!-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  §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  §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.

## 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}
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    "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)  [doi:10.1016/0304-3975(87)90045-4, [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]], §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]], §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]], §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]]:

* {#MihályiNovitzká13} Daniel Mihályi, Valerie Novitzká, 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]], §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 [doi:10.1016/0168-0072(93)90093-S]

* {#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 846–858 ([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]], §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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[[!redirects exponential conjunction]]
[[!redirects exponential modality]]
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[[!redirects exponential modalities]]
[[!redirects storage modalities]]

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[[!redirects dereliction rules]]


$\infty$-category > history

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't Hooft anomaly

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## 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

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## Idea

In [[Yang-Mills theory]]/[[QCD]] with [[coupling constant]] $g_{YM}$ and with $N_c$ [[colour charge|colours]], the _’t 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 ’t 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), doi:10.1016/0550-3213(74)90154-0)

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) $[$doi:10.1016/S0370-1573(99)00083-6, [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

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## Idea

### Double line notation

What is called _’t Hooft double line notation_ (following [’t 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

{#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:

{#Example} For example, for $\mathfrak{g} = \mathfrak{so}(N)$ a [[special orthogonal Lie algebra]], the ’t Hooft double line notation of a single trivalent vertex inside a single Wilson loop is the following:

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 [’t 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 [Cvitanović 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 [Cvitanović 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:

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:

{#SurfaceExample} For example:

> 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)):

> 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 ’t Hooft double line construction applies not just to [[Jacobi diagrams]] but also to [[Sullivan chord diagrams]]:

> 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 ’t Hooft]], _A Planar Diagram Theory for Strong Interactions_, Nucl. Phys. B72 (1974) 461 ([spire:80491](http://inspirehep.net/record/80491), doi:10.1016/0550-3213(74)90154-0)

reviewed in

* [[Gerard ’t 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 Cvitanović]], _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 Cvitanović]], _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 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

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) 43–52  ([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

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## 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_Hooft–Polyakov_monopole)_

[[!redirects 't Hooft-Polyakov monopoles]]


(-1)-category

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## 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 [[(−2)-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 (−1)-category]]
[[!redirects (−1)-categories]]


(-1)-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 (−1)-functor]]
[[!redirects (−1)-functors]]


(-1)-groupoid

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## 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 [[∞Grpd]].  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

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 [[(−1)-category]] for references on this sort of [[negative thinking]].

[[!redirects (−1)-poset]]


(-2)-category

There is just one _$(-2)$-category_, namely the truth value [[True]]. Compare the concepts of [[(−1)-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 [[(−1)-poset]]. If you think of a $(-1)$-category as a [[(−1)-groupoid]], then this makes a $(-2)$-category a [[(−2)-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 (−2)-category]]
[[!redirects (−2)-categories]]


(-2)-groupoid

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=--
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#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 [[∞Grpd]].

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

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# 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 [[(∞,1)-category]] such that every [[hom-space|hom ∞-groupoid]] is [[n-truncated|(n-1)-truncated]].

* a [[0-truncated]] [[∞-groupoid]] is equivalently a set;

* a [[(-1)-truncated]] [[∞-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]]

* [[∞-groupoid]]

* [[(0,0)-category]]

* [[(1,0)-category]]

* [[(n,0)-category]]

* [[(∞,0)-category]]

* [[0-category]]

* [[1-category]]

* [[2-category]]

* [[3-category]]

* [[4-category]]

* [[n-category]]

* [[∞-category]]

* **(0,1)-category**

* [[(1,1)-category]]

* [[(2,1)-category]]

* [[(n,1)-category]]

* [[(∞,1)-category]]

* [[(1,2)-category]]

* [[(∞,2)-category]]

* [[(∞,n)-category]]

* [[(n,r)-category]]

[[!redirects (0,1)-category]]
[[!redirects (0,1)-categories]]


(0,1)-category theory

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#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

**[[(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

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#### Topos theory
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[[!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]] [[(∞,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]]

* [[(∞,1)-presheaf]]

* [[(∞,n)-presheaf]]

[[!redirects (0,1)-presheaves]]

[[!redirects proposition-valued presheaf]]
[[!redirects proposition-valued presheaves]]


(0,1)-topos

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[[!include (0,1)-category theory - contents]]
=--
#### Topos Theory
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[[!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

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#### Higher category theory
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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

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In the pattern of [[(n,r)-categories]] a **$(1,1)$-category** is just an ordinary [[category]].

[[!redirects (1,1)-categories]]


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#### Functorial quantum field theory
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[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
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[[!include supergeometry - contents]]
=--
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=--

+-- {: .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]]: Spectral standard model and gravity

* **(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

+-- {: .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

[[!redirects (2,1)-algebraic theeory of E-infinity algebras]]

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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[(∞,1)-algebraic theory]] whose [[∞-algebra over an (∞,1)-algebraic theory|algebras]] are [[E-∞ 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 [[(∞,1)-category]] with [[(∞,1)-limit|(∞,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 [[(∞,1)Cat]] is
a [[symmetric monoidal (∞,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-∞ 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 [[∞Grpd]] on a single generator is $2Comm(*, -) : 2Comm \to \infty Grpd$. Its underlying [[∞-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-∞-algebra]], on the nose so if we use the [[Barratt-Eccles operad]] $P$ as our model for the [[E-∞-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-∞ algebras]]
[[!redirects (2,1)-algebraic theory of E-∞-algebras]]


(2,1)-category

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###Context###
#### 2-Category theory
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[[!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 [[(∞,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

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###Context###
#### Functorial quantum field theory
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[[!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

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###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 [[(∞,1)-categories]], then $(2,1)$-functors are equivalently the [[(∞,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

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###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]] [[(∞,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 [[∞-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]]

* [[(∞,1)-presheaf]]

* [[(∞,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

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###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]] [[(∞,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]]

* [[(∞,1)-sheaf]] / [[∞-stack]]

* [[(∞,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

[[!redirects (2,1)-sheafififcation]]

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+--{: .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**

* [[(∞,1)-sheafification]]

[[!redirects stackification]]

[[!redirects stackifications]]


(2,1)-site

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### Context
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Definition

A _$(2,1)$-site_ is an [[(∞,1)-site]]. whose underlying [[(∞,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 [[(∞,1)-category of (∞,1)-sheaves]] on a $(2,1)$-site is an [[n-localic (∞,1)-topos|2-localic (∞,1)-topos]].

## Related concepts

* [[site]]

* [[2-site]], **(2,1)-site**

  * [[(2,1)-sheaf]]

* [[(∞,1)-site]]

  * [[model site]], [[simplicial site]]

[[!redirects (2,1)-site]]
[[!redirects (2,1)-sites]]


(2,1)-topos of (2,1)-sheaves

## Idea

A [[(2,1)-topos]] of [[(2,1)-sheaves]]

## Related concepts

* [[sheaf topos]]

* [[(infinity,1)-category of (infinity,1)-sheaves]]