993) 201-217 [<a href="https://doi.org/10.1016/0168-0072(93)90093-S">doi:10.1016/0168-0072(93)90093-S</a>]
* {#Plotkin93} [[Gordon Plotkin]], *Type theory and recursion*, in: Proceedings of the Eigth Symposium of Logic in Computer Science, Montreal , IEEE Computer Society Press (1993) 374 [[doi:10.1109/LICS.1993.287571](https://doi.org/10.1109/LICS.1993.287571)]
* {#Wadler93} [[Philip Wadler]], *A syntax for linear logic*, in: *Ninth International Coference on the Mathematical Foundations of Programming Semantics*, Lecture Notes in Computer Science **802** Springer (1993) [[doi:10.1007/3-540-58027-1_24](https://doi.org/10.1007/3-540-58027-1_24)]
* [Benton 1995](#Benton95)
* {#Barber96} Andrew Barber, *Dual Intuitionistic Linear Logic*, Technical Report ECS-LFCS-96-347, University of Edinburgh (1996), [[web](http://www.lfcs.inf.ed.ac.uk/reports/96/ECS-LFCS-96-347/), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/96/ECS-LFCS-96-347/ECS-LFCS-96-347.pdf), [[Barber-DualIntLinLogic.pdf:file]]]
A [[quantum programming language]] based on this linear/non-linear type theory adunction is [[QWIRE]]:
* [[Jennifer Paykin]], [[Robert Rand]], [[Steve Zdancewic]], *QWIRE: a core language for quantum circuits*, POPL 2017: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming LanguagesJanuary 2017 Pages 846β858 ([doi:10.1145/3009837.3009894](https://doi.org/10.1145/3009837.3009894))
theoretical background:
* [[Jennifer Paykin]], *Linear/non-Linear Types For Embedded Domain-Specific Languages*, 2018 ([upenn:2752](https://repository.upenn.edu/edissertations/2752))
applied to [[verified programming]] after implementation in [[Coq]]:
* [[Robert Rand]], [[Jennifer Paykin]], [[Steve Zdancewic]], *QWIRE Practice: Formal Verification of Quantum Circuits in Coq*, EPTCS 266, 2018, pp. 119-132 ([arXiv:1803.00699](https://arxiv.org/abs/1803.00699))
and using ambient [[homotopy type theory]]:
* [[Jennifer Paykin]], [[Steve Zdancewic]], *A HoTT Quantum Equational Theory*, [talk at QPL2019](http://qpl2019.org/a-hott-quantum-equational-theory/) ([arXiv:1904.04371](https://arxiv.org/abs/1904.04371))
### In dependent linear type theory
Discussion of the exponential modality via [[stabilization]] in [[dependent linear homotopy type theory]]:
* {#PontoShulman12} [[Kate Ponto]], [[Mike Shulman]], Ex. 4.2 in: *Duality and traces in indexed monoidal categories*, Theory and Applications of Categories **26** 23 (2012) [[arXiv:1211.1555](http://arxiv.org/abs/1211.1555), [tac:26-23](http://www.tac.mta.ca/tac/volumes/26/23/26-23abs.html), [blog](http://golem.ph.utexas.edu/category/2011/11/traces_in_indexed_monoidal_cat.html)]
* {#Schreiber14} [[Urs Schreiber]], Sec. 4.2 of: *[[schreiber:Quantization via Linear homotopy types]]* [[arXiv:1402.7041](http://arxiv.org/abs/1402.7041)]
* {#Riley22Thesis} [[Mitchell Riley]], Β§2.1.2 in: *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139)]
The $\Omega^\infty \Sigma^\infty$ [[comonadic functor|comonadicity]] of simply connected pointed spaces over spectra:
* {#BlomquistHarper16} Jacobson R. Blomquist, John E. Harper, Thm. 1.8 in: *Suspension spectra and higher stabilization* [[arXiv:1612.08623](https://arxiv.org/abs/1612.08623)]
* {#HessKedziorek17} [[Kathryn Hess]], [[Magdalena Kedziorek]], Thm. 3.11 in: *The homotopy theory of coalgebras over simplicial comonads*, Homology, Homotopy and Applications **21** 1 (2019) [[arXiv:1707.07104](https://arxiv.org/abs/1707.07104), [doi:10.4310/HHA.2019.v21.n1.a11](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a11)]
category: logic
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$\infty$-category > history | https://ncatlab.org/nlab/source/%24%5Cinfty%24-category+%3E+history | < [[$\infty$-category]]
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't Hooft anomaly | https://ncatlab.org/nlab/source/%27t+Hooft+anomaly |
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### Context
#### Quantum systems
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#### Physics
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#Contents#
* table of contents
{:toc}
## Idea
As described in ['t Hooft (1980)](#tHooft1980), a [[global symmetry]] of a [[quantum field theory]] is said to have a **'t Hooft anomaly** if it is non-anomalous as a global symmetry but has a [[quantum anomaly]] if one attempts to turn it into a [[gauge symmetry]].
Informally, a global symmetry described by the [[action]] of a [[group]] $G$ may be "coupled" to a [[gauge potential]] $A$ (a [[connection on a bundle]]). Denoting the [[partition function]] coupled to such a connection as $Z(A)$, the partition function of the gauged theory is, modulo normalization factors, $Z=\sum_A Z(A).$ If the symmetry has a 't Hooft anomaly, then perfoming a [[gauge transformation]] on all connections results in a nontrivial phase multiplying each partition function $Z_g=\sum_A \phi(A,g) Z(g\cdot A)$ where each phase $\phi(A,g)$ depends on the particular background field and gauge transformation, so that generally $Z\neq Z_g$. This is problematic since the partition function is supposed to be gauge invariant.
For $G$ a [[finite group]], and when the $n$-dimensional spacetime $\Sigma$ is the boundary of an $(n+1)$-dimensional space $X$, this situation may be remedied by "coupling" the $n$-dimensional theory with symmetry $G$ to a $(n+1)$-dimensional [[topological quantum field theory]] (a [[Dijkgraaf-Witten theory]] classified by $H^{n+1}(G,U(1))$) on $X$ such that the phase contributions of a gauge transformation of both cancel each other. This is known as *anomaly inflow* (see e.g. [Freed, Hopkins, Lurie, and Teleman (2009)](#FHLT09) and [Freed (2014)](#Freed14)).
Any [[generalized global symmetry]] is also thought to potentially exhibit 't Hooft anomalies described by a TQFT. In the literature, this is referred to as the *Anomaly TFT*, an [[invertible field theory]] (cf. [Freed 2014](#Freed14)), but little is known about what this TQFT is supposed to be for cases not equivalent to group-like cases (for which the TQFT is DW).
## References
{#References}
* {#tHooft1980} [[Gerard 't Hooft]], *Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking* In: G. 'Hooft et al., *Recent Developments in Gauge Theories*, NATO Advanced Study Institutes Series **59**, Springer (1980) [[doi:10.1007/978-1-4684-7571-5_9](https://doi.org/10.1007/978-1-4684-7571-5_9)]
* {#FHLT09} [[Daniel Freed]], [[Michael Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], *[[Topological Quantum Field Theories from Compact Lie Groups]]*, in P. R. Kotiuga (ed.) *A celebration of the mathematical legacy of Raoul Bott* AMS (2010) [[arXiv:0905.0731](https://arxiv.org/abs/0905.0731), [ISBN:978-0-8218-4777-0](https://bookstore.ams.org/view?ProductCode=CRMP/50)]
* {#Freed14} [[Daniel Freed]]. *Anomalies and Invertible Field Theories*, talk at [StringMath2013](http://scgp.stonybrook.edu/events/event-pages/string-math-2013) [[arXiv:1404.7224](https://arxiv.org/abs/1404.7224)]
[[!redirects 't Hooft anomalies]]
|
't Hooft coupling | https://ncatlab.org/nlab/source/%27t+Hooft+coupling |
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#Contents#
* table of contents
{:toc}
## Idea
In [[Yang-Mills theory]]/[[QCD]] with [[coupling constant]] $g_{YM}$ and with $N_c$ [[colour charge|colours]], the _βt Hooft coupling_ is the expression
$$
\lambda
\;\coloneqq\;
g_{YM}^2 N_c
\,.
$$
## Properties
### In the large $N$ limit
The study of [[Yang-Mills theory]] in the [[large N limit]] but at fixed βt Hooft coupling controls the [[AdS/CFT correspondence]]:
[[large N limit]] $\;N_c \to \infty$ at fixed [['t Hooft coupling]] $\lambda = g_{YM}^2 N_c$
| $\lambda \lt 1$ | $\lambda \gt 1$ |
|--|--|
| [[perturbative quantum field theory|small coupling limit]] of [[Yang-Mills theory]] | low-energy [[supergravity]] limit of [[string theory]] on asymptotically [[AdS spacetime]]
(e. g. [Aharony-Gubser-Maldacena-OoguriOz 99, p. 11, p. 60](#AharonyGubserMaldacenaOoguriOz99), [Nastase 07, Section 8](#Nastase07))
(...)
## Related concepts
* [[large N limit]]
* [[AdS/CFT]]
## References
The original article
* [[Gerard 't Hooft]], _A Planar Diagram Theory for Strong Interactions_, Nucl. Phys. B72 (1974) 461 ([spire:80491](http://inspirehep.net/record/80491), <a href="https://doi.org/10.1016/0550-3213(74)90154-0">doi:10.1016/0550-3213(74)90154-0</a>)
Review:
* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys. Rept. **323** 183-386 (2000) $[$<a href="https://doi.org/10.1016/S0370-1573(99)00083-6">doi:10.1016/S0370-1573(99)00083-6</a>, [arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111)$]$
* {#Nastase07} [[Horatiu Nastase]], _Introduction to AdS-CFT_ ([arXiv:0712.0689](http://arxiv.org/abs/0712.0689)) |
't Hooft double line notation | https://ncatlab.org/nlab/source/%27t+Hooft+double+line+notation |
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#### Lie theory
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#### Quantum Field Theory
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#### String theory
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[[!include string theory - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
### Double line notation
What is called _βt Hooft double line notation_ (following [βt Hooft 74](#tHooft74)) is the observation that for the [[fundamental representations]] $V$ of [[semisimple Lie algebras]] $\mathfrak{g}$. the [[Lie algebra weight system|Lie algebra weight]] of a $\mathfrak{g}$-[[Yang-Mills theory]] [[Feynman diagram]] with internal/[[virtual particle|virtual]] [[gluon]] lines is equivalent to one without any [[virtual particle|virtual]] [[gluon]]-lines, obtained by:
{#JacobiIdentity} 1) using the [[Jacobi identity]] to replace all internal gluon vertices by ([[linear combinations]] of) quark-gluon vertices
<center>
<img src="https://ncatlab.org/nlab/files/QuarkGluonJacobiIdentity.jpg" width="580">
</center>
{#MetricContractionOfFundamentalLieAction} 2) replacing any remaining internal gluon line together with the [[quark]] [[interaction]] [[vertices]] at its ends (equivalently: an [[M2-brane 3-algebra]] [[tensor]])
by (a [[linear combination]] of) _double_ [[quark]]-lines, according to the following rules:
<center>
<img src="https://ncatlab.org/nlab/files/MetricActionContraction.jpg" width="600">
</center>
{#Example} For example, for $\mathfrak{g} = \mathfrak{so}(N)$ a [[special orthogonal Lie algebra]], the βt Hooft double line notation of a single trivalent vertex inside a single Wilson loop is the following:
<center>
<img src="https://ncatlab.org/nlab/files/tHooftDoubleLineTrivalentVertex.jpg" width="640">
</center>
Here we are using the [[string diagram]]/[[Penrose notation]] from _[[metric Lie representations]]_.
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
{#Attribution} **Attribution.** In [βt Hooft 74](#tHooft74) this is observed for the [[unitary Lie algebra]] $\mathfrak{g} = \mathfrak{u}(N)$ (see also [Bar-Natan 95 (34)](#BarNatan95), [Chmutov-Duzhin-Mostovoy 11, p. 177](#ChmutovDuzhinMostovoy11), in which case only the first summand of the expressions [above](#MetricContractionOfFundamentalLieAction) appears. The generalization to arbitrary [[semisimple Lie algebras]] is observed in [CvitanoviΔ 76, Fig. 14](#Cvitanovic76), reviewed for [[special unitary Lie algebra|su(N)]], [[special orthogonal Lie algebra|so(N)]] and [[symplectic Lie algebra|sp(N)]] in [CvitanoviΔ 08 (9.57), (10.13) and (12.9)](#Cvitanovic08) and for [[special orthogonal Lie algebra|so(N)]] in [Chmutov-Duzhin-Mostovoy 11, 6.2.6](#ChmutovDuzhinMostovoy11).
For the case $\mathfrak{so}(N)$ see also
[Bar-Natan 95, Section 6.3](#BarNatan95) (with an eye towards [[Vassiliev knot invariants]]) and
[Cicuta 82](#Cicuta82), [Ita-Nieder-Oz 02, Figure 3](#ItaNiederOz02), [McGreevy-Swingle 08, Figure 10](#McGreevySwingle08) (with more discussion of the [[large N limit]]). The case of [[general linear Lie algebra|gl(N)]] is considered in [Bar-Natan 96, Section 2.2](#BarNatan96), [Chmutov-Duzhin-Mostovoy 11, 6.2.5](#ChmutovDuzhinMostovoy11), and the case [[special linear Lie algebra|sl(N)]] in [Chmutov-Duzhin-Mostovoy 11, 6.1.8](#ChmutovDuzhinMostovoy11), [Jackson-Moffat 19, Section 14.4](#JacksonMoffat19).
### Surface notation
Furthermore, one may regard the resulting double-line diagrams as a [[ribbon graph]] thickening of the original [[Feynman diagrams]], and thus as [[surfaces]] [[manifold with boundary|with boundary]] and with markings on the [[boundary]].
In the case $\mathfrak{g} = \mathfrak{so}(N)$ this means that a single [[virtual particle|virtual]] [[gluon]] line is represented by the [[formal linear combination]] of a strip (marked [[disk]]) and a twisted strip:
<center>
<img src="https://ncatlab.org/nlab/files/tHooftDoubleLineToSurfaces.jpg" width="600">
</center>
Using this on the reduction of internal gluon vertices by the [[Jacobi identity]] as [above](#JacobiIdentity) one finds that a single [[gluon]] vertex turns into the following linear combination of marked surfaces:
<center>
<img src="https://ncatlab.org/nlab/files/tHooftSurfacesForGluonVertex.jpg" width="700">
</center>
{#SurfaceExample} For example:
<center>
<img src="https://ncatlab.org/nlab/files/RisingSuntHooftConstruction.jpg" width="620">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
This gives [[open string]] [[worldsheets]]. Regarding them as such in the case of [[Chern-Simons theory]] exhibits [[Chern-Simons theory as an open topological string theory]] ([Witten 92, see Figures 1 & 2](#Witten92)) even for [[small N limit|small N]].
For [[AdS/CFT duality]] relating [[super Yang-Mills theory]] to _[[closed strings|closed]]_ [[string theory]] and open to _closed_ [[topological string theory]] ([Gopakumar-Vafa 98](#GopakumarVafa98)) there is an operation of gluing in [[faces]] to turn these [[open strings]] into [[closed string]] [[worldsheets]], see
[Gaiotto-Rastelli 03, Section 1.1](#GaiottoRastelli03)
and see [Marino 04, Section III, p. 14](#Marino04) for a clear statement).
Here [[open/closed string duality]] plays a subtle role in interpreting the [['t Hooft double line notation]] of [[gauge theory]] [[Feynman diagrams]] in the [[large N limit]] alternatively as [[open string]] or as [[closed string]] [[worldsheets]], see also [Gopakumar 04](#Gopakumar04).
### As a surface-valued weight system
After averaging over the $n!$ permutations of the ordering of the $n$ external vertices (those on the Wilson line) of a Jacobi diagram, this construction respects the [[STU relation]] on [[Jacobi diagrams]] and hence gives a [[weight system]] with values in marked surfaces ([Bar-Natan 95, Theorem 10 with Theorem 8](#BarNatan95)):
<center>
<img src="https://ncatlab.org/nlab/files/AtHooftConstrictionIII.jpg" width="650">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
This means that [[stringy weight systems]] pull back to ordinary [[weight systems]] along this map, corresponding to taking their [[worldline formalism|point-particle limit]]. Under this map [[stringy weight systems span classical Lie algebra weight systems]].
\linebreak
Of course, the βt Hooft double line construction applies not just to [[Jacobi diagrams]] but also to [[Sullivan chord diagrams]]:
<center>
<img src="https://ncatlab.org/nlab/files/ClosingHorizontalChordsToSullivanChordsIII.jpg" width="800">
</center>
> from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
## Applications
### Large $N$ Limit and holography
One upshot of this double-line reformulation is that it makes the dependence of the [[Feynman amplitude]] on the [[natural number]] $N$ fully explicit, since a double line diagram simply contributes one factor of $N$ for each closed [[quark]] line (this being the [[trace]] over the [[fundamental representation]]).
In fact, by regarding the resulting double-line diagrams as a [[ribbon graph]] thickenings the original [[Feynman diagrams]], and regarding these [[ribbon graphs]] in turn as [[surfaces]], the $N$-dependence is proportional to the [[genus of a surface|genus]] of these surfaces.
One finds this way that in the _[[large N limit]]_, i.e. the [[limit of a sequence|limit]] where $N \to \infty$, precisely only the [[planar graphs]] contribute to the [[Yang-Mills theory]] [[Feynman amplitudes]], corresponding to [[surfaces]] of [[genus]] 0 (i.e. [[tree level]] [[string scattering amplitudes]] see at _[[planar limit]]_ for more on this), while $1/N$-corrections correspond to higher [[worldsheet]] [[topology]].
Last not least, these [[surfaces]] have the interpretation of [[open string]] [[worldsheets]] of a [[string theory]] which is "[[duality in string theory|dual]]" to the original ([[super Yang-Mills theory|super]]) in the sense made precise by [[AdS-CFT duality]].
### Classification of weight systems
Later the same double line technique was used (without any reference to the earlier physics articles(?)) in [Bar-Natan 95, Section 6](#BarNatan95) for discussion of the classification of [[Lie algebra weight systems]] and [[stringy weight systems]] with an eye towards discussion of [[Vassiliev knot invariants]].
## Related concepts
* [[large N limit]], [[small N limit]]
* [[AdS/CFT correspondence]]
## References
### General
The original article is:
* {#tHooft74} [[Gerard βt Hooft]], _A Planar Diagram Theory for Strong Interactions_, Nucl. Phys. B72 (1974) 461 ([spire:80491](http://inspirehep.net/record/80491), <a href="https://doi.org/10.1016/0550-3213(74)90154-0">doi:10.1016/0550-3213(74)90154-0</a>)
reviewed in
* [[Gerard βt Hooft]], _Large $N$_, workshop lecture ([hep-th/0204069](http://arxiv.org/abs/hep-th/0204069))
* Markus Gross, _Large $N$_, 2006 ([[GrossLargeN.pdf:file]])
Generalization to arbitrary [[semisimple Lie algebras]] ([[semisimple Lie groups]]) is due to:
* {#Cvitanovic76} [[Predrag CvitanoviΔ]], _Group theory for Feynman diagrams in non-Abelian gauge theories_, Phys. Rev. D14 (1976) 1536-1553 ([doi:10.1103/PhysRevD.14.1536](https://doi.org/10.1103/PhysRevD.14.1536), [spire:108133](http://inspirehep.net/record/108133), [[CvitanovicWeights76.pdf:file]])
with a textbook account in
* {#Cvitanovic08} [[Predrag CvitanoviΔ]], _Group Theory: Birdtracks, Lie's, and Exceptional Groups_, Princeton University Press July 2008 ([PUP](https://press.princeton.edu/books/paperback/9780691202983/group-theory), [birdtracks.eu](http://birdtracks.eu/), [pdf](http://www.birdtracks.eu/version9.0/GroupTheory.pdf))
Discussion in the context of [[Vassiliev invariants]] and the abstract classification of [[Lie algebra weight systems]] and [[stringy weight systems]]:
* {#BarNatan95} [[Dror Bar-Natan]], Section 6 of: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))
of which textbook accounts are in
* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))
* {#JacksonMoffat19} [[David Jackson]], [[Iain Moffat]], _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))
Further discussion of the case of $\mathfrak{so}(N)$ in the context of the [[large N limit]]:
* {#Cicuta82} G.M. Cicuta, _Topological Expansion for $SO(N)$ and $Sp(2n)$ Gauge Theories_, Lett. Nuovo Cim. 35 (1982) 87 ([spire:177713](http://inspirehep.net/record/177713), [doi:10.1007/BF02754653](https://doi.org/10.1007/BF02754653))
* {#ItaNiederOz02} Harald Ita, Harald Nieder, [[Yaron Oz]], _Perturbative Computation of Glueball Superpotentials for $SO(N)$ and $USp(N)$_, JHEP 0301:018, 2003 ([arXiv:hep-th/0211261](https://arxiv.org/abs/hep-th/0211261))
* {#McGreevySwingle08} McGreevy, Swingle, _Large $N$ counting_, 2008 ([[GreevySwingle.pdf:file]])
On the [[logical equivalence]] between the [[four-colour theorem]] and a statement about transition from the [[small N limit]] to the [[large N limit]] for [[Lie algebra weight systems]] on [[Jacobi diagrams]] via the [['t Hooft double line construction]]:
* [[Dror Bar-Natan]], _Lie Algebras and the Four Color Theorem_, Combinatorica 17-1(1997) 43β52 ([arXiv:q-alg/9606016](https://arxiv.org/abs/q-alg/9606016), [doi:10.1007/BF01196130](https://doi.org/10.1007/BF01196130))
### For Chern-Simons theory
Discussion of [['t Hooft double line notation]] for [[Chern-Simons theory]], exhibiting [[Chern-Simons theory as topological string theory]]:
* {#Witten92} [[Edward Witten]], _Chern-Simons Gauge Theory As A String Theory_, Prog. Math. 133: 637-678, 1995 ([arXiv:hep-th/9207094](https://arxiv.org/abs/hep-th/9207094))
* S. Sinha, [[Cumrun Vafa]], _$SO$ and $Sp$ Chern-Simons at Large $N$_ ([arXiv:hep-th/0012136](https://arxiv.org/abs/hep-th/0012136))
### Open/closed string duality
On the role of [[open/closed string duality]] in interpreting the [[large N limit]] of the [['t Hooft double line notation]]:
* {#GopakumarVafa98} [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))
* {#GaiottiRastelli05} [[Davide Gaiotto]], [[Leonardo Rastelli]], _A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model_, JHEP 0507:053,2005 ([hep-th/0312196](https://arxiv.org/abs/hep-th/0312196))
> Nowadays we interpret $[$ the [['t Hooft double line notation]] $]$ quite literally as the perturbative expansion of an open string theory, either because the full open string theory is just equal to the gauge
theory (as e.g. for Chern-Simons theory [27]), or because we take an appropriate
low-energy limit (as e.g. for N = 4 SYM [31]).
> The general speculation [1] is that upon summing over the number of holes, (1.1) can be recast as the genus expansion for some closed string theory of coupling $g_s = g_{YM}^2$. This speculation is sometimes justified by appealing to the intuition that diagrams with
a larger and larger number of holes look more and more like smooth closed Riemann surfaces. This intuition is perfectly appropriate for the double-scaled matrix models, where the finite N theory is interpreted as a discretization of the closed Riemann surface; to recover the continuum limit, one must send $N\to \infty$ and tune $t$ to the
critical point $t_c$ where diagrams with a diverging number of holes dominate.
> However, in AdS/CFT, or in the Gopakumar-Vafa duality [2], $t$ is a free parameter, corresponding on the closed string theory side to a geometric modulus. The intuition described above clearly goes wrong here.
> A much more fitting way in which the open/closed duality may come about in these cases is for each fatgraph of genus g and
with h holes to be replaced by a closed Riemann surface of the same genus g and with h punctures: each hole is filled and replaced by a single closed string insertion.
* {#Gopakumar04} [[Rajesh Gopakumar]], _Free Field Theory as a String Theory?_, Comptes Rendus Physique 5 (2004) 1111-1119 ([hep-th/0409233](https://arxiv.org/abs/hep-th/0409233))
* {#Marino04} [[Marcos Marino]], _Chern-Simons Theory and Topological Strings_, Rev. Mod. Phys. 77:675-720, 2005 ([arXiv:hep-th/0406005](https://arxiv.org/abs/hep-th/0406005))
[[!redirects 't Hooft double line notations]]
[[!redirects 't Hooft double line construction]]
[[!redirects 't Hooft double line constructions]]
|
't Hooft-Polyakov monopole | https://ncatlab.org/nlab/source/%27t+Hooft-Polyakov+monopole |
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A type of [[monopole]] solution of [[Yang-Mills theory]] with [[Higgs field]] included.
## Properties
### In string theory
In [[string theory]], in the context of [[intersecting D-brane models]], the 't Hooft-Polyakov monopole has been identified with the [[black brane|black]] "half" [[NS5-brane]] coincident on an [[O8-plane]] ([Hanany-Zaffaroni 99](#HananyZaffaroni99)). See there at _[Intersection of D6s with O8s](intersecting+D-brane+model#IntersectionOfD6WithO8)_.
## Related concepts
* [[Yang monopole]]
* [[Dirac monopole]]
## References
The original articles are
* [[Gerard 't Hooft]], _Magnetic Monopoles in Unified Gauge Theories_, Nucl.Phys. B79 (1974) 276-284 ([spire](http://inspirehep.net/record/89705?ln=en))
* [[Alexander Polyakov]], _Particle Spectrum in the Quantum Field Theory_, JETP Lett. 20 (1974) 194-195 ([spire](http://inspirehep.net/record/90679/))
Interpretation in [[string theory]] in terms of "half" [[NS5-branes]] at [[O8-planes]] is due to
* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arXiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))
See also
* Wikipedia _['t Hooft-Polyakov monopole](https://en.wikipedia.org/wiki/'t_HooftβPolyakov_monopole)_
[[!redirects 't Hooft-Polyakov monopoles]] |
(-1)-category | https://ncatlab.org/nlab/source/%28-1%29-category | +-- {: .rightHandSide}
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### Context ###
#### Higher category theory
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[[!include higher category theory - contents]]
=--
=--
=--
## Idea
As a degenerate case of the general notion of *[[n-category]]*, _$(-1)$-categories_ may be understood as [[truth value]]. Compare the concept of [[0-category]] (a [[set]]) and [[(β2)-category]] (which is trivial). The point of $(-1)$-categories (a kind of [[negative thinking]]) is that they complete some patterns in the [[periodic table]] of $n$-categories. (They also shed light on the theory of [[homotopy group]]s and [[n-stuff]].)
For example, there should be a $0$-category of $(-1)$-categories; this is the set of truth values, classically
$$
(-1)Cat := \{true, false\}
\,.
$$
Similarly, $(-2)$-categories form a $(-1)$-category (specifically, the true one).
If we equip the category of $(-1)$-categories with the monoidal structure of [[logical conjunction|conjunction]] (the logical AND operation), then a [[enriched category|category enriched]] over this is a [[partial order|poset]]; an enriched groupoid is a [[set]]. Notice that this doesn\'t fit the proper patterns of the [[periodic table]]; we see that $(-1)$-categories work better as either $0$-[[0-poset|poset]]s or as $(-1)$-[[(-1)-groupoid|groupoid]]s. Nevertheless, there is no better alternative for the term '$(-1)$-category'.
For an introduction to $(-1)$-categories and $(-2)$-[[(-2)-category|categories]] see [page 11](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=11) and [page 34](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=34) of
* John C. Baez, Michael Shulman, _[[Lectures on n-Categories and Cohomology]]_ ([arXiv](http://arxiv.org/abs/math.CT/0608420)).
$(-1)$-categories and $(-2)$-categories were discovered (or invented) by [[James Dolan]] and [[Toby Bartels]]. To witness these concepts in the process of being discovered, read the discussion here:
* John Baez, Toby Bartels, David Corfield and James Dolan,
[Property, structure and stuff](http://math.ucr.edu/home/baez/qg-spring2004/discussion.html). See also [[stuff, structure, property]] for more on that material.
[[!redirects (-1)-category]]
[[!redirects (-1)-categories]]
[[!redirects (β1)-category]]
[[!redirects (β1)-categories]]
|
(-1)-functor | https://ncatlab.org/nlab/source/%28-1%29-functor |
As a $(-1)$-[[(-1)-category|category]] is simply a [[truth value]], so a __$(-1)$-functor__ is simply [[implication]]. See also $n$-[[n-functor|functor]].
[[!redirects (-1)-functor]]
[[!redirects (-1)-functors]]
[[!redirects (β1)-functor]]
[[!redirects (β1)-functors]]
|
(-1)-groupoid | https://ncatlab.org/nlab/source/%28-1%29-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A __$(-1)$-groupoid__ or **[[homotopy n-type|(-1)-type]]** is a [[truth value]], or equivalently an [[n-truncated object of an (infinity,1)-category|(-1)-truncated object]] in [[βGrpd]]. By [[excluded middle]], this is either [[the]] empty groupoid (false) or the [[terminal category|terminal groupoid]] (true, the [[point]]).
## Remarks
Compare the concept of [[0-groupoid]] (a [[set]]) and [[(-2)-groupoid]] (which is trivial). The point of $(-1)$-groupoids is that they complete some patterns in the [[periodic table]] of $n$-categories. (They also shed light on the theory of [[homotopy group]]s and [[n-stuff]].)
For example, there should be a $0$-[[0-category|category]] of $(-1)$-groupoids; a $0$-category is also a set, and this set is the set of [[truth value|truth values]]: classically
$$
(-1)Grpd := \{\bot, \top\}
$$
Actually, since for other values of $n$, [[n-groupoid]]s form not just an $(n+1)$-category but an $(n+1,1)$-category, we should expect the $0$-category of $(-1)$-groupoids to be a $(0,1)$-category, or $1$-[[1-poset|poset]]. This simply means a [[partial order|poset]], and indeed truth values do always form a poset, classically ($\bot \leq \top$).
If we equip the category of $(-1)$-groupoids with the [[monoidal category|monoidal structure]] of [[logical conjunction|conjunction]] (the logical AND operation), then a [[enriched category|groupoid enriched]] over this is a [[symmetric proset]], and a category enriched over it is a [[preorder|proset]]. Up to [[equivalence of categories]], these are the same as a [[set]] (a $0$-[[0-groupoid|groupoid]]) and a [[partial order|poset]] (a (0,1)-[[1-poset|category]]); this fits the patterns of the periodic table.
See [[(-1)-category]] for more on this sort of [[negative thinking]].
## Related concepts
[[!include homotopy n-types - table]]
[[!redirects (-1)-groupoid]]
[[!redirects (-1)-groupoids]]
[[!redirects (-1)-groupoid]]
[[!redirects (-1)-groupoids]]
[[!redirects (-1)-type]]
[[!redirects (-1)-types]]
[[!redirects (-1)-type]]
[[!redirects (-1)-types]]
|
(-1)-poset | https://ncatlab.org/nlab/source/%28-1%29-poset | There is just one __$(-1)$-poset__, namely the [[point]]. Compare the concepts of $0$-[[0-poset|poset]] (a [[truth value]]) and $1$-[[1-poset|poset]] (a [[partial order|poset]]). Compare also with $(-2)$-[[(-2)-category|category]] and $(-2)$-[[(-2)-groupoid|groupoid]], which mean the same thing for their own reasons.
The point of $(-1)$-posets is that they complete some patterns in the [[periodic table]]s and complete the general concept of $n$-[[n-poset|poset]]. For example, there should be a $0$-[[0-poset|poset]] $(-1)\Pos$ of $(-1)$-posets; a $0$-poset is simply a truth value, and $(-1)\Pos$ is the [[true]] truth value.
As a category, $(-1)\Pos$ is a [[monoidal category]] in a unique way, and a [[enriched category|category enriched]] over this should be (at least up to equivalence) a $0$-poset, which is a truth value; and indeed, a category enriched over $(-1)\Pos$ is a category in which any two objects are isomorphic in a unique way, which is [[equivalence of categories|equivalent]] to a truth value.
See [[(β1)-category]] for references on this sort of [[negative thinking]].
[[!redirects (β1)-poset]]
|
(-2)-category | https://ncatlab.org/nlab/source/%28-2%29-category | There is just one _$(-2)$-category_, namely the truth value [[True]]. Compare the concepts of [[(β1)-category]] (a [[truth value]] in general) and [[0-category]] (a [[set]]). The point of $(-2)$-categories is that they complete some patterns in the [[periodic table]] of $n$-categories. (They also shed light on the theory of [[homotopy group]]s and [[n-stuff]].)
For example, there should be a $(-1)$-category of $(-2)$-categories; this is the true truth value. The category of $(-2)$-categories is a [[monoidal category]] in a unique way; then a [[enriched category|category enriched]] over this is a $(-1)$-category; such is necessarily an enriched groupoid. If you think of a $(-1)$-category as a [[0-poset]], then this makes a $(-2)$-category a [[(β1)-poset]]. If you think of a $(-1)$-category as a [[(β1)-groupoid]], then this makes a $(-2)$-category a [[(β2)-groupoid]].
For an introduction to $(-1)$-[[(-1)-category|categories]] and $(-2)$-categories see [page 11](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=11) of
* John C. Baez, Michael Shulman, _[[Lectures on n-Categories and Cohomology]]_ ([arXiv](http://arxiv.org/abs/math.CT/0608420)).
$(-1)$-categories and $(-2)$-categories were discovered (or invented) by [[James Dolan]] and [[Toby Bartels]]. To witness these concepts in the process of being discovered, read the discussion here:
* John Baez, Toby Bartels, David Corfield and James Dolan,
[Property, structure and stuff](http://math.ucr.edu/home/baez/qg-spring2004/discussion.html). See also [[stuff, structure, property]] for more on that material.
[[!redirects (-2)-category]]
[[!redirects (-2)-categories]]
[[!redirects (β2)-category]]
[[!redirects (β2)-categories]]
|
(-2)-groupoid | https://ncatlab.org/nlab/source/%28-2%29-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A **$(-2)$-groupoid** or **[[homotopy n-type|(-2)-type]]** is a [[n-truncated object of an (infinity,1)-category|(-2)-truncated object]] in [[βGrpd]].
There is, up to equivalence, just one $(-2)$-groupoid, namely the [[point]].
## Remarks
Compare the concepts of $(-1)$-[[(-1)-groupoid|groupoid]] (a [[truth value]]) and $0$-[[0-groupoid|groupoid]] (a [[set]]). Compare also with $(-2)$-[[(-2)-category|category]] and $(-1)$-[[(-1)-poset|poset]], which mean the same thing for their own reasons.
The point of $(-2)$-groupoids is that they complete some patterns in the [[periodic table]]s and complete the general concept of $n$-[[n-groupoid|groupoid]]. For example, there should be a $(-1)$-[[(-1)-groupoid|groupoid]] $(-2)\Grpd$ of $(-2)$-groupoids; a $(-1)$-groupoid is simply a truth value, and $(-2)\Grpd$ is the [[true]] truth value.
As a category, $(-2)\Grpd$ is a [[monoidal category]] in a unique way, and a [[enriched category|groupoid enriched]] over this should be (at least up to equivalence) a $(-1)$-groupoid, which is a truth value; and indeed, a groupoid enriched over $(-2)\Grpd$ is a groupoid in which any two objects are isomorphic in a unique way, which is [[equivalence of categories|equivalent]] to a truth value.
See [[(-1)-category]] for references on this sort of [[negative thinking]].
## Related concepts
* [[true]]
[[!include homotopy n-types - table]]
[[!redirects (?2)-groupoid]]
[[!redirects (-2)-type]]
[[!redirects (-2)-types]]
|
(0,1)-category | https://ncatlab.org/nlab/source/%280%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
{#Idea}
Following the general concept of *[[(n,r)-category|$(n,r)$-category]]*, a *$(0,1)$-category* is a [[category]] whose [[hom-objects]] are [[(-1)-groupoids]], hence which for every [[pair]] of [[objects]] $a,b$ have either no morphism $a \to b$ or an [[essentially unique]] one.
More in detail, recall that:
* an [[(n,1)-category]] is an [[(β,1)-category]] such that every [[hom-space|hom β-groupoid]] is [[n-truncated|(n-1)-truncated]].
* a [[0-truncated]] [[β-groupoid]] is equivalently a set;
* a [[(-1)-truncated]] [[β-groupoid]] is either [[contractible]] or [[empty set|empty]].
Therefore:
+-- {: .num_remark}
###### Remark
**([[relation between preorders and (0,1)-categories]])**
\linebreak
An $(0,1)$-category is [[equivalence of categories|equivalently]] a [[proset]] ([[relation between preorders and (0,1)-categories|hence]] a [[poset]]).
We may without restriction assume that every hom-$\infty$-groupoid is just a set. Then since this is [[(-1)-truncated]] it is either empty or the singleton. So there is at most one morphism from any object to any other.
=--
## Extra stuff, structure, property
* A $(0,1)$-category with the structure of a [[site]] is a [[(0,1)-site]]: a [[posite]].
* A $(0,1)$-category with the structure of a [[topos]] is a [[(0,1)-topos]]: a [[Heyting algebra]].
* A $(0,1)$-category with the structure of a [[Grothendieck topos]] is a [[Grothendieck (0,1)-topos]]: a [[frame]] or [[locale]].
* A $(0,1)$-category which is also a [[groupoid]] (that is, every morphism is an isomorphism) is a $(0,0)$-category (which may think of as either a $0$-category or as a $0$-groupoid), which is the same as a [[set]] (up to equivalence) or a [[symmetric proset]] (up to isomorphism).
## Related concepts
* [[relation between preorders and (0,1)-categories]]
* [[(-2)-groupoid]]
* [[(-1)-groupoid]]
* [[0-groupoid]]
* [[1-groupoid]]
* [[2-groupoid]]
* [[3-groupoid]]
* [[4-groupoid]]
* [[n-groupoid]]
* [[β-groupoid]]
* [[(0,0)-category]]
* [[(1,0)-category]]
* [[(n,0)-category]]
* [[(β,0)-category]]
* [[0-category]]
* [[1-category]]
* [[2-category]]
* [[3-category]]
* [[4-category]]
* [[n-category]]
* [[β-category]]
* **(0,1)-category**
* [[(1,1)-category]]
* [[(2,1)-category]]
* [[(n,1)-category]]
* [[(β,1)-category]]
* [[(1,2)-category]]
* [[(β,2)-category]]
* [[(β,n)-category]]
* [[(n,r)-category]]
[[!redirects (0,1)-category]]
[[!redirects (0,1)-categories]]
|
(0,1)-category theory | https://ncatlab.org/nlab/source/%280%2C1%29-category+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the context of [[higher category theory]] / [[(n,r)-categories]], a [[poset]] is equivalently regarded as a [[(0,1)-category]].
$(0,1)$-categories play a major role in [[logic]], where their objects are interpreted as [[propositions]], their morphisms as [[implications]] and [[limits]]/[[products]] and [[colimits]]/[[coproducts]] as [[logical conjunctions]] _[[and]]_ and _[[or]]_, respectively.
[[Stone duality|Dually]], $(0,1)$-categories play a major role in [[topology]], where they are interpreted as [[categories of open subsets]] of a [[topological spaces]], or, more generally, of [[locales]].
Clearly, much of [[category theory]] simplifies drastically when restricted to $(0,1)$-categories, but it is often most useful to make the parallel explicit.
## Related concepts
[[!include table of category theories]]
|
(0,1)-category theory - contents | https://ncatlab.org/nlab/source/%280%2C1%29-category+theory+-+contents |
**[[(0,1)-category theory]]**: [[logic]], [[order theory]]
**[[(0,1)-category]]**
* [[relation between preorders and (0,1)-categories]]
* [[proset]], [[partially ordered set]] ([[directed set]], [[total order]], [[linear order]])
* [[top]], [[true]],
* [[bottom]], [[false]]
* [[monotone function]]
* [[implication]]
* [[filter]], [[interval]]
* [[lattice]], [[semilattice]]
* [[meet]], [[logical conjunction]], [[and]]
* [[join]], [[logical disjunction]], [[or]]
* [[compact element]]
* [[lattice of subobjects]]
* [[complete lattice]], [[algebraic lattice]]
* [[distributive lattice]], [[completely distributive lattice]], [[canonical extension]]
* [[hyperdoctrine]]
* [[first-order hyperdoctrine|first-order]], [[Boolean hyperdoctrine|Boolean]], [[coherent hyperdoctrine|coherent]], [[tripos]]
**[[(0,1)-topos]]**
* [[Heyting algebra]]
* [[regular element]]
* [[Boolean algebra]]
* [[frame]], [[locale]]
## Theorems
* [[Stone duality]]
|
(0,1)-presheaf | https://ncatlab.org/nlab/source/%280%2C1%29-presheaf | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### (0,1)-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Topos theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **(0,1)-presheaf** is a [[presheaf]] with values in the [[(0,1)-category]]
of [[truth values]]. A 0-[[truncated]] [[(β,1)-presheaf]].
## Definition
A **(0,1)-presheaf** on a [[poset]] or [[proset]] $P$ is an [[antitone function|antitone]] [[predicate]]
$$F:P \rightarrow \Omega$$
from $P$ to the poset $\Omega$ of truth values, or equivalently, a [[monotone]] predicate
$$F:P^\op \rightarrow \Omega$$
from the [[opposite poset]] of $P$ to $\Omega$.
More generally, for a poset $S$, a **S-valued (0,1)-presheaf** on $P$ is just an antitone
$$f:P \rightarrow S$$
so (0,1)-presheaves are just antitones.
## (0,1)-category of (0,1)-presheaves
The [[(0,1)-category]] of a (0,1)-presheaf on a [[(0,1)-site]] forms a [[(0,1)-topos]]. In traditional order theoretic language, the poset (or proset) of (0,1)-presheaves on a [[posite]] forms a [[locale]].
## Related concepts
* [[lower set]]
* **(0,1)-presheaf**, [[(0,1)-sheaf]]
* [[presheaf]]
* [[(2,1)-presheaf]]
[[presheaf of groupoids]]
[[(2,1)-sheaf]], [[2-sheaf]], [[indexed category]], [[stack]]
* [[(β,1)-presheaf]]
* [[(β,n)-presheaf]]
[[!redirects (0,1)-presheaves]]
[[!redirects proposition-valued presheaf]]
[[!redirects proposition-valued presheaves]] |
(0,1)-topos | https://ncatlab.org/nlab/source/%280%2C1%29-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The notion of $(0,1)$-topos is that of [[topos]] in the context of [[(0,1)-category theory]].
The notion of $(0,1)$-topos is essentially equivalent to that of [[Heyting algebra]]; similarly, a [[Grothendieck topos|Grothendieck]] $(0,1)$-topos is a [[locale]].
Notice that every $(1,1)$-[[Grothendieck topos]] comes from a [[localic groupoid]], i.e. a [[groupoid]] [[internal category|internal to]] locales, hence a groupoid internal to $(0,1)$-toposes. See [[classifying topos of a localic groupoid]] for more.
## Related concepts
[[!include flavors of higher toposes -- list]]
## References
* [[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); section 5.3)
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ , Princeton UP 2009. (section 6.4.2)
[[!redirects Grothendieck (0,1)-topos]]
[[!redirects 0-topos]]
[[!redirects Grothendieck 0-topos]]
[[!redirects (0,1)-toposes]].
|
(1,0)-category | https://ncatlab.org/nlab/source/%281%2C0%29-category | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
By the general rules of $(n,r)$-[[(n,r)-category|categories]], a $(1,0)$-category is an $\infty$-[[infinity-category|category]] such that
* any $j$-morphism is an [[equivalence]], for $j \gt 0$;
* any two parallel $j$-morphisms are equivalent, for $j \gt 1$.
You can start from any notion of $\infty$-category, strict or weak; up to [[equivalence of categories|equivalence]], the result is always the same as a [[groupoid]].
[[!redirects (1,0)-categories]] |
(1,1)-category | https://ncatlab.org/nlab/source/%281%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
In the pattern of [[(n,r)-categories]] a **$(1,1)$-category** is just an ordinary [[category]].
[[!redirects (1,1)-categories]]
|
(1,1)-dimensional Euclidean field theories and K-theory | https://ncatlab.org/nlab/source/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]
See there for background and context.
This entry here indicates how 1-dimensional [[FQFT]]s (the [[superparticle]]) may be related to [[topological K-theory|topological]] [[K-theory]].
=--
> **raw material**: this are notes taken more or less verbatim in a seminar -- needs polishing
Previous:
* [[Axiomatic field theories and their motivation from topology]].
Next
* [[(2,1)-dimensional Euclidean field theories and tmf]]
#Contents#
* table of contents
{:toc}
## $(1,1)d$ EFTs
recall the **commercial for supergeometry** with which we ended [[Axiomatic field theories and their motivation from topology|last time]]: the grading introduced by supergeometry makes it possible to have push-forward diagrams of the kind:
$$
\array{
(0|1)TFTs^n(X)/\simeq &\leftarrow& H^n_{dR}(X)
\\
\downarrow && \downarrow
\\
(0|1)TFT^0(X)/\simeq &\leftarrow& H^0_{dR}(pt)
}
$$
**Example** of 1-EFT
$$
\sigma_1(M^n) = E : 1-EB \to tV
$$
$$
pt \mapsto \Gamma M
$$
$$
(pt \stackrel{[0,t]}{\to})
\mapsto
e^{- t \Delta}
$$
**Example** of $(1|1)-EFT$ associated to a [[Spin structure|spin manifold]], there is the [[spinor bundle]]
$$
S = S^+ \oplus S^-
$$
a $\mathbb{Z}/2$-graded [[vector bundle]] and on this there is the [[Dirac operator]]
$$
D : \Gamma(S) \to \Gamma(S)
$$
where $\Gamma(S) = \Gamma(S^+) \oplus \Gamma(S^-)$. So we can write
$$
D =
\left(
\array{
0 & D_-
\\
D+ & 0
}
\right)
$$
$$
\sigma_{1|1}(M) : Bord_{1|1} \to TV
$$
$$
\mathbb{R}^{0|1} \mapsto E(\mathbb{R}^{0|1}) = \Gamma(S)
$$
there is an involution $invol : \mathbb{R}^{0|1} \to \mathbb{R}^{0|1}$. It maps to
$$
invol \mapsto grading involution
$$
we have the following [[moduli space]] of super [[interval]]s (super 1d-bordisms)
$$
\mathbb{R}^{1|1}_+ \simeq
\{super intervals I_{t,\theta}\}/\sim
$$
and these are mapped by the EFT as
$$
I_{t,\theta} \mapsto e^{-t D^2 + \theta D}
$$
(here we are implicitly working in the [[topos]] of [[sheaf|sheaves]] on the category of [[supermanifold]]s and these equations have to be interpreted in that topos-logic, mapping [[generalized element]]s to [[generalized element]]s).
So we have for $E$ a $1|1$ EFT a _reduced_ non-susy field theory
$$
\array{
(1|1)EBord &\stackrel{E}{\to}& TV
\\
\uparrow & \nearrow_{E_{red}}
\\
EBord_1^{spin}
}
$$
**Definition** $E \in (1|1)EFT$, the [[partition function]] $Z_E$ of $E$ is the function
$$
Z_E : \mathbb{R}_+ \to \mathbb{C}
$$
$$
t \mapsto Z_{E_{red}}(t) = E_{red}(S^1_t)
$$
that sends a length to the value of the EFT on the circle of that circumferene.
**Example** Consider from above the EFT
$$
E = \sigma_{1|1}(M)
$$
look at its reduced part
$$
z_E(t) = E_{red}(S^1_t)
$$
notice that by the above this assigns
$$
[0,t] \stackrel{E_{red}}{\mapsto} e^{-t D^2}
$$
$$
S^1_t \mapsto str(e^{-t D^2})
=
tr(e^{-t D^2})|_{even} - tr(e^{-t D^2})|_{odd}
$$
where on the right we have the [[super trace]].
This evaluates to
$$
str(e^{-t D^2})
=
\sum_{\lambda \in Spec(D^2)}
e^{-t \lambda}
sdim E_{\lambda}
$$
where the [[super dimension]] of the [[eigenspace]] $E_\lambda$ is
$$
dim E^+_\lambda - dim E^-_\lambda
$$
and this vanishes for $\lambda \neq 0$ since there $D : E_\lambda^+ \stackrel{\simeq}{\to} E_\lambda^-$
is an [[isomorphism]].
So further in the computation we have
$$
\cdots = dim ker D_+ - dim coker D_+
=
\hat A(M)
$$
where the last step is the [[Atiyah-Singer index theorem]].
So **due to supersymmetry** , the [[partition function]] has two very special properties:
* it is constant -- in that it does not depend on $t$,
* it takes integer values $\in \mathbb{N} \subset \mathbb{R}$.
**recall** from $V \to X$ a [[vector bundle]] [[connection on a bundle|with connection]] $\nabla$ we get a 1d EFT
$$
E_{(V,\nabla)} \in 1d EFT(X)
$$
given by the assignment
$$
E_{(V,\nabla)} : 1s EB(X) \to TV
$$
$$
(x : pt \to X) \mapsto V_x = fiber of V over x
$$
a morphism is an [[interval]] $[0,t]$ of length $t$ equipped with a map $\gamma : [0,t] \to X$, this is sent to the [[parallel transport]] associated with the [[connection on a bundle]]
$$
\gamma \mapsto (V_{\gamma_x} \to V_{\gamma_y})
$$
Now refine this example to super-dimension $(1|1)$:
**example** of a $(1|1)$-EFT over $X$ consider
$$
EBord_{(1|1)} \to EBord_{1}(X) \stackrel{E_{(V,\nabla)}}{\to}
TV
$$
given by the assignment
$$
(\Sigma^{(1|1)} \to X)(
\mapsto
(\Sigma^{(1|1)}_{red} \to X)
\mapsto
parallel transport as before
$$
so we just forget the super-part and consider the same [[parallel transport]] as before.
now to [[K-theory]]:
$
KO^0(X) =
$ [[Grothendieck group]] of real [[vector bundle]]s over $X$
$$
KO^{-n}(pt) =
\left\{
\array{
\mathbb{Z} & n = 0 mod 4
\\
\mathbb{Z}_2 & n = 1,2 mod 8
\\
0 & otherwise
}
\right.
$$
there is a [[Bott element]] $\beta \in KO^{-8}(pt)$
such that
$$
KO^*(pt) \stackrel{\simeq_{\mathbb{Q}}}{\to}
\mathbb{Z}[u,u^{-1}]
$$
$$
\beta \mapsto u^2
$$
now the **push-forward in [[topological K-theory]]**
$$
p : X^n \to pt
$$
for $X$ a closed [[spin structure]] manifold
then there exists an embedding $X \hookrightarrow S^{n+m}$. Let $\nu$ be the [[normal bundle]] to this embedding.
then we define
$$
\int_X : KO^k(X) \to KO^{k-n}(pt)
$$
as follows:
let $D(\nu)$ be the [[disk bundle]] and $S(\nu)$ be the [[sphere bundle]] of $\nu$. Then the [[Thom bundle]] is
$$
T(\nu) := D(\nu)/S(\nu)
$$
we get a map
$$
S^{n+m} \stackrel{C}{\to} T(\nu) := D(\nu)/S(\nu)
$$
involving the [[Thom isomorphism]]
$$
C(X) = \left\{
\array{
X & if x \in D(\nu)
\\
* & otherwise
}
\right.
$$
then we set
$$
\array{
KO^k(X)
&& \stackrel{\int_X}{\to}&& KO^{k-n}(pt)
\\
& {}_{Thom iso}\searrow
&&&
\downarrow^{\simeq}_{suspension}
\\
&& \tilde KO^{k+m}(T(\nu))
&\stackrel{C^*}{\to}&
}
$$
***
now start with $X^n$ again a [[spin structure|spin]] [[manifold]]
then
**theorem** (Stolz-Teichner): we have the horizontal isomorphism in the following diagram:
$$
\array{
&& [E_{(V,\nabla)}]&& \stackrel{}{\leftarrow}
&& [V^+ - V^-]
\\
1 \in
&&(1|1)EFT^0(X)/_{conc}
&&\stackrel{\simeq}{\to}&& KO^0(X) && \ni 1
\\
\downarrow &&\downarrow^{quantization}
&&&& \downarrow^{\int_X} && \downarrow
\\
\sigma_{(1|1)}(X)
&&EFT^{-n}(pt)/_{conc}
&&\stackrel{\simeq}{\to}&& KO^{-n} && \alpha(X)
\\
&\searrow&&{}_{partition func}\searrow&&
\swarrow_{\simeq} && \swarrow_{Atiyah's \alpha invariant}
\\
&&&&
(\mathbb{Z}[u,u^{-1}])^{-n}
\\
&&&&
index D = \hat A(X) u^{n/4}
}
$$
**question** if we don't divide out [[concordance]], do we get [[differential K-theory]] on the right?
**answer** presumeably, but not worked out yet
## Related concepts
* [[supersymmetric quantum mechanics]]
* [[Euclidean quantum field theory]]
* [[spectral triple]]
* [[spectral action]]
* [[higher category theory and physics]]: <a href="http://ncatlab.org/nlab/show/higher+category+theory+and+physics#SpecStandModAndGravity">Spectral standard model and gravity</a>
* **(1,1)-dimensional Euclidean field theories and K-theory**
* [[(2,1)-dimensional Euclidean field theories and tmf]]
* [[2-spectral triple]]
## References
{#References}
* [[Stephan Stolz]], [[Peter Teichner]], _[[What is an elliptic object?]]_ in _Topology, geometry and quantum field theory_ , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343. ([pdf](http://web.me.com/teichner/Math/Reading_files/Elliptic-Objects.pdf))
* Pokman Cheung, _Supersymmetric field theories and cohomology_ ([arXiv:0811.2267](http://arxiv.org/abs/0811.2267))
* {#Stolz} [[Stefan Stolz]] (notes by Arlo Caine), _Supersymmetric Euclidean field theories and generalized cohomology_ Lecture notes (2009) ([pdf](http://www.cpp.edu/~jacaine/pdf/Lectures_complete.pdf))
* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric Euclidean field theories and generalized cohomology_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) |
(1,2)-topos | https://ncatlab.org/nlab/source/%281%2C2%29-topos | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of *(1,2)-topos* should be the notion of [[higher toposes]] among [[(1,2)-categories]] or [[2-posets]].
## Examples
\begin{example}\label{The12ToposOfPosets}
The [[(1,2)-category]] [[Pos]] of [[posets]] and monotone maps should be the archetypal $(1,2)$-topos.
The poset of [[truth values]]
$$
\big( \bot \to \top \big)
\;\in\;
Pos
$$
should play the role of the "sub-poset classifier" in $Pos$, the (1,2)-analog of the [[subobject classifier]] in a [[1-topos]].
Here, morphisms into it
classify [[monic]] [[fibrations]] of posets, namely [[sieves]] (e.g. [Exp. 9.26 here](https://www.andrew.cmu.edu/course/80-413-713/notes/chap09.pdf#page=31))
\end{example}
## Related concepts
[[!include flavors of higher toposes -- list]]
|
(2,1)-algebraic theory of E-infinity algebras | https://ncatlab.org/nlab/source/%282%2C1%29-algebraic+theory+of+E-infinity+algebras | [[!redirects (2,1)-algebraic theeory of E-infinity algebras]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[(β,1)-algebraic theory]] whose [[β-algebra over an (β,1)-algebraic theory|algebras]] are [[E-β algebra]]s is the [[(2,1)-category]] of [[span]]s of [[finite set]]s.
## Definition
+-- {: .num_defn}
###### Definition
Let
$$
2Comm := Span(FinSet)
$$
be the [[(2,1)-category]] of [[span]]s of [[finite set]]s:
* [[object]]s are finite sets;
* [[morphism]]s are [[span]]s $X_1 \leftarrow Y \to X_1$ in [[FinSet]];
* [[2-morphism]]s are diagrams
$$
\array{
&& Y
\\
& \swarrow && \searrow
\\
X_0 &&\downarrow^{\mathrlap{\simeq}}&& X_1
\\
& \nwarrow && \nearrow
\\
&& Y'
}
$$
in [[FinSet]] with the vertical morphism an [[isomorphism]].
=--
+-- {: .num_prop}
###### Observation
The [[homotopy category of an (infinity,1)-category|homotopy category]] of $2Comm$ is the category $Comm$ that is the [[Lawvere theory]] of commutative [[monoid]]s.
=--
+-- {: .proof}
###### Proof
The Lawvere theory of commutative monoids has as objects the free commutative monoids $F[k]$ on $k \in \mathbb{N}$ generators, and as morphisms monoid homomorphisms.
By the [[free functor|free property]], morphisms
$$
f : F[k] \to F[l]
$$
are in natural bijection to $k$-tuples of elements of $F[l]$. Such elements in turn are sums $a_1 + a_1 + \cdots + a_1 + a_2 + a_2 + \cdots + a_2 + a_3 + \cdots$ of copies of the $l$ generators, hence are in bijection to sequences of natural numbers $(n_{1}, \cdots, n_l)$. Hence morphisms $f : F[k] \to F[l]$ are in bijection to $k \times l$-[[matrices]] with entries in the [[natural number]]s.
One checks that under this identification composition of morphisms corresponds to matrix multiplication.
=--
+-- {: .num_remark}
###### Remark
For instance the spans
$$
\{1,2\} \stackrel{id}{\leftarrow} \{1,2\} \to \{1\}
$$
and
$$
\{1,2\} \stackrel{\simeq}{\leftarrow} \{2,1\} \to \{1\}
$$
describe the operation
$$
(a,b) \mapsto a + b
$$
and the operation
$$
(a,b) \mapsto b + a
\,,
$$
respectively. Clearly, in $Comm$ both these operations are identified. In $2Comm$ however they the are only equivalent
$$
\array{
&& \{1,2\}
\\
& {}^{\mathllap{id}}\swarrow && \searrow
\\
\{1,2\} &&\downarrow^{\mathrlap{\simeq}}&& \{1\}
\\
& {}_{\mathllap{\simeq}}\nwarrow && \nearrow
\\
&& \{2,1\}
}
\,.
$$
=--
## Properties
+-- {: .num_lemma}
###### Observation
Let $Comm$ be the ordinary [[Lawvere theory]] of commutative monoids. There is a forgetful 2-functor
$$
2Comm \to Comm
\,.
$$
This exhibits $2Comm$ as being like $Comm$ but with some additional auto-2-morphisms of the morphism of $Comm$.
=--
This is discussed in ([Cranch, beginning of section 5.2](#Cranch)).
+-- {: .num_prop}
###### Proposition
The $(\infty,1)$-category $2Comm$ has finite [[product]]s. The products of objects $A, B$ in $2Comm$ is their [[coproduct]] $A \coprod B$ in [[FinSet]].
=--
This appears as ([Cranch, prop. 4.7](#Cranch)).
+-- {: .num_prop}
###### Proposition
An [[(β,1)-category]] with [[(β,1)-limit|(β,1)-product]] is naturally an algebra over the $(2,1)$-theory $2Comm$.
=--
This appears as ([Cranch, theorem 4.26](#Cranch)).
+-- {: .num_theorem}
###### Theorem
An algebra over the $(2,1)$-theory $2Comm$ in [[(β,1)Cat]] is
a [[symmetric monoidal (β,1)-category]].
=--
This appears as ([Cranch, theorem 5.3](#Cranch)).
+-- {: .num_theorem}
###### Theorem
There is a $(2,1)$-algebraic theory $E_\infty$ whose algebras
are at least in parts like [[E-β algebra]]s.
=--
This is ([Cranch](#Cranch)), prop. 6.12, theorem 6.13 and section 8.
## Examples {#Examples}
### Free algebras
The free algebra over $2Comm$ in [[βGrpd]] on a single generator is $2Comm(*, -) : 2Comm \to \infty Grpd$. Its underlying [[β-groupoid]] is therefore
$$
2Comm(*,*) = Core(FinSet)
\,,
$$
the [[core]] groupoid of the category [[FinSet]]. This is equivalent to
$$
\cdots \simeq \coprod_{n \in \mathbb{N}} \mathbf{B} \Sigma_n
\,,
$$
where $\Sigma_n$ is the [[symmetric group]] on $n$ elements and $\mathbf{B}\Sigma_n$ its one-object [[delooping]] groupoid.
Notice that this is indeed the free [[E-β-algebra]], on the nose so if we use the [[Barratt-Eccles operad]] $P$ as our model for the [[E-β-operad]]: that has $P_n = \mathbf{E} \Sigma_n$. The free [[algebra over an operad]] is given by $\coprod_{n \in \mathbb{N}} P_n/\Sigma_n$, which here is $\cdots = \coprod_{n \in \mathbb{N}} \mathbf{E}\Sigma_n/\Sigma_n = \coprod_n \mathbf{B} \Sigma_n$.
## References
* [[James Cranch]], _Algebraic Theories and $(\infty,1)$-Categories_ ([arXiv](http://arxiv.org/abs/1011.3243))
{#Cranch}
[[!redirects (2,1)-algebraic theory of E-β algebras]]
[[!redirects (2,1)-algebraic theory of E-β-algebras]] |
(2,1)-category | https://ncatlab.org/nlab/source/%282%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By the general rules of $(n,r)$-[[(n,r)-category|categories]], a __$(2,1)$-category__ is an $\infty$-[[infinity-category|category]] such that
* any $j$-morphism is an [[equivalence]], for $j \gt 1$;
* any two parallel $j$-morphisms are equivalent, for $j \gt 2$.
You can start from any notion of $\infty$-category, strict or weak; up to [[equivalence of categories|equivalence]], the result can always be understood as a [[locally groupoidal 2-category|locally groupoidal]] $2$-[[2-category|category]].
## Models
So, a (2,1)-category is in particular modeled by
* a [[2-category]] in which all [[2-morphisms]] are invertible;
* an [[(β,1)-category]] that is 2-[[truncated]].
## Properties
The [[oidification]] of a [[monoidal groupoid]] is a (2,1)-category.
## Related concepts
* [[hom-groupoid]]
* [[strict (2,1)-category]]
* [[equivalence of (2,1)-categories]]
* [[concrete (2,1)-category]]
* [[monoidal (2,1)-category]], [[symmetric monoidal (2,1)-category]]
* [[monoidal groupoid]]
## References
The special case of [[strict (2,1)-categories]], motivated from the [[homotopy 2-category]] of [[topological spaces]]:
* [[Peter H. H. Fantham]], [[Eric J. Moore]], *Groupoid Enriched Categories and Homotopy Theory*, Canadian Journal of Mathematics **35** 3 (1983) 385-416 ([doi:10.4153/CJM-1983-022-8](https://doi.org/10.4153/CJM-1983-022-8))
[[!include oidification - table]]
[[!redirects (2,1)-categories]]
|
(2,1)-dimensional Euclidean field theories and tmf | https://ncatlab.org/nlab/source/%282%2C1%29-dimensional+Euclidean+field+theories+and+tmf | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]
See there for background and context.
This entry here indicates how 2-dimensional [[FQFT]]s may be related to [[tmf]].
=--
> **raw material**: this are notes taken more or less verbatim in a seminar -- needs polishing
Previous:
* [[Axiomatic field theories and their motivation from topology]].
* [[(1,1)-dimensional Euclidean field theories and K-theory]]
recall the big diagram from the end of the [[(1,1)-dimensional Euclidean field theories and K-theory|previous entry]].
The **goal** now is to replace everywhere [[topological K-theory]] by [[tmf]].
previously we had assumed that $X$ has [[spin structure]]. Now we assume [[String structure]].
So we are looking for a diagram of the form
$$
\array{
1
&&
(2|1)EFT^0(X)/\sim
&& \stackrel{\simeq conjectural}{\leftarrow}&&
tmf^0(X)
&& \ni 1
\\
\downarrow && \downarrow^{quantization}
&&&&
\downarrow^{\int_X}
&&
\downarrow
\\
\sigma_{(2|1)(X)}&& (2|1)EFT^{-n}(X)/\sim
&&\stackrel{\simeq conjectural }{\leftarrow}&&
tmf^{-n}(pt)
&&
\\
&\searrow & \searrow &&& \swarrow& \swarrow
\\
&&&& mf^{-n}
\\
&&&&
index^{S^1}(D_{L X})
= W(X)
}
$$
the vertical maps here are due to various theorems by various people -- except for the "physical quantization" on the left, that is used in physics but hasn't been formalized
the **horizontal maps are the conjecture we are after** in the Stolz-Teichner program: The top horizontal map will involve making the notion of $(2|1)$EFT _local_ by refining it to an _extended_ [[FQFT]]s. This will not be considered here.
we will explain the following items
* the [[ring]] $mf^\bullet$ of [[integral modular form]]s
$$
mf^\bullet \simeq \mathbb{Z}[c_4, c_6, \Delta, \Delta^{-1}]/(c_4^{3}- c_6^{2} - 1728 \Delta)
$$
one calls $w = -\frac{n}{2}$ the _weight_ . We have degree of $\Delta$ is $deg(\Delta) = -24$, hence $w(\Delta) = 12$.
* $W(X)$ is the [[Witten genus]]
$$
W(X) = \sum_{k \in \mathbb{Z}} a_k \cdot q^k
\,,
a_k \in \mathbb{Z}
$$
where $a_k = index(D_X \otimes E_k)$ where $E_k$ is some explicit vector bundle over $X$.
## modular forms ##
**definition** An **(integral) [[modular form]]** of weight $w$ is a [[holomorphic function]] on the [[upper half plane]]
$$
f : (\mathbb{R}^2)_+ \hookrightarrow \mathbb{C}
$$
(complex numbers with strictly positive imaginary part)
such that
1. if $A = \left( \array{a & b \\ c& d}\right) \in SL_2(\mathbb{Z})$ acting by $A : \tau \mapsto = \frac{a \tau + b }{c \tau + d}$ we have
$$
f(A(\tau)) = (c \tau + d)^w f(\tau)
$$
**note** take $A = \left( \array{1 & 1 \\ 0& 1}\right)$ then we get that $f(\tau + 1) = f(\tau)$
1. $f$ has at worst a pole at $\{0\}$ (for _weak_ modular forms this condition is relaxed)
it follows that $f = f(q)$ with $q = e^{2 \pi i \tau}$ is a meromorphic funtion on the open disk.
1. **integrality** $\tilde f(q) = \sum_{k = -N}^\infty a_k \cdot q^k$ then $a_k \in \mathbb{Z}$
by this definition, modular forms are not really functions on the upper half plane, but functions on a [[moduli space]] of [[torus|tori]]. See the following definition:
if the weight vanishes, we say that modular form is a **[[modular function]]** .
**definition (2|1)-dim [[partition function]]**
Let $E$ be an EFT
$$
(2|1)EFT^0 \stackrel{S}{\to}
2 EFT \ne E
$$
$$
E \mapsto E_{red}
$$
then the [[partition function]] is the map $Z_E : \mathbb{C} \to \mathbb{R}$
$$
Z_E : \tau \mapsto E_{red}(T_\tau)
$$
where
$$
T_\tau := \mathbb{C}/{\mathbb{Z} \times \mathbb{Z} \cdot \tau}
$$
is thee standard torus of modulus $\tau$.
then the central **theorem** that we are after here is
**therorem (Stolz-Teichner)** (after a suggestion by [[Edward Witten]])
There is a precise definition of $(2|1)$-EFTs $E$ such that the [[partition function]] $Z_E$ is an integral [[modular function]]
(so this is really four theorems: the function is holomorphic, integral, etc.)
moreover, every [[integral modular function]] arises in this way.
## Related concepts
A concrete relation between [[2d SCFT]] and [[tmf]] is the lift of the [[Witten genus]] to the [[string orientation of tmf]]. See there fore more.
## References
* [[Stephan Stolz]], [[Peter Teichner]], _[[What is an elliptic object?]]_ in _Topology, geometry and quantum field theory_ , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343. ([pdf](http://web.me.com/teichner/Math/Reading_files/Elliptic-Objects.pdf))
* Pokman Cheung, _Supersymmetric field theories and cohomology_ ([arXiv:0811.2267](http://arxiv.org/abs/0811.2267))
* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric Euclidean field theories and generalized cohomology_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_
A hint supporting the conjectured relation of [[2d SCFT]] to [[tmf]], vaguely in line with the lift of the [[Witten genus]] to the [[string orientation of tmf]]:
* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], _Holomorphic SCFTs with small index_ ([arXiv:1811.00589](https://arxiv.org/abs/1811.00589))
|
(2,1)-functor | https://ncatlab.org/nlab/source/%282%2C1%29-functor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of _$(2,1)$-functors_ is that of the natural kind of [[morphisms]] between [[(2,1)-categories]].
If [[(2,1)-categories]] are regarded as special cases of [[2-categories]], then $(2,1)$-functors are equivalently the [[2-functors]] between [[(2,1)-categories]].
If [[(2,1)-categories]] are regarded as special cases of [[(β,1)-categories]], then $(2,1)$-functors are equivalently the [[(β,1)-functors]] between [[(2,1)-categories]].
## Examples
* The construction of [[groupoid convolution algebras]] constitutes a $(2,1)$-functor from [[differentiable stacks]] with proper maps between them and the [[opposite 2-category|opposite]] $(2,1)$-category of [[C*-algebras]] with [[Hilbert bimodules]] between them. For details see [there](#category+algebra#GroupoidConvolutionIs2Functor).
[[!redirects (2,1)-functors]]
|
(2,1)-presheaf | https://ncatlab.org/nlab/source/%282%2C1%29-presheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **(2,1)-presheaf** is a [[presheaf]] with values in the [[(2,1)-category]] [[Grpd]]. A 2-[[truncated]] [[(β,1)-presheaf]].
Sometimes this is also called a **prestack**. Other times a prestack is more specifically taken to be a [[separated (infinity,1)-presheaf|separated (2,1)-presheaf]]: a $(2,1)$-presheaf such that the functors into its [[descent]] objects are [[full and faithful functor]]s.
The [[β-stackification]] of a $(2,1)$-presheaf is a certain [[2-sheaf]] or [[stack]].
## Related concepts
* [[(0,1)-presheaf]]
* [[presheaf]]
* **(2,1)-presheaf**
[[presheaf of groupoids]]
[[(2,1)-sheaf]], [[2-sheaf]], [[indexed category]], [[stack]]
* [[(β,1)-presheaf]]
* [[(β,n)-presheaf]]
[[!redirects (2,1)-presheaves]]
[[!redirects prestack]]
[[!redirects prestacks]]
[[!redirects pre-stack]]
[[!redirects pre-stacks]]
[[!redirects groupoid-valued presheaf]]
[[!redirects groupoid-valued presheaves]] |
(2,1)-sheaf | https://ncatlab.org/nlab/source/%282%2C1%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A $(2,1)$-sheaf is a [[sheaf]] with values in [[groupoid]]s. This is traditionally called a [[stack]].
## Definition
Let $C$ be a [[(2,1)-site]]. Write [[Grpd]] for the [[(2,1)-category]] of [[groupoid]]s, [[functor]]s and [[natural isomorphism]]s.
A **$(2,1)$-sheaf** on $C$ is equivalently
* a [[2-functor]] $C^{op} \to Grpd$ that satisfies [[descent]];
* a [[2-sheaf]] with values in [[groupoid]]s;
* a [[1-truncated]] [[(β,1)-sheaf]] on $C$.
## The $(2,1)$-category of $(2,1)$-sheaves
The [[(2,1)-category]] of a $(2,1)$-sheaves on a [[(2,1)-site]] forms a [[(2,1)-topos]].
There are [[model category]] [[presentable (infinity,1)-category|presentations]] of this $(2,1)$-topos. See [[model structure for (2,1)-sheaves]].
## Related concepts
* [[sheaf]]
* [[2-sheaf]] / **$(2,1)$-sheaf** / [[stack]]
* [[(β,1)-sheaf]] / [[β-stack]]
* [[(β,n)-sheaf]]
* [[descent]]
* [[cover]]
* [[cohomological descent]]
* [[descent morphism]]
* [[monadic descent]],
* [[Sweedler coring]]
* [[higher monadic descent]]
* [[descent in noncommutative algebraic geometry]]
[[!include homotopy n-types - table]]
[[!redirects (2,1)-sheaves]]
[[!redirects (2,1)-category of (2,1)-sheaves]] |
(2,1)-sheafification | https://ncatlab.org/nlab/source/%282%2C1%29-sheafification | [[!redirects (2,1)-sheafififcation]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The higher analogue of [[sheafification]].
A [[stack]]/[[2-sheaf]] [[2-topos]] on a [[site]]/[[2-site]] $C$ is
$$
St(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSt(C)
\,.
$$
The [[left adjoint]] $L$ to the inclusion of [[stack]]s into all [[2-functor]]s on $C^{op}$ is _stackification_ .
## Related concepts
* [[sheafification]]
* **$(2,1)$-sheafification**
* [[(β,1)-sheafification]]
[[!redirects stackification]]
[[!redirects stackifications]]
|
(2,1)-site | https://ncatlab.org/nlab/source/%282%2C1%29-site |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Definition
A _$(2,1)$-site_ is an [[(β,1)-site]]. whose underlying [[(β,1)-category]] is a [[(2,1)-category]]. Equivalently, it is a [[2-site]] whose underlying [[2-category]] is a [[(2,1)-category]].
## Properties
The [[(2,1)-category]] of [[(2,1)-sheaves]] on a (2,1)-site is a Grothendieck-[[(2,1)-topos]].
The [[(β,1)-category of (β,1)-sheaves]] on a $(2,1)$-site is an [[n-localic (β,1)-topos|2-localic (β,1)-topos]].
## Related concepts
* [[site]]
* [[2-site]], **(2,1)-site**
* [[(2,1)-sheaf]]
* [[(β,1)-site]]
* [[model site]], [[simplicial site]]
[[!redirects (2,1)-site]]
[[!redirects (2,1)-sites]]
|
(2,1)-topos of (2,1)-sheaves | https://ncatlab.org/nlab/source/%282%2C1%29-topos+of+%282%2C1%29-sheaves |
## Idea
A [[(2,1)-topos]] of [[(2,1)-sheaves]]
## Related concepts
* [[sheaf topos]]
* [[(infinity,1)-category of (infinity,1)-sheaves]]
|
(2|1)-dimensional Euclidean field theory | https://ncatlab.org/nlab/source/%282%7C1%29-dimensional+Euclidean+field+theory | [[!redirects (2,1)-dimensional Euclidean field theory]]
[[!redirects (2,1)-dimensional Euclidean field theory]]
[[!redirects (2,1)-dimensional Euclidean field theory]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Supergeometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]
See there for background and context.
This entry here is about the definition of $(2|1)$-dimensional [[super-cobordism]] categories where cobordisms are [[Euclidean supermanifold]]s, and about $the (2|1)$-dimensional [[FQFT]]s given by functors on these.
=--
Previous:
* [[Axiomatic field theories and their motivation from topology]]
* [[(1,1)-dimensional Euclidean field theories and K-theory]]
* [[(2,1)-dimensional Euclidean field theories and tmf]]
* [[bordism categories following Stolz-Teichner]]
#Contents#
* tic
{:toc}
#Idea#
Previously we had defined smooth categories of [[Riemannian cobordism]]s. Now we pass from [[Riemannian manifold]]s to [[Euclidean supermanifold]]s and define the corresponding smooth [[cobordism category]]. Then we define $(d|\delta)$-dimensional Euclidean field theories to be smooth representations of these categories.
As described at [[(2,1)-dimensional Euclidean field theories and tmf]], the idea is that $(2|1)$-dimensional Euclidean field theories are a geometric model for [[tmf]] [[cohomology theory]]. While there is no complete proof of this so far, in the next and final session
* [[modular forms from partition functions]]
it will be shows that the claim is true at least for the [[cohomology ring]] over the point: the [[partition function]] of a $(2|1)$-dimensional EFT is a modular form. Hence $(2|1)$-dimensional EFTs do yield the correct [[cohomology ring]] of [[tmf]] over the point.
#Details#
Let [[SDiff]] be the [[category]] of [[supermanifold]]s.
We will define a [[stack]]/[[fibered category]] on $SDiff$ called $E Bord_{2|1}$ whose morphisms are smooth families of (2|1)-dimensional [[super-cobordism]]s, and a [[stack]]/[[fibered category]] $sTV^{fam}$ of topological super vector bundles.
So recall
* [[supergeometry - contents|supergeometry]].
**question**: What is the right notion of Riemannian or Euclidean structure on [[super-cobordism]]s?
**strategy**: From the [[path integral]] perspective we need some structure on $\Sigma$ such that the "space" of maps $Maps(\Sigma,X)$ naturally carries some measure that allows to perform a [[path integral]].
This perspective suggests certain generalizations of the notion of [[Riemannian manifold]] to [[supermanifold]]s which may be a little different than what one might have thought of naively.
We want to define [[Euclidean supermanifold]]s as a generalization of [[Riemannian manifold]] with _flat_ Riemannian metric.
notice that there is a canonical bijection between
* flat [[Riemannian metric]]s on a $d$-dimensional [[manifold]] $X$
* a maximal [[atlas]] on $X$ consisting of charts such that all transition functions belong the the **Euclidean group** or **Galileo group**
$$
Eucl(\mathbb{R}^d)
:=
\mathbb{R}^d \rtimes O(\mathbb{R}^d)
$$
(rigid translations and rotations)
In analogy to that we define:
Similarly a **Euclidean structure** on a $(d|\delta)$-dimensional [[supermanifold]] is defined using the Euclidean [[super Lie group]] given by the [[semidirect product]]
$$
Eucl(\mathbb{R}^{d|\delta})
:=
\mathbb{R}^{d|\delta}
\rtimes
Spin(\mathbb{R}^d)
$$
where the [[Spin]] group acts on the translations in $\mathbb{R}^{d|\delta}$ in a way to be specified.
first recall the notion of
* [[complex supermanifold]]
**goal** replace the standard Euclidean group $(\mathbb{R}^d, Eucl(\mathbb{R}^d))$ by the [[super Euclidean group]]
$(X,G)$ where $X$ is a suitable [[supermanifold]]
and $G$ a suitable [[super Lie group]].
This leads to the notion of
* [[Euclidean supermanifold]].
The morphisms of the category $E Bord_{(2|1)}$ will be [[cobordism]]s that are [[Euclidean supermanifold]]s.
**goal** define the [[fibered category]]
$$
\array{
E Bord_{d|\delta}^{sfam}
\\
\downarrow
\\
cSDiff
}
$$
where $cSDiff$ is the category of [[complex supermanifold]]s.
The objects of this fibered category are
$$
\array{
Y^{\pm} &\to& Y &\leftarrow& Y^c
\\
& \searrow & \downarrow & \swarrow
\\
&& S
}
$$
where $Y \to S$ is a family of [[Euclidean supermanifold]]s of dimension $(d|\delta)$.
For the non-super, non-family version of **Euclidean bordism** we require that the core $Y^c$ is totally geodesic in $Y$.
now for the superversion we require that there exist charts (in the open atlas) of $Y \to S$ covering all of $Y^c$ such that
$$
\array{
&& S
\\
& \nearrow && \nwarrow
\\
Y \supset_{open} U &&\stackrel{\phi}{\to}&&
V \subset S \times \mathbb{R}^{d|\delta}_{cs}
\\
\downarrow^{\supset} &&&& \downarrow^{\supset}
\\
Y^c \supset U \cap Y^c
&&\stackrel{\simeq}{\to}&&
V \cap S \times \mathbb{R}^{d-1|\delta}
\subset S \times \mathbb{R}^{d-1|\delta}_{cs}
}
$$
next, a **Euclidean superbordism** from $Y_0 \to S$ to $Y_1 \to S$ is a diagram
$$
\array{
Y_1
&\stackrel{i_1}{\to}&
\Sigma
&\stackrel{i_0}{\leftarrow}& Y_1
\\
& \searrow & \downarrow & \swarrow
\\
&& S
}
$$
where $i_0, i_1$ are morphisms (of families of $(X,G)$-spaces) satisfying the (+)-condition and the (c)-condition described at [[bordism categories following Stolz-Teichner]].
Now a morphism in $E Bord^{sfam}_{d|\delta}$ from $Y_0 \to S_0$ to $Y_1 \to S_1$ is a bordism fitting into a diagram
$$
\array{
\Sigma &\stackrel{i_1}{\leftarrow}&
f^* Y_1 &\to& Y_1
\\
\uparrow^{i_0} &\searrow& \downarrow && \downarrow
\\
Y_1 &\to & S_0 &\stackrel{f}{\to}& S1
}
$$
and we identify bordisms $\Sigma, \Sigma'$ if they are isometric -- namely isomorphic in the category of [[Euclidean supermanifold]]s -- "rel boundary".
**definition** A **$(d|\delta)$-dimensional Euclidean field theory** is a [[symmetric monoidal functor]]
$$
E \in Fun_{csDiff}^\otimes(E Bord_{(d|\delta)}^{sfam},
TV^{sfam})
$$
of [[symmetric monoidal category|symmetric monoidal]] [[fibered category|fibered cateories]] (i. [[symmetric monoidal functor]] as well as [[cartesian functor]] ) over the category $cSDiff$ of [[complex supermanifold]]s.
**Definition** (roughly) $TV^{sfam}$ is the category of families of topological vector spaces parameterized by [[complex supermanifold]]s.
Recall that the ordinary category $TV$ is the category of complete [[Hausdorff space|Hausdorff]], locally convex [[topological vector space]].
define the [[projective tensor product]] of two such $V, W \in TV$. This is a certain completion of the algebraic tensor product $V \otimes_{alg} W$ with respect to the projective topology on $V \otimes_{alg} W$.
This will be the coarsest [[topology]] (the one with the least open sets) making the following maps $f'$
$$
\array{
V \otimes_{alg} W &\to& Z
\\
& \nearrow_{f'}
\\
V \times W
}
$$
continuous.
**Remark**
$$
\array{
C^\infty(M \times N)
&\leftarrow&
C^\infty(M) \otimes_{alg} C^\infty(N)
\\
& {}_{\simeq}\nwarrow & \downarrow^{\subset}
\\
&& C^\infty(M) \otimes C^\infty(N)
}
$$
**Definition** the objects of $TV^{sfam}$ are pairs $(S,V)$ for $S$ a [[supermanifold]] and $V$ is a [[sheaf]] of locally complex $\mathbb{Z}_2$-graded [[topological vector space]] with the structure of a sheaf of modules of the [[structure sheaf]] $O_S$.
**goal** define the [[partition function]] of of a $(2|1)$-dimensional Euclidean field theory.
**definition** Let $E$ be an EFT as above.
We may think of $\mathbb{R}_+ \times h$ (positive axis times upper half plane) as moduli space of Euclidean tori, where for $(\ell, \tau) \in \mathbb{R}_+ \times h$ we get a torus (regarded as a [[cobordism]]) denoted $T_{\ell,\tau}$. This is the torus given by the lattice spanned by $(1,0)$ and $\ell Re(\tau) + Im(\tau)$ in the upper half plane. Then for the ordinary EFT we would define
$$
Z_E : \mathbb{R}_+ \times h \to \mathbb {C}
$$
$$
(\ell,\tau) \mapsto E(T_{\ell,\tau})
$$
For the superversion we put
$$
Z_{E} := Z_{E_{red}}
$$
where
$$
\array{
&& E Bord_{2|1}^{sfam}
\\
& {}^{\rho}\nearrow
&&
\searrow^{E}
\\
E Bord_{2, Spin}^{fam}
&\stackrel{E_{red}}{\to}&
TV^{fam}
&
\hookrightarrow
&
TV^{sfam}
}
$$
# Examples #
## explicit description of $E Bord_{1}^{fam}$ ##
See [[bordism categories following Stolz-Teichner]].
The category $E Bord_1^{fam}$ is generated from
* the _family of right elbows__
$$
\array{
1-E Bord_{\mathbb{R}_+}^{fam}(\emptyset, pt \coprod pt)
& \ni& R & := \mathbb{R}_+ \times \mathbb{R}
\\
&& \downarrow
\\
&& S := \mathbb{R}_+
}
$$
* the point-family of the left elbox
$$
\array{
L_0
\\
\downarrow
\\
S := pt
}
$$
* the family of intervals in $E Bord^{fam}_{\mathbb{R}^+}(pt,pt)$
$$
\array{
I
\\
\downarrow
\\
\mathbb{R}_{\geq 0}
}
$$
Because:
$
E \in Fun^{\otimes}_{Diff}(E Bord^{fam}, TV^{fam})
$
is determined by
$$
E(pt) =: V \in TV
$$
$$
E(L_0) =: \lambda : V \otimes V \to \mathbb{R}
$$
$$
E(R) =: \rho \in TV_{\mathbb{R}^+}(\mathbb{R}, V \otimes V) \simeq C^\infty(\mathbb{R}_+, V \otimes V)
$$
$$
E(I) =: e^{-t H} \in C^\infty(\mathbb{R}_{\geq}, End(V))
$$
forms a _smooth_ semigroup under composition generated by
$$
H \in End(V)
$$
(the [[Hamiltonian operator]])
$$
\array{
V \otimes V &&\stackrel{\lambda}{\hookrightarrow}&&
End(V)
\\
& {}_{\rho}\nwarrow && \nearrow_{e^{- t H}}
\\
&& \mathbb{R}_+
}
$$
so due to smoothness the data collapses to the infinitesimal data
$$
(V, \lambda, H)
$$
**example -- ordinary quantum mechanics** Let $M$ be a [[Riemannian manifold]]. Then set
* $H:= \Delta$ the corresponding [[Laplace operator]];
* $V := C^\infty(M) \subset L^2(M)$;
* $\lambda$ is the restriction of the $L^2(M)$ inner product to $V$
where $e^{-t H}$ is "trace class" in the non-standard sense described above in that it makes the above diagram commute.
So everything as known from standard [[quantum mechanics]] textbooks, except that we don't use the full [[Hilbert space]] of states, but just the [[Frechet space]] of smooth functions.
## explicit description of $E Bord_{2}$ ##
The category $E Bord_{2}_{oriented}^{fam}$ has the following generators:
objects are generated from
* the **circle** $K_\ell := \mathbb{R}^2/\mathbb{Z}\cdot \ell$ of length $\ell \gt 0$ (with collars!! that's why it looks like a cylinder of circumference $\ell$)
notice that we may think of $\ell $ as parameteriing translation by $\ell$ in $\mathbb{R}^2 \rtimes SO(2) = Eucl_{or}(\mathbb{R}^2)$
and the circle with $(+)$/$(-)$-collars reversed
morphisms are generated from
* **cylinders** $C_{\ell,\tau}$ which as a manifold is $\simeq \mathbb{R}^2/\mathbb{Z}\cdot \ell$ where $\tau$ parameterizes the embedding of the outgoing circle: the incoming circle is embedded in the canonical way (the identity map on the cylinder, really), while the outgoing circle is embedded by translating the cylinder by $\ell \cdot Re(\tau)$ upwards and rotated by $\ell \cdots Im(\tau)$.
* **right elbows** which are the same as the cylinder, except that now the second circle is embedded after reflection so that it becomes an ingoing circle.
* the **thin left elbow** $L_\ell$, similar to the above, with arbitrary $\ell$ but $\tau = 0$
* the **torus** $T_\tau$ obtained from the cylinder by gluing incoming and outgoing
**notice** the **pair of pants** is not a morphism in the category at all! since, recall, we require all bordisms to be _flat_ and all boundaries to be _geodesics_ . There is no way to put such a flat metric on the trinion.
satisfying the relations
$$
L_\ell \circ R_{\ell, \tau}
=
T_{\ell, \tau}
$$
as for the non-family version, but now also with the new relations
$$
T_{\ell', \tau'} = T_{\ell, \tau}
$$
whenever $\ell' = \ell \cdot|c \tau + d|$ and $\tau' = \frac{a \tau + b }{c \tau + d}$
for $\left(\array{a & b \\ c & d }\right) \in SL_2(\mathbb{Z})$.
Notice that $SL_2(\mathbb{Z})$ is generated by
* translation $(\ell, \tau) \mapsto (\ell, \tau + 1)$
* S-matrix $(\ell, \tau) \mapsto (\ell \cdot |\tau|, - \frac{1}{\tau})$
and now there is **one more relation**
$$
T_{\ell, \tau}
=
T_{\ell |\tau|, - \frac{1}{\tau}}
$$
as usual write $q := e^{2 \pi i \tau}$ which is on the pointed unit disk since $\tau$ is half plane since $\tau$
## explicit description of $E Bord_{2}^{fam}$ ##
thwe category $E Bord_{2, oriented}^{fam}$ is generated from
objects:
* $\array{K \\ \downarrow \\ S = \mathbb{R}_+}$
* morphisms
$$
\array{
L \\ \downarrow \\ \mathbb{R}_+
}
$$
$$
\array{
R \\ \downarrow \\ \mathbb{R}_+ \times h/\mathbb{Z}
}
$$
$$
\array{
C \\ \downarrow \\ \mathbb{R}_+ \times (h \cup \mathbb{R})/\mathbb{Z}
}
$$
subject to the relations
... as before (homework 3, problem 4).. and the furhter one:
for
$$
\array{
T && \alpha^* T
\\
\downarrow && \downarrow
\\
\mathbb{R}_+ \times h
&\stackrel{\alpha}{\leftarrow}&
\mathbb{R}_+ \times h
\\
\\
(\ell\cdot |\tau|, -\frac{1}{\tau})
&\stackrel{}{\lt\leftarrow}&
(\ell, \tau)
}
$$
the relation is
$$
\alpha^* T = T
\,.
$$
## References
* [[Stephan Stolz]], [[Peter Teichner]], *[[What is an elliptic object?]]* in: _Topology, geometry and quantum field theory_, London Math. Soc. LNS **308**, Cambridge Univ. Press (2004) 247-343 [[pdf](https://math.berkeley.edu/~teichner/Papers/Oxford.pdf), [[Stolz-Teichner_EllipticObject.pdf:file]]]
* {#Stolz} [[Stefan Stolz]] (notes by Arlo Caine), _Supersymmetric Euclidean field theories and generalized cohomology_ Lecture notes (2009) ([pdf](http://www.nd.edu/~jcaine1/pdf/Lectures_complete.pdf))
* [[Stefan Stolz]], [[Peter Teichner]], _Supersymmetric field theories and generalized cohomology_ , in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), *[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]*, Symposia in Pure Mathematics (2011) [[arXiv:1108.0189](http://arxiv.org/abs/1108.0189)]
* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?* [[arXiv:2303.09138](https://arxiv.org/abs/2303.09138)]
* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?*, 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#BerwickEvansMar23)]
[[!redirects (2,1)-dimensional Euclidean field theories]] |
(bo, ff) factorization system | https://ncatlab.org/nlab/source/%28bo%2C+ff%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
There is an [[orthogonal factorization system]] on the category [[StrCat]], whose left class is the class of [[bo functor|bijective-on-objects functors]], or "bo functors" and whose right class is the class of [[full and faithful functor|full and faithful functors]], or "ff functors".
This means that each functor $f$ decomposes as a composition of the form $j e$, where $e$ is bijective on objects and $j$ fully faithful; and if
$$\array{
A &\overset{u}{\longrightarrow}& C
\\
\mathllap{{}^{e}}\big\downarrow &&\big\downarrow \mathrlap{{}^{j}}
\\
B &\underset{v}\longrightarrow& D
}$$
is a commutative diagram with $e$ bijective on objects and $j$ fully faithful, then there is a unique functor $h \colon B\to C$ such that $h e = u$ and $j h = v$. The object through which $f$ factors is called the [[full image]] of $f$.
In fact, this can be generalized to a square commuting up to invertible [[natural transformation]], in which case one still concludes that $h e = u$ but that $j h \cong v$, with the isomorphism composing with $e$ to give the original isomorphism. This means that this is an [[enhanced factorization system]].
## Properties
This factorization system can be constructed using [[generalized kernels]].
For [[essentially surjective functors]], one can relax both the commuting and the uniqueness to obtain a [[factorization system in a 2-category]].
[[!redirects (bo,ff) factorization system]]
[[!redirects bo-ff factorization system]]
[[!redirects (bo,ff) factorization]]
[[!redirects bo-ff factorization]]
[[!redirects (bo,ff) factorisation system]]
[[!redirects bo-ff factorisation system]]
[[!redirects (bo,ff) factorisation]]
[[!redirects bo-ff factorisation]]
|
(classical) axiom of multiple choice | https://ncatlab.org/nlab/source/%28classical%29+axiom+of+multiple+choice |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# The axiom of multiple choice
* table of contents
{: toc}
This article is about a [[classical mathematics|classical]] [[set theory]] [[axiom]]. Some literature instead uses this name for an unrelated weakening of [[axiom of choice|AC]]. For that notion, see _[[axiom of multiple choice]]_.
## Idea
The **axiom of multiple choice** (AMC) weakens the [[axiom of choice]] by allowing [[choice functions]] to choose [[finite sets]], rather than particular [[elements]].
##Statement
Recall that one common statement of the [[axiom of choice]] is:
:For every [[set]] $S$ of [[inhabited set|non-empty sets]], there is a [[function]] $f$ defined on $S$ such that $\forall X\in S$, $f(X)\in X$.
Such an $f$ is called a __choice function__ for $S$.
The axiom of multiple choice weakens the axiom of choice by allowing choice functions to pick out finite [[subsets]], rather than finite sets. It says:
:For every set $S$ of non-empty sets, there is a function $f$ defined on $S$ such that $\forall X\in S$, $f(X)\subseteq X$ and $f(X)$ is finite and non-empty.
## Relationships to other axioms
The axiom of multiple choice is [[logical equivalence|equivalent]] to the axiom of choice modulo [[ZF]] set theory. However, it is strictly weaker in [[ZFA]] and other similar set theories. AMC holds in any [[permutation model]] with finite supports where each atom has a finite orbit. For a detailed proof, see [Jech's "The Axiom of Choice"](https://www.gwern.net/docs/math/1973-jech-theaxiomofchoice.pdf), chapter 9.
The [[axiom of multiple choice|constructive axiom by the same name]] is not historically related, and the two axioms are independent. Any permutation model will satisfy [[SVC]], which [Rathjen](#Rathjen) proves implies the constructive axiom, but this AMC can fail in a permutation model.
## References
* Jech, _The Axiom of Choice_ (1973), ISBN : 0444104844 (New York)
* A. LΓ©vy. Axioms of multiple choice. Fundamenta mathematicae, vol. 50 no. 5 (1962), pp. 475β483
The constructive axiom by the same name is discussed in:
* {#Rathjen} Rathjen, "Choice principles in constructive and classical set theories"
category: foundational axiom
|
(co)isotropic subspaces - table | https://ncatlab.org/nlab/source/%28co%29isotropic+subspaces+-+table |
| type of [[subspace]] $W$ of [[inner product space]] | condition on [[orthogonal]] space $W^\perp$ | |
|--|--|--|
| [[isotropic subspace]] | $W \subset W^\perp $ | |
| [[coisotropic subspace]] | $W^\perp \subset W$ | |
| [[Lagrangian subspace]] | $W = W^\perp$ | (for [[symplectic form]]) |
| [[symplectic vector space|symplectic space]] | $W \cap W^\perp = \{0\}$ | (for [[symplectic form]])|
|
(d,d) Klein space | https://ncatlab.org/nlab/source/%28d%2Cd%29+Klein+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
> not to be confused with [[Klein geometry]]
#Contents#
* table of contents
{:toc}
## Idea
The [[Euclidean space|flat space]] $\mathbb{R}^{d,d}$ of $2d$ dimensions equipped with a [[pseudo-Riemannian metric]] of $(d,d)$ signature.
## Related concepts
* [[Klein geometry]]
* [[pseudo-Riemannian manifold]]
* [[smooth Lorentzian space]]
## References:
* [[John Barrett]], [[Gary W. Gibbons]], M. J. Perry, [[Christopher N. Pope]], P. J. Ruback, *Kleinian Geometry and the $N=2$ Superstring*, Int. J. Mod. Phys. **A9** (1994) 1457-1494 [[doi:10.1142/S0217751X94000650](https://doi.org/10.1142/S0217751X94000650), [arXiv:hep-th/9302073](https://arxiv.org/abs/hep-th/9302073)] |
(dense,closed)-factorization | https://ncatlab.org/nlab/source/%28dense%2Cclosed%29-factorization |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[topology|point set topology]], every subspace $X$ of a space $A$ has a unique largest subspace in which $X$ is [[dense subspace|dense]], namely simply the closure $\overline{X}$. Using maps, this amounts to say that $X\hookrightarrow A$ factors as $X\hookrightarrow\overline{X}$ followed by $\overline{X}\hookrightarrow A$.
The **(dense,closed)-factorization** generalizes this idea from topology to [[topos theory]]. It can be viewed as a way to associate to every [[subtopos]] $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ a closure $\overline{Sh_{j}(\mathcal{E})}$.
## Statement
A [[geometric embedding]] of [[elementary toposes]]
$$
Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}
$$
factors as
$$
Sh_j(\mathcal{E})
\hookrightarrow
Sh_{c(ext(j))}(\mathcal{E})
\hookrightarrow
\mathcal{E}
$$
where $ext(j)$ (the "exterior" of $j$) denotes the $j$-closure of $\emptyset \rightarrowtail 1$ and
$$
\bar j \coloneqq c(ext(j))
$$
the [[closed subtopos|closed topology]] that corresponds to the [[subterminal object]] $ext(j)$.
Here the first inclusion exhibits a [[dense subtopos]] and the second a [[closed subtopos]].
## Remark
$Sh_{c(ext(j))}(\mathcal{E})$ can be viewed as the 'best approximation' of $Sh_j(\mathcal{E})$ by a closed subtopos and therefore might be called the _closure_ $Cl(Sh_j(\mathcal{E}))$ of $Sh_j(\mathcal{E})$.[^sga4]
[^sga4]: ([SGA4](#SGA4), p.461) uses the term '_l'adhérence_' for it.
Its complement, the [[open subtopos]] $Ext(Sh_j(\mathcal{E}))$ corresponding to the [[subterminal object]] $ext(j)$ deserves in turn to be called the _exterior_ of $Sh_j(\mathcal{E})$.
## The (dominant,closed)-factorization
The (dense,closed)-factorization is a special case for inclusions of a slightly more general factorization which attaches to a general [[geometric morphism]] the closure of its image.
Recall that an inclusion is [[dense subtopos|dense]] precisely if it is a [[dominant geometric morphism]], hence the following is pertinent for the (dense,closed)-factorization as well.
+-- {: .num_prop #dense-closed}
###### Proposition
Let $i:Sh_{c(U)}(\mathcal{E})\hookrightarrow\mathcal{E}$ be dominant and a [[closed subtopos|closed inclusion]] at the same time. Then $i$ is an isomorphism.
=--
**Proof**: Recall that $X\in\mathcal{E}$ are in the [[closed subtopos]] precisely when they satisfy $X\times U\cong U$ with $U$ the [[subterminal object]] associated to $i$. But $i$ is dominant, or what comes to the same for inclusions: dense, hence $\emptyset$ is in $Sh_{c(U)}(\mathcal{E})$ and therefore $\emptyset\times U\cong U$ . From this follows $U\cong\emptyset$, which in turn implies that all $X\in\mathcal{E}$ are in $Sh_{c(U)}(\mathcal{E})$ . $\qed$
+-- {: .num_prop #dominant-closed}
###### Proposition
Let $f:\mathcal{F}\to\mathcal{E}$ be a geometric morphism. Then $f$ factors as a [[dominant geometric morphism]] $d$ followed by a [[closed subtopos|closed inclusion]] $c$.
=--
**Proof**: Let $i\circ d_1$ be the [[(geometric surjection, embedding) factorization system|surjection-inclusion factorization]] of $f$. Since $d_1$ is surjective, it is dominant (cf. [this proposition](dominant+geometric+morphism#dominant_surjection)). Then we use the (dense,closed)-factorization to factor $i$ into $c\circ d_2$. Since both $d_i$ are dominant, so is $d:=d_2\circ d_1$ and $c\circ d$ yields the demanded factorization of $f$. $\qed$
## Related entries
* [[dense]]
* [[dense subtopos]]
* [[dominant geometric morphism]]
* [[(geometric surjection, embedding) factorization system]]
## References
* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (Exposé IV 9.3.4-9.4., pp.456ff)
* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol.I_ , Oxford UP 2002. (around Lemma A 4.5.19, p. 219)
* {#Caramello09} [[Olivia Caramello]], _Lattices of theories_ , [arXiv:0905.0299](http://arxiv.org/abs/0905.0299) (2009). (section 8)
|
(epi, mono) factorization system | https://ncatlab.org/nlab/source/%28epi%2C+mono%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An **epi-mono factorization system** is an [[orthogonal
factorization system]] in which the left class is the class of [[epimorphisms]] and the right class is the class of [[monomorphisms]]. Such a factorization system exists on any (elementary) [[topos]], and indeed on any [[pretopos]]. It provides the factorization through the [[image]] of any morphism.
## Properties
Note that any category which admits an epi-mono factorization system is necessarily [[balanced category|balanced]]. This excludes many commonly occurring categories. More common are ([[strong epimorphism|strong epi]], mono) and (epi, strong mono) factorization systems; the former exists in any [[regular category]] and the latter in any [[quasitopos]], as well as in other categories such as [[Top]].
The epi-mono factorization system in a topos is the special case of the [[n-connected/n-truncated factorization system]] in an [[(β,1)-topos]] for the case that $(n = -1)$ and restricted to [[0-truncated]] [[object]]s.
## Related concepts
* **(epi, mono) factorization system**
* [[(eso+full, faithful) factorization system]]
* [[n-connected/n-truncated factorization system]].
## References
For instance:
* {#Borceux94} [[Francis Borceux]], vol 1, section 4.4. of: _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)
[[!redirects (epi, mono) factorization system]]
[[!redirects (epi,mono)-factorization]]
[[!redirects (epi,mono) factorization]]
[[!redirects (epi, mono)-factorization]]
[[!redirects (epi, mono) factorization]]
[[!redirects (epi, mono)-factorization system]]
[[!redirects (epi, mono)-factorization systems]]
[[!redirects (epi,mono) factorization system]]
[[!redirects (epi,mono) factorization systems]]
[[!redirects (epi,mono)-factorization system]]
[[!redirects (epi,mono)-factorization systems]]
[[!redirects epi/mono factorization system]]
[[!redirects epi/mono factorization systems]]
[[!redirects epi-mono factorization system]]
[[!redirects epi-mono factorization systems]]
[[!redirects (effective epi, mono) factorization system]]
[[!redirects (effective epi, mono) factorization systems]]
[[!redirects (effective epi,mono) factorization system]]
[[!redirects (effective epi,mono) factorization systems]]
[[!redirects epi-mono factorization]]
[[!redirects epi-mono factorizations]]
|
(eso and full, faithful) factorization system | https://ncatlab.org/nlab/source/%28eso+and+full%2C+faithful%29+factorization+system | [[!redirects (eso+full, faithful) factorization system]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In the [[2-topos]] [[Cat]], the [[pair]] of [[classes]] of [[morphisms]]
* left class: [[essentially surjective and full functors]]
* right class: [[faithful functors]]
forms a [[factorization system in a 2-category]]. This factorization system can also be restricted to the [[(2,1)-topos]] [[Grpd]].
In fact, an analogous factorization system exists in any [[2-exact 2-category]] and any [[(2,1)-exact (2,1)-category]], including any [[Grothendieck 2-topos]] or [[(2,1)-topos]]; see [[michaelshulman:full morphism|here]].
## Properties
* When restricted to [[Grpd]], this is the special case of the [[n-connected/n-truncated factorization system]] in the [[(β,1)-topos]] [[βGrpd]] for the case that $(n = 0)$ and restricted to [[1-truncated]] [[objects]].
* For $f : X \to Y$ a functor between groupoids, its factorization is through a groupoid $im_2 f$ which is, up to equivalence, given as follows;
* [[objects]] are those of $X$;
* a [[morphism]] $[\phi] : x_1 \to x_2$ is an [[equivalence class]] of morphisms in $X$ where $[\phi] = [\phi']$ if $f(\phi) = f(\phi')$.
More on this is at _[infinity-image -- Of Functors between groupoids](infinity-image#NImagesOf1FunctorsBetweenGroupoids)_.
## Related concepts
* [[epi-mono factorization system]]
* [[(eso, fully faithful) factorization system]]
* **(eso+full, faithful)-factorization system**
* [[n-connected/n-truncated factorization system]].
[[!redirects essentially surjective and full/faithful factorization system]]
|
(eso, fully faithful) factorization system | https://ncatlab.org/nlab/source/%28eso%2C+fully+faithful%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In the [[2-topos]] [[Cat]], the pair of classes of morphisms ([[essentially surjective functor|essentially surjective]] functors, [[fully faithful functors]]) form a [[factorization system in a 2-category]]. This factorization system can also be restricted to the [[(2,1)-topos]] [[Grpd]].
e.g. ([Dupont-Vitale 03, 7.9, example 2](#DupontVitale03))
In fact, an analogous factorization system exists in any regular 2-category and any (2,1)-exact (2,1)-category, including any Grothendieck 2-topos or (2,1)-topos; see [[michaelshulman:regular morphism|here]].
## Properties
* The factorization of a functor in this factorization system is the construction of its [[full image]].
* More on this for the case of functors between [[groupoids]] is at _[infinity-image -- Of Functors between groupoids](infinity-image#NImagesOf1FunctorsBetweenGroupoids)_.
## Related concepts
* [[(epi, mono) factorization system]]
* **(eso, fully faithful)-factorization system**
* [[(eso+full, faithful) factorization system]]
* [[ternary factorization system]]
## References
* {#DupontVitale03} M. Dupont, [[Enrico Vitale]], _Proper factorization systems in 2-categories_, Journal of pure and applied algebra, 179 (2003), pp.65-86
[[!redirects essentially surjective/fully faithful factorization system]]
[[!redirects essentially surjective-fully faithful factorization system]]
[[!redirects eso/ff factorization system]]
[[!redirects eso/fully faithful factorization system]]
[[!redirects eso-ff factorization system]]
[[!redirects eso-fully faithful factorization system]]
[[!redirects (eso, ff) factorization system]]
|
(g-2) anomaly | https://ncatlab.org/nlab/source/%28g-2%29+anomaly |
#Contents#
* table of contents
{:toc}
## Idea
For the moment see at _[[anomalous magnetic moment]]_ _[this section](https://ncatlab.org/nlab/show/anomalous+magnetic+moment#Anomalies)_
## Related concepts
* [[flavour anomaly]]
* [[Cabibbo anomaly]]
* [[V_cb puzzle]]
[[!redirects (g-2)-anomaly]]
|
(geometric surjection, embedding) factorization system | https://ncatlab.org/nlab/source/%28geometric+surjection%2C+embedding%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Every [[geometric morphism]] between [[toposes]] factors into a [[geometric surjection]] followed by a [[geometric embedding]]. This exhibits an [[image]] construction in the [[topos theory|topos-theoretic]] sense, and gives rise to a [[factorization system in a 2-category]] for [[Topos]].
## Statement
+-- {: .num_prop}
###### Proposition
There is a [[factorization system on a 2-category|factorization system]] on the [[2-category]] [[Topos]] whose left class is the
[[surjective geometric morphism]]s and whose right class is the [[geometric embeddings]].
Moreover, the factorization of a given geometric morphism $f : \mathcal{E} \to \mathcal{F}$ is, up to [[equivalence of categories|equivalence]], through the canonical surjection onto the [[topos of coalgebras]] $f^* f_* CoAlg(\mathcal{E})$ of the [[comonad]] $f^* f_* : \mathcal{E} \to \mathcal{E}$:
$$
\array{
\mathcal{E} &&\stackrel{f}{\to}&& \mathcal{F}
\\
& {}_{\mathllap{F}}\searrow && \nearrow
\\
&& f^* f_* CoAlg(\mathcal{E})
}
\,E.
$$
=--
This appears for instance as ([MacLaneMoerdijk, VII 4., theorem 6](#MacLaneMoerdijk)).
We use the following lemma
+-- {: .num_lemma #ConditionForSheafFactorization}
###### Lemma
Let $j$ be a [[Lawvere-Tierney topology]] on a [[topos]] $\mathcal{E}$ and write $i : Sh_j(\mathcal{E}) \to \mathcal{E}$ for the corresponding [[geometric embedding]].
Then a [[geometric morphism]] $f : \mathcal{F} \to \mathcal{E}$ factors through $i$ precisely if
* the [[direct image]] $f_*$ takes values in $j$-sheaves;
or, equivalently
* the [[inverse image]] $f^*$ sends $j$-[[dense monomorphism]]s to [[isomorphism]]s.
=--
This appears as ([MacLaneMoerdijk, VII 4. prop. 2](#MacLaneMoerdijk)).
+-- {: .proof}
###### Proof of the lemma
We first show the first statement, that for $f$ to factor it is sufficient for $f_*$ to take values in $j$-sheaves: in that case, set
$$
p_* := i^* f_*: \mathcal{F} \to Sh_j(\mathcal{E})
\,.
$$
Since by assumption the [[unit of an adjunction|unit]] map $x \to i_* i^* x$ is an [[isomorphism]] on the image of $f_*$ this indeed serves to factor $f_*$:
$$
i_* p_* \simeq i_* i^* f_* \simeq f_*
\,.
$$
The [[left adjoint]] to $p_*$ is then
$$
p^* \simeq f^* i_*
\,,
$$
because
$$
\begin{aligned}
\mathcal{F}(g^* E, F)
& \simeq \mathcal{F}(f^* i_* E, F)
\\
& \simeq \mathcal{E}(i_* E, f_* F)
\\
& \simeq \mathcal{E}(i_* E, i_* i^* f_* F)
\\
& \simeq Sh_j\mathcal{E} (E, i^* f_* F)
\\
& \simeq Sh_j(E, p_* F)
\end{aligned}
\,,
$$
where in the middle steps we used that $f_* F$ is a $j$-sheaf, by assumption, and that $i_*$ is full and faithful.
It is clear that $p^*$ is left exact, and so $(p^* \dashv p_*)$ is indeed a factorizing geometric morphism.
We now show that $f_*$ taking values in sheaves is equivalent to $f^*$ mapping dense monos to isos.
Let $u : U \hookrightarrow X$ be a $j$-[[dense monomorphism]] and $A \in \mathcal{E}$ any object. Consider the induced naturality square
$$
\array{
\mathcal{E}(X, f_* A) &\stackrel{\simeq}{\to}& \mathcal{F}(f^* X, A)
\\
{}^{\mathllap{\mathcal{E}(u, f_* A)}}\downarrow
&&
\downarrow^{\mathrlap{\mathcal{F}(f^* u, A)}}
\\
\mathcal{E}(U, f_* A)
&\stackrel{\simeq}{\to}&
\mathcal{F}(f^* U, A)
}
$$
of the adjunction [[natural isomorphism]]. If now $f_* A$ is a $j$-sheaf and $u$ a [[dense monomorphism]], then by definition the left vertical morphism is also an isomorphism and so is the right one. By the [[Yoneda lemma]] this being an iso for all $A$ is equivalent to $f^* u$ being an iso. And conversely.
=--
+-- {: .proof}
###### Proof of the proposition
Let $f : \mathcal{F} \to \mathcal{E}$ be any [[geometric morphism]].
We first discuss the existence of the factorization, then its uniqueness.
To construct the factorization, we shall give a [[Lawvere-Tierney topology]] on $\mathcal{E}$ and factor $f$ through the [[geometric embedding]] of the corresponding [[sheaf topos]].
Take the closure operator $\overline{(-)} : Sub(-)_{\mathcal{E}} \to Sub(-)_{\mathcal{E}}$ to be given by sending $U \hookrightarrow X$ to the [[pullback]]
$$
\array{
\overline{U} &\to& f_* f^* U
\\
\downarrow && \downarrow
\\
X &\to& f_* f^* X
}
\,,
$$
where the bottom morphism is the $(f^* \dashv f_*)$-[[unit of an adjunction|unit]]. One checks that this is indeed a closure operator by the fact that $f^*$ preserves both pullbacks and pushouts.
Notice that this implies that for two [[subobject]]s $U_1, U_2 \hookrightarrow X$ we have
\[
\label{ASubobjectRelation}
(U_1 \subset \overline{U_2})
\;\;\;
\Leftrightarrow
\;\;\;
(f^* U_1 \subset f^* U_2)
\]
Write $j$ for the corresponding [[Lawvere-Tierney topology]] and
$$
i : Sh_j(\mathcal{E}) \to \mathcal{E}
$$
for the corresponding [[geometric embedding]].
By lemma \ref{ConditionForSheafFactorization} we get a factorization through $I$ if $f^*$ sends $j$-[[dense monomorphism]]s to [[isomorphism]]s. But if $U \hookrightarrow X$ is dense so that $X \subset \overline{U}$ then, by (eq:ASubobjectRelation), $f^* X \subset f^* U$ and hence $f^* X = f^* U$.
Write
$$
f : \mathcal{F} \stackrel{p}{\to} Sh_j(\mathcal{E}) \stackrel{i}{\to}
\mathcal{E}
$$
for the factorization thus established. It remains to show that $p$ here is a [[geometric surjection]]. By one of the equivalent characterizations discussed there, this is the case if $p^*$ induces an injective homomorphism of subobject lattices.
So suppose that for subobjects $U_1, U_2 \subset X$ we have $p^* U_1 \simeq p^* U_2$. Observe that then also $f^* i_* U_1 \simeq f^* i_* U_2$, because
$$
\begin{aligned}
f^* i_* U_1 & \simeq p^* i^* i_* U_1
\\
& \simeq p^* U_1
\\
& \simeq p^* U_2
\\
& \simeq p^* i^* i_* U_2
\\
& \simeq f^* i:* U_2
\end{aligned}
$$
by the fact that $i_*$ is [[full and faithful functor|full and faithful]]. With (eq:ASubobjectRelation) it follows that also
$$
i_* U_1 \simeq \overline{i_* U_2}
$$
and hence
$$
\cdots \simeq i_* U_2
$$
by the very fact that $i_*$ includes $j$-sheaves in general, hence $j$-closed subobjects in particular. Finally since $i_*$ if a [[full and faithful functor]] this means that
$$
U_1 \simeq U_2
\,.
$$
So $p^*$ is indeed injective on subobjects and so $p$ is a [[geometric surjection]].
This establishes the existence of a surjection/embedding factorization. Next we discss that this is unique.
So consider two factorizations
$$
\array{
&& \mathcal{A}
\\
& {}^{\mathllap{p_1}}\nearrow &\Downarrow^\simeq& \searrow^{\mathrlap{i_1}}
\\
\mathcal{F}
&&\stackrel{f}{\to}&&
\mathcal{E}
\\
& {}_{\mathllap{p_2}}\searrow &\downarrow^{\simeq}& \nearrow_{\mathrlap{i_2}}
\\
&& \mathcal{B}
}
$$
into a geometric surjection followed by a geometric embedding.
We will now argue that $i_1$ factors -- essentially uniquely -- through $i_2$ in a way that makes
$$
\array{
&& \mathcal{A}
\\
& {}^{\mathllap{p_1}}\nearrow && \searrow^{\mathrlap{i_1}}
\\
\mathcal{F}
&&\downarrow^g&&
\mathcal{E}
\\
& {}_{\mathllap{p_2}}\searrow && \nearrow_{\mathrlap{i_2}}
\\
&& \mathcal{B}
}
$$
commute up to natural isomorphisms. By the same argument for the upside-down situation we find an essentially unique middle vertical morphism $h : \mathcal{B} \to \mathcal{A}$ the other way round. Then essential uniqueness of these factorizations implies that $g \circ h \simeq Id$ and $h \circ g \simeq Id$. This means that the original two factorizations are equivalent.
To find $g$ and $h$, use again that every [[geometric embedding]] (by the discussion there) is, up to equivalence, an inclusion of $j$-sheaves for some $j$. Find such a $j$ the bottom morphism and then use again lemma \ref{ConditionForSheafFactorization} that $i_1$ factors through $i_2$ -- essentially uniquely -- precisely if $i_1^*$ sends [[dense monomorphism]]s to isomorphisms.
To see that it does, let $IU \to X$ be a dense mono and consider the naturality square
$$
\array{
p_2^* i_2^* U &\stackrel{\simeq}{\to}& p_1^* i_1^* U
\\
\downarrow && \downarrow
\\
p_2^* i_2^* X &\stackrel{\simeq}{\to}& p_1^* i_1^* X
}
\,.
$$
Since $i_2^*(U \to X)$ is an iso by definition, the left vertical morphism is, and thus so is the right vertical morphism. But since $p_1$ is a [[geometric surjection]] we have (by the discussion there) that $p_1^*$ is [[conservative functor|conservative]], and hence also $i_1^* U \to i_1^* X$ is an isomorphism.
Hence $i_1$ factors via some $g$ through $i_2$ and the proof is completed by the above argument.
=--
## Examples
* For $f : X \to Y$ a [[continuous function]] between [[topological space]]s and $X \to im(f) \to Y$ its ordinary [[image]] factorization through an [[embedding]], the corresponding composite of geometric morphisms of [[sheaf topos]]es
$$
Sh(X) \to Sh(im(f)) \to Sh(Y)
$$
is a geometric surjection/geometric embedding factorization.
* For $\mathcal{E}$ any topos, $f : X \to Y$ any morphism in $\mathcal{E}$, and $X \to im(f) \to Y$ its [[image]] factorization, the corresponding composite of [[base change geometric morphisms]]
$$
\mathcal{E}/X \to \mathcal{E}/im(f) \to \mathcal{E}/Y
$$
is a geometric surjection/embedding factorization.
* For $f : C \to D$ any [[functor]] between [[categories]], write $C \to im(f) \to D$ for its [[essential image]] factorization. Then the induced composite <a href="http://ncatlab.org/nlab/show/geometric%20morphism#BetweenPresheafToposes">geometric morphism of presheaf toposes</a>
$$
[C^{op}, Set] \stackrel{}{\to} [im(f)^{op}, Set] \to
[D^{op}, Set]
$$
is a geometric surjection/embedding factorization.
See ([MacLaneMoerdijk, p. 377](#MacLaneMoerdijk)).
## Properties
### As idempotent approximation
A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ induces via the adjunction $f^\ast\vdash f_\ast$ a [[monad]] on $\mathcal{E}$. Due to a general result by S. Fakir this induces an associated [[idempotent monad]] on $\mathcal{E}$ and this idempotent approximation coincides with the monad induced by $i^\ast\vdash i_\ast$ given by the inclusion $i$ from the factorization $f=i\circ q$.
For references and further details on the idempotent approximation see at [[idempotent monad]].
### A logical description
Let $T$ be a [[geometric theory]] over a signature $\Sigma$ and $f:\mathcal{E}\to Set[T]$ a geometric morphism to its [[classifying topos]]. Then by the general properties of a classifying topos, $f$ corresponds to a certain $T$-model $M$ in $\mathcal{E}$.
Notice that every geometric morphism $f$ between [[Grothendieck toposes]] is of this form for some geometric theory $T$ and hence corresponds to some model $M$ ! This model permits to attach a geometric theory to $f$ as well:
The **theory of M** $Th(M)$ consists of all geometric sequents $\sigma$ over $\Sigma$ such that $M\models \sigma$.
Then the following holds ([Caramello 2009](#Caramello09), p.57):
+-- {: .num_prop}
###### Proposition
The topos occurring in the middle of the surjection-embedding factorization of $f$ is precisely the classifying topos for $Th(M)$: $\mathcal{E}\twoheadrightarrow Set[Th(M)]\hookrightarrow Set[T]$.
=--
## Related entries
* [[open subtopos]]
* [[(dense,closed)-factorization]]
## References
* {#Johnstone77} [[Peter Johnstone]], _Topos Theory_ , Academic Press 1977 (Dover reprint 2014). (section 4.1, pp.103-107)
* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_, Oxford UP 2002. (section A4.2, pp.172ff)
* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (section VII.4)
{#MacLaneMoerdijk}
* [[Olivia Caramello|O. Caramello]], _Lattices of theories_ , arXiv:0905.0299v1 (2009). ([pdf](http://arxiv.org/pdf/0905.0299v1)) {#Caramello09}
[[!redirects geometric surjection/embedding factorization]]
[[!redirects geometric surjection/inclusion factorization]]
[[!redirects geometric surjection/embedding factorization system]]
[[!redirects geometric surjection/inclusion factorization system]]
[[!redirects (geometric surjection, inclusion) factorization system]]
|
(hyperconnected, localic) factorization system | https://ncatlab.org/nlab/source/%28hyperconnected%2C+localic%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+-- {: .hide}
[[!include factorization systems - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[hyperconnected geometric morphism|Hyperconnected geometric morphisms]] are the left class of a 2-categorical [[orthogonal factorization system]] on the [[2-category]] [[Topos]] of [[toposes]]; the right class is the class of [[localic geometric morphisms]].
## References
* [[Peter Johnstone]], _Factorization theorems for geometric morphisms_ Cahiers, 22, no1 (1981) ([numdam](http://www.numdam.org/item?id=CTGDC_1981__22_1_3_0))
{#Johnstone}
[[!redirects (hyperconnected, localic) factorization system]]
[[!redirects (hyperconnected,localic) factorization system]] |
(infinity,0)-category | https://ncatlab.org/nlab/source/%28infinity%2C0%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Following the terminology of *[[(n,r)-category|(n,r)-categories]]*, an __$(\infty,0)$-category__ is an [[β-category]] in which every $j$-morphism (for $j \gt 0$) is an [[equivalence]].
So in an $(\infty,0)$-category _every_ morphism is an [[equivalence]]. Such [[β-categories]] are usually called _[[β-groupoid]]s_.
This is directly analogous to how a [[0-category]] is equivalent to a [[set]], a [[(1,0)-category]] is equivalent to a [[groupoid]], and so on. (In general, an [[(n,0)-category]] is equivalent to an [[n-groupoid]].)
The term "$(\infty,0)$-category" is rarely used, but does for instance serve the purpose of amplifying the generalization from [[Kan complex]]es, which are one model for [[β-groupoid]]s, to [[quasi-category|quasi-categories]], which are a model for [[(β,1)-categories]].
## References
On [[model categories]] [[presentable (infinity,1)-category|presenting]] $(\infty,0)$-categories, namely models for [[infinity-groupoids|$\infty$-groupoids]] (such as the [[model structure on simplicial groupoids]]) akin to corresponding models for [[(infinity,1)-categories|$(\infty,1)$-categories]] (such as the [[model structure on simplicial categories]]):
* [[Julia E. Bergner]], *Adding inverses to diagrams II: Invertible homotopy theories are spaces*, Homology, Homotopy Appl. **10** 2 (2008) 175-193 [[doi:10.4310/HHA.2008.v10.n2.a9](https://dx.doi.org/10.4310/HHA.2008.v10.n2.a9), [doi:0710.2254](https://arxiv.org/abs/0710.2254), erratum:[doi:10.4310/HHA.2012.v14.n1.a15](https://dx.doi.org/10.4310/HHA.2012.v14.n1.a15)]
[[!redirects (infinity,0)-categories]]
[[!redirects (β,0)-category]]
[[!redirects (β,0)-categories]]
|
(infinity,1)-bimodule | https://ncatlab.org/nlab/source/%28infinity%2C1%29-bimodule |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
## Definition
### $(\infty,1)$-Category of $(\infty,1)$-Bimodules and intertwiners
Write $BMod^\otimes$ for the [[(β,1)-category of operators]] of the [[(β,1)-operad]] [[operad for bimodules]]. Write
$$
\iota_{\pm} \colon Assoc \to BMod
$$
for the two canonical inclusions of the [[associative operad]] (as discussed at _[operad for bimodules - relation to the associative operad](#RelationToTheAssociativeOperad)_).
+-- {: .num_defn #NotationForWeaklyBiEnrichedInfinityCategory}
###### Definition (Notation)
For $p \colon \mathcal{C}^\otimes \to BMod^\otimes$ a [[fibration of (β,1)-operads]], write
$$
\mathcal{C}^\otimes_{\pm}
\coloneqq
\mathcal{C}^\otimes \underset{BMod^\otimes}{\times}^\pm Assoc^\otimes
$$
for the two [[fiber products]] of $p$ with the inclusions $\iota_\pm$. The canonical [[projection]] maps
$$
\mathcal{C}^\otimes_{\pm}
\to
Assoc^\otimes
$$
exhibit these as two [[planar (β,1)-operads]].
Finally write
$$
\mathcal{C}
\coloneqq
\mathcal{C}^\otimes \underset{BMod^\otimes}{\times} \{\mathfrak{n}\}
$$
for the [[(β,1)-category]] over the object labeled $\mathfrak{n}$.
=--
([Lurie, notation 4.3.1.11](#Lurie)).
+-- {: .num_remark }
###### Remark
This exhibits $\mathcal{C}$ as equipped with [[weak tensoring]] over $\mathcal{C}_-$ and reverse weak tensoring over $\mathcal{C}_+$.
=--
The most familiar special case of these definitions to keep in mind is the following.
+-- {: .num_remark #MonoidalCategoryAsBitensoredOverItself}
###### Remark
For $\mathcal{C}^\otimes \to Assoc^\otimes$ a [[coCartesian fibration of (β,1)-operads]], hence exhibiting $\mathcal{C}^\otimes$ as a [[monoidal (β,1)-category]], [[pullback]] along the canonical map $\phi \colon BMod^\otimes \to Assoc^\otimes$ gives a fibration
$$
\phi^* \mathcal{C}^\otimes \to BMod^\otimes
$$
as in def. \ref{NotationForWeaklyBiEnrichedInfinityCategory} above. In the terminology there this exhibts $\mathcal{C}$ as weakly enriched (weakly [[tensoring|tensored]]) over itself from the left and from the right.
This is the special case for which bimodules are traditionally considered.
=--
([Lurie, example 4.3.1.15](#Lurie))
+-- {: .num_defn }
###### Definition
For $\mathcal{C}^\otimes \to BMod^\otimes$ a [[fibration of (β,1)-operads]] we say that the corresponding [[(β,1)-category]] of [[(β,1)-algebras over an (β,1)-operad]]
$$
BMod(\mathcal{C}) \coloneqq Alg_{/BMod}(\mathcal{C})
$$
is the **$(\infty,1)$-category of $(\infty,1)$-bimodules** in $\mathcal{C}$.
Composition with the two inclusions $\iota_{1,2}\colon Assoc BMod$ of the [[associative operad]] yields a [[fibration]] in the [[model structure for quasi-categories]] $BMod(\mathcal{C}) \to Alg(\mathcal{C}_-)\times Alg(\mathcal{C}_+)$. Then for $A_- \in Alg_{\mathcal{C}_-}$ and $A_+ \in Alg_{\mathcal{C}_+}$ two algebras the [[fiber product]]
$$
{}_A BMod_{B}(\mathcal{C})
\coloneqq
\{A\}
\underset{Alg(\mathcal{C}_-)}{\times}
BMod(\mathcal{C})
\underset{Alg(\mathcal{C}_-)}{\times}
\{B\}
$$
we call the **$(\infty,1)$-category of $A$-$B$-bimodules**.
=--
([Lurie, def. 4.3.1.12](#Lurie))
+-- {: .num_example}
###### Example
For the special case of remark \ref{MonoidalCategoryAsBitensoredOverItself} where the bitensored structure on $\mathcal{C}$ is induced from a monoidal structure $\mathcal{C}^\otimes \to Asoc^\otimes$, we have by the [[universal property]] of the [[pullback]] that
$$
BMod(\mathcal{C})
\simeq
{Alg_{BMod}}_{/Assoc}(\mathcal{C})
\simeq
\left\{
\array{
&& \mathcal{C}
\\
&{}^{\mathllap{(A,B,N)}}\nearrow& \downarrow
\\
BMod^\otimes &\to& Assoc^\otimes
}
\right\}
$$
=--
+-- {: .num_remark}
###### Remark
Let $\mathcal{C}$ be a [[1-category]], for simplicity.
Then a [[morphism]]
$$
(A_1,B_1,N_1) \to $(A_2,B_2,N_2)$
$$
in $BMod(\mathcal{C})$ is a pair $\phi_1 \colon A_1 \to A_1$, $\rho \colon B_1 \to B_2$ of algebra homomorphisms and a morphism $\kappa \colon N_1 \to N_2$ which is "linear in both $A$ and $B$" or "is an [[intertwiner]]" with respect to $\phi$ and $\rho$ in that for all $a \in A$, $b \in B$ and $n \in N$ we have
$$
\kappa(a \cdot n \cdot b) = \phi(a) \cdot \kappa(n)
\,.
$$
It is natural to depict this by the square diagram
$$
\array{
A_1 &\stackrel{N_1}{\to}& B_1
\\
{}^{\mathllap{\phi}}\downarrow
&
\Downarrow^{\kappa}
&
\downarrow^{\mathrlap{\rho}}
\\
A_2 &\underset{N_2}{\to}& B_2
}
\,.
$$
This notation is naturally suggestive of the existence of the further [[horizontal composition]] by [[tensor product of (β,1)-modules]], which we come to [below](#TensorProductOfBimodules).
On the other hand, a morphism $N_1 \to N_2$ in ${}_A BMod(\mathcal{C})_B$ is given by the special case of the above for $\phi = id$ and $\rho = id$.
=--
### Tensor products of $(\infty,1)$-Bimodules
{#TensorProductOfBimodules}
+-- {: .num_defn #NotationForTensS}
###### Definition (Notation)
Write $Tens^\otimes$ for the [[generalized (β,1)-operad]]
discussed at _[[tensor product of β-modules]]_.
For $S \to \Delta^{op}$ an [[(β,1)-functor]] (given as a map of simplicial sets from a [[quasi-category]] $S$ to the [[nerve]] of the [[simplex category]]), write
$$
Tens^\otimes_{S}
\coloneqq
Tens^\otimes \underset{\Delta^{op}}{\times} S
$$
for the [[fiber product]] in [[sSet]].
Moreover, for $\mathcal{C}^\otimes \to Tens^\otimes_S$ a [[fibration]] in the [[model structure for quasi-categories]] which exhibits $\mathcal{C}^\otimes$ as an $S$-[[family of (β,1)-operads]], write
$$
Alg_S(\mathcal{C}) \hookrightarrow Fun_{Tens^\otimes_S}(Step_S, \mathcal{C}^\otimes)
$$
for the full [[sub-(β,1)-category]] on those [[(β,1)-functors]] which send inert morphisms to inert morphisms.
=--
([Lurie, notation 4.3.4.15](#Lurie))
### The $(\infty,2)$-Category of $(\infty,1)$-algebras and -bimodules
We discuss the generalization of the notion of bimodules to [[homotopy theory]], hence the generalization from [[category theory]] to [[(β,1)-category theory]]. ([Lurie, section 4.3](#Lurie)).
Let $\mathcal{C}$ be [[monoidal (β,1)-category]] such that
1. it admits [[geometric realization]] of [[simplicial objects in an (β,1)-category]] (hence a [[left adjoint|left]] [[adjoint (β,1)-functor]] ${\vert-\vert} \colon \mathcal{C}^{\Delta^{op}} \to \mathcal{C}$ to the constant simplicial object functor), true notably when $\mathcal{C}$ is a [[presentable (β,1)-category]];
1. the [[tensor product]] $\otimes \colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$ preserves this geometric realization separately in each argument.
Then there is an
[[(β,2)-category]] $Mod(\mathcal{C})$ which given as an [[(β,1)-category object]] internal to [[(β,1)Cat]] has
* $(\infty,1)$-category of objects
$$
Mod(\mathcal{C})_{[0]} \simeq Alg(\mathcal{C})
$$
the [[A-β algebras]] and [[β-algebra]] [[homomorphisms]] in $\mathcal{C}$;
* $(\infty,1)$-category of morphisms
$$
Mod(\mathcal{C})_{[1]} \simeq BMod(\mathcal{C})
$$
the $\infty$-bimodules and bimodule homomorphisms ([[intertwiners]]) in $\mathcal{C}$
This is ([Lurie, def. 4.3.6.10, remark 4.3.6.11](#Lurie)).
Morover, the [[horizontal composition]] of bimodules in this [[(β,2)-category]] is indeed the relative
[[tensor product of β-modules]]
$$
\circ_{A,B,C}
=
(-) \otimes_B (-)
\;\colon\;
{}_A Mod_{B} \times {}_{B}Mod_C \to {}_A Mod_C
\,.
$$
This is ([Lurie, lemma 4.3.6.9 (3)](#Lurie)).
Here are some steps in the construction:
The **idea** of the following constructions is this:
we start with a [[generalized (β,1)-operad]]
$Tens^\otimes \to FinSet_* \times \Delta^{op}$ which is such that
the [[(β,1)-algebras over an (β,1)-operad]] over its fiber over $[k] \in \Delta^{op}$ is equivalently the collection of $(k+1)$-tuples of
[[A-β algebras]] in $\mathcal{C}$ together with a string of $k$ $\infty$-bimodules between them. Then we turn that into a [[simplicial object in an (β,1)-category|simplicial object]] in [[(β,1)Cat]]
$$
Mod(\mathcal{C}) \in ((\infty,1)Cat)^{\Delta^{op}}
\,.
$$
This turns out to be an [[internal (β,1)-category]] object in [[(β,1)Cat]], hence an [[(β,2)-category]] whose object of objects is the category $Alg(\mathcal{C})$ of [[A-β algebras]] and [[homomorphisms]] in $\mathcal{C}$ between them, and whose object of morphisms is the category $BMod(\mathcal{C})$ of $\infty$-bimodules and [[intertwiners]].
+-- {: .num_defn}
###### Definition
Define $Mod(\mathcal{C}) \to \Delta^{op}$ as the map of [[simplicial sets]] with the [[universal property]] that for every other map of simplicial set $K \to \Delta^{op}$ there is a canonical bijection
$$
Hom_{sSet/\Delta^{op}}(K, Mod(\mathcal{C}))
\simeq
Alg_{Tens_K / Assoc}( \mathcal{C} )
\,,
$$
where
* on the left we have the hom-simplicial set in the [[slice category]]
* on the right we have the [[(β,1)-category]] of [[(β,1)-algebras over an (β,1)-operad]] given by lifts $\mathcal{A}$ in
$$
\array{
&& \mathcal{C}^\otimes
\\
&{}^{\mathcal{A}}\nearrow& \downarrow
\\
Tens_K &\to& Assoc
}
\,.
$$
=--
This is ([Lurie, cor. 4.3.6.2](#Lurie)) specified to the case of ([Lurie, lemma 4.3.6.9](#Lurie)). Also ([Lurie, def. 4.3.4.19](#Lurie))
## References
The general theory in terms of [[higher algebra]] of [[(β,1)-operads]] is discussed in section 4.3 of
* [[Jacob Lurie]], _[[Higher Algebra]]_
Specifically the homotopy theory of [[A-infinity bimodules]] is discussed in
* Volodymyr Lyubashenko, Oleksandr Manzyuk, _A-infinity-bimodules and Serre A-infinity-functors_ ([arXiv:math/0701165](http://arxiv.org/abs/math/0701165))
and section 5.4.1 of
* [[Boris Tsygan]], _Noncommutative calculus and operads_ in
Guillermo Cortinas (ed.) _Topics in Noncommutative geometry_, Clay Mathematics Proceedings volume 16
The generalization to [[(infinity,n)-modules]] is in
* {#Haugseng14} [[Rune Haugseng]], _The higher Morita category of $E_n$-algebras_ ([arXiv:1412.8459](http://arxiv.org/abs/1412.8459))
[[!redirects (infinity,1)-bimodules]]
[[!redirects β-bimodule]]
[[!redirects β-bimodules]]
[[!redirects (β,1)-bimodule]]
[[!redirects (β,1)-bimodules]]
[[!redirects (β,1)-category of (β,1)-bimodules]]
[[!redirects (infinity,1)-category of (β,1)-bimodules]]
[[!redirects (β,1)-categories of (β,1)-bimodules]]
[[!redirects (infinity,1)-categories of (β,1)-bimodules]]
[[!redirects infinity-bimodule]]
[[!redirects infinity-bimodules]] |
(infinity,1)-categorical hom-space | https://ncatlab.org/nlab/source/%28infinity%2C1%29-categorical+hom-space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Where an ordinary [[category]] has a [[hom-set]], an [[(β,1)-category]] has an [[β-groupoid]] of morphisms between any two objects, a _hom-space_.
There are several ways to _present_ an [[(β,1)-category]] $\mathbf{C}$ by an ordinary [[category]] $C$ equipped with some extra structure: for instance $C$ may be a [[category with weak equivalences]] or a [[model category]] or even a [[simplicial model category]]. In all of these presentations, given two objects $X, Y \in C$, there is a way to construct a [[simplicial set]] $\mathbb{R}\mathbf{C}(X,Y)$ that presents the hom-[[β-groupoid]] $\mathbf{C}(X,Y)$. This simplicial set -- or rather its [[homotopy type]] -- is called the
_derived hom space_ or _homotopy function complex_ and denoted $\mathbf{R}Hom(X,Y)$ or similarly.
## Presentations
There are many ways to present an [[(β,1)-category]] by [[category theory|category theoretic data]], and for each of these there are corresponding tools for explicitly computing the derived hom spaces.
The most basic data is that of a [[category with weak equivalences]]. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called _[[simplicial localization]]_. This we discuss below in _[For a category with weak equivalences](#ForACategoryWithWeakEquivalences)_.
Particularly useful extra structure on a [[category with weak equivalences]] that helps with computing the derived hom spaces is the structure of a _[[model category]]_. Using this one can construct simplicial resolutions of objects -- called _framings_ -- that generalize [[cylinder objects]] and [[path objects]], and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in _[For a model category](#Framings)_.
Still a bit more helpful structure on top of a bare model category is that of a [[simplicial model category]]. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the [[sSet]]-[[hom objects]]. This we discuss below in _[For a simplicial model category](#EnrichedHomsCofToFib)_.
### For a category with weak equivalences
{#ForACategoryWithWeakEquivalences}
Let $(C,W \subset Mor(C))$ be a [[category with weak equivalences]].
+-- {: .num_defn #ZigZagCategories}
###### Definition
Fix $n \in \mathbb{N}$. For $X,Y \in Obj(C)$, define a category $wMor_C^n(X,Y)$
* whose objects are [[zig-zag]]s of morphisms in $C$ of length $n$
$$
X = X_0 \leftarrow X_1 \to X_2 \leftarrow \cdots \to X_{n-1} \leftarrow X_n = Y
$$
such that each morphism going to the left, $X_{2k}\leftarrow X_{2k +1}$, is a [[weak equivalence]], an element in $W$;
* morphisms between such objects $(X,X_i,Y) \to (X',X'_i,Y')$ are collections of weak equivalences $(X_i \to X'_i)$ for all $0 \lt i \lt n $ such that all triangles and squares commute.
=--
+-- {: .num_defn #HammockLocalization}
###### Definition
Write $N(wMor_C^n(X,Y))$ for the [[nerve]] of this category, a [[simplicial set]].
The _[[hammock localization]]_ $L_W^H C$ of $C$ is the [[simplicially enriched category]] with objects those of $C$ and [[hom-objects]] given by the [[colimit]] over the length of these hammock hom-categories
$$
L^H C(X,Y) := \lim_{\to_n} N(wMor_C^n(X,Y))
\,.
$$
The [[Kan fibrant replacement]] of this simplicial set is the derived hom-space between $X$ and $Y$ of the $(\infty,1)$-category modeled by $(C,W)$.
=--
### For a model category
{#Framings}
The derived hom spaces of a model category $C$ may always be computed
in terms of simplicial resolutions with respect to the [[Reedy model structure]]
$[\Delta^{op}, C]_{Reedy}$. These resolutions are often called
_framings_ ([Hovey](#Hovey)). These constructions are originally due to
([Dwyer-Hirschhorn-Kan](#DHK)).
Let $C$ be any [[model category]].
+-- {: .num_prop }
###### Observation
There is an [[adjoint triple]]
$$
(const \dashv ev_0 \dashv (-)^{\times^\bullet})
:
C
\stackrel{\overset{const}{\longrightarrow}}{\stackrel{\overset{ev_0}{\longleftarrow}}{\underset{(-)^{\times^\bullet}}{\longrightarrow}}}
\,,
[\Delta^{op}, C]
\,,
$$
where
1. $const X : [n] \mapsto X$;
1. $ev_0 X_\bullet = X_0$;
1. $X^{\times^\bullet} : [n] \mapsto X^{\times^{n+1}}$.
=--
+-- {: .num_remark #CoDiscreteIsReedyFibrant}
###### Remark
For $X \in C$ fibrant, $X^{\times^\bullet}$ is fibrant in
the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$.
=--
+-- {: .proof}
###### Proof
The matching morphisms are in fact [[isomorphisms]].
=--
+-- {: .num_defn}
###### Definition
Let $C$ be a model category.
1. For $X \in C$ any object, a _simplicial frame_ on $X$ is a factorization of $const X \to X^{\times^\bullet}$ into a weak equivalence followed by a fibration in the [[Reedy model structure]] $[\Delta^{op}, C]_{Reedy}$.
1. A _right framing_ in $C$ is a functor $(-)_\bullet : C \to [\Delta^{op}, C]$
with a [[natural isomorphism]] $(X)_0 \simeq X$ such that $X_\bullet$ is a simplicial
frame on $X$.
Dually for _cosimplicial frames_.
=--
This appears as ([Hovey, def. 5.2.7](#Hovey)).
+-- {: .num_remark}
###### Remark
By remark \ref{CoDiscreteIsReedyFibrant} a simplicial frame $X_\bullet$ in the above is in particular fibrant in $[\Delta^{op}, C]_{Reedy}$.
=--
+-- {: .num_prop #SimplicialFunctionComplexes}
###### Proposition
For $X \in C$ cofibrant and $A \in C$ fibrant, there
are weak equivalences in $sSet_{Quillen}$
$$
Hom_C(X^\bullet, A)
\stackrel{\simeq}{\to}
diag Hom_C(X^\bullet, A_\bullet)
\stackrel{\simeq}{\leftarrow}
Hom_C(X, A_\bullet)
\,,
$$
(where in the middle we have the diagonal of the [[bisimplicial set]] $Hom(X^\bullet, A_\bullet)$).
=--
This appears as ([Hovey, prop. 5.4.7](#Hovey)).
Either of these simplicial sets is a model for the derived hom-space $\mathbb{R}Hom(X,A)$.
+-- {: .num_remark }
###### Remark
By developing these constructions further, one obtains
a canonical [[simplicial model category]]-resolution of
(left proper and combinatorial) model categories $C$, such that
the simplicial resolutions given by framings are just the
cofibrant$\to$fibrant $sSet$-hom objects as discussed
[below](#EnrichedHomsCofToFib).
This is discussed at _[Simplicial Quillen equivalent models](http://ncatlab.org/nlab/show/simplicial+model+category#SimpEquivMods)_.
=--
+-- {: .num_prop }
###### Proposition
Let $C$ be a model category, let $\mathrm{c}_\mathrm{w} C$ be the full subcategory of $[\Delta, C]$ spanned by the cosimplicial objects whose coface and codegeneracy operators are weak equivalences, and let $\mathrm{s}_\mathrm{w} C$ be the full subcategory of $[\Delta^{op}, C]$ spanned by the simplicial objects whose face and degeneracy operators are weak equivalences.
1. $const : C \to \mathrm{c}_\mathrm{w} C$ is the right half of an adjoint homotopical equivalence of [[homotopical category|homotopical categories]], and $const : C \to \mathrm{s}_\mathrm{w} C$ is the left half of an adjoint homotopical equivalence of homotopical categories.
2. The functor $\operatorname{diag} Hom_C : (\mathrm{c}_\mathrm{w} C)^{op} \times \mathrm{s}_\mathrm{w} C \to sSet$ admits a right [[derived functor]].
3. The induced functor $(\operatorname{Ho} C)^{op} \times \operatorname{Ho} C \to \operatorname{Ho} sSet$ is the derived hom-space functor.
=--
### For a simplicial model category
{#EnrichedHomsCofToFib}
We describe here in more detail properties of [[derived hom-functors]] (see there for more) in a [[simplicial model category]].
The crucial axiom used for this is the axiom of an [[enriched model category]] $C$ which says that
* the [[copower|tensor operation]]
$$
\cdot : C \times SSet \to C
$$
is a [[Quillen bifunctor]];
* or equivalently that for $X \to Y$ a cofibration
and $A \to B$ a fibration the induced morphism
$$
C(Y, A) \to C(X,A) \times_{C(X,B)} C(Y,B)
$$
is a fibration, which is acyclic if either
$X \to Y$ or $A \to B$ is.
First of all the first statement directly implies that for $\emptyset \in C$ the [[initial object]] and $A \in C$ any object, the simplicial set $C(\emptyset,A) = {*}$ is the terminal simplicial set (see also [this Prop.](powered+and+copowered+category#InTensoredCotensoredCategoryInitialObjectIsEnrichedInitial)): because for any simplicial set $S$
$$
\begin{aligned}
SSet(S,C(\emptyset, A))
& = Hom_C(\emptyset \cdot S, A)
\\
& = Hom_C(colim_{\emptyset} \cdot S, A)
\\
& = Hom_C(\emptyset, A)
\\
&= {*}
\end{aligned}
\,,
$$
where we use that the [[copower|tensor]] [[Quillen bifunctor]] is required to respect [[colimit]]s and that the empty colimit is the [[initial object]]. (All equality signs here denote [[isomorphisms]], to distinguish them from weak equivalences.)
Similarly one has for all $X$ that $C(X,{*}) = {*}$.
Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.
+-- {: .num_lemma }
###### Lemma
In a [[simplicial model category]] $C$, for $X \in C$ cofibrant and $A \in C$ fibrant, the [[simplicial set]] $C(X,A)$ is a [[Kan complex]].
=--
+-- {: .proof}
###### Proof
We apply the [[enriched model category]] axiom to the cofibration $\emptyset \to X$ and the fibration $A \to {*}$ to obtain a fibration
$$
C(X,A) \to C(\emptyset, A) \times_{C(\emptyset,{*})} C(X,{*})
\,.
$$
The right hand is the [[pullback]] of the terminal simplicial set ${*} = \Delta^0$ to itself, hence is itself the point. So we have a fibration $ C(X,A) \to {*}$
and $C(X,A)$ is a fibrant object in the standard [[model structure on simplicial sets]], hence a [[Kan complex]].
.
=--
+-- {: .num_lemma }
###### Lemma
In a [[simplicial model category]] $C$, for $X \in C$ cofibrant and $f : A \to B$ a fibration, the morphism of [[simplicial set]]s $C(X,f) : C(X,A) \to C(X,B)$ is a [[Kan fibration]] that is a [[weak homotopy equivalence]] if $f$ is acyclic.
Dually, for $i : X \to Y$ a cofibration and $A$ fibrant, the morphism $C(i,A) : C(X,A) \to C(Y,A)$ is a cofibration of simplicial sets.
=--
+-- {: .proof}
###### Proof
This is as before. Explicity, consider the first case, the second one is the formal dual of that:
We enter the enrichment axiom with the morphisms $\emptyset \to X$ and $A \to B$ and find for the required [[pullback]] that
$$
C(\emptyset,A) \times_{C(\emptyset, B)} C(X,B)
=
{*} \times_{*} C(X,B)
=
C(X,B)
$$
and hence that $C(X,A) \to C(X,B)$ is, indeed, a fibration, which is acyclic if $A \to B$ is.
=--
+-- {: .num_proposition }
###### Proposition
Let $C$ be a [[simplicial model category]].
Then for $X$ a cofibant object and
$$
f : A \stackrel{\simeq}{\to} B
$$
a weak equivalence between fibrant objects, the [[enriched functor|enriched]] [[hom-object|hom-functor]]
$$
C(X,f) : C(X,A) \to C(X,B)
$$
is a [[weak homotopy equivalence]] of [[Kan complex]]es.
Similarly, for $A$ a fibrant object and $j : X \stackrel{\simeq}{\to} Y$ a weak equivalence between cofibrant objects, the morphism
$$
C(j,A) : C(X,A) \to C(Y,A)
$$
is a [[weak homotopy equivalence]] of [[Kan complex]]es.
=--
+-- {: .proof}
###### Proof
The second case is formally dual to the first, so we restrict attention to the first one.
By the above, the axioms of an [[enriched model category]] ensure that the above statement is true when $f$ is in addition a fibration. So we reduce the situation to that case.
This is possible because both $A$ and $B$ are assumed to be fibrant. This allows to apply the _factorization lemma_ that is described in great detail at [[category of fibrant objects]]. By this lemma, for every morphism $f : A \to B$ between fibrant objects there is a commutative diagram
$$
\array{
&& \mathbf{E}_f B
\\
& {}^{\mathllap{\in fib \cap W}}\swarrow && \searrow^{\mathrlap{\in fib}}
\\
A &&\stackrel{\simeq}{\to}&& B
}
$$
Since $f$ is assumed a weak equivalence it follows by [[category with weak equivalences|2-out-of-3]] that $\mathbf{E}_f B$ is also a weak equivalence.
Therefore by the above properties of simpliciall enriched categories we obtain a [[span]] of acyclic fibrations of [[Kan complex]]es
$$
C(X,A) \stackrel{\simeq}{\leftarrow}
C(X, \mathbf{E}_f B)
\stackrel{\simeq}{\to}
C(X,B)
\,.
$$
By the [[Whitehead theorem]] every weak equivalence of Kan complexes is a [[homotopy equivalence]], hence there is a weak equivalence
$$
C(X,A) \stackrel{\simeq}{\to} C(X,\mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B)
$$
that is homotopic to our $C(X,f)$. Therefore this is also a weak equivalence.
=--
### Comparison
Let $C$ be a [[model category]]. We discuss how its
simplicial function complexes from prop. \ref{SimplicialFunctionComplexes}
are related to the simplicial localization from def. \ref{ZigZagCategories} and def. \ref{HammockLocalization}.
Suppose now that $Q : C \to C$ is a [[cofibrant replacement functor]] and $R : C \to C$ a [[fibrant replacement functor]], $\Gamma^\bullet : C \to (cC)_c$ a [[cosimplicial resolution functor]] and $\Lambda_\bullet : C \to (sC)_f$ a [[simplicial resolution functor]] in the [[model category]] $C$.
+-- {: .num_theorem #DKTheorem}
###### Theorem
**(Dwyer--Kan)**
There are natural weak equivalences between the following equivalent realizations of this [[SSet]] [[hom-object]]:
$$
\array{
Mor_C(\Gamma^\bullet X, R Y)
&\stackrel{\simeq}{\to}&
diag Mor_C(\Gamma^\bullet X, \Lambda_\bullet Y)
&\stackrel{\simeq}{\leftarrow}&
Mor_C(Q X, \Lambda_\bullet Y)
\\
&&
\uparrow^\simeq
\\
&&
hocolim_{p,q \in \Delta^{op} \times \Delta^{op}}
Mor_C(\Gamma^p X, \Lambda_q Y)
\\
&&\downarrow^\simeq
\\
&&N wMor_C^3(X,Y)
\\
&&\downarrow^\simeq
\\
&&Mor_{L^H C}(X,Y)
}
\,.
$$
=--
The top row weak equivalences are those of prop. \ref{SimplicialFunctionComplexes}
### In a category of fibrant objects
{#InACategoryOfFibrantObjects}
There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a [[category of fibrant objects]].
This is described in ([Cisinksi 10](#Cisinski10)) and ([Nikolaus-Schreiber-Stevenson 12, section 3.6.2](#NSS12)).
## Properties
### Hom-spaces of equivalences
{#SpacesOfEquivalences}
+-- {: .num_theorem #DKTheorem}
###### Theorem
For $C$ a [[simplicial model category]] and $X$ an object, the [[delooping]] of the [[automorphism β-group]]
$$
Aut_W(X) \subset \mathbb{R}Hom(X,X)
$$
has the [[homotopy type]] of the component on $X$ of the [[nerve]] $N(C_W)$ of the [[subcategory]] of weak equivalences:
$$
\mathbf{B} Aut_W(X) \simeq N(C_W)_X
\,.
$$
The equivalence is given by a finite sequence of [[zig-zags]] and is natural with respect to [[sSet]]-[[enriched functors]] of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for $X$ again to a fibrant cofibrant object.
=--
This is [Dwyer-Kan 84, 2.3, 2.4](#DK84).
+-- {: .num_cor }
###### Corollary
For $C$ a model category, the simplicial set $N(C_W)$ is a model for the
[[core]] of the [[(β,1)-category]] determined by $C$.
=--
+-- {: .proof}
###### Proof
That core, like every [[β-groupoid]] is equivalent to the disjoint union over its connected components of the deloopings of the automorphism $\infty$-groups of any representatives in each connected component.
=--
## Related concepts
* [[hom-object]]
* [[hom-set]], [[hom-functor]]
* [[hom-category]]
* [[hom-space]], [[cocycle space]]
* [[simplicial mapping complex]]
[[!include homotopy-homology-cohomology]]
## References
For some original references by [[William Dwyer]] and [[Dan Kan]] see [[simplicial localization]]. For instance
* {#DK84} [[William Dwyer]], [[Dan Kan]], _A classification theorem for diagrams of simplicial sets_, Topology 23 (1984), 139-155.
On the derived [[function complexes]] in a [[projective model structure on simplicial presheaves]]:
* [[William Dwyer]], [[Daniel Kan]], *Function complexes for diagrams of simplicial sets*, Indagationes Mathematicae (Proceedings) **86** 2 (1983) 139-147 [<a href="https://doi.org/10.1016/1385-7258(83)90051-3">doi:10.1016/1385-7258(83)90051-3</a>, [pdf](https://core.ac.uk/download/pdf/82652265.pdf)]
Discussion in terms of [[quasi-categories]]:
* [[Jacob Lurie]], Section 1.2.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))
* [[Dan Dugger]], [[David Spivak]], *Mapping spaces in quasi-categories*, Algebraic & Geometric Topology **11** (2011) 263β325 [[arXiv:0911.0469](http://arxiv.org/abs/0911.0469), [doi:10.2140/agt.2011.11.263](http://dx.doi.org/10.2140/agt.2011.11.263)]
The theory of _framings_ is due to
* {#DHK} [[William Dwyer]], [[Philip Hirschhorn]], [[Dan Kan]], _Model categories and general abstract homotopy theory_, (1997) ([pdf](http://www.mimuw.edu.pl/~jacho/literatura/ModelCategory/DHK_ModelCateogories1.pdf))
and in parallel section 5 of
* {#Hovey} [[Mark Hovey]], _Model categories_ ([ps](http://math.unice.fr/~brunov/SecretPassage/Hovey-Model%20Categories.ps))
and in sections 16, 17 of
* [[Philip Hirschhorn]], _Model categories and their localization_ .
A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of
* [[Clark Barwick]], _On (enriched) left Bousfield localization of model categories_ ([arXiv](http://arxiv.org/abs/0708.2067))
Discussion of derived hom spaces for [[categories of fibrant objects]] is in
* {#Cisinski10} [[Denis-Charles Cisinski]], _Invariance de la K-théorie par equivalences dérivées_, J. K-theory, 6 (2010), 505–546.
and section 3.6.2 of
* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal β-bundles -- theory, presentations and applications|Principal β-bundles -- Presentations]]_ ([arXiv:1207.0249](http://arxiv.org/abs/1207.0249))
[[!redirects (infinity,1)-categorical hom-spaces]]
[[!redirects (β,1)-categorical hom-space]]
[[!redirects (β,1)-categorical hom-spaces]]
[[!redirects (infinity,1)-categorical hom space]]
[[!redirects (β,1)-categorical hom space]]
[[!redirects (infinity,1)-categorial hom-space]]
[[!redirects (β,1)-categorial hom-space]]
[[!redirects (infinity,1)-categorial hom space]]
[[!redirects (β,1)-categorial hom space]]
[[!redirects quasi-categorical hom-space]]
[[!redirects quasi-categorical hom-spaces]]
[[!redirects derived hom space]]
[[!redirects derived hom-space]]
[[!redirects derived hom spaces]]
[[!redirects derived hom-spaces]]
[[!redirects hom-space]]
[[!redirects hom-spaces]]
[[!redirects hom-β-groupoid]]
[[!redirects hom-infinity-groupoid]]
[[!redirects hom-(β,0)-category]]
[[!redirects hom-(infinity,0)-category]]
[[!redirects hom β-groupoid]]
[[!redirects hom infinity-groupoid]]
[[!redirects hom (β,0)-category]]
[[!redirects hom (infinity,0)-category]]
[[!redirects hom-β-groupoids]]
[[!redirects hom-infinity-groupoids]]
[[!redirects hom-(β,0)-categories]]
[[!redirects hom-(infinity,0)-categories]]
[[!redirects hom β-groupoids]]
[[!redirects hom infinity-groupoids]]
[[!redirects hom (β,0)-categories]]
[[!redirects hom (infinity,0)-categories]]
[[!redirects homotopy function complex]]
[[!redirects homotopy function complexes]]
[[!redirects (β,1)-hom (β,1)-functor)]]
[[!redirects (β,1)-categorical hom]]
[[!redirects (β,1)-categorical hom-spaces]]
[[!redirects derived mapping space]]
[[!redirects derived mapping spaces]]
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