simplicial type theory

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### Context
#### Directed Type Theory
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[[!include directed homotopy type theory - contents]]
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## Idea

*Simplicial type theory* ([Riehl & Shulman 2017, §3](#RiehlShulman17)) is a [[homotopy type theory]] which introduces additional non-fibrant layers on top of [[Martin-Löf type theory]] to provide a logical calculus for a [[directed homotopy theory|directed]] [[interval type]] and with it for [[geometric shape for higher structures|geometric shapes]], consisting of [[Segal space]]-types, [[Rezk completion|Rezk complete]]-types, and covariant fibrations.

## Formal presentations

There are many different ways to formalize the directed interval primitive in simplicial type theory:

* one could simply postulate via inference rules and axioms a directed interval primitive in a separate layer.

* one could use [[two-level type theory]] consisting of a non-fibrant layer, and then axiomatise a directed interval primitive in the non-fibrant layer

* one could use [[type theory with shapes]] consisting of a non-fibrant layer of cubes which is a [[simple type theory]] and a non-fibrant layer of topes which is a [[propositional logic]] over the cube layer, and then define the directed interval primitive using the cube and tope layers

### Two-level type theory

We use $\Xi$ to represent the context in the non-fibrant layer, and $\Gamma$ to represent the context in the fibrant layer. Type judgments in the non-fibrant layer are represented by $A \; \mathrm{exotype}$, while type judgments in the fibrant layer are just represented by $A \; \mathrm{type}$.

#### Directed interval primitive

The directed interval primitive $\mathbb{2}$ in simplicial type theory is a non-trivial bounded [[total order]]. Bounded total orders have many definitions in mathematics, such as a bounded [[partial order]] satisfying totality, or a [[lattice]] satisfying totality. As such, there are multiple possible sets of inference rules one could use to present the directed interval in the two-level type theory formalization.

The inference rules for the directed interval primitive $(\mathbb{2}, \leq)$ as a [[partial order]] in the two-level type theory formalization are as follows:

$$\frac{\Xi \; \mathrm{ctx}}{\Xi \vdash \mathbb{2} \; \mathrm{exotype}} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi \vdash 0:\mathbb{2}} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi \vdash 1:\mathbb{2}} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash x \leq y \; \mathrm{exotype}} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash \tau_\leq(x, y):\mathrm{isProp}(x \leq y)}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{refl}_\leq(x):x \leq x} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2}, z:\mathbb{2}, p:x \leq y, q:y \leq z \vdash \mathrm{trans}_\leq(x, y, z, p, q):(x \leq z)}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{antisym}_\leq(x, y):\mathrm{isEquiv}(idtoleqgeq(x, y))} \quad \mathrm{where} \quad
\begin{array}{l}
idtoleqgeq(x, y):(x =_\mathbb{2} y) \to (x \leq y) \times (y \leq x) \\
idtoleqgeq(x, x)(\mathrm{id}_\mathbb{2}(x)) \coloneqq (\mathrm{refl}_\leq(x), \mathrm{refl}_\leq(x))
\end{array}
$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{totl}_\leq(x, y):[(x \leq y) + (y \leq x)]} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{bot}_\leq(x):0 \leq x} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{top}_\leq(x):x \leq 1}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi \vdash \mathrm{nontriv}_\leq:(0 =_\mathbb{2} 1) \to \mathbb{0}}$$

The inference rules for the directed interval primitive $\mathbb{2}$ as a [[lattice]] in the two-level type theory formalization are as follows:

$$\frac{\Xi \; \mathrm{ctx}}{\Xi \vdash \mathbb{2} \; \mathrm{exotype}} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi \vdash \tau_0:\mathrm{isSet}(\mathbb{2})} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi \vdash 0:\mathbb{2}} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi \vdash 1:\mathbb{2}} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash x \wedge y:\mathbb{2}} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash x \vee y:\mathbb{2}}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{idem}_\wedge(x):x \wedge x =_{\mathbb{2}} x} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{comm}_\wedge(x, y):x \wedge y =_{\mathbb{2}} y \wedge x}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2}, z:\mathbb{2} \vdash \mathrm{assoc}_\wedge(x, y, z):(x \wedge y) \wedge z =_{\mathbb{2}} x \wedge (y \wedge z)}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{lunit}_\wedge(x):1 \wedge x =_{\mathbb{2}} x} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{runit}_\wedge(x):x \wedge 1 =_{\mathbb{2}} x}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{idem}_\vee(x):x \vee x =_{\mathbb{2}} x} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{comm}_\vee(x, y):x \vee y =_{\mathbb{2}} y \vee x}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2}, z:\mathbb{2} \vdash \mathrm{assoc}_\vee(x, y, z):(x \vee y) \vee z =_{\mathbb{2}} x \vee (y \vee z)}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{lunit}_\vee(x):0 \vee x =_{\mathbb{2}} x} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2} \vdash \mathrm{runit}_\vee(x):x \vee 0 =_{\mathbb{2}} x}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{asorp}_\wedge(x, y):x \wedge (x \vee y) =_{\mathbb{2}} x} \qquad \frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{asorp}_\vee(x, y):x \vee (x \wedge y) =_{\mathbb{2}} x}$$

$$\frac{\Xi \; \mathrm{ctx}}{\Xi, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{totl}(x, y):[(x \wedge y =_{\mathbb{2}} x) + (x \wedge y =_{\mathbb{2}} y)]} \quad \frac{\Xi \; \mathrm{ctx}}{\Xi \vdash \mathrm{nontriv}_\leq:(0 =_\mathbb{2} 1) \to \mathbb{0}}$$

#### Simplices and subshapes

We inductively define the $n$-simplices as the following family of non-fibrant types:

* The $0$-simplex is the unit type in the non-fibrant layer
$$\Delta^0 \coloneqq \mathbf{1}$$

* In the non-fibrant layer, let $\mathrm{Fin}(n) \coloneqq \sum_{i:\mathbb{N}} i \lt n$ be the canonical [[finite type]] with $n$ elements. The $n+1$-simplex is defined as the type
$$n:\mathbb{N} \vdash \Delta^{n + 1} \coloneqq \sum_{t:\mathrm{Fin}(n + 1) \to \mathbb{2}} \prod_{i:\mathrm{Fin}(n)} t(\pi_1(i + 1)) \leq t(\pi_1(i))$$

The first few simplicies can be explicitly written as

$$\Delta^1 \coloneqq \mathbb{2}$$
$$\Delta^2 \coloneqq \sum_{t:\mathrm{Fin}(2) \to \mathbb{2}} t(1) \leq t(0)$$
$$\Delta^3 \coloneqq \sum_{t:\mathrm{Fin}(3) \to \mathbb{2}} (t(2) \leq t(1)) \times (t(1) \leq t(0))$$

Subshapes of the simplicies could also be defined.

* The boundary of the $1$-simplex is defined as
$$\partial \Delta^1 \coloneqq \sum_{t:\Delta^1} [(t =_{\Delta^1} 0) + (t =_{\mathbb{2}} 1)]$$

* The boundary of the $2$-simplex is defined as
$$\partial \Delta^2 \coloneqq \sum_{t:\mathrm{Fin}(2) \to \mathbb{2}} [(t(0) =_{\mathbb{2}} 0) \times (t(0) \leq t(1)) + (t(0) =_{\mathbb{2}} t(1)) + (t(0) \leq t(1)) \times (t(1) =_{\mathbb{2}} 1)]$$

### Type theory with shapes

...

## See also

* [[Reedy model structure]]

* [[bisimplicial set]]

* [[synthetic (infinity,1)-category theory|synthetic $(\infty,1)$-category theory]]

* [[type theory with shapes]]

* [[two-level type theory]]

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Mike Shulman]], *A type theory for synthetic ∞-categories*, Higher Structures **1** 1 (2017) [[arxiv:1705.07442](https://arxiv.org/abs/1705.07442), [published article](https://higher-structures.math.cas.cz/api/files/issues/Vol1Iss1/RiehlShulman)]

* [[Jonathan Weinberger]], *A Synthetic Perspective on $(\infty,1)$-Category Theory: Fibrational and Semantic Aspects* $[$[arXiv:2202.13132](https://arxiv.org/abs/2202.13132)$]$

* [[Jonathan Weinberger]], *Strict stability of extension types* $[$[arXiv:2203.07194](https://arxiv.org/abs/2203.07194)$]$

A talk on [[synthetic (infinity,1)-category theory]] in [[simplicial type theory]] and [[infinity-cosmos]] theory:

* [[Emily Riehl]], *The synthetic theory of infinity-categories vs the synthetic theory of infinity-categories*, [[Homotopy Type Theory Electronic Seminar Talks]], 1 March 2018, ([video](https://www.youtube.com/watch?v=ge-9m1SsEmc), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Riehl-2018-03-01-HoTTEST.pdf))

A [[proof assistant]] implementing simplicial type theory:

* [[Rzk]] [[github.com/fizruk/rzk](https://github.com/fizruk/rzk)]

Formalization of the [[(infinity,1)-Yoneda lemma|$(\infty,1)$-Yoneda lemma]] via [[simplicial homotopy type theory]] (in [[Rzk]]):

* [[Nikolai Kudasov]], [[Emily Riehl]], [[Jonathan Weinberger]]. *Formalizing the $\infty$-categorical Yoneda lemma* (2023) [[arXiv:2309.08340](https://arxiv.org/abs/2309.08340)]

[[!redirects simplicial homotopy type theory]]


simplicial vector space

A **simplicial vector space** is a [[simplicial object]] in the category [[Vect]] of vector spaces over some field.


simplicial weak equivalence

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## Idea

A _simplicial weak equivalence_ is a [[weak equivalence]] in the standard [[homotopy theory]] of [[simplicial sets]], hence
with respect to the [[classical model structure on simplicial sets]].

## Definition

A [[morphism]] $f$ between [[simplicial sets]] is a __simplicial weak equivalence__ if any of the following equivalent conditions is satisfied (where  $Ex^\infty$ denotes [[Kan fibrant replacement]]):

* $Ex^\infty(f)$ is a [[simplicial homotopy equivalence]].

* $Ex^\infty(f)$ has the [[right homotopy lifting property]] with respect to $\partial\Delta^n\to\Delta^n$.

* $Ex^\infty(f)$ induces [[isomorphisms]] on [[simplicial homotopy groups]],
i.e., $Ex^\infty(f)$ induces an [[isomorphism]] on $\pi_0$ and all [[homotopy groups]] for any choice of basepoints.

* $Hom(f, A)$ is a [[simplicial homotopy equivalence]] for every [[Kan complex]] $A$ (where $Hom$ denotes the [[hom-functor]]).

* $f$ has the [[right homotopy lifting property]] with respect to $Sd^i \partial\Delta^n \to Sd^i \Delta^n$ (allowing [[subdivisions]] for homotopies also).

* $f$ belongs to the class of [[weak equivalences]] in the [[classical model structure on simplicial sets]], whose [[cofibrations]] are [[monomorphisms]] and [[fibrant objects]] are [[Kan complexes]].

* The morphism $f$ is a [[composition]] of a [[trivial cofibration]] and a [[trivial fibration]] on the [[classical model structure on simplicial sets]], both of which are defined using [[lifting properties]].

* Applying the [[category of elements]] [[functor]] produces a [[Thomason weak equivalence]] of categories. The class of [[Thomason weak equivalences]] forms the smallest [[basic localizer]], i.e., the smallest class of [[functors]] between [[small categories]] that contains identities, is closed under [[retracts]] and the 2-out-of-3 property, contains all functors $A \to 1$ for which the category A has a [[terminal object]], and is locally determined: if $u\colon A\to B$ and $w\colon B\to C$ are functors, with $v=w\circ u\colon A\to C$, and for any $c\in C$ the induced functor of [[comma categories]] $v/c\to w/c$ is a [[Thomason weak equivalence]], then so is $u$.

## Related concepts

* [[simplicial homotopy equivalence]]

* [[weak equivalence]]

* [[weak homotopy equivalence]]

[[!redirects simplicial weak equivalences]]

[[!redirects weak equivalence of simplicial sets]]
[[!redirects weak equivalences of simplicial sets]]

[[!redirects simplicial weak homotopy equivalence]]
[[!redirects simplicial weak homotopy equivalences]]


simplicial weak omega-category

a "simplicial weak $\omega$-category" can be used to mean two different things:

* a [[simplicial object]] in some category of weak [[omega-category|∞-categories]];

* a [[simplicial model for weak ∞-categories]].

However, when you run into the term in the literature, chances are high that what is meant is the second sense. See there for references.

[[!redirects simplicial weak omega-categories]]
[[!redirects simplicial weak ∞-category]]
[[!redirects simplicial weak ∞-categories]]


simplicially enriched category

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#### Enriched category theory
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#### Homotopy theory
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#### $(\infty,1)$-Category theory
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# Contents
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## Idea

A _simplicially enriched category_ is a [[category]] with a [[simplicial set]] of [[morphism]]s between any two objects.

One may think of the 1-cells in a hom-simplicial set as a [[2-morphism]], the 2-cells as a [[3-morphism]] and generally a $(k-1)$-cell as a [[k-morphism]]. Therefore simplicially enriched categories may serves as models for [[∞-categories]]. Precisely which notion of $\infty$-category depends on which extra [[stuff, structure, property|structure and property]] one imposes.

For instance

* requiring the hom-simplicial sets to be [[Kan complex]]es makes simplicially enriched categories a model for [[(∞,1)-categories]];

* similary, equipping the $sSet$-enriched category with the structure of a $sSet_{Quillen}$-[[enriched model category]] -- a [[simplicial model category]] -- makes it a model for an $(\infty,1)$-category.

  This is discussed in more detail at [[relation between quasi-categories and simplicial categories]].

* on the other hand, equipping the $sSet$-enriched category with the structure of an $sSet_{Joyal}$-enriched model category over the Joyal-[[model structure for quasi-categories]] makes it a model for an [[(∞,2)-category]].

## Definition

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###### Definition

A _simplicially enriched category_ is a [[enriched category|category enriched over]] the [[cartesian monoidal category]] [[sSet]] of [[simplicial sets]].

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###### Remark

These $sSet$-enriched categories are sometimes, somewhat imprecisely, called just _[[simplicial categories]]_.

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There is a related notion of [[simplicial groupoid]] with the added requirement that all arrows in the categories concerned are [[isomorphisms]].

## As models for $(\infty,1)$-categories

Since simplicial sets that are [[Kan complex]]es are an incarnation of [[∞-groupoid]]s, an $sSet$-category all whose [[hom-object]]s happen to be [[Kan complex]]es may be regarded as a category enriched in [[∞-groupoid]]s. By the logic of [[(n,r)-category]] theory this should be a model for an [[(∞,1)-category]].

Treating simplicial categories this way as models for $(\infty,1)$-categories is one of the central tools in [[homotopy coherent category theory]].

Indeed, there is a [[model structure on simplicial categories]] whose fibrant objects are [[Kan complex|Kan-complex]]-enriched categories, and which is one model for the [[(∞,1)-category of (∞,1)-categories]].

By a web of [[Quillen equivalence]]s this is related to the other incarnations of $(\infty,1)$-categories. Notably to [[quasi-categories]] and [[complete Segal space]]s. For more on this see

* [[relation between quasi-categories and simplicial categories]].

### Properties

#### Simplicial localization

To every [[category with weak equivalences]] $(C,W)$ is associated its [[simplicial localization]] $L_W C$, which is an $sSet$-category with the property that its [[homotopy category of an (∞,1)-category]] coincides with the [[homotopy category]] $Ho_W(C)$.

#### Model structure

There is a [[model structure on sSet-categories]] that presents the [[(∞,1)-category]] [[(∞,1)Cat]].

#### Homotopy Kan extension

The notion of [[homotopy Kan extension]] and hence in particular that of [[homotopy limit]] and [[homotopy colimit]] has a direct formulation in terms of [[Kan complex|Kan-complex]]-enriched categories.  See [[homotopy Kan extension]] for more.

#### Presentation of $(\infty,1)$-topos theory

All of [[(∞,1)-topos theory]] can be modeled in terms of $sSet$-categories. ([ToënVezzosi](#Toenvezzosie)). There is a notion of [[sSet-site]] $C$ that models the notion of [[(∞,1)-site]] and a [[model structure on sSet-enriched presheaves]] on $sSet$-sites that is a [[presentable (∞,1)-category|presentation]] for the [[∞-stack]] [[(∞,1)-topos]]es on $C$.

## As models for $(\infty,2)$-categories

See [[(∞,2)-category]].

## Related concepts

* [[simplicial category]]

  * **simplicially enriched category**

  * [[simplicial object in Cat]]

* [[simplicial groupoid]]

* [[relation between quasi-categories and simplicial categories]]

* [[topologically enriched category]]

## References
 {#References}

Discussion the context of ([[simplicial model category|simplicial]]) [[model category]] theory:

* {#Quillen67} [[Daniel Quillen]], II §1 of: *[[Homotopical Algebra]]*, Lecture Notes in Mathematics **43**, Springer (1967) [[doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)]

* {#Hirschhorn02} [[Philip Hirschhorn]], Section 9.1.1 of: _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

* {#ToenVezzosi} [[Bertrand Toën]], [[Gabriele Vezzosi]], Section 2 of: *Topos theory*, Advances in Mathematics **193** 2 (2005) 257-372 [[arXiv:0207028](http://arxiv.org/abs/math/0207028), [doi:10.1016/j.aim.2004.05.004](https://doi.org/10.1016/j.aim.2004.05.004)]

* [[Jacob Lurie]], Section A.3 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

* [[Emily Riehl]], §3.6 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

The original references in the context of [[simplicial localization]]:

* {#DwyerKanLocalizations} [[William Dwyer]], [[Daniel Kan]], _Simplicial localizations of categories_, J. Pure Appl. Algebra **17** 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3]

* {#DwyerKanEquivalences} [[William Dwyer]], [[Daniel Kan]], *Equivalences between homotopy theories of diagrams*, in: *Algebraic topology and algebraic K-theory*, Ann. of Math. Stud. **113**, Princeton University Pr
Simplicially enriched categories as models for $(\infty,1)$-categories are also discussed in

Homotopy coherent category theory on $sSet$-categories is discussed in

*  [[Jean-Marc Cordier]] and [[Timothy Porter]], _Homotopy Coherent Category Theory_, Trans. Amer. Math. Soc. 349 (1997) 1-54, ([pdf](http://www.ams.org/journals/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf))2-2.pdf))

which describes resolutions of the simplicial functor categories between two simplicial categories and

* [[Michael Batanin]], _Homotopy coherent category theory and $A_\infty$-structures in monoidal categories_ ([pdf](http://dodo.pdmi.ras.ru/~topology/books/batanin.pdf))

which shows that these resolved functor categories are in fact $sSet$-[[A-∞ categories]].

[[!redirects simplicially enriched category]]
[[!redirects simplicially enriched categories]]
[[!redirects sSet-enriched category]]
[[!redirects sSet-enriched categories]]
[[!redirects SSet-enriched category]]
[[!redirects SSet-enriched categories]]
[[!redirects SimpSet-enriched category]]
[[!redirects SimpSet-enriched categories]]
[[!redirects sSet-category]]
[[!redirects sSet-categories]]
[[!redirects SSet-category]]
[[!redirects SSet-categories]]
[[!redirects SimpSet-category]]
[[!redirects SimpSet-categories]]


simply connected space

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## Definition

A __simply connected topological space__ is a [[n-connected space|1-connected]] [[topological space]] $X$: a [[connected space]] whose [[fundamental group]] is the [[trivial group]]:

$\pi_1(X) = \{\mathrm{e}\}$.

Equivalently: if it is a [[n-connected object of an (infinity,1)-topos|simply connected object]] of the [[(∞,1)-topos]] [[Top]].

See [[n-connected object of an (infinity,1)-topos]] for more.

## Related concepts

* [[connected topological space]]

* [[simple topological space]]

* [[nilpotent topological space]]

[[!redirects simply connected space]]
[[!redirects simply connected spaces]]
[[!redirects simply-connected space]]
[[!redirects simply-connected spaces]]

[[!redirects simply connected topological space]]
[[!redirects simply connected topological spaces]]
[[!redirects simply-connected topological space]]
[[!redirects simply-connected topological spaces]]

[[!redirects simply connected]]
[[!redirects simply-connected]]

[[!redirects simply connected homotopy type]]
[[!redirects simply connected homotopy types]]

[[!redirects simply-connected homotopy type]]
[[!redirects simply-connected homotopy types]]


simply sorted set theory

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## Overview

Simply sorted set theory is a [[set theory]] which is a [[first order theory]] without [[dependent type|dependent]] [[sorts]]; i.e. written in [[first order logic]] and whose [[domains of discourse]] (called [[sorts]] here) do not on other domains of discourse. In addition, membership $a \in A$ is a [[relation]], rather than a [[judgment]] as in [[dependently sorted set theory]]. Simply sorted set theories come in both [[material set theories]] and [[structural set theories]].

Simply sorted set theory could be distinguished between how many sorts the theory has; these include

* one-sorted set theory, which include certain presentations of [[ZFC]], [[Mostowski set theory]], [[fully formal ETCS]], and [[ZFA]]
* [[two-sorted set theory]], which include certain presentations of [[ETCS]] and [[ZFA]],
* [[three-sorted set theory]], which include [[three-sorted SEAR]], [[three-sorted structural ZFC]] as well as certain presentations of [[ETCS with elements]].
* four-sorted set theory, which include certain presentations of [[SEPS]].

## Notions of sets and elements

\begin{definition}
A notion of set and element in a simply sorted first-order theory consists of:

* A sort $S$ of probable sets. By probable set, we mean that certain terms of $S$ are to be considered as the sets in the theory.

* A sort $E$ of probable elements. By probable element, we mean that certain terms of $E$ are to be considered as the elements in the theory.

* A binary predicate in $S$, $(-)=_S(-)$, called *[[definitional equality]] of sets*

* A binary predicate in $E$, $(-)=_E(-)$, called *[[equality]] of elements*

* A unary predicate in $S$, $\mathrm{set}(-)$, called *being a set*

* A unary predicate in $E$, $\mathrm{element}(-)$, called *being an element*

* A binary predicate in $E$ on the left and $S$ on the right, $(-)\in(-)$, called *membership*

such that

* $(-)=_E(-)$ is an [[equivalence relation]]

* if $a \in b$, then $a$ is an element and $b$ is a set

$$\forall a:E.\forall b:S.(a \in b) \to \mathrm{element}(a) \wedge \mathrm{set}(b)$$

* weak extensionality for sets and elements: for all $a:S$ and for all $b:S$, if a and b are sets, then $a =_S b$ if and only if, for all $c:E$, if $c$ is an element, then $c \in a$ if and only if $c \in b$

$$\forall a:S.\forall b:S.(\mathrm{set}(a) \wedge \mathrm{set}(b)) \implies ((a =_S b) \iff \forall c:E.(\mathrm{element}(c) \implies ((c \in a) \iff (c \in b))))$$
\end{definition}

In one-sorted set theory, the sorts $E$ and $S$ are the same sort.

## See also

* [[unsorted set theory]], [[simply sorted set theory]], [[dependently sorted set theory]]
* [[material set theory]], [[structural set theory]]

[[!redirects simply sorted set theory]]
[[!redirects simply sorted set theories]]

[[!redirects simply-sorted set theory]]
[[!redirects simply-sorted set theories]]


SimpSet

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## Definition

The [[category]] $SimpSet$, or $sSet$ for short, is the category whose [[objects]] are [[simplicial sets]] and whose [[morphisms]] are [[simplicial maps]].

[[equivalence of categories|Equivalently]], this is the [[functor category]] from the [[opposite category]] $\Delta^{op}$ of the [[simplex category]] $\Delta$ to the [[category of sets]]:

$$
  SimpSet \simeq [\Delta^{op}, Set]
  \,.
$$

## Properties

### Basic properties

Many basic properties of the category of simplicial sets follow as it being a special case of a [[category of presheaves]] over a [[small category]] (namely over the [[simplex category]]), and hence in particular a [[Grothendieck topos]].

For example, this immediately implies (see [there](category+of +presheaves#GeneralProperties)) that and how $sSet$ is [[complete category|complete]] and [[cocomplete category|cocomplete]] (with [[limits]] and [[colimits]] constructed levelwise) and [[cartesian closed category|cartesian closed]].

In this vein:

\begin{prop}\label{EpimorphismsOfSimplicialSetsDetectedDegreewise}
  A morpism $X_\bullet \xrightarrow{f_\bullet} Y_\bullet$ of simplicial sets is

* a [[monomorphism]] precisely if all component [[functions]] $f_n \colon X_n \to Y_n$ are [[injections]];

* an [[epimorphism]] precisely if all component [[functions]] $f_n \colon X_n \to Y_n$ are [[surjections]].

\end{prop}
\begin{proof}
  This is the immediate specialization of [this Proposition](category+of+presheaves#EpimorphismsOfPresheavesAreDetectedObjectwise) for general presheaves.
\end{proof}

### Cartesian closed monoidal structure

As described at [[closed monoidal structure on presheaves]]
the cartesian tensor product $S \otimes T = S \times T$ of simplicial sets $S$ and $T$ is the simplicial set
$$
  (S \otimes T) : [n] \mapsto S_n \times T_n
  \,,
$$
where the product on the right is the cartesian product in [[Set]].

One central reason why simplicial sets are useful and important is that this simple monoidal structure ("disturbingly simple minded" in the words of [Friedman08, p. 24](http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.4221v1.pdf#page=24)) actually does fully capture the standard monoidal structure on [[topological spaces]] under [[geometric realization]] $|\cdot| : SSet \to Top$

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

For $S$ and $T$ simplicial sets, we have
$$
  |S \times T| \simeq |S| \times |T|
  \,,
$$
where on the right the cartesian product is in the [[nice category of spaces|nice category]] of compactly generated Hausdorff spaces.
=--

See also at *[[products of simplices]]*.

\linebreak

As described at [[closed monoidal structure on presheaves]],
the [[internal hom]]  $[S,T]$ of simplicial sets is the simplicial set
$$
  [S,T] : [n] \mapsto Hom_{SSet}(S \times \Delta[n], T)
  \,,
$$
where $\Delta[n] = Hom_{\mathbf{\Delta}}(-,[n])$ is the standard simplicial $n$-[[simplex]], the image of $[n] \in \mathbf{\Delta}$ under the [[Yoneda embedding]]. This formula is clearly representing a Kan extension.

### Adjunctions

The maps $N: \Cat \rightarrow \Simp\Set$ and $S: \Top \rightarrow \Simp\Set$ described in the examples are actually functors, both of which have left adjoints. These adjoint pairs are examples of a very general sort of adjunction involving simplicial sets, of which there are many examples.

Let $E$ be any cocomplete category and let $F: \Delta \rightarrow E$ be a functor. We define the right adjoint $R : E \rightarrow \Simp\Set$ as follows. Given an object $e \in E$ the $n$-simplices of $Re$ are defined to be the set $E(F[n],e)$ of morphisms in $E$ from $F[n]$ to $e$. Face and degeneracy maps are given by precomposition by the appropriate (dual) maps in the image of $F$. $R$ is defined on morphisms by postcomposition.

The left adjoint $L$ is defined to be the left [[Kan extension]] of $F$ along the Yoneda embedding $y: \Delta \rightarrow \Simp\Set$. Because the $y$ is full and faithful, we will have $Ly = F$, i.e., $L (\Delta[n]) = F[n]$. By specifying $F$, we have already defined a functor to $E$ on the represented simplicial sets; $L$ is the unique cocontinuous extension of this functor to $\Simp\Set$. It can be described explicitly on objects as a [[end|coend]], or as a [[weighted limit|weighted colimit]].

(Easy) [[category theory|abstract nonsense]] shows that $L$ and $R$ form an [[adjunction|adjoint pair]] $L \dashv R$.

Here are some examples:

* Let $E = \Cat$ and $F$ be the functor $[n] \mapsto [n]$ (the inclusion of posets into categories). The right adjoint is the nerve functor $N$ described above. The left adjoint ${\tau}_1$ takes a simplicial set to its **fundamental category**.

* Let $E = \Top$ and $F$ be the functor $[n] \mapsto {\Delta}_n$. The right adjoint is the total singular complex functor $S$ described above. The left adjoint $|-|$ is called **[[geometric realization]]**.  As a consequence of the Kan extension construction, the geometric realization of the represented simplicial set $\Delta[n]$ is the standard $n$-simplex ${\Delta}^n$.

* (Barycentric) subdivision and extension $\sd: \Simp\Set \leftrightarrow \Simp\Set :\ex$.

* The [[homotopy coherent nerve]] functor and its left adjoint $\Simp\Set \leftrightarrow \Simp\Cat$ where [[SimpCat]] denotes the category of [[simplicially enriched category|simplicially enriched categories]], i.e., categories enriched in $\Simp\Set$.

* The adjunction $- \times X: \SimpSet \leftrightarrow \SimpSet :(-)^X$ between the product with a simplicial set $X$ and the internal-hom, which makes $\Simp\Set$ into a [[cartesian closed category]].

* Let $E$ be a [[Grothendieck topos]] equipped with an "interval" $I$, i.e. a [[total order|totally ordered object]] in the [[internal logic]] equipped with distinct top and bottom elements.  Then we have the functor $\Delta \to E$ sending $[n]$ to the subobject $ \{ (x_1,x_2,\dots,x_n) \;|\; x_1 \le x_2 \le \dots \le x_n \} \hookrightarrow I^n $ which gives rise to a [[geometric morphism]] $E\to \SimpSet$.  Therefore, $\SimpSet$ is the [[classifying topos]] of such "intervals".

### Model category structures

There are important [[model category]] structures on $sSet$.

* The standard [[model structure on simplicial sets]] [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category]] [[∞Grpd]] of [[∞-groupoid|∞-groupoids]].

* The [[model structure for quasi-categories]] on $sSet$ presents the [[(∞,2)-category]] of [[(∞,1)-categories]] [[(∞,1)Cat]].

### Internal logic

Like any [[elementary topos]], $\SimpSet$ has an [[internal logic]].  Here we list some properties of this logic.

* It is a [[two-valued topos]], i.e. the only subobjects of $1 = \Delta^0$ are $0$ and $1$.  (This is not really a property of the internal logic, but we include it to contrast with the next point.)

* It is not [[Boolean topos|Boolean]].  In general, the [[complement]] of a simplicial subset $A\subseteq B$ is the *full* simplicial subset on the vertices of $B$ not contained in $A$ ("full" meaning it contains a simplex of $B$ as soon as it contains all its vertices).  Thus, $A\cup \neg A = B$ only if $A$ is a [[connected component]] of $B$, i.e. any simplex with at least one vertex in $A$ lies entirely in $A$.

* By [[Diaconescu's theorem]], $\SimpSet$ therefore does not satisfy the [[axiom of choice]].

* Like any presheaf topos, it satisfies [[dependent choice]] (assuming
 it holds in the metatheory); see [Fourman and Scedrov](#FourmanScedrov). Moreover, [[natural numbers object]] is simply the discrete simplicial set of ordinary natural numbers.

* Similarly, it satisfies [[Markov's principle]].

* Less obviously, it satisfies the [[Kreisel-Putnam axiom]] that $(\neg p \to (q\vee r)) = ((\neg p \to q) \vee (\neg p \to r))$; see [this MO question](http://mathoverflow.net/questions/159989/internal-logic-of-the-topos-of-simplicial-sets/) and answers.

## Related concepts

* [[asSet]] ([[category]] of [[augmented simplicial sets]])

* [[Ho(sSet)]] (the [[homotopy category]] of [[simplicial sets]])

* [[dSet]] ([[category]] of [[dendroidal sets]])

## References
* {#FourmanScedrov} [[Michael Fourman]] and [[Andrej Ščedrov]], The "world's simplest axiom of choice" fails, manuscripta mathematica 1982, Volume 38, Issue 3, pp 325-332 [PDF](http://link.springer.com/content/pdf/10.1007%2FBF01170929.pdf)

category: category

[[!redirects SSet]]
[[!redirects sSet]]
[[!redirects Simp Set]]

[[!redirects category of simplicial sets]]
[[!redirects categories of simplicial sets]]

[[!redirects SimplicialSets]]


Simpson's conjecture

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

[[Carlos Simpson]] conjectured, in the context of [[semi-strict infinity-categories]] that every [[weak omega-category]] is [[equivalence|equivalent]] to one that is a [[strict omega-category]] except for the units.

## Statement

In

* {#Simpson98} [[Carlos Simpson]], _Homotopy types of strict 3-groupoids_ ([arXiv:9810059](http://arxiv.org/abs/math/9810059))

it was conjectured ([page 27](http://arxiv.org/PS_cache/math/pdf/9810/9810059v1.pdf#page=27)) that

**Simpson's Conjecture:** _Every weak $\infty$-[[infinity-category|category]] is equivalent to an $\infty$-category in which [[strict omega-category|composition and exchange laws are strict]] and only the unit laws are allowed to hold weakly._

For weak unit laws in $\infty$-categories see

* Joachim Kock,  _Weak identity arrows in higher categories_ ([arXiv](http://arxiv.org/abs/math.CT/0507116))

In

* {#JoyalKock06} [[Andre Joyal]], Kock, _Weak units and homotopy 3-types_ ([arXiv](http://arxiv.org/abs/math.CT/0602084))

Simpson's conjecture is proven up to the case of 3-categories with a single object.

## Remarks

One expects several alternative such [[semi-strict infinity-category|semi-strictification]] statements. Eugenia Cheng and Nick Gurski write the following at the end of their paper:

+--{style="font-size: 90%; margin-left:2em;"}
Finally we consider the question of eliminating the distinguished invertible
elements by using a stricter form of $n$-category. Generalising from the previous
sections, we see that we do not need to restrict all the way to strict $n$-categories
– a semistrict version will suffice. One form of semistrictness has everything strict
except interchange (cf. Gray-categories and see Crans 2000b, Crans 2000a); another has everything strict except units (Koch 2005, Simpson 1998). These have both been proposed
as solutions to the coherence problem for $n$-categories.

However, there are other possible "shades" of semistrictness and the above
notions do not appear to be right for the present purposes. Instead, we need a
form of semistrict $n$-category in which the units and interchange for $(n-1)$-cells
are strict, but everything else can be weak. This is to eliminate the constraint
$n$-cells that become distinguished invertible elements in our $n$-degenerate situation;
we expect that as in the case $n = 2$ the associator is automatically forced to be the
identity.

**Hypothesis 5.3. Semistrictness**
Let $n \ge 3$. Then an $n$-degenerate semistrict $n$-category in the above sense is precisely a commutative monoid.
=--
E. Cheng and N. Gurski's,  _The periodic table of $n$-categories for low dimensions I_ ([arXiv](http://arxiv.org/abs/0708.1178))

## History
 {#History}

The conjecture was initially triggered by the claim in

* {#KapranovVoevodsky91} [[Mikhail Kapranov]],  [[Vladimir Voevodsky]],  _$\infty$-Groupoids and homotopy types_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 32 no. 1 (1991), p. 29-46  ([Numdam](http://www.numdam.org/item?id=CTGDC_1991__32_1_29_0))

that "[[semi-strict infinity-category|semi-strict infinity-groupoids]]", in which only the [[inverses]] could be weak and everything else strict, model all [[homotopy types]] (hence solve the [[homotopy hypothesis]])

In ([Simpson 98](#Simpson98)) it was pointed out that this seems to be wrong for the [[homotopy type]] of the [[sphere|2-sphere]].

The issue, however, is quite subtle, as highlighted by Voevodsky [here](http://ncatlab.org/nlab/show/homotopy+type+theory#VoevodskyIASTalk2014))

(For instance  Corollary 5.2 in ([Simpson 98](#Simpson98)) leaves open the possibility that the [[geometric realization]] in  ([Kapranov-Voevodsky 91](#KapranovVoevodsky91)) is essentially surjective, only that the homotopy groups of the resulting space could not be read off from the groupoid in the obvious way.

Examining [KV 91](#KapranovVoevodsky91)'s motivation, Simpson noticed a singularity at the origin of [[Moore paths]], which led him to conjecture that weakening invertibility and units (while keeping strict associativity and interchange) would be enough to get at all homotopy types. Specifically, he writes on p. 27

> I think that the argument of (K. and V.) (which is unclear on the question of identity elements) actually serves to prove the above statement. I have called the above statement a "conjecture" because I haven't checked this.

[[!redirects Simpson conjecture]]
[[!redirects Simpson's conjecture]]
[[!redirects Simpson's conjecture]]
[[!redirects Simpson\'s conjecture]]
[[!redirects Simpson's conjecture]]
[[!redirects Simpson's conjecture]]


simulation

# Simulations
* table of contents
{: toc}

## Idea

A simulation is a [[morphism]] of [[coalgebra for an endofunctor|coalgebras for an endofunctor]].

## Definition

Let $C$ be a [[category]], and let $F\colon C \to C$ be an [[endofunctor]].  Recall that a [[coalgebra for an endofunctor|coalgebra]] of $F$ is an [[object]] $X$ of $C$ equipped with a [[morphism]] $\theta\colon X \to F(X)$.

+-- {: .un_defn}
###### Definition

Given two coalgebras $X, X'$ of $F$, a __simulation__ of $X$ in $X'$ is a morphism $s\colon X \to X'$ in $C$ such that $F(s) \circ \theta = \theta' \circ s$.  That is, this [[commutative diagram|diagram commutes]]:
$$ \array {
            X & \overset{\theta}\to & F(X) \\
            \llap{s}\downarrow & & \downarrow\rlap{F(s)} \\
            X' & \underset{\theta'}\to & F(X')
          } $$
=--

The terminology comes from [[computer science]], and we should probably explain what it's about.

## Examples

### Relations

Consider the [[category of sets]] and the (covariant) [[power set]] functor $\mathcal{P}$.  A coalgebra of $\mathcal{P}$ is a [[set]] $X$ equipped with a [[function]] $\theta\colon X \to \mathcal{P}(X)$, which is the same thing as a [[binary relation]] $\prec$ on $X$.  To fix the notation, we write $a \prec b$ ($a$ __precedes__ $b$) if $a \in \theta(b)$.

Then in concrete terms, a simulation of $X$ in $X'$ is a function $s\colon X \to X'$ such that

* $s(a) \prec' s(b)$ whenever $a \prec b$, and
* $t = s(a)$ for some $a \prec b$ whenever $t \prec' s(b)$.

Simulations are commonly used as morphisms between sets equipped with [[well founded relation|well-founded]] or [[extensional relation|extensional]] relations.  In particular, between [[well-ordered sets]] (sets equipped with relations that are both well-founded and extensional, as well as [[transitive relation|transitive]]), a simulation (if it exists) is unique, and we recover the usual [[poset|ordered]] [[proper class|class]] of [[ordinal numbers]] as a [[thin category]].

### Open maps in algebraic set theory

In their book _Algebraic Set Theory_, Joyal and Moerdijk present the notion of a pretopos equipped with a class of open maps. As they remark, every presheaf category carries a canonical class of open maps. We reformulate their notion of canonical open map for an example of primary interest to them, the category of forests, in coalgebraic terms.

The [[category of forests]] is by definition the [[presheaf|category of presheaves]] over the first infinite ordinal $\omega$. Concretely, a forest $F$ is a diagram of sets and functions

$$\ldots \to F_{n+1} \to F_n \to \ldots \to F_1 \to F_0.$$

If $y \in F_{n+1}$ maps to $x \in F_n$, we call $y$ a **predecessor** of $x$.

The successor map $s: \omega \to \omega$ induces a pullback functor

$$s^\ast = Set^{s^{op}}: Set^{\omega^{op}} \to Set^{\omega^{op}}.$$

Let $P$ be the covariant power-object functor on the presheaf [[topos]]. Each forest $F$ carries a canonical coalgebra structure over the endofunctor $P s^\ast$. Namely, the structure is a presheaf map $\theta_F \to P s^\ast F$ that corresponds to the subobject

$$R \hookrightarrow F \times s^\ast F$$

where the subset $R_n$ is the set of pairs $(x, y) \in F_n \times F_{n+1}$ such that $y$ is an immediate predecessor of $x$.

If $\phi: F \to G$ is a map of forests, in other words if the functions $\phi_n: F_n \to G_n$ collectively preserve the predecessor relation, then we have an inclusion

$$\array{
F & \stackrel{\phi}{\to} & G \\
 ^\mathllap{\theta_F} \downarrow & \leq & \downarrow^\mathrlap{\theta_G} \\
P s^\ast F & \underset{P s^\ast \phi}{\to} & P s^\ast G
}$$

where $\leq$ comes from the internal poset structure of $P s^\ast G$. If this inclusion is an equality; in other words, if $\phi$ is a coalgebra map or simulation over $P s^\ast$, then we also say (following Joyal and Moerdijk) that $\phi$ is _open_. Alternatively, we could say that $\phi$ is open if the maps $\phi_n$ collectively preserve and reflect the predecessor relation.

Pursuing this more concretely: the composite $\theta_G \circ \phi$ corresponds to the subobject $S$ of $F \times s^\ast G$ occurring in the pullback

$$\array{
S & \to & R_G \\
\downarrow & & \downarrow \\
F \times s^\ast G & \underset{\phi \times 1}{\to} & G \times s^\ast G
}$$

so that $S_n \subseteq F_n \times G_{n+1}$ consists of pairs $(x, y)$ such that $y$ is a predecessor of $\phi_n(x)$ in $G$. The composite $P s^\ast(\phi) \circ \theta_F$ corresponds to the subobject $S'$ of $F \times s^\ast G$ occurring in an image factorization

$$\array{
R_F & \twoheadrightarrow & S' \\
\downarrow & & \downarrow \\
F \times s^\ast F & \underset{1 \times s^\ast \phi}{\to} & F \times s^\ast G
}$$

so that (taking advantage of the fact that image factorizations in a presheaf topos are computed objectwise) $S_{n}' \subseteq F_n \times G_{n+1}$ consists of pairs $(x', y')$ such that $y' = \phi_{n+1} y$ for some predecessor $y$ of $x$ in $F$. The equality of subobjects $S = S'$ matches the notion of open map given in _Algebraic Set Theory_.

**Remark:** Joyal and Moerdijk actually refine this idea as follows. Instead of taking forests to be set-valued presheaves, they take them to be presheaves over $\omega$ valued in a given [[pretopos]] $C$ equipped with a class of _small maps_ (which are defined axiomatically). Then, a _small forest_ is a diagram in $C$ of the form

$$\ldots \to F_{n+1} \to F_n \to \ldots \to F_1 \to F_0$$

where each map $F_{n+1} \to F_n$ and $F_0 \to 1$ is small. On the other hand, the small map axioms guarantee that small subobjects are representable by a functor $P_{small}: C \to C$. This induces a functor $P_{small}: C^{\omega^{op}} \to C^{\omega^{op}}$ on the category of forests, and analogous to what we described above, a small forest can be construed in terms of a coalgebra structure $F \to P_{small} (s^\ast F)$.

## Related Concepts

* [[bisimulation]]

[[!redirects simulation]]
[[!redirects simulations]]


Sina Hazratpour

* [Webpage](https://sinhp.github.io/)

## Selected writings

On [[Kripke-Joyal semantics]] and [[forcing]] in [[homotopy type theory]]:

* {#AGH21} [[Steve Awodey]], [[Nicola Gambino]], [[Sina Hazratpour]], *Kripke-Joyal forcing for type theory and uniform fibrations* (2021) [[arXiv:2110.14576](https://arxiv.org/abs/2110.14576)]

category: people


sind-object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **sind-object** of a [[category]] $\mathcal{C}$ is a **formal [[sifted colimit]]** of objects of $\mathcal{C}$.  Here "formal" means that the colimit is taken in the [[category of presheaves]] of $\mathcal{C}$ (the [[free cocompletion]] of $\mathcal{C}$). The category of sind-objects of $\mathcal{C}$ is written $sind$-$\mathcal{C}$ or $SInd(\mathcal{C})$. It is the sifted colimit completion of $\mathcal{C}$.

There is an analogy here with [[ind-objects]], which are formal filtered colimits, and where the category of ind-objects $Ind(\mathcal{C})$ is the filtered colimit completion of $\mathcal{C}$.

Typically the main case of interest here is the situation where $\mathcal{C}$ is small with finite coproducts. If a small category $\mathcal{C}$ has finite coproducts, the sifted colimit completion comprises the finite-product-preserving presheaves:

 $$Sind(\mathcal{C})\cong FP(\mathcal{C}^{op},Set)$$

The analogy here is that if a small category $\mathcal{C}$ has finite [[colimits]], the filtered colimit completion comprises the finite limit preserving presheaves:

 $$Ind(\mathcal{C})\cong Lex(\mathcal{C}^{op},Set)$$

One reason this is often relevant because if $\mathcal{C}^\op$ has finite [[products]], it can be regarded as an [[algebraic theory]], and then the sifted colimit completion comprises the category of models of the theory.

Another point of relevance is that this is a canonical way of building a [[cartesian closed category]] out of a [[distributive category]].

## Definition

If $\mathcal{C}$ has finite coproducts then consider the category of finite product preserving functors

$$Sind(\mathcal{C})\cong  FP(\mathcal{C}^{op},Set)$$

The representable objects always preserves finite products, and so the Yoneda embedding $\mathcal{C}\to [\mathcal{C}^{op},Set]$ factors through $Sind(\mathcal{C})$. This exhibits $Sind(\mathcal{C})$ as the free sifted colimit completion of $\mathcal{C}$.

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

If $\mathcal{D}$ is has sifted colimits then the Yoneda embedding induces a natural bijection between functors $\mathcal{C}\to \mathcal{D}$ and sifted-colimit-preserving functors $Sind(\mathcal{C})\to \mathcal{D}$.

=--

The construction has another universal property, as the free colimit completion of a category respecting the existing finite coproducts.

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

If $\mathcal{D}$ is has colimits then the Yoneda embedding induces a natural bijection between finite-coproduct-preserving functors $\mathcal{C}\to \mathcal{D}$ and colimit-preserving-functors $Sind(\mathcal{C})\to \mathcal{D}$.

=--

## Distributive categories and cartesian closure

The sifted-colimit-completion also gives a way of [[full and faithful|full and faithfully]] embedding a [[distributive category]] in a [[cartesian closed]] category, while preserving the structure. From a type-theoretic perspective, this shows that function types are a conservative extension of finite type theory.

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

Let $\mathcal{C}$ be a small category with finite coproducts and products. The following are equivalent.

* $\mathcal{C}$ is a [[distributive category]].

* $Sind(\mathcal{C})$ is [[cartesian closed]].

=--

## References

*  {#AdamekRosickyVitale10} [[Jiri Adamek]], [[Jiri Rosicky]], [[Enrico Vitale]], _What are sifted colimits?_, TAC **23** (2010) pp. 251–260. ([web](http://www.tac.mta.ca/tac/volumes/23/13/23-13abs.html))

* {#AdamekRosicky01}[[Jiri Adamek]], [[Jiri Rosicky]], _On sifted colimits and generalized varieties_, TAC **8** (2001) pp 33–53. ([web](http://www.tac.mta.ca/tac/volumes/8/n3/8-03abs.html))

The idea of product-preserving-functors as a cartesian closed category is in various sources, including:

* [[Marcelo Fiore]], "Enrichment and representation theorems for categories of domains and partial functions". 1996. [web](http://www.cl.cam.ac.uk/~mpf23/papers/ADT/rep.ps.gz).

* [[John Power]], "Generic models for computational effects",  Theor. Comput. Sci. 364(2): 254-269 (2006)

* Younesse Kaddar, "Ideal distributors", Section 1. 2020. [report](https://younesse.net/assets/ARPE_report.pdf).


sine

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Trigonometry
+-- {: .hide}
[[!include trigonometry -- contents]]
=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The sine function $\sin$ is one of the basic [[trigonometric functions]].

It may be thought of as assigning to any [[angle]] the [[distance]] to a chosen axis of the point on the [[unit circle]] with that angle to that axis.

## Definitions

The _sine function_ is the [[function]] $\sin \;\colon\; \mathbb{R} \to \mathbb{R}$ from the [[real numbers]] to themselves which is characterized by the following equivalent conditions:

1. $\sin$ is the unique [[solution]] among [[smooth functions]] to the [[differential equation]]/[[initial value problem]]

   $$
     sin'' = -sin
   $$

   (where a prime indicates the [[derivative]]) subject to the initial conditions

   $$
     \begin{aligned}
       sin(0) &= 0
       \\
       sin'(0) & = 1
       \,.
     \end{aligned}
   $$

1. ([[Euler's formula]]) $\sin$ is the [[imaginary part]] of the [[exponential function]] with imaginary argument

   $$
     \begin{aligned}
       \sin(x) & = Im\left( \exp(i x) \right)
       \\
       & = \frac{1}{2 i}\left( \exp(i x) - \exp(- i x)\right)
     \end{aligned}
     \,.
   $$

1. $\sin$ is the unique function which

   1. is [[continuous function|continuous]] at $0$

   1. satisfies the [[inequality]]

      $$
        1 - x^2 \;\leq\; \frac{\sin(x)}{x} \;\leq\; 1
      $$

      and the [[equation]]

      $$
        \sin(3 x) \;=\; 3 \sin(x) - 4 (\sin(x))^3
      $$

   (see [Trimble](#Trimble): *[[toddtrimble:Characterization of sine]]*. Some additional discussion at the [nForum](https://nforum.ncatlab.org/discussion/5773/a-short-note-for-todd-trimble/).)

1. $\sin$ is the unique function given by the infinite series

$$\sin(x) \coloneqq \sum_{i = 0}^{\infty} \frac{(-1)^i x^{2 i + 1}}{(2 i + 1)!}$$

### In arbitrary Archimedean ordered fields

In general, Archimedean ordered fields which are not [[sequentially Cauchy complete]] do not have an sine function. Nevertheless, the sine map is still guaranteed to be a [[partial function]], because every Archimedean ordered field is a [[Hausdorff space]] and thus a [[sequentially Hausdorff space]]. Thus, an axiom could be added to an Archimedean ordered field $F$ to ensure that the sine partial function is actually a total function:

**Axiom of sine function:** _For all elements $x \in F$, there exists a unique element $\sin(x) \in F$ such that for all positive elements $\epsilon \in F_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then  $-\epsilon \lt \left(\sum_{i = 0}^{n} \frac{(-1)^i x^{2 i + 1}}{(2 i + 1)!}\right) - \sin(x) \lt \epsilon$._

There is another axiom which uses the fact that derivatives of functions are well defined in the [[ordered local ring|ordered local]] [[local Artin algebra|Artin $F$-algebra]] $F[\epsilon]/\epsilon^2$ by the equation $f(x + \epsilon) = f(x) + f'(x) \epsilon$:

**Axiom of sine and cosine functions:** _Let $F[\epsilon]/\epsilon^2$ be the ordered local Artin $F$-algebra, with non-zero non-positive non-negative [[nilpotent element]] $\epsilon \in F[\epsilon]/\epsilon^2$ where $\epsilon^2 = 0$ and canonical $F$-algebra homomorphism $h:F \to F[x]/x^2$. There exists unique functions $\sin:F \to F$ and $\cos:F \to F$ and functions $\sin':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ and $\cos':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ such that for every element $x \in F$, $h(\sin(x)) = \sin'(h(x))$, $h(\cos(x)) = \cos'(h(x))$, $\sin'(x + \epsilon) = \sin'(x) + \cos'(x) \epsilon$, $\cos'(x + \epsilon) = \cos'(x) - \sin'(x) \epsilon$, $\sin'(0) = 0$, and $\cos'(0) = 1$._

### In the dual numbers

The [[dual number]] real algebra $\mathbb{D} \coloneqq \mathbb{R}[\epsilon]/\epsilon^2$ has a notion of sine and cosine function, which are the solutions to the system of functional equations $\sin(x + \epsilon) = \sin(x) + \cos(x) \epsilon$ and $\cos(x + \epsilon) = \cos(x) - \sin(x) \epsilon$ with $\sin(0) = 1$ and $\cos(0) = 1$.

## In constructive mathematics

In classical mathematics, one could prove that the [[modulated Cantor real numbers]] $\mathbb{R}_C$ are [[sequentially Cauchy complete]] and equivalent to the [[HoTT book real numbers]] $\mathbb{R}_H$. However, in constructive mathematics, the above cannot be proven; while the HoTT book real numbers $\mathbb{R}_H$ are still sequentially Cauchy complete, the modulated Cantor real numbers $\mathbb{R}_C$ in general cannot be proven to be sequentially Cauchy complete. In particular, this means that the sequence

$$\sum_{i = 0}^{n} \frac{(-1)^i x^{2 i + 1}}{(2 i + 1)!}$$

does not have a limit for all modulated Cantor real numbers $x \in \mathbb{R}_C$. However, the sequence, by definition of $\mathbb{R}_C$, does have a limit for all [[rational numbers]] $x \in \mathbb{Q}$; this means that one could restrict the [[domain]] of the sine function to the rational numbers $\sin:\mathbb{Q} \to \mathbb{R}_C$, and define it in the usual manner:

* For all rational numbers $x \in \mathbb{Q}$, there exists a unique modulated Cantor real number $\sin(x) \in \mathbb{R}_C$ such that for all positive rational numbers $\epsilon \in \mathbb{Q}_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then $-\epsilon \lt \sum_{i = 0}^{n} \frac{(-1)^i x^{2 i + 1}}{(2 i + 1)!} - \sin(x) \lt \epsilon$.

## Properties

### Relation to other functions

See at _[[trigonometric identity]]_.

### Roots

The [[roots]] of the sine function, hence the argument where its value is [[zero]], are the [[integer]] multiples of [[pi]] $\pi \in \mathbb{R}$.

## Related concepts

* [[periodic function]]

* [[cosine]]

* [[arcsin]]

* [[hyperbolic sine]], [[hyperbolic cosine]]

* [[exponential]], [[exponential ring]], [[Euler's formula]]

* [[trigonometric identity]]

* [[pi]], [[circle constant]]

## References

Discussion in [[constructive analysis]]:

* [[Errett Bishop]], §7 in: *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* [[Errett Bishop]], [[Douglas Bridges]] §7 in: *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

See also

* Wikipedia, _[Sine and cosine](https://en.wikipedia.org/wiki/Sine_and_cosine)_

* {#Trimble} [[Todd Trimble]], _[[toddtrimble:Characterization of sine]]_

[[!redirects sin]]

[[!redirects sine function]]
[[!redirects sine functions]]


sine-Gordon equation

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### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is known as the *sine-Gordon equation* is the  [[partial differential equation]] on a [[scalar field]] on 1+1-[[dimension of a manifold|dimensional]] [[Minkowski spacetime]] which asserts that the value of its [[wave operator]] equals (minus) its [[sine]].

As such the name is a pun on the *[[Klein-Gordon equation]]* which  (also in higher [[spacetime]] dimension) has this same form except that in place of the [[sine]] there is a [[mass term]] (and possibly in addition a [[potential energy]], [[Lorentz force|gauge coupling]], etc.).

The sine-Gordon equation is an [[integrable model]] which admits [[soliton]] solutions. It is [[duality in physics|dual]], in some sense, to the [[Thirring model]].

## Related concepts

* [[integrable model]]

## References

* Jesús Cuevas-Maraver, Panayotis G. Kevrekidis, Floyd Williams (eds.), *The sine-Gordon Model and its Applications*, Spinger (2014) [[doi:10.1007/978-3-319-06722-3](https://doi.org/10.1007/978-3-319-06722-3)]

* [[Alessandro Torrielli]], *LonTI Lectures on Sine-Gordon and Thirring* [[arXiv:2211.01186](https://arxiv.org/abs/2211.01186)]

See also:

* Wikipedia, *[Sine-Gordon equation](https://en.m.wikipedia.org/wiki/Sine-Gordon_equation)*

[[!redirects sine-Gordon equations]]

[[!redirects sine-Gordon model]]
[[!redirects sine-Gordon models]]


single trace operator

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###Context###
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] and [[statistical physics]], a _single trace operator_ is an [[observable]]/[[correlator]] on [[square matrix]]- or rather [[adjoint representation]]-valued [[scalar fields]]/[[random variables]] $\{\phi_i\}_{i \in I}$ (notationally subsuming [[derivatives]] of fields) which is built from the elementary [[field observables]] $\mathbf{\Phi}_i$ as a [[trace]] over the [[matrix product]] of a finite string of them:

\[
  \label{ASingleTraceObservable}
  \mathcal{O}_{i_1, \cdots i_n}
  \;=\;
  Tr\big(
   \mathbf{\Phi}_{i_1}
   \cdot
   \mathbf{\Phi}_{i_2}
    \cdot
    \cdots
    \cdot
   \mathbf{\Phi}_{i_n}
  \big)
\]

or more generally as [[linear combinations]] thereof.

If the [[field (physics)|fields]] $\phi_i$ indeed take values in the [[adjoint representation]] of a given [[gauge group]], then the [[trace]] in the expression (eq:ASingleTraceObservable) ensures that this [[observable]] is [[gauge invariant]]. More general [[gauge invariant]] [[observables]] are hence _[[multi-trace operators]]_ which are [[linear combinations]] of products of single-trace observables (i.e. [[polynomials]] of single-trace observables).

## Properties

### Correspondence to strings under AdS/CFT
 {#CorrespondenceToStringsUnderAdSCFT}

Single trace operators/observables in [[conformal field theories]] such as [[super Yang-Mills theories]] play a special role in the [[AdS-CFT correspondence]]: They correspond to single [[string]] excitations on the [[AdS spacetime|AdS]]-[[supergravity]] side of the correspondence, where, curiously, the "string of characters/letters" in the argument of the trace gets literally mapped to a [[superstring]] in [[spacetime]] (see the references [below](#ReferencesRelationToStringExcitations)).

From [Polyakov 02](#Polyakov02), referring to gauge fields and their single trace operators as _letter_ and _words_, respectively:

> The picture which slowly arises from the above considerations is that of the space-time gradually disappearing in the regions of large curvature. The natural description in this case is provided by a gauge theory in which the basic objects are the texts formed from the gauge-invariant words. The theory provides us with the expectation values assigned to the various texts, words and sentences.

> These expectation values can be calculated either from the gauge theory or from the strongly coupled 2d sigma model. The coupling in this model is proportional to the target space curvature. This target space can be interpreted as a usual continuous space-time only when the curvature is small. As we increase the coupling, this interpretation becomes more and more fuzzy and finally completely meaningless.

From [Berenstein-Maldacena-Nastase 02](#BerensteinMaldacenaNastase02), who write $Z$ for the elementary [[field observables]] ("letters") $\mathbf{\Phi}$ above:

> In summary, the "string of $Z$s" becomes the physical string and that each $Z$ carries one unit of $J$ which is one unit of $p_+$. Locality along the worldsheet of the string comes from the fact that planar diagrams allow only contractions of neighboring operators. So the Yang Mills theory gives a [[string bit model]] where each bit is a $Z$ operator.

On the [[CFT]] side these _[[BMN operators]]_ of fixed length (of "letters") are usefully identified as  [[spin chains]] which, with the [[dilatation operator]] regarded as their [[Hamiltonian]], are [[integrable systems]] ([Minahan-Zarembo 02](#MinahanZarembo02), [Beisert-Staudacher 03](#BeisertStaudacher03)).

This integrability allows a detailed matching between

* [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]]

* the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] $\times$ [[5-sphere|S5]]

under [[AdS/CFT duality]] ([Beisert-Frolov-Staudacher-Tseytlin 03](#BeisertFrolovStaudacherTseytlin03), ...). For review see [BBGK 04](#BBGK04), [Beisert et al. 10](#BeisertEtAl10).

(...)

### Relation to weight systems on chord diagrams

For [[field (physics)|fields]] $\phi$ valued in a [[Lie algebra representations]] $C \in \mathfrak{g}Mod$ of a [[metric Lie algebra]], with $\{\phi_a\}$ a [[linear basis]] for $C$ and $\{\phi^a\}$ denoting the dual basis, those single trace operators involving only metric pairs of fields, such as

$$
  \mathcal{O}
  \;\coloneqq\;
  Tr_C
  \big(
    \mathbf{\Phi}_{a_1}
    \cdot
    \mathbf{\Phi}_{a_2}
    \cdot
    \mathbf{\Phi}^{a_1}
    \cdot
    \mathbf{\Phi}^{a_2}
  \big)
$$

(using [[Einstein summation convention]]) are equivalently values of [[Lie algebra weight systems]] on the [[chord diagram]] that expresses the index pairing.

Such binary pairings inside the trace are induced notably after statistical averaging in  the [[SYK model]] and related systems, and has been argued to appear generically for [[AdS/CFT|dual]] [[observables]] of [[black hole thermodynamics]] ([Berkooz-Narayan-Simón 18, Section 2.1](#BerkoozNarayanSimon18)).

For more on this see at _[[weight systems on chord diagrams in physics]]_.

## Examples

### Shape of D$p$ $\bot$ D$(p+2)$-fuzzy funnels
  {#SingleTraceObservablesAsWeightSystemsOnChordDiagrams}

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

We discuss how the [[single trace observables]] on the [[fuzzy 2-sphere]]-sections  of [[Dp-D(p+2) brane intersection]] [[fuzzy funnels]] are given by [[su(2)]]-[[Lie algebra weight systems]] on [[chord diagrams]] (following [Ramgoolam-Spence-Thomas 04](#RamgoolamSpenceThomas04), [McNamara-Papageorgakis 05](#McNamaraPapageorgakis05), see [McNamara 06, Section 4](#McNamara06) for review).

\linebreak

While in the commutative [[large N limit]], all powers of the [[radius]] function on the [[fuzzy 2-sphere]] are equal

$$
  \underset{N\to \infty}{\lim}
  \int_{S^2_N} R^{2 k}
  \;=\;
  4 \pi
  \,;
$$

for [[finite number|finite]] $N$ there is an ordering ambiguity: In fact, the number of functions on the [[fuzzy 2-sphere]] at [[finite number|finite]] $N$ that all go to the same function $R^{2k}$ in the [[large N limit]] grows rapidly with $k$.

At $k = 1$ there is the single radius observable (eq:RadiusOfFuzzy2Sphere)

$$
  \int_{S^2_N} R^2
  \;=\;
  \int_{S^2_N}
  \underset{i}{\sum} X_i \cdot X_i
  \;=\;
  4 \pi \tfrac{ N }{ \sqrt{N^2 -1} }
$$

At $k = 2$ there are, under the integral (eq:FuzzyS2Integration), two radius observables:

1. $ \int_{S^2_N} \underset{i,j}{\sum} X_i X_i X_j X_j$

1. $\int_{S^2_N} \underset{i,j}{\sum} X_i X_j X_j X_i$

(Here we are using that under the integral/trace, a [[cyclic permutation]] of the factors in the integrand does not change the result).

Similarly for higher $k$, where the number of possible orderings increases rapidly. The [[combinatorics]] that appears here is familiar in [[knot theory]]:

Every ordering of operators, up to cyclic permutation, in the [[single trace observable]] $Tr(R^2)^n$ is encoded in a [[chord diagram]] and the value of the corresponding [[single trace observable]] is the value of the [[su(2)]]-[[Lie algebra weight system]] on this chord diagram.

## Related concepts

* [[string bit model]]

* [[multi-trace operator]]

## References

### General

(...)

### Correspondence to string excitations under AdS/CFT
 {#ReferencesRelationToStringExcitations}

The correspondence of single trace operators to [[superstring]] excitations under the [[AdS-CFT correspondence]] originates with these articles:

* {#Polyakov02} [[Alexander Polyakov]], _Gauge Fields and Space-Time_, Int. J. Mod. Phys. A17S1 (2002) 119-136 ([arXiv:hep-th/0110196](https://arxiv.org/abs/hep-th/0110196))

* {#BerensteinMaldacenaNastase02} [[David Berenstein]], [[Juan Maldacena]], [[Horatiu Nastase]], _Strings in flat space and pp waves from $\mathcal{N} = 4$ Super Yang Mills_, JHEP 0204 (2002) 013 ([arXiv:hep-th/0202021](https://arxiv.org/abs/hep-th/0202021))

  (whence "[[BMN operators]]")

* [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], _A semi-classical limit of the gauge/string correspondence_, Nucl. Phys. B636 (2002) 99-114 ([arXiv:hep-th/0204051](https://arxiv.org/abs/hep-th/0204051))

* {#Kruczenski04} [[Martin Kruczenski]], _Spiky strings and single trace operators in gauge theories_, JHEP 0508:014, 2005 ([arXiv:hep-th/0410226](https://arxiv.org/abs/hep-th/0410226))

The identification of the relevant [[single trace operators]] with [[integrable system|integrable]] [[spin chains]] is due to

* {#MinahanZarembo02}  J. A. Minahan, [[Konstantin Zarembo]], _The Bethe-Ansatz for $\mathcal{N} = 4$ Super Yang-Mills_, JHEP 0303 (2003) 013 ([arXiv:hep-th/0212208](https://arxiv.org/abs/hep-th/0212208))

* {#BeisertStaudacher03} [[Niklas Beisert]], [[Matthias Staudacher]], _The $\mathcal{N}=4$ SYM Integrable Super Spin Chain_,  Nucl. Phys. B 670:439-463, 2003 ([arXiv:hep-th/0307042](https://arxiv.org/abs/hep-th/0307042))

which led to more detailed matching of [[single trace operators]] to rotating string excitations in

* {#BeisertFrolovStaudacherTseytlin03} [[Niklas Beisert]], [[Sergey Frolov]], [[Matthias Staudacher]], [[Arkady Tseytlin]], _Precision Spectroscopy of AdS/CFT_, JHEP 0310:037, 2003 ([arXiv:hep-th/0308117](https://arxiv.org/abs/hep-th/0308117))

Review includes

* {#BBGK04} A. V. Belitsky, [[Volker Braun]], A. S. Gorsky, G. P. Korchemsky, _Integrability in QCD and beyond_, Int. J. Mod. Phys. A19:4715-4788, 2004 ([arXiv:hep-th/0407232](https://arxiv.org/abs/hep-th/0407232))

* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schafer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))

### Relation to weight systems on chord diagrams

A general relation between [[single trace operators]] for [[observables]] on [[black hole thermodynamics]] and [[Lie algebra weight systems]] on [[chord diagrams]] (though not using the term "weight system") is proposed in

* {#BerkoozNarayanSimon18} [[Micha Berkooz]], [[Prithvi Narayan]], [[Joan Simón]], Section 2.1 of: _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))

For more on this see at _[[weight systems on chord diagrams in physics]]_.

#### For JT-gravity/SYK model

Discussion of ([[Lie algebra weight system|Lie algebra]]-)[[weight systems]] on [[chord diagrams]] encoding [[SYK model]] [[single trace observables]]:

* Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, _Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$_, JHEP 04 (2018) 146 ([arXiv:1801.02696](https://arxiv.org/abs/1801.02696))

* [[Micha Berkooz]], Prithvi Narayan, [[Joan Simón]], _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))

following:

* László Erdős, Dominik Schröder, _Phase Transition in the Density of States of Quantum Spin Glasses_, D. Math Phys Anal Geom (2014) 17: 9164 ([arXiv:1407.1552](https://arxiv.org/abs/1407.1552))

which in turn follows

* Philippe Flajolet, Marc Noy, _Analytic Combinatorics of Chord Diagrams_, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), _Formal Power Series and Algebraic Combinatorics_, Springer 2000 ([doi:10.1007/978-3-662-04166-6_17](https://doi.org/10.1007/978-3-662-04166-6_17))

With emphasis on [[AdS-CFT|holographic content]]:

* [[Micha Berkooz]], Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: _Towards a full solution of the large $N$ double-scaled SYK model_, JHEP 03 (2019) 079 ([arxiv:1811.02584](https://arxiv.org/abs/1811.02584))

* Vladimir Narovlansky, Slide 23 (of 28) of: _Towards a Solution of Large $N$ Double-Scaled SYK_, 2019 ([[NarovlanskySYK19.pdf:file]])

and specifically in relation to [[Jackiw-Teitelboim gravity]]:

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _The Schwarzian Theory - A Wilson Line Perspective_, JHEP 1812 (2018) 022 ([arXiv:1806.07765](https://arxiv.org/abs/1806.07765))

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _Fine Structure of Jackiw-Teitelboim Quantum Gravity_, JHEP 1909 (2019) 066 ([arXiv:1812.00918](https://arxiv.org/abs/1812.00918))

#### For D$p$-D$(p+2)$-brane intersections

Discussion of [[weight systems]] on [[chord diagrams]] as [[single trace observables]] on the [[fuzzy funnel]]/[[fuzzy sphere]] [[non-commutative geometry]] of [[Dp-D(p+2)-brane intersections]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]):

* [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

[[!redirects single trace operators]]

[[!redirects single-trace operator]]
[[!redirects single-trace operator]]

[[!redirects single trace observable]]
[[!redirects single trace observables]]

[[!redirects single-trace observable]]
[[!redirects single-trace observables]]

[[!redirects BMN operator]]
[[!redirects BMN operators]]


single-sorted definition of categories

[[!redirects single-sorted definition of a category]]

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

There are two common ways to define a [[category]]:

1. As a [[collection]] $C_0$ of [[objects]] and a collection $C_1$ of [[morphisms]], together with [[source]] and [[target]] maps $C_1\to C_0$, a composition map $C_1\times_{C_0} C_1\to C_1$, and an [[identity assigning map]] $C_0\to C_1$, satisfying axioms.

1. As a collection $C_0$ of objects, together with for each pair $x,y$ of objects, a collection $hom_C(x,y)$ of morphisms, together with identities $1_x\in hom_C(x,x)$ and composition maps $hom_C(y,z)\times hom_C(x,y)\to hom_C(x,z)$, satisfying axioms.

In [[logic|logical]] terms, the first is formulated as a $2$-sorted theory in [[first-order logic]], while the second is a [[dependent type theory|dependently typed theory]]; the usual way of interpreting any dependently typed theory as an independently sorted theory turns the latter into the former.

However, there is a third way of defining a category which uses only one collection (representing the collection of morphisms) and thus is formulated as an untyped (or $1$-sorted) first-order theory.  Note that this is *not* the usual way of replacing sorts with predicates, but instead a slightly clever trick.  The basic idea is that an [[object]] can be identified with its [[identity morphism]].  This reformulation is occasionally useful, but mostly for technical reasons.

## Definition

A **category** (single-sorted version) is a [[collection]] $C$, whose elements are called _morphisms_, together with two functions $s,t:C\to C$ and a [[partial function]] $\circ:C\times C\to C$, such that:

1. $s(s(x)) = s(x) = t(s(x))$
2. $t(t(x)) = t(x) = s(t(x))$
3. $x\circ y$ is defined if and only if $s(x)=t(y)$.
4. If $x\circ y$ is defined, then $s(x\circ y) = s(y)$ and $t(x\circ y)=t(x)$.
5. $x\circ s(x)=x$ and $t(x)\circ x=x$ (both composites are always defined, because of the first two axioms)
6. $(x\circ y)\circ z = x\circ (y\circ z)$, if either is defined (in which case the other is defined by the axiom 3).

The first two axioms say that $s$ and $t$ are [[idempotent]] [[endofunctions]] on $C$ which have the same [[image]].  The elements of their common image (the $x$ such that $s(x)=x$, or equivalently $t(x)=x$) are called _identities_ or _objects_.  Once that is done, the rest of the identification is straightforward.

A **[[functor]]** between single-sorted categories is just a function $f:C\to D$ such that $f(s(x)) = s(f(x))$, $f(t(x)) = t(f(x))$, and $f(x\circ y)= f(x)\circ f(y)$ whenever $x\circ y$ is defined (which, by the first two axioms of a functor and axiom (3) of a category, implies that $f(x)\circ f(y)$ is defined).

Finally, a **[[natural transformation]]** between functors $f,g:C\to D$ of single-sorted categories is a function $\alpha:C\to D$ such that $s(\alpha(x)) = s(f(x))$, $t(\alpha(x)) = t(g(x))$, and $\alpha(x) \circ f(y) = g(x) \circ \alpha(y)$ whenever $x\circ y$ is defined (which implies that both composites in this identity are defined).  Note that while a natural transformation is ordinarily defined to consist of a component $\alpha(x)$ only when $x$ is an _object_, this definition supplies a component to each _morphism_.  In terms of the usual definition, the component of $\alpha$ at a morphism is the diagonal of the corresponding naturality square.

It can now be proved that single-sorted categories, functors, and natural transformations form a [[2-category]] which is (strictly) [[equivalence of categories|equivalent]] to the usual $2$-category [[Cat]].

## Remarks

### Specializations

A [[monoid]] is a single-sorted category in which $s$ is a constant function (hence so is $t$, and they are equal).  This works up to [[isomorphism of categories]], not merely equivalence, so single-sorted categories may seem to be a more direct [[oidification]] of monoids than the usual categories.

### Internalization

The usual definition of an [[internal category]] is two-sorted, but the one-sorted definition can also be interpreted [[internalization|internally]].  While the usual notion of internal category requires the [[ambient category]] only to have [[pullbacks]], the one-sorted version appears to require one to make sense of an "internal partial binary operation."  However, since in this case the domain of $\circ$ is specified explicitly in the definition, one can just require $\circ$ to be an ordinary morphism whose domain is the pullback of $s$ and $t$; thus only pullbacks are required for the single-sorted definition as well.

It is easy to see that any internal two-sorted category gives an internal one-sorted category (consider the object of arrows).  The converse is true as long as the ambient category has [[split idempotents]], for then given an internal one-sorted category we can split either $s$ or $t$ to obtain an object of objects.  In general, however, the two concepts are not equivalent.

### Generalizations

There exist similar single-sorted definitions of $n$-[[n-categories|categories]] and [[∞-categories]].  The single sort in the definition of $n$-category is the set of $n$-morphisms, but you can also think of this as the union (over all $k \leq n$) of the sets of $k$-morphisms, as long as you identify each $k$-morphism (for $k \lt n$) with its identity $(k+1)$-morphism.  In the the definition of $\infty$-category, there is no notion of $\infty$-morphism to take care of everything at once, but the single sort can still be understood as this union (now over all $k$).

## References

* [[Saunders MacLane]], §I.1 and §XII.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects single-sorted definition of category]]
[[!redirects single-sorted definition of a category]]
[[!redirects single-sorted definition of categories]]
[[!redirects single-sorted definitions of category]]
[[!redirects single-sorted definitions of a category]]
[[!redirects single-sorted definitions of categories]]


singleton

+-- {: .rightHandSide}
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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[set]] $X$ and an element $a$ of $X$, the __singleton__ $\{a\}$ is that [[subset]] of $X$ whose only element is $a$.  Strictly speaking, we are considering here a __singleton subset__; we could also consider a __singleton list__ or __$1$-tuple__ $(a)$, but this is an equivalent concept.

Here, $\{a\}$ is [[subobject classifier|classified]] by the characteristic map $c: X \to \Omega$ (where $\Omega$ is the set of [[truth value]]s) given by
$$ c(b) = (a = b) .$$
As an [[injection]] to $X$, $\{a\}$ is precisely the same map $1 \to X$ as $a$ itself is as a [[generalized element]] of $X$; the same goes for the $1$-tuple $(a)$ as a map from $[1]$.  One can take this to justify the common abuse of notation (as it would normally be considered) in which $\{a\}$ or $(a)$ is written as $a$ when no confusion can result.

Note that the set of all singletons of elements of $X$ is [[bijection|isomorphic]] to $X$ itself.  In this way, the entire concept can be seen as a triviality.

A [[subset]] of a singleton is called a _[[subsingleton]]_.  In [[classical mathematics]] (using the [[principle of excluded middle]]), the only subsingletons are the singletons and the [[empty subset]], but in [[constructive mathematics]], this is an important concept.

Everything above can be generalised from the [[Set|category of sets]] to any [[topos]].

Singleton subsets are important in distinguishing between two kinds of [[categorical set theories]]; there are the [[categorical set theories]] like [[ETCS]] where elements *are* singleton subsets; and then there are the categorical set theories like [[structural ZFC]] where elements are different from singleton subsets but have a reflection into singleton subsets.

## Related concepts

* [[point]], [[point space]]

* [[empty set]]

[[!redirects singleton]]
[[!redirects singletons]]

[[!redirects singleton set]]
[[!redirects singleton sets]]
[[!redirects one-element set]]
[[!redirects one-element sets]]
[[!redirects 1-element set]]
[[!redirects 1-element sets]]

[[!redirects singleton subset]]
[[!redirects singleton subsets]]
[[!redirects one-element subset]]
[[!redirects one-element subsets]]
[[!redirects 1-element subset]]
[[!redirects 1-element subsets]]

[[!redirects singleton list]]
[[!redirects singleton lists]]
[[!redirects one-tuple]]
[[!redirects one-tuples]]
[[!redirects 1-tuple]]
[[!redirects 1-tuples]]


singleton pretopology

A __singleton Grothendieck pretopology__ is a [[Grothendieck pretopology]] whose all covers are consisting of precisely one morphism, i.e. are [[singleton]]s.

[[!redirects singleton Grothendieck pretopology]]


singleton representation

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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[anti de Sitter group]] $SO(2,D-1)$ has "ultra-short" [[unitary representations]] that admit no [[Minkowski spacetime]]-limit (along the [[Lie algebra contraction]]). For $D = 4$ these were first discussed by [Dirac 63](#Dirac63) and named _singletons_.

More generally, given a (non-[[compact Lie group|compact]]) [[Lie group]] $G$ admitting [[lowest weight representation]] with [[maximal compact subgroup]] of the form $H \times U(1)$ for some other [[compact Lie group]] $H$, the _oscillator method_ is to describe [[unitary representation]] by realizing the generators of $G$ as [[creation and annihilation operators]] on some [[Fock space]], and have them transform in the (anti-)[[fundamental representation]] of $H$. Then a _singleton representation_ is one where the generators are realized as the creators and annihilators of a single oscillator, and there are typically two of these (usually the scalar and spinor representations, in a physical language).

A _doubleton representation_ is one where the generators are realized as two sets of oscillators, and so forth.

## In AdS/CFT

In ([Fronsdal 81](#Fronsdal81), [Flato-Fronsdal 81](#FlatoFronsdal81), [Angelopoulos-Flato-Fronsdal-Sternheimer 81](#AngelopoulosFlatoFronsdalSternheimer81)) it was observed that these representations may naturally be understood as arising in [[free field theory]] on the [[asymptotic boundary]] of [[anti de Sitter spacetime]].

Analogous statements hold true for the [[super anti de Sitter group]] (e.g. [Gunyadin 89](#Gunyadin89))

Based on this observation, it was conjectured in [Duff 88, p. 29-30](#Duff88) that the singleton representation of $SO(3,2)$ is realized by the [[field (physics)|field]] content of the [[worldvolume]]-theory of a [[Green-Schwarz sigma model|fundamental]] [[M2-brane]] stretched along the [[asymptotic boundary]] of [[anti de Sitter spacetime]] factor $AdS_{4}$ in a [[Freund-Rubin compactification]] of [[11-dimensional supergravity]]. [^Dirac ]

[^Dirac]: Curiously, also the idea of a fundamental [[membrane]] is due to [[Paul Dirac]] ([Dirac 62](supermembrane#Dirac62))

This conjecture was shown to be true in ... and is a pre-cursor of what is now known as the _[[AdS-CFT correspondence]]_ (see [Duff 98](#Duff98) for review). See also at _[super p-brane -- As part of the AdS-CFT correspondence](Green-Schwarz+action+functional#AsPartOfTheAdSCFTCorrespodence)_.

From [Gunyadin 98, p. 2](#Gunyadin98)

> The ultra-short singleton supermultiplet sits at the bottom of this infinite tower of Kaluza-Klein modes and decouple from the spectrum as local gauge degrees of freedom [25]. However , even though the singleton supermultiplet decouples from the spectrum as local gauge modes, one can generate the entire spectrum of 11-dimensional supergravity over $S^7$ by tensoring the singleton supermultiplets repeatedly and restricting oneself
to "CPT self-conjugate" vacuum supermultiplets.

## References

* {#Dirac63} [[Paul Dirac]], _A remarkable representation of the 3+2 de Sitter group_, Journal of Mathematical Physics, 4(7), 901-909, 1963 ([doi:10.1063/1.1704016](https://aip.scitation.org/doi/pdf/10.1063/1.1704016))

* {#Fronsdal81} [[Christian Fronsdal]], _Dirac supermultiplet_, Phys. Rev. D **26** (1982) 1988 [[doi:10.1103/PhysRevD.26.1988](https://doi.org/10.1103/PhysRevD.26.1988)]

* {#FlatoFronsdal81} [[Moshé Flato]], [[Christian Fronsdal]], _Quantum field theory of singletons. The rac_, J. Math. Phys. **22** (1981) 1100 [[doi:10.1063/1.524993](https://doi.org/10.1063/1.524993)]

* {#AngelopoulosFlatoFronsdalSternheimer81} E. Angelopoulos, [[Moshe Flato]], [[Christian Fronsdal]], [[Daniel Sternheimer]], _Massless Particles, Conformal Group, and De Sitter Universe_, Phys. Rev. D23 (1981) 1278

* [[Sergio Ferrara]], [[Christian Fronsdal]], _Conformal Maxwell theory as a singleton field theory on $AdS_5$, IIB three-branes and duality_, Class.Quant.Grav.15:2153-2164, 1998 ([arXiv:hep-th/9712239](https://arxiv.org/abs/hep-th/9712239))

* {#Gunyadin89} [[Murat Günaydin]], _Singleton And Doubleton Supermultiplets Of Space-time Supergroups And Infinite Spin Superalgebras_, 1989 ([spire:282501](http://inspirehep.net/record/282501/))

* {#Duff88} [[Mike Duff]], _Supermembranes: The First Fifteen Weeks_, Class.Quant.Grav. 5 (1988) 189 ([spire:248034](http://inspirehep.net/record/248034))

* [[Eric Bergshoeff]], [[Mike Duff]], [[Christopher Pope]] and [[Ergin Sezgin]], _Supersymmetric supermembrane vacua and singletons_, Phys. Lett. B199, 69 (1988)

* [[Gianguido Dall'Agata]], Davide Fabbri, Christophe Fraser, [[Pietro Fré]], Piet Termonia, Mario Trigiante, _The $Osp(8|4)$ singleton action from the supermembrane_, Nucl.Phys.B542:157-194, 1999 ([arXiv:hep-th/9807115](http://arxiv.org/abs/hep-th/9807115))

* [[Paolo Pasti]], [[Dmitri Sorokin]], Mario Tonin, _Branes in Super-AdS Backgrounds and Superconformal Theories_ ([arXiv:hep-th/9912076](http://arxiv.org/abs/hep-th/9912076))

* {#Duff98} [[Mike Duff]], _Anti-de Sitter space, branes, singletons, superconformal field theories and all that_, Int.J.Mod.Phys.A14:815-844, 1999 ([arXiv:hep-th/9808100](https://arxiv.org/abs/hep-th/9808100))

* {#Gunyadin98} [[Murat Günaydin]], _Unitary Supermultiplets of $OSp(1 \vert32,\mathbb{R}) and M-theory_, Nucl.Phys.B528:432-450, 1998 ([arXiv:hep-th/9803138](https://arxiv.org/abs/hep-th/9803138))

[[!redirects singleton representations]]


singular cohesion

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###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called *singular cohesion* in [SaSc 2020](#SaSc20), following observations in [Rezk 2014](#Rezk14), is a form of [[cohesion]] [[cohesive (infinity,1)-toposes|on $\infty$-toposes]] which enhances any [[base (infinity,1)-topos|base]] notion of "smooth" cohesive geometry to a "singular-smooth" cohesive geometry which reflects the possible presence of [[orbi-singularities]].

For example, where [[smooth manifolds]] are faithfully embedded into the [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]] of [[smooth infinity-groupoids|smooth $\infty$-groupoids]], so their [[orbifolds]] are faithfully embedded among [[singular-smooth infinity-groupoid|singular-smooth $\infty$-groupoids]] such that, for [[global quotient orbifolds]], their [[cohesive shape]] is the homotopy type of the underlying [[G-spaces]] in [[proper equivariant homotopy theory|proper]] [[equivariant homotopy theory]].

For the [[base (infinity,1)-topos|base $\infty$-topos]] [[Infinity-Grpd|$Grpd_\infty$]], embodying plain [[homotopy theory]], its singular-cohesive enhancement is [[global equivariant homotopy theory]] and hence every singular-cohesive $\infty$-topos is cohesive over (in particular) [[global equivariant homotopy theory]], hence may be thought of as embodying a "globally equivariant and cohesive" homotopy theory.

In fact, every [[slice (infinity,1)-topos|slice]] of a singular-cohesive $\infty$-topos over the canonical $G$-[[orbi-singularity]] $\boxed{\prec}\mathbf{B}G$ [[cohesion of global- over G-equivariant homotopy theory|is cohesive over]] the $G$-[[equivariant homotopy theory]] over the given smooth cohesive base, with the [[inverse image]] of the cohesive [[geometric morphism]] being the embedding of cohesive $G$-[[orbispaces]].

### Globally equivariant geometric homotopy theory

Consider the [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of [[Infinity-Grpd|$Grp_\infty$]] on the [[connected object in an (infinity,1)-topos|connected]], [[truncated object in an (infinity,1)-category|1-truncated]] [[pi-finite homotopy types|$\pi$-finite homotopy types]]

\[
  \label{CategoryOfSingularities}
  \array{
    Singularities
    &\coloneqq&
    Grpd^{fin}_{1, \geq 1}
    &\xhookrightarrow{\;}&
    Grp_\infty
    \\
    \boxed{\prec}\mathbf{B}G
    &
    \mapsto
    &
    B G
  }
\]

(sometimes called the "[[global orbit category]]"), hence the [[full sub-(infinity,1)-category|full sub]] [[(2,1)-category]] of [[groupoids]] on the [[delooping groupoids]] of [[finite groups]] $G$.

Then for $\mathbf{H}_{\subset}$ any [[base (infinity,1)-topos|base $\infty$-topos]],
the [[(infinity,1)-category of (infinity,1)-presheaves|$\infty$-category of $\infty$-presheaves]] over $Singularities$ (eq:CategoryOfSingularities)

$$
  \mathbf{H}
  \,\coloneqq\,
  Glo \mathbf{H}_{\subset}
  \;\coloneqq\;
  PSh_\infty
  \big(
    Singularities
    \,,
    \mathbf{H}_{\subset}
  \big)
  \,,
$$

which may be called the *[[globally equivariant homotopy theory]]* over the [[geometric homotopy theory]] $\mathbf{H}_{\subset}$,
is [[cohesive (infinity,1)-topos|cohesive]] over $\mathbf{H}_{\subset}$ (as is any $\infty$-presheaf $\infty$-topos over an [[(infinity,1)-site|$\infty$-site]] with a [[terminal object]], see e.g. at *[[adjoint quadruple]]* the section *[Cohesion](adjoint+quadruple#CohesiveToposes)*).

In fact, for any $\boxed{\prec}\mathbf{B}G \,\in\, Singularities$, the [[slice (infinity,1)-topos|slice $\infty$-topos]] $Glo(\mathbf{H}_{\subset})_{/\boxed{\prec}\mathbf{B}G}$ is [[cohesive (infinity,1)-topos|cohesive]] over the $G$-[[equivariant homotopy theory]], hence over the [[(infinity,1)-category of (infinity,1)-presheaves|$\infty$-topos of $\infty$-presheaves]] over the actual [[orbit category]] of $G$:

$$
  G \mathbf{H}_{\subset}
  \;\coloneqq\;
  Psh_\infty
  \big(
    G Orbits
    ,\,
    \mathbf{H}_{\subset}
  \big)
  \,.
$$

\begin{tikzcd}
  \mathllap{
    \mathbf{H}_{/\boxed{\prec}\mathbf{B}G}
    \;\coloneqq\;
  }
  (\mathrm{Glo} \mathbf{H}_{\subset})_{/\boxed{\prec}\mathbf{B}G}
      \ar[
        rr,
        "{ G \mathrm{Smth}    }"{description}
      ]
      \ar[
        rr,
        shift left=40pt,
        "{ G \mathrm{Cncl} }"{description, pos=.5},
        "\mathclap{\times}"{description, pos=0}
      ]
      &&
     G \mathbf{H}_{\subset}
      \ar[
        ll,
        hook',
        shift right=20pt,
        "{ G \mathrm{Spc} }"{description}
      ]
      \ar[
        ll,
        hook',
        shift right=-20pt,
        "{ G \mathrm{Snglrt} }"{description, pos=.5}
      ]
      \ar[
        ll,
        phantom,
        shift right=10pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
      \ar[
        ll,
        phantom,
        shift right=30pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
      \ar[
        ll,
        phantom,
        shift right=-10pt,
        "{\scalebox{.6}{$\bot$}}"
      ]
\end{tikzcd}

For the discrete equivariance groups considered here, this follows on general abstract grounds (see at *[Adjoint quadruple -- Cohesion](adjoint+quadruple#CohesiveToposes)*) from the [[full sub-(infinity,1)-category|full sub]] [[(2,1)-category]] inclusion

$$
  \Singularities_{/\boxed{\prec}\mathbf{B}G}
  \underoverset
    {\hookleftarrow}
    {\overset{\tau_0}{\longrightarrow}}
    {\;\;\; \bot \;\;\;}
  G Orbits
  \,,
$$

as discussed at *[[cohesion of global- over G-equivariant homotopy theory]]* (highlighted in [Rezk 2014, Sec. 7](#Rezk14)).

### Globally equivariant cohesive homotopy theory

This means that if $\mathbf{H}_{\subset}$ is itself [[cohesive (infinity,1)-topos|cohesive]] over [[Infinity-Grpd|$Grpd_\infty$]] then $\mathbf{H} = Glo(\mathbf{H}_{\subset})$ carries the following system of [[adjoint quadruples]]:

\begin{imagefromfile}
    "file_name": "SingularCohesionAdjoints_20211026.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [SaSc 21](https://ncatlab.org/schreiber/show/Equivariant+principal+infinity-bundles)"
\end{imagefromfile}

### Singular cohesion

From this is obtained, in particular, a pair of [[adjoint triples]] of [[adjoint modalities]] on $\mathbf{H} \,\coloneqq\, Glo(\mathbf{H}_{\subset})$ which jointly reflect that its objects are [[geometric homotopy types]] whose geometric nature is "cohesive with [[orbi-singularities]]":

\begin{imagefromfile}
    "file_name": "SingularCohesionModalities_20211026.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

### Orbifolds among singular-smooth $\infty$-groupoids

For example, when $\mathbf{H}_{\subset} \,=\,$ [[smooth infinity-groupoids|$SmthGrpd_\infty$]],  containing [[diffeological spaces]] as the [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of [[n-truncated object of an (infinity,1)-category|0-truncated]] [[concrete objects]]

$$
  DiffSpc
  \xhookrightarrow{ i_{\tau_0, \sharp_1} }
  SmthGrpd_\infty
  \,,
$$

then the singular-cohesive $\infty$-topos $\mathbf{H} \,\coloneqq\, Glo(SmthGrpd_\infty)$ of *[[singular-smooth infinity-groupoids|singular-smooth $\infty$-groupoids]]* contains diffeological [[orbifolds]] as a [[full sub-(infinity,1)-category|full sub]] [[(2,1)-category]] of orbi-singular-,  hence of $\boxed{\prec}$-[[modal objects]]

$$
  \array{
    DiffOrbfld
    &\xhookrightarrow{\;}&
    LieGrpd
    &\xhookrightarrow{ \;}&
    Glo(SmthGrpd_\infty)
    \\
    &&
    X \sslash G
    &\mapsto&
    \boxed{\prec}(X \sslash G)
    \mathrlap{\,,}
  }
$$

and in such a way that their [[shape modality|shape]]

$$
  \esh
  \,
  \prec
  \,
  (X \!\sslash\! G)
  \;\;\in\;\;
  G Grpd_\infty
   \xhookrightarrow{ Dsc }
  (G SmthGrpd_\infty)_{/\boxed{\prec}\mathbf{B}G}
    \xhookrightarrow{ G Spc }
  \mathbf{H}_{/\boxed{\prec}\mathbf{B}G}
$$

is the incarnation of the [[D-topological space]] underlying $X$ as a [[G-space]] in $G$-[[equivariant homotopy theory]] (via [[Elmendorf's theorem]]).

## Related concepts

* [[orbifold cohomology]]

* [[equivariant homotopy theory]]

  * [[globally equivariant homotopy theory]]

  * [[cohesion of global- over G-equivariant homotopy theory]]

## References

The [[cohesion of global- over G-equivariant homotopy theory]] was first observed in

* {#Rezk14} [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_, 2014  ([pdf](http://www.math.uiuc.edu/~rezk/global-cohesion.pdf), [[Rezk14.pdf:file]])

with emphasis on characterization of the full image under $G Spc$ of [[orbispaces]] in sliced [[global homotopy theory]].

The above combination with smooth cohesion to singular cohesion was considered in

* {#SaSc20} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

in discussion of [[orbifold cohomology]], where the above notation is taken from, and in

* {#SaSc20} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Equivariant principal infinity-bundles|Equivariant principal $\infty$-bundles]]*

in discussion of [[equivariant bundle|equivariant]] [[principal infinity-bundle|principal $\infty$-bundles]], where the above diagrams are taken from.

[[!redirects singular-cohesive (infinity)-topos]]
[[!redirects singular-cohesive (infinity)-toposes]]

[[!redirects singular-cohesive (infinity,1)-topos]]
[[!redirects singular-cohesive (infinity,1)-toposes]]