+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

For $\mathcal{C}$ an [[(∞,1)-category]], a [[simplicial object]] in $\mathcal{C}$ is an [[(∞,1)-functor]]

$$
  X \colon \Delta^{op} \to \mathcal{C}
$$

from the [[opposite category]] of the [[simplex category]] into $\mathcal{C}$.

The $(\infty,1)$-category of simplicial objects in $\mathcal{C}$ and morphisms between them is the [[(∞,1)-category of (∞,1)-functors]]

$$
  \mathcal{C}^{\Delta^{op}}
  =
  Func_\infty(\Delta^{op}, \mathcal{C})
  \,.
$$

=--

For instance ([Lurie, def. 6.1.2.2](#Lurie)).

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

For $\mathcal{C}$ a [[1-category]] a simplicial object in $\mathcal{C}$ is a [[simplicial object]] in the traditional sense of [[category theory]].

=--

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

A _cosimplicial object_ in $\mathcal{C}$ is a simplicial object in the [[opposite category]] $\mathcal{C}^{op}$.

=--

## Properties

### Powering over simplicial sets
 {#Powering}

Assume that $\mathcal{C}$ has all [[(∞,1)-limits]].
The following is a model for the [[powering]] of simplicial objects in $\mathcal{C}$ by simplicial sets.

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

Let $\mathcal{C} \in QCat \hookrightarrow sSet$ be an [[(∞,1)-category]] incarnated as a [[quasi-category]], and let $X \colon \Delta^{op} \to \mathcal{C}$ be a simplicial object. Then for $K \in sSet$ any [[simplicial set]], write

$$
  X[K]
  \;\colon\;
  \Delta_{/K}^{op}
  \longrightarrow
  \Delta^{op}
  \stackrel{X}{\longrightarrow}
  \mathcal{C}
$$

for the the [[composition|composite]] [[(∞,1)-functor]] of $X_\bullet$ with the projection from (the [[opposite category]] of) the [[category of simplices]] of $K$, and write

$$
  X(K)
    \;\colon\;
  \underset{\leftarrow}{\lim}
  \left(
    \Delta_{/K}^{op}
    \longrightarrow
    \Delta^{op}
    \stackrel{X}{\longrightarrow}
    \mathcal{C}
  \right)
$$

for the [[(∞,1)-limit]] over it (if it exists).

=--

This is discussed in ([Lurie HTT 4.2.3, notation 6.1.2.5](#Lurie)). See also around ([Lurie 2, notation 1.1.7](#LurieGood)).

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

The inclusion $\Delta_{/K}^{nd} \hookrightarrow \Delta_{K}$ of the [[full subcategory]] on non-degenerate simplicies is a [[homotopy cofinal functor]] (as discussed there). Therefore the $(\infty,1)$-limit in def. \ref{PoweringByLimitOverCategoryOfSimplices} may equivalently be taken over this category of non-degenerate simplices.

=--

+-- {: .num_example}
###### Example

For $K = \Delta^1 \coprod_{\Delta^0} \Delta^1$ the simplicial set consisting of two consecutive edges, we have for $X_\bullet \in \mathcal{C}^{\Delta^\bullet}$ that

$$
  X(K) \simeq X_1 \underset{X_0}{\times} X_1
$$

is the [[homotopy fiber product]] in

$$
  \array{
    &&
    X(K)
    \\
    & \swarrow && \searrow
    \\
    X_1 &&&& X_1
    \\
    & {}_{\mathllap{\partial_1}}\searrow && \swarrow_{\mathrlap{\partial_0}}
    \\
    && X_0
  }
  \,.
$$

=--

+-- {: .num_example}
###### Example

For $K = \Delta^n$ itself an $n$-[[simplex]], for some $n \in \mathbb{N}$ the powering reduces to evaluation on that simplex:

$$
  X(\Delta^n) \simeq X_n
  \,.
$$

This is because the [[category of non-degenerate simplices]] of an $n$-simplex has a [[terminal object]] (namely that $n$-simplex itself), and so its [[opposite category]] has an [[initial object]] and the $(\infty,1)$-limit over a [[diagram]] with initial object is given by evaluation at that initial object.

=--

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

For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ and $K \to K'$ the following are equivalent

1. the induced morphism of cone $(\infty,1)$-categoris $\mathcal{C}_{X[K]} \to \mathcal{C}_{X[K']}$ is an [[equivalence of (∞,1)-categories]];

1. the induced morphism of [[(∞,1)-limits]] $X(K) \to X(K')$ is an [[equivalence in an (∞,1)-category|equivalence]].

=--

(The first perspective is used in ([Lurie](#Lurie)), the second in ([Lurie2](#Lurie2)).)

+-- {: .proof}
###### Proof

In one direction: the limit is the [[terminal object in an (∞,1)-category|terminal object]] in the cone category, and so is preserved by equivalences of cone categories. (This direction appears as ([Lurie, prop. 4.1.1.8](#Lurie))).  Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.

=--

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

Let $X \colon \Delta^{op} \to \mathcal{C}$ be a simplicial object which is a [[groupoid object in an (∞,1)-category]].

If $K \to K'$ is a morphism in [[sSet]] which is a [[weak homotopy equivalence]] and a [[bijection]] on [[vertices]], then the induced morphism on [[slice-(∞,1)-categories]]

$$
  \mathcal{C}_{/X[K]}
  \to
  \mathcal{C}_{/X[K']}
$$

is an [[equivalence of (∞,1)-categories]] (a [[weak equivalence]] in the [[model structure for quasi-categories]]).

Equivalently, by remark \ref{SlicingOverPoweringOfSimplicialObjects}, we have an equivalence

$$
  X(K) \to X(K')
  \,.
$$

=--

This is ([Lurie, prop. 6.1.2.6](#Lurie)).

### Cohesion

If $\mathcal{C} = \mathbf{H}$ is an [[(∞,1)-topos]] then $\mathcal{C}^{\Delta^{op}}$ is a [[cohesive (∞,1)-topos]] over $\mathbf{H}$. For more see at _[cohesive (∞,1)-topos - Examples - Simplicial objects](cohesive+%28infinity%2C1%29-topos#SimplicialObjctsInACohesiveTopos)_.

### Internal language

If $\mathcal{C}$ is a [[locally cartesian closed (∞,1)-category]] whose [[internal language]] is [[homotopy type theory]], then the internal language of $\mathcal{C}^{\Delta^{op}}$ is that homotopy type theory equipped with the axioms for a [[linear interval]] object. (...)

### Geometric realization and filtering

The _[[geometric realization]]_ ${\vert X_\bullet \vert}$ of a simplicial object $X_\bullet$  is, if it exists, the [[(∞,1)-colimit]] over the corresponding [[(∞,1)-functor]] $X_\bullet \;\colon\; \Delta^{op} \to \mathcal{C}$.

$$
  {\vert X_\bullet \vert}
  \coloneqq
  \underset{\longrightarrow}{\lim}_n X_n
  \,.
$$

Hence the geometric realization of a cosimplicial object $\Delta^{op} \to \mathcal{C}^{op}$ -- called its [[totalization]] -- is the [[(∞,1)-limit]] over $\Delta \to \mathcal{C}$.

The [[geometric realization]] of the [[simplicial skeleta]] of $X_\bullet$

$$
  {\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots
  \,.
$$

constitutes a [[filtered object in an (infinity,1)-category|filtering]] on the [[geometric realization]] of $X_\bullet$ itself

$$
  {\vert X_\bullet \vert}
  \simeq
  \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert}
  \,.
$$

If $\mathcal{C}$ is a [[stable (∞,1)-category]], then the
the corresponding [[spectral sequence of a filtered stable homotopy type]] is the [[spectral sequence of a simplicial stable homotopy type]].

### $\infty$-Dold-Kan correspondence

The following statement is the _[[infinity-Dold-Kan correspondence]]_.

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

Let $\mathcal{C}$ be a [[stable (∞,1)-category]]. Then the [[(∞,1)-categories]] of non-negatively graded [[sequential diagram|sequences]] in $C$ is [[equivalence of (∞,1)-categories|equivalent]] to the [[(∞,1)-category]] of [[simplicial objects in an (∞,1)-category]] in $\mathcal{C}$

$$
  Fun(N(\mathbb{Z}_{\geq 0}), C)
  \simeq
  Fun(N(\Delta)^{op}, C)
  \,.
$$

Under this equivalence, a [[simplicial object]] $X_\bullet$ is sent to the [[sequential diagram|sequence]] of [[geometric realizations]] ([[(∞,1)-colimits]]) of its [[simplicial skeleta]]

$$
  {\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots
  \,.
$$

This constitutes a [[filtered object in an (infinity,1)-category|filtering]] on the [[geometric realization]] of $X_\bullet$ itself

$$
  {\vert X_\bullet \vert}
  \simeq
  \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert}
  \,.
$$

=--

([[Higher Algebra|Higher Algebra, theorem 1.2.4.1]])

## Examples

### Internal category objects

* A [[pre-category object in an (∞,1)-category]] $\mathcal{C}$ is a simplicial object which satisfies the [[Segal conditions]];

* a [[category object in an (∞,1)-category]] is a pre-category object which also satisfies the [[univalence axiom]];

* a [[groupoid object in an (∞,1)-category]] is a category object all of whose [[morphisms]] are [[equivalences]] under composition.

## Related concepts

* [[simplicial object]]

  * [[simplicial set]]

* [[semi-simplicial object]]

  * [[semisimplicial set]]

* [[simplicial homotopy theory]]

* [[homotopy Kan fibration]]

## References

Simplicial objects in general [[(∞,1)-categories]] are discussed in

* [[Jacob Lurie]], section 6.1.2 of _[[Higher Topos Theory]]_
 {#Lurie}

Related discussion is also in

* [[Jacob Lurie]], _[[(∞,2)-Categories and the Goodwillie Calculus]]_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462))
 {#LurieGood}

Simplicial obects in [[stable (∞,1)-categories]] are discussed in

* [[Jacob Lurie]], section 1.2.4 of _[[Higher Algebra]]_

[[!redirects simplicial objects in an (∞,1)-category]]
[[!redirects simplicial objects in an (infinity,1)-category]]

[[!redirects simplicial object in an (∞,1)-category]]

[[!redirects cosimplicial object in an (∞,1)-category]]
[[!redirects cosimplicial objects in an (∞,1)-category]]

[[!redirects cosimplicial object in an (infinity,1)-category]]
[[!redirects cosimplicial objects in an (infinity,1)-category]]


simplicial object in Cat

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

#Contents#
* table of contents
{:toc}

## Definition

A **simplicial object in  $Cat$** is a [[simplicial object]], [[internalization|in]] to the [[1-category]] [[Cat]] of [[categories]] and [[functors]] between them, equivalently an [[internal category]] [[internalization|in]] [[sSet]].

This would deserve to be and sometimes is called a *[[simplicial category]]*, but more often the latter terminology is used to refer to [[simplicially enriched categories]] only (which may be regarded as the special case of simplicial objects in $Cat$ whose simplicial set of objects happens to be simplicially constant).

## Related concepts

* [[simplicial category]]

  * [[simplicially enriched category]]

  * **simplicial object in Cat**

* [[simplicial groupoid]]

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

## References

One of the few references that discusses a [[model category]] structure on $Cat^{\Delta^{op}}$ as opposed to just $sSet Cat$ (but see at *[[canonical model structure on Cat]]*) is:

* [[William Dwyer]], [[Dan Kan]], [[Jeff Smith]], Prop. 7.2 in: _Homotopy commutative diagrams and their realizations_, Journal of Pure and Applied Algebra **57** (1989) 5-24 [doi:10.1016/0022-4049(89)90023-6]

A [[model category]] stucture on $Cat(sSet)$ presenting [[(infinity,1)-categories]]:

* {#Horel15} [[Geoffroy Horel]], *A model structure on internal categories*, Theory and Applications of Categories, **30** 20 (2015) 704-750 [[tac:30-20](http://www.tac.mta.ca/tac/volumes/30/20/30-20abs.html), [arXiv:1403.6873](http://arxiv.org/abs/1403.6873)]

[[!redirects simplicial objects in Cat]]


simplicial operad

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

#Contents#
* table of contents
{:toc}

## Definition

A **simplicial operad** is an [[operad]] over [[sSet]]. It has for each $k \in \mathbb{N}$ a [[simplicial set]] of $k$-ary operations.

## Structures on the category of simplicial operads

### Model structure

There is a [[model category]] structure on the [[category]] of simplicial operads that makes them [[presentable (∞,1)-category|present]] [[(∞,1)-operad]]s. See [[model structure on operads]].

There is a [[Quillen equivalence]] between this model structure and the [[model structure on dendroidal sets]], via the [dendroidal homotopy coherent nerve](http://ncatlab.org/nlab/show/dendroidal+set#HomotopyCoherentNerve).

## Examples

### Classes of examples

* Every operad over [[Set]] may be regarded as a simplicial operad, via the [[discrete object]] embedding $Disc : Set \to sSet$.

* Every [[topological operad]] induces a simplicial operad by applying the [[singular simplicial complex]] functor $Sing : Top \to sSet$ degreewise.

* Every [[dendroidal set]] induces a simplicial operad via the [[right adjoint]] to the [dendroidal homotopy coherent nerve](http://ncatlab.org/nlab/show/dendroidal+set#HomotopyCoherentNerve).

### Specific examples

* The [[Barratt-Eccles operad]] has, as a simplicial operad, a single color, and its simplicial set of $n$-ary operations is the [[nerve]] $N(\Sigma_n//\Sigma_n)$ of the [[action groupoid]] $\Sigma_n // \Sigma_n$ of the [[symmetric group]] $\Sigma_n$ acting on itself. This is a [[contractible]] simplicial set in each degree, and, indeed, this operad is a [[resolution]] of [[Comm]], which has the point in each degree.

## Related concepts

* [[topological operad]]

* [[dg-operad]]

* [[dendroidal set]]

[[!redirects simplicial operads]]


simplicial presheaf

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

_Simplicial presheaves_ over some [[site]] $S$ are

* [[presheaf|Presheaves]] with values in the category [[SimpSet]] of simplicial sets, i.e., functors $S^{op} \to \Simp\Set$, i.e., functors $S^{op} \to [\Delta^{op}, \Set]$;

or equivalently, using the Hom-[[adjoint functor|adjunction]] and symmetry of the [[closed monoidal category|closed monoidal structure]] on [[Cat]]

* simplicial objects in the category of presheaves, i.e. functors $\Delta^{op} \to [S^{op},\Set]$.

## Interpretation as $\infty$-stacks ##

Regarding $\Simp\Set$ as a [[model category]] using the standard [[model structure on simplicial sets]] and inducing from that a model structure on $[S^{op}, \Simp\Set]$ makes simplicial presheaves a model for $\infty$-[[infinity-stack|stacks]], as described at [[infinity-stack homotopically]].

In more illustrative language this means that a simplicial presheaf on $S$ can be regarded as an $\infty$-[[infinity-groupoid|groupoid]] (in particular a [[Kan complex]]) whose space of $n$-morphisms is modeled on the objects of $S$ in the sense described at [[space and quantity]].

## Examples ##

* Notice that most definitions of $\infty$-[[infinity-category|category]] the $\infty$-category is itself defined to be a [[simplicial set]] with extra structure (in a [[geometric definition of higher category]]) or gives rise to a simplicial set under taking its [[nerve]] (in an [[algebraic definition of higher category]]). So most notions of presheaves of higher categories will naturally induce presheaves of simplicial sets.

* In particular, regarding a [[group]] $G$ as  a one object category $\mathbf{B}G$ and then taking the nerve $N(\mathbf{B}G) \in \Simp\Set$ of these (the "classifying simplicial set of the group whose [[geometric realization]] is the [[classifying space]] $\mathcal{B}G$), which is clearly a functorial operation, turns any presheaf with values in groups into a simplicial presheaf.

## Remarks ##

* There are various useful [[model category]] structures on the category of simplicial presheaves. See [[model structure on simplicial presheaves]].

## Properties ##

Here are some basic but useful facts about simplicial presheaves.

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

Every simplicial presheaf $X$ is a
[[homotopy limit|homotopy colimit]] over a [[diagram]]
of [[Set]]-valued sheaves regarded as discrete simplicial sheaves.

More precisely, for $X : S^{op} \to SSet$ a simplicial presheaf,
let $D_X : \Delta^{op} \to [S^{op},Set] \hookrightarrow [S^{op},SSet]$ be
given by $D_X : [n] \mapsto X_n$. Then there is a weak equivalence
$$
  hocolim_{[n] \in \Delta} D_X([n]) \stackrel{\simeq}{\to} X
  \,.
$$
=--

+--{: .proof}
######Proof
See for instance [remark 2.1, p. 6](http://www.math.uiuc.edu/K-theory/0563/spre.pdf#page=6)

* [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 136 Issue 1, 2004 ([arXiv:math/0205027](https://arxiv.org/abs/math/0205027), [K-theory:0563](http://www.math.uiuc.edu/K-theory/0563), [doi:10.1017/S0305004103007175](https://doi.org/10.1017/S0305004103007175))

(which is otherwise about [[descent for simplicial presheaves]]).
=--

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

Let $[-,-] : (SSet^{S^{op}})^{op} \times SSet^{S^{op}} \to SSet$ be the canonical $SSet$-enrichment of the category of simplicial presheaves (i.e. the assignment of [[SSet]]-[[enriched functor category|enriched functor categories]]).

It follows in particular from the above that every such [[hom-object]] $[X,A]$ of simplical presheaves can be written as a [[homotopy limit]] (in [[SSet]] for instance realized as a [[weighted limit]], as described there) over evaluations of $A$.
=--

+--{: .proof}
######Proof

First the above yields
$$
  \begin{aligned}
     [X, A ]  & \simeq [ hocolim_{[n] \in \Delta} X_n , A ]
  \\
      & holim_{[n] \in \Delta} [X_n, A]
  \end{aligned}
  \,.
$$

Next from the [[co-Yoneda lemma]] we know that the [[Set]]-valued presheaves $X_n$ are in turn colimits over representables in $S$, so that
$$
  \begin{aligned}
     \cdots & \simeq
     holim_{[n] \in \Delta}
     [ colim_i U_{i}, A]
     \\
     & \simeq
     holim_{[n] \in \Delta} lim_i
     [  U_{i}, A]
  \end{aligned}
  \,.
$$

And finally the [[Yoneda lemma]] reduces this to
$$
  \begin{aligned}
     \cdots
      &
     holim_{[n] \in \Delta} lim_i
     A(U_i)
  \end{aligned}
  \,.
$$
=--

Notice that these kinds of computations are in particular often used when checking/computing [[descent|descent and codescent]] along a [[cover]] or [[hypercover]]. For more on that in the context of simplicial presheaves see [[descent for simplicial presheaves]].

## Related entries ##

* [[model structure on simplicial presheaves]]

* [[descent for simplicial presheaves]]

* [[presheaf of spectra]]

Applications appear for instance at

* [[geometric infinity-function theory]]

## References ##

The original articles are

* [[Kenneth S. Brown]], _Abstract homotopy theory and generalized sheaf cohomology_.  Transactions of the American Mathematical Society 186 (1973), 419-419.  [doi](http://dx.doi.org/10.1090/s0002-9947-1973-0341469-9).

* [[Kenneth S. Brown]], [[Stephen M. Gersten]], _Algebraic K-theory as generalized sheaf cohomology_.  In: Higher K-Theories.  Lecture Notes in Mathematics (1973), 266–292.  [doi](http://dx.doi.org/10.1007/bfb0067062).

The Joyal model structure on [[simplicial sheaves]] was constructed in 1984 by [[Joyal]] in

* [[André Joyal]], _Lettre d’André Joyal à Alexandre Grothendieck_, April 11, 1984, [PDF](https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf).

Simplicial sheaves were endowed with a structure of a [[category of fibrant objects]] in

* [[J. F. Jardine]], _Simplicial objects in a Grothendieck topos_.  In: Applications of algebraic K-theory to algebraic geometry and number theory.  Contemporary Mathematics (1986), 193-239.  [doi](http://dx.doi.org/10.1090/conm/055.1/862637)

The model structure on simplicial presheaves is due to [[Jardine]]:

* [[John F. Jardine]], _Simplical presheaves_, Journal of Pure and Applied Algebra **47** 1 (1987) 35-87 [doi:10.1016/0022-4049(87)90100-9]

A modern expository account is in

* [[John F. Jardine]], _Local Homotopy Theory_, Springer, 2015.  [doi](http://dx.doi.org/10.1007/978-1-4939-2300-7).

The local [[projective model structure]] (as opposed to the injective model structures used above) on [[simplicial presheaves]] and [[simplicial sheaves]] is due to

* [[Benjamin A. Blander]], _Local Projective Model Structures on Simplicial Presheaves_, K-Theory 24:3, 283-301.  [doi](http://dx.doi.org/10.1023/a:1013302313123).

Further articles:

* [[J. F. Jardine]], _Stacks and the homotopy theory of simplicial sheaves_.  Homology, Homotopy and Applications 3:2 (2001), 361-384.  [doi](http://dx.doi.org/10.4310/hha.2001.v3.n2.a5).

* [[J. F. Jardine]], _Fields Lectures: Simplicial presheaves_.
[PDF](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf).

For their interpretation in the more general context of [[(infinity,1)-category of (infinity,1)-sheaves|(infinity,1)-sheaves]] see Section 6.5.2 of

*  [[Jacob Lurie]], [[Higher Topos Theory]].

[[!redirects simplicial presheaves]]

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

[[!redirects SimplicialPresheaves]]


simplicial presheaves > history

< [[simplicial presheaves]]
[[!redirects simplicial presheaves -- history]]


simplicial principal bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[simplicial group|Simplicial groups]] model all [[∞-group]]s in [[∞Grpd]]. Accordingly  [[principal ∞-bundle]]s in [[∞Grpd]] (a [[discrete ∞-groupoid|discrete]] $\infty$-bundle) should be modeled by [[Kan complex]]es $E \to X$ equipped with a principal [[action]] by a [[simplicial group]]. It is suficient to assume the action to be strict. This yields the notion of _simplicial principal bundles_ .

## Definition

+-- {: .num_defn}
###### Definition
**(strict principal simplicial bundle)**

A strict simplicial principal bundle is...

=--

+-- {: .num_defn #WeaklyPrincipalSimplicialBundle}
###### Definition
**(weakly principal simplicial bundle)**

For $G$ a [[simplicial group]], a [[Kan fibration]] $P \to X$ of [[Kan complexes]] equipped with a $G$-[[action]] on $P$ over $X$

$$
  \array{
    P \times G
    &&
    \overset{\rho}{\longrightarrow}
    &&
    P
    \\
    & \searrow && \swarrow
    \\
    &&
    X
  }
$$

is a _weakly $G$-principal simplicial bundle_ if the _shear map_

$$
  P \times G
   \overset{ (p_1, \rho) }{\longrightarrow}
  P \times_X P
$$

(to the [[fiber product]] of $P$ with itself over $X$) is a [[simplicial weak equivalence]].

=--

(e.g. [NSS 12b, Def. 3.79](#NSS12b))

## Properties

### General

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

A simplicial $G$-principal bundle $P \to X$ is necessarily a [[Kan fibration]].

=--

+-- {: .proof}
###### Proof

This appears as ([May, Lemma 18.2](#May)).

=--

### Twisted cartesian products

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

Let $E \to B$ be a [[twisted cartesian product]] of the [[simplicial set]] $B$ with a [[simplicial group]] $G$. Then with respect to the canonical $G$-[[action]] this is a simplicial principal bundle.

=--

This is ([May, prop. 18.4](#May)).

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

It is simplicial principal bundles of this form that one is mainly interested in. These are the objects that are classified by the evident [[classifying space]] $\overline{W} G$. This is discussed [below](UniversalSimplicialBundle).

=--

## The universal simplicial $G$-principal bundle
 {#UniversalSimplicialBundle}

Recall from [[generalized universal bundle]] that a universal $G$-principal simplicial bundle should be a principal bundle $\mathbf{E}G \to \mathbf{B}G$ such that every other $G$-principal simplicial bundle $P \to X$ arises up to equivalence as the [[pullback]] of $\mathbf{E}G$ along a morphism $X \to \mathbf{B}G$.

A standard model for the [[delooping]] [[Kan complex]] $\mathbf{B}G$ for $G$ a simplicial group goes by the simplicial set, $
  \overline{W} G
  \,.
$

This is described at simplicial group - delooping. The following establishes a model for the universal simplicial bundle over this model of $\mathbf{B}G$.

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

For $G$ a simplicial group, define the [[simplicial set]] $W G$ to be the [[decalage]] of $\overline{W}G$

$$
  W G \coloneqq Dec \overline{W}G
  \,.
$$

=--

By the discussion at [[homotopy pullback]] this means that for $X_\bullet$ any [[Kan complex]], an ordinary [[pullback]] diagram

$$
  \array{
    P_\bullet &\to& W G
    \\
    \downarrow && \downarrow
    \\
    X_\bullet &\stackrel{g}{\to}& \overline{W}G
  }
$$

in [[sSet]] exhibits $P_\bullet$ as the [[homotopy pullback]] in $sSet_{Quillen}$ / [[(∞,1)-pullback]] in [[∞Grpd]]

$$
  \array{
    P_\bullet &\to& *
    \\
    \downarrow &\swArrow& \downarrow
    \\
    X_\bullet &\stackrel{g}{\to}& \overline{W}G
  }
  \,,
$$

i.e. as the [[homotopy fiber]] of the cocycle $g$.

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

We call $P_\bullet := X_\bullet \times^g W G$ the simplicial $G$-[[principal bundle]] corresponding to $g$.

=--

### Properties

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

Let $\{\phi : X_n \to G_{(n-1)}\}$ be the [[twisting function]] corresponding to $g : X_\bullet \to \overline{W}G$ by the above discussion.

Then the simplicial set  $P_\bullet := X_\bullet \times_{g} W G$ is explicitly given by the formula called the [[twisted Cartesian product]] $X_\bullet \times^\phi G_\bullet$:

its cells are

$$
  P_n  = X_n \times G_n
$$

with face and degeneracy maps

*  $d_i (x,g) = (d_i x , d_i g)$ if $i \gt 0$

* $d_0 (x,g) = (d_0 x, \phi(x) d_0 g)$

* $s_i (x,g) = (s_i x, s_i g)$.

=--

## Related notions

* [[simplicial classifying space]]

* [[principal bundle]]

* [[torsor]] and [[heap]]

## References

Here are some pointers on where precisely in the literature the above statements can be found.

One useful reference is

* [[Peter May]], _Simplicial Objects in Algebraic Topology_ ([djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu)).
 {#May}

There the abbreviation PCTP ( _principal twisted cartesian product_ ) is used for what above we called just [[twisted Cartesian product]]s.

The fact that every PTCP $X \times_\phi G \to X$ defined by a [[twisting function]] $\phi$ arises as the pullback of $W G \to \overline{W}G$ along a morphism of simplicial sets $X \to \overline{W}G$ can be found there by combining

1. the last sentence on p. 81 which asserts that pullbacks of PTCPs $X \times_\phi G \to X$ along morphisms of simplicial sets $f : Y \to X$ yield PTCPs corresponding to the composite of $f$ with $\phi$;

1. section 21 which establishes that $W G \to \bar W G$ is the PTCP for some universal twisting function $r(G)$.

1. lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle $Y \to \bar W G$ with the universal twisting function $r(G)$. In view of the relation to pullbacks in item 1, this yields the statement in the form we stated it above.

An explicit version of the statement that [[twisted Cartesian product]]s are nothing but pullbacks of a [[generalized universal bundle]] is on [page 148](http://ncatlab.org/timporter/files/menagerie10.pdf#page=148) of

* [[Tim Porter]], _[Crossed Menagerie](http://ncatlab.org/timporter/show/crossed+menagerie)_

On [page 239](http://ncatlab.org/timporter/files/menagerie10.pdf#page=239) there it is mentioned that
$$
  G \to W G \to \overline{W}G
$$
is a model for the [[loop space object]] [[fiber sequence]]
$$
  G \to * \to \mathbf{B}G.
$$

One place in the literature where the observation that $W G $ is the [[decalage]] of $\overline{W}G$ is mentioned fairly explicitly is page 85 of

* [[John Duskin]], _Simplicial methods and the interpretation of "triple" cohomology_, number  163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc. (1975)

Discussion of _topological_ simplicial principal bundles is in

* [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundle in parameterized spaces_ ([arXiv:1203.2460](http://arxiv.org/abs/1203.2460))
 {#RobertsStevenson}

* [[Danny Stevenson]], _Classifying theory for simplicial parametrized groups_ ([arXiv:1203.2461](http://arxiv.org/abs/1203.2461))
 {#Stevenson}

Discussion of principal simplicial bundles in more general [[categories of simplicial presheaves]] (presenting [[(infinity,1)-toposes]]):

* {#NSS12b} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Section 3.7.2 of: _[[schreiber:Principal ∞-bundles -- models and general theory|Principal $\infty$-bundles -- Presentations]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 ([doi:10.1007/s40062-014-0077-4](http://link.springer.com/article/10.1007/s40062-014-0077-4), [arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

[[!redirects simplicial principal bundles]]

[[!redirects principal simplicial bundle]]
[[!redirects principal simplicial bundles]]

[[!redirects universal simplicial principal bundle]]
[[!redirects universal simplicial principal bundles]]

[[!redirects simplicial fiber bundle]]
[[!redirects simplicial fiber bundles]]

[[!redirects simplicial fibre bundle]]
[[!redirects simplicial fibre bundles]]


simplicial profinite space

A _simplicial profinite space_ is a simplicial object in the category of [[profinite spaces]] (i.e. of [[Stone spaces]]).

Simplicial profinite spaces are not equivalent to pro-objects in the category of finite simplicial sets, as was pointed out by Isaksen.

The homotopy theory of simplicial profinite spaces has been explored by [[Fabien Morel]] and [[Gereon Quick]].

For Morel's theory see

* [[F. Morel]], _Ensembles profinis simpliciaux et interprétation géométrique du foncteur $T$_, Bull. Soc. Math. France, 124, (1996), 347–373,

A reference to Quick's work is in

* [[G. Quick]], _Profinite homotopy theory_, [pdf](http://www.math.uiuc.edu/documenta/vol-13/17.pdf)

but there was an error in the proof of one of the main results which has been corrected in

* [[G. Quick]],  _Continuous group actions on profinite spaces_, J. Pure Appl. Algebra 215 (2011),
1024-1039.

(Note there are several incorrect statements, notably with respect to the construction of non-abelian cohomology, in these papers, but the essentials of the proofs would seem to be correct.)


simplicial Quillen adjunction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _simplicial Quillen adjunction_ is an [[sSet]]-[[enriched Quillen adjunction]]: an [[enriched adjunction]]

$$
  (L \dashv R) : C \stackrel{\leftarrow}{\to} D
$$

of [[sSet]]-[[enriched functor]]s between [[simplicial model categories]] $C$ and $D$, such that the underlying adjunction of ordinary functors is a [[Quillen adjunction]] between the [[model category]] structures underlying the simplicial model categories.

## Properties

### Presentation of $\infty$-adjunctions

Simplicial Quillen adjunctions model pairs of [[adjoint (∞,1)-functor]]s in a fairly immediate manner: their restriction to fibrant-cofibrant objects is the [[sSet]]-[[enriched functor]] that presents the $(\infty,1)$-[[derived functor]] under the model of [[(∞,1)-categories]] by [[simplicially enriched categories]].

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

Let $C$ and $D$ be [[simplicial model categories]] and let

$$
  (L \dashv R)
  \;\colon\;
  C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D
$$

be an [[sSet]]-[[enriched category theory|enriched]] [[adjunction]] whose underlying ordinary adjunction is a Quillen adjunction. Let $C^\circ$ and $D^\circ$ be the [[(∞,1)-categories]] presented by $C$ and $D$ (the [[Kan complex]]-enriched full [[sSet]]-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of [[adjoint (∞,1)-functors]]

$$
  (\mathbb{L} \dashv \mathbb{R})
  \;\colon\;
  C^\circ \stackrel{\leftarrow}{\to} D^{\circ}
  \,.
$$

On the [[decategorification|decategorified]] level of the [[homotopy category|homotopoy categories]] these are the total left and right [[derived functors]], respectively, of $L$ and $R$.

=--

+-- {: .proof}
###### Proof

This is proposition 5.2.4.6 in [[Higher Topos Theory|HTT]].

=--

### Recognition {#Recognition}

The following proposition states conditions under which a simplicial Quillen adjunction may be detected already from knowing of the right adjoint only that it preserves fibrant objects (instead of all fibrations).

\begin{proposition}\label{RecognitionOfSimplicialQuillenAdjunctions}
**(recognition of simplicial Quillen adjunctions)**
\linebreak
If

1. $\mathcal{C}$ and $\mathcal{D}$ are [[simplicial model categories]]

1. $\mathcal{D}$ is a [[left proper model category]],

then for an [[sSet]]-[[enriched adjunction]]

$$
  (L \dashv R)
  \;\colon\;
  \mathcal{C}
  \underoverset
    {\underset{R}{\longrightarrow}}
    {\overset{L}{\longleftarrow}}
    {\;\;\;\bot\;\;\;}
  \mathcal{D}
$$

to be a Quillen adjunction it is already sufficient that

1. $L$ preserves [[cofibrations]]

1. $R$ preserves [[fibrant objects]].

\end{proposition}

This appears as [[Higher Topos Theory|HTT, cor. A.3.7.2]].

\begin{remark}

Prop. \ref{RecognitionOfSimplicialQuillenAdjunctions} is particularly useful for finding simplicial Quillen adjunctions into [[Bousfield localization of model categories|left Bousfield localizations]] of left proper model categories: the left Bousfield localization keeps the cofibrations unchanged and preserves left properness, and the fibrant objects in the Bousfield localized structure have a good characterization: they are the fibrant objects in the original model structure that are also [[local object]]s with respect to the set of morphisms at which one localizes.

Therefore for $D$ the left Bousfield localization of a simplicial left proper model category $E$ at a [[class]] $S$ of morphisms, for checking the Quillen adjunction property of $(L \dashv R)$ it is sufficient to check that $L$ preserves cofibrations, and that $R$ takes fibrant objects $c$ of $C$ to such fibrant objects of $E$ that have the property that for all $f \in S$ the [[derived hom-space]] map $\mathbb{R}Hom(f,R(c))$ is a weak equivalence.

\end{remark}

## Related concepts

* [[Quillen adjunction]]

* **simplicial Quillen adjunction**

* [[Quillen equivalence]]

* [[monoidal Quillen adjunction]]

## References

On the enhancement of plain [[Quillen adjunctions]] between [[left proper model category|left proper]] [[combinatorial model categories]] to simplicial Quillen adjunctions:

* {#BlumbergRiehl14} [[Andrew Blumberg]], [[Emily Riehl]], Appendix A of: *Homotopical resolutions associated to deformable adjunctions*, Algebr. Geom. Topol. 14 (2014) 3021-3048 ([arXiv:1208.2844](https://arxiv.org/abs/1208.2844))

[[!redirects simplicial Quillen adjunctions]]


simplicial resolution

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of _[[resolution]]_ in [[homotopy theory]] specialized to [[simplicial homotopy theory]] is "simplicial resolution".  Simplicial resolutions can be constructed in various ways, for instance, by a comonad, or by a step-by-step method, developed by Michel André, that resembles the construction of a Eilenberg-Mac Lane space from a [[group presentation]], followed by adding cells to 'kill' the higher homotopy groups.

> the following needs attention

##Remark

The term _simplicial resolution_ is also used more generally.

In any ambient [[category]] or $\infty$-[[infinity-category|category]] $C$ that admits a notion of
[[colimit]] or [[weak limit|weak colimit]], a _simplicial resolution_
of an object $c \in C$ is a [[simplicial object]]
$y_\bullet : \Delta^{op} \to C$ together with a realization of $c$ as a colimit

$$
  c \simeq colim_{[k]} y_k
  \,.
$$

The term is also used for a [[Reedy model structure|Reedy fibrant]] replacement of a constant simplicial object in a [[model category]]; see also [[resolution]].

[[cocone|Cocones]] under a simplicial diagram are exactly the [[augmented simplicial objects]], so a simplicial resolution of $c$ can alternately be described as an augmented simplicial object $y_\bullet : \Delta^{op}_+ \to C$ that is a colimiting cocone, and such that $y_{-1} = c$.

## Examples

### Free simplicial resolution of a group.

There is the obvious adjoint pair of functors,

$$U:Groups\to Sets,  F:Sets\to Groups.$$

 Writing $\eta : Id \to UF$ and $\varepsilon :FU\to Id$ for the unit and counit of this adjunction, we have a comonad on $Groups$, the free  group comonad, $( FU, \varepsilon, F\eta U)$.

We write $L = FU$, $\delta = F\eta U$, so that

$$\varepsilon : L \to I$$

is the counit of the comonad whilst

$$\delta: L \to L^2$$

is the comultiplication.

Now suppose $G$ is a  group and set $F(G)_i = L^{i+1}(G)$, so that $F(G)_0$ is the free  group on the underlying set of $G$ and so on.

The counit (which is just the augmentation morphism from $FU(G)$ to $G$) gives, in each dimension, face morphisms

$$d_i = L^{n-i}\varepsilon L^{i}(G) : L^{n+1}(G) \to L^n(G),$$
for $i = 0, \ldots, n$,
whilst the comultiplication gives degeneracies,

$$s_i : L^n(G) \to L^{n+1}(G)$$

$$s_i = L^{n-1-i}\delta L^{i}, $$
for $i = 0, \ldots, n-1$,

satisfying the simplicial identities.

The resulting object is an augmented [[simplicial group]] that is free in all non-negative dimensions and is acyclic. It has zeroth homotopy equal to $G$ and all homotopy groups are trivial.

Of course this construction did not depend on the fact that we were handling groups, so we could apply it to any [[comonad]] on any category (within reason!) This has the advantage of providing simplicial resolutions that are functorial. These are sometimes called _comonadic resolutions_.  Working with [[monad]]s on a category gives, a cosimplicial object which is a _cosimplicial resolution_ given any object.

This leads to the subject of [[monadic cohomology]].

A *cautionary note* is in order. The simplicial resolution of an object derived from a [[comonad]] is sometimes presented in an opposite form, so

$$d_i = L^i\varepsilon L^{n-i}(G) : L^{n+1}(G) \to L^n(G),$$
for $i = 0, \ldots, n$, and similarly for the degeneracies.  This is more or less equivalent to the form given here as it just uses the opposite simplicial object.

### Cech nerve

In an [[(infinity,1)-topos]] [[Čech cover]] $C(U) \stackrel{\simeq}{\to} X$
  induced by a [[cover]] $(U = \coprod_i U_i) \to X$ is a simplicial resolution of $X$.

## Related concepts

* [[homological resolution]]

* [[monadic cohomology]]

* [[canonical resolution]]

## References and Literature

Classical soureces for the comonadic form include [[Roger Godement|Godement]]'s book,

* [[Roger Godement]], _Topologie Algébrique et Théorie des Faisceaux_. Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13 Hermann, Paris 1958.

The theory is explained in many sources including

* [[M. Barr]] and [[J. Beck]], 1969, _Homology and Standard Constructions_, in _Seminar on triples and categorical homology_, number 80 in Lecture Notes in Maths., Springer-Verlag, Berlin,

and in Jack Duskin's  monograph:

*  [[J. Duskin]], 1975, Simplicial methods and the interpretation of "triple" cohomology , number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc.

The step-by-step method of producing a simplicial resolution was developed by [[Michel André]] in the texts:

* [[Michel André]], _Méthode simpliciale en algèbre homologique et algèbre commutative_, Springer Lecture Notes in Mathematics, Vol 32, 1967.

* [[Michel André]], _Homologie des algèbres commutatives_ Grundlehren der mathematischen Wissenschaften, Band 206. Springer. 1974

with a view to applications in commutative algebra and in particular in the development of the cohomology known as [[André-Quillen cohomology]] and the study of the [[cotangent complex]].

Simplicial resolutions in the context of
[[presentable (infinity,1)-category|presentable (infinity,1)-categories]] are discussed in section 6.1.4 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

(below lemma 6.1.4.3)

[[!redirects simplicial resolutions]]

[[!redirects comonadic resolution]]


simplicial ring

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

#Contents#
* automatic table of contents
{:toc}

## Idea

A _simplicial ring_ is a [[simplicial object]] in the [[category]] [[Ring]] of [[rings]].

This may be understood conceptually as follows:

* as ordinary rings are algebras over the ordinary [[algebraic theory]] $T$ of rings, if we regard this as an [[(∞,1)-algebraic theory]] then simplicial rings model the $(\infty,1)$-algebras over that;

* the category [[Ring]]${}^{op}$ is naturally equipped with the structure of a [[geometry (for structured (infinity,1)-toposes)|pregeometry]]. The corresponding [[geometry (for structured (∞,1)-toposes)]] is $sRing^{op}$, the opposite of the category of simplicial rings.

## Definition

A **simplicial ring** is a [[simplicial object]] in the [[category]] [[Ring]] of [[rings]].

There is an evident notion of [[(∞,1)-category]] of [[module]]s over a simplicial ring. The corresponding [[bifibration]] $sMod \to sRing$ of modules over simplicial ring is equivalently to the [[tangent (∞,1)-category]] of the [[(∞,1)-category]] of simplicial rings.

## Properties

### Connected components

Given a simplicial ring $A = A_{\bullet}$, its [[connected components]] (the 0th "[[homotopy group]]")) is an ordinary [[ring]]

$$
  \pi_0(A) \in Ring
  \,.
$$

Forming connected components is a [[functor]] from simplicial rings to plain rings, which is [[left adjoint]] to the inclusion of ordinary rings as simplicially constant simplicial rings, exhibiting a [[reflective subcategory]] inclusion

$$
  Ring \stackrel{\stackrel{\pi_0}{\longleftarrow}}{\hookrightarrow} Ring^{\Delta^{op}}
  \,.
$$

([Toën 14, section 2.2](Toen14))

So on [[formal duals]] of [[commutative ring|commutative]] (simplicial) rings this is a [[coreflection]] of [[affine schemes]] in affine [[derived schemes]]

$$
  CRing^{op}
    \stackrel{\hookrightarrow}{\longleftarrow}
  (CRing^{\Delta^{op}})^{op}
  \,.
$$

Notice that coreflective embeddings are also given for instance by the inclusion of [[manifolds]] into [[formal manifolds]]. This is one way in which formal duals of simplicial rings manifest themselves as [[infinitesimal neighbourhoods]] of formal duals of plain rings.

### Higher homotopy groups

The higher [[homotopy groups]] $\pi_n(A)$ of a simplicial ring $A_\bullet$ are naturally [[modules]] over the ring $\pi_0(A)$ of connected components, so $A_\bullet$ is weakly [[contractible]], as a [[simplicial set]], iff $\pi_0(A)=0$.

(Again this is a manifestation of the simplicial ring being just an infinitesimal thickening of its connected components.)

Notice that $\pi_0(A)$ is equivalently the [[cokernel]] of $d_0-d_1 \colon A_1 \longrightarrow A_0$.  Accordingly, any chain of [[face maps]] $A_n \longrightarrow A_0$ compose with the projection to $\pi_0(A)$ is independent of the choices.  These maps

$$
  A_n \longrightarrow \pi_0(A)
$$

give a [[surjective]] map from $A_\bullet$ to the constant simplicial ring $\pi_0(A)$.

(This is just the simplest piece of the [[Postnikov tower]].) If $\pi_0(A)\neq 0$ we can then compose with a surjective map to a constant simplicial field.

### Model category structure

There is a [[model category]] structure on simplicial rings that presents $\infty$-rings. See [[model structure on simplicial T-algebras]] for more.

We describe here the [[model category]] presentation of the [[(∞,1)-category]] of modules over simplicial rings.

Let $A$ be a simplicial commutative algebra. Write $A SMod$ for the [[category]] which, with $A$ regarded as a [[monoid]] in the category $SAb$ of abelian [[simplicial group]]s is just the category of $A$-[[module]]s in $SAb$. This means that

* objects are abelian [[simplicial group]]s $N$ equipped with
  an [[action]] [[morphism]] $A \otimes N \to N$ of simplicial
  abelian groups;

Equip $A SMod$ with the structure of a [[model category]] by setting:

* a morphism $N_1\to N_2$ of $A$-[[modules]] is a weak equivalence
  resp. a fibration precisely if the  underlying morphism of [[simplicial set]]s is a
  weak equivalence, resp. fibration, in the standard
  [[model structure on simplicial sets]].

**Proposition** This defines a [[model category]] structure which is

* [[cofibrantly generated model category|cofibrantly generated]];

* [[proper model category|proper]];

* [[cellular model category|cellular]].

## Examples

### Simplicial fields

All simplicial [[fields]] are simplicially constant.  This is because the composite $A_0\xrightarrow{s_0^n}A_n\xrightarrow{d_0^n}A_0$ is the identity, so $d_0^n$ is surjective, but all field homomorphisms are injective, so $d_0^n$ is an isomorphism.

## Related concepts

* [[simplicial algebra]]

* [[model structure on simplicial T-algebras]]

## References

Introduction and survey includes

* [[Bertrand Toën]], chapter 4 of _Simplicial presheaves and derived algebraic geometry_ , lecture at [Simplicial methods in higher categories](http://www.crm.es/HigherCategories/)  ([pdf](http://www.crm.cat/HigherCategories/hc1.pdf))

* {#Toen14} [[Bertrand Toën]], _Derived Algebraic Geometry_ ([arXiv:1401.1044](http://arxiv.org/abs/1401.1044))

See [[model structure on simplicial algebras]] for references on the model structure discussed above.

Some of the above material is taken from this MO entry.

Discussion in the context of [[homotopy theory]], hence for simplicial [[ring spectra]] includes

* {#GoerssHopkins99} [[Paul Goerss]], [[Michael Hopkins]], _Simplicial Structured Ring Spectra_ (1999) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.9216))

[[!redirects simplicial rings]]
[[!redirects simplicial commutative ring]]
[[!redirects simplicial commutative rings]]


simplicial scheme

#Contents#
* table of contents
{:toc}

## Definition

A *simplicial scheme* is a [[simplicial object]] in a [[category]] of [[schemes]].

## References

* Bojan Mohar, _Simplicial schemes_, Journal of Combinatorial Theory, Series B
Volume 42, Issue 1, February 1987, Pages 68–86, doi: 10.1016/0095-8956(87)90064-5, [ScienceDirect](http://www.sciencedirect.com/science/article/pii/0095895687900645)

[[homotopy theory]] of simplicial schemes is discussed in

* [[David A. Cox]], _Homotopy theory and simplicial schemes_, [pdf-slides](http://www.math.northwestern.edu/efconf/cox.pdf)

* David A. Cox, _Homotopy theory of simplicial schemes_. Compositio Mathematica, 39 no. 3 (1979), p. 263-296
Full text djvu | pdf | Reviews MR 550644 | Zbl 0385.14004
stable [URL](http://www.numdam.org/item?id=CM_1979__39_3_263_0)

[[étale homotopy]] of simplicial schemes is discussed in

* [[Eric Friedlander]], 1982, _Étale homotopy of simplicial schemes_ , volume 104 of Annals of Mathematics Studies , Princeton University Press, Princeton, N.J.

[[motives]] in the context of simplicial schemes are discussed in

* [[V.  Voevodsky]], _Motives over simplicial schemes_, Journal of K-Theory, Volume 5, Issue 01 , pp 1 - 38,
(preliminary version in K-theory preprint archive: [here](http://www.math.uiuc.edu/K-theory/0638/))

* Carlo Mazza, [[V. Voevodsky|Vladimir Voevodsky]], Charles Weibel, Lecture Notes on Motivic Cohomology, http://www.claymath.org/library/monographs/cmim02.pdf

[[!redirects simplicial schemes]]


simplicial set

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Content#
* table of contents
{:toc}

## Idea

Simplicial sets generalize the idea of [[simplicial complexes]]: a _simplicial set_ is like a combinatorial space built up out of gluing abstract [[simplex|simplices]] to each other. Equivalently, it is an object equipped with a rule for how to consistently map the objects of the [[simplex category]] into it.

More concretely, a simplicial set $S$ is a collection of [[sets]] $S_n$ for $n \in \mathbb{N}$, so that elements in $S_n$ are to be thought of as $n$-[[simplex|simplices]], equipped with a rule that says:

* which $(n-1)$-simplices in $S_{n-1}$ are faces of which elements of $S_n$;
* which $(n+1)$-simplices are [[thin element|thin]] in that they are really just $n$-simplices regarded as degenerate $(n+1)$-simplices.

One of the main uses of simplicial sets is as combinatorial _models_ for  the (weak) [[homotopy type]] of [[topological spaces]]. They can also be taken as models for [[∞-groupoids]]. This is encoded in the [[model structure on simplicial sets]]. For more reasons why simplicial sets see MathOverflow [here](http://mathoverflow.net/questions/58497/is-there-a-high-concept-explanation-for-why-simplicial-leads-to-homotopy-theor).

## Definition
 {#Definition}

The quick abstract definition of a simplicial set goes as follows:

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

A **simplicial set**  is a [[presheaf]] on the [[simplex category]] $\Delta$, that is, a [[functor]] $X : \Delta^{op} \to Sets$ from the [[opposite category]] of the [[simplex category]] to the category [[Set]] of [[sets]]; equivalently this a [[simplicial object]] in [[Set]].

Equipped with the standard [[homomorphisms]] of [[presheaf|presheaves]] as morphisms (namely [[natural transformation|natural transformations]] of the corresponding [[functors]]), simplicial sets form the category [[sSet]] (denoted both  $SSet$ or $sSet$).

=--

Explicitly this means the following.

+-- {: .num_defn}
###### Definition
**(simplicial set)**

A **simplicial set** $X \in sSet$ is

* for each $n \in \mathbb{N}$ a [[set]] $X_n \in Set$ -- the **set of $n$-[[simplices]]**;

* for each [[injective map]] $\;\delta_i :\: [n-1] \to [n]\;$ of [[totally ordered sets]] ($[n] \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}$)

  a [[function]] $\;d_i :\: X_{n} \to X_{n-1}\;$ -- the $i$th _[[face map]]_ on $n$-simplices ($n \gt 0$ and $0 \leq i \leq n$);

* for each [[surjective map]] $\;\sigma_i :\: [n+1] \to [n]\;$ of totally ordered sets

  a function $\;s_i :\: X_{n} \to X_{n+1}\;$ -- the $i$th _[[degeneracy map]]_ on $n$-simplices ($n \geq 0$ and $0 \leq i \leq n$);

such that these functions satisfy the _[[simplicial identities]]_.

=--

{#SimplicialSetIdea} $\,$

## Remarks

### Simplicial sets as spaces built of simplices

* The definition is to be understood from the point of view of [[space and quantity]]: a **simplicial set** is a space characterized by the fact that it may be _probed_ by mapping standard simplices into it: the set $S_n$ assigned by a simplicial set to the standard $n$-simplex $[n]$ is the **set of $n$-simplices** in this space, hence the way of mapping a standard $n$-simplex into this space.

* For $S$ a simplicial set, the **face map**
  $$
    d_i \coloneqq S(\delta^i): S_n \rightarrow S_{n-1}
  $$
  is dual to the unique injection $\delta^i : [n-1] \rightarrow [n]$ in the category $\Delta$ whose image omits the element $i \in [n]$.

* Similarly, the **degeneracy map**
  $$
    s_i \coloneqq S(\sigma^i) : S_n \rightarrow S_{n+1}
  $$
  is dual to the unique surjection $\sigma^i : [n+1] \rightarrow [n]$ in $\Delta$ such that $i \in [n]$ has two elements in its preimage.

* The maps $\delta^i$ and $\sigma^i$ satisfy certain obvious relations -- the [[simplicial identities]] -- dual to those  spelled out at [[simplex category]].

### Visualisation
(based on [[cubical set]])

The **face maps**  go from sets $S_{n+1}$ of $(n+1)$-dimensional simplices to the corresponding set $S_{n}$ of $n$-dimensional simplices and can be thought of as sending each simplex in the simplicial set to one of its faces, for instance for $n=1$ the set $S_2$ of 2-simplices would be sent in three different ways by three different face maps to the set of $1$-simplices, for instance one of the face maps would send

$$
  \left(
  \array{
       &          & b
     \\
       & \nearrow & \Downarrow^F & \searrow
     \\
     a &          & \rightarrow  &          & c
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \array{
       &          & b
     \\
       & \nearrow
     \\
     a
  }
  \right)
$$

another one would send

$$
  \left(
  \array{
       &          & b
     \\
       & \nearrow & \Downarrow^F & \searrow
     \\
     a &          & \rightarrow  &          & c
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \array{
     a &          & \rightarrow  &          & c
  }
  \right)
  \,.
$$

On the other hand, the **degeneracy maps** go the other way round and send sets $S_n$ of $n$-simplices to sets $S_{n+1}$ of $(n+1)$-simplices by regarding an $n$-simplex as a degenerate or "thin" $(n+1)$-simplex in the various different ways that this is possible. For instance, again for $n=1$, a degeneracy map may act by sending

$$
  \left(
  \array{
     a
     &\stackrel{f}{\to}&
     b
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \array{
       &            & b
     \\
       & \nearrow_f & \Downarrow^{Id} & \searrow^{Id}
     \\
     a &            & \stackrel{f}\to &               & b
  }
  \right)
  \,.
$$

Notice the $Id$-labels, which indicate that the edges and faces labeled by them are "[[thin element|thin]]" in much the same way as an [[identity morphism]] is thin. They depend on lower dimensional features,  (notice however that a simplicial set by itself is not equipped with any notion of composition of simplices, nor really, therefore, of identities.  See [[quasicategory]] for a kind of simplicial set which does have such notions and [[simplicial T-complex]] for more on the intuitions behind this idea of "thinness").

## Examples
 {#Examples}

### The empty simplicial set

The [[empty simplicial set]] is a simplicial set.

### $n$-simplices (Yoneda embeddings)

Let $[n]$ denote the object of the [[simplex category]] $\Delta$ corresponding to the totally ordered set $\{ 0, 1, 2,\ldots, n\}$. Then the represented presheaf $\Delta(-, [n])$, typically written as $\Delta[n]$ is an example of a simplicial set. In particular we have $\Delta[n]_m=Hom_\Delta([m],[n])$ and hence $\Delta[n]_m$ is a finite set with $\binom{n+m+1}{n}$ elements.

By the Yoneda lemma, the $n$-simplices of a simplicial set $X$ are in natural bijective correspondence to maps $\Delta[n] \rightarrow X$ of simplicial sets.

### The minimal simplicial circle

See at *[[minimal simplicial circle]]*.

### Cartesian products of simplices

The non-degenerate [[simplices]] in the [[Cartesian product of simplices]]

$$
  \Delta[1] \times \Delta[2]
$$

of the [[1-simplex]] $\Delta[1]$ with the [[2-simplex]] $\Delta[2]$ (i.e. the canonical simplicial [[cylinder object]] over the [[2-simplex]]) is the simplicial set which looks as follows:

> graphics grabbed from [Friedman 08, p. 33](#Friedman08)

For more on this and related examples see at *[[product of simplices]]*.

### Simplicial complexes

Every [[simplicial complex]] can be viewed a simplicial set (in several different ways).

> graphics grabbed form [arXiv:1710.06129](https://arxiv.org/abs/1710.06129)

> graphics grabbed from Maletic, 2013

In particular any graph is thought of as being built of vertices and edges and so is a (1-dimensional) simplicial complex. Choosing a direction on the edges then gives a directed graph and that gives a simplicial set, as follows.

### Directed graphs

A [[directed graph]] (with loops and multiple edges allowed, i.e., a [[quiver]]) $E \rightrightarrows V$ is essentially the same thing as a 1-dimensional simplicial set, by taking $S_0 \coloneqq V$ to be the set of vertices and $S_1 \coloneqq E \uplus V$ to be the disjoint union of the set of edges with the set of vertices (the latter corresponding to the degenerate 1-simplices).

### Nerve of a category

If $C$ is a small category, the **nerve** of $C$ is a simplicial set which we denote $NC$. If we intepret the poset $[n]$ defined above as a category, we define the $n$-simplices of $NC$ to be the set of functors $[n] \rightarrow C$. Equivalently, the $0$-simplices of $NC$ are the objects of $C$, the $1$-simplices are the morphisms, and the $n$-simplices are strings of $n$ composable arrows in $C$. Face maps are given by composition (or omission, in the case of $d_0$ and $d_n$) and degeneracy maps are given by inserting identity arrows.

### Singular simplicial complex of a topological space

Recall from [[simplex category]] or [[geometric realization]] the standard functor $\Delta \to Top$ which sends $[n] \in \Delta$ to the standard topological $n$-simplex $\Delta^n$.  This functor induces for every [[topological space]] $X$ the simplicial set
$$
  S X : [n] \mapsto Hom_{Top}(\Delta^n, X)
$$
called the **simplicial singular complex** of $X$. This simplicial set is always a [[Kan complex]] and may be regarded as the [[fundamental ∞-groupoid]] of $X$.

Following up on the idea of ''thinness'', a singular simplex $f: \Delta^n \to X$ may be called **thin** if it factors through a [[retraction]] $r: \Delta^n \to \Lambda^{n-1}_i$ to some horn of $\Delta^n$,  then the well known [[Kan complex|Kan]] condition on $S X$ can be strengthened to say that every horn in $S X$ has a _thin_ filler. This also helps to give some intuitive underpinning to the idea of [[thin element]] in this simplicial context.

### Bar construction

For the moment see _[[bar construction]]_.

## Properties

### Basic properties
 {#BasicProperties}

\begin{proposition}\label{SimplicialSetIsColimitOfItsSimplices}
**([[simplicial set]] is [[colimit]] of its [[category of elements|elements]])**
\linebreak
  Every simplicial set $X \in sSet$ is the [[colimit]] of its simplices, in that
  $$
    \begin{aligned}
      X
      &
      \;\simeq\;
      \underset{
        \Delta[k] \in el(X)
      }{lim}
      \Delta[k]
      \\
      & \;\simeq\;
      \int^{[k] \in \Delta} \Delta[k] \times X_k
    \end{aligned}
  $$
\end{proposition}
Here the first line shows a [[colimit]] over the [[category of elements]] of $X$ (e.g. [Goerss & Jardine, I, Lemma 2.1](#GoerssJardine09)) and the second line shows the corresponding [[coend]]-expression (by [this Prop.](end#CoendAsColimitOverCategoryOfElements)).
\begin{proof}
Since [[SimplicialSets]] is a [[category of presheaves]] (over the [[simplex category]] $\Delta$) this is a special case of the general fact that every presheaf is a colimit of [[representable functor|representables]] ([this Prop.](presheaf#EveryPresheafIsColimitOfRepresentables), the "[[co-Yoneda lemma]]").
\end{proof}

\begin{remark}
In fact, the colimit in Prop. \ref{SimplicialSetIsColimitOfItsSimplices} is a [[homotopy colimit]], see [there](homotopy+limit#SimplicialSetIsHomotopyColimitOverItself).
\end{remark}

### Classifying topos

#### Simplicial sets

The category of simplicial sets is a [[presheaf category]], and so in particular a [[Grothendieck topos]].  In fact, it is the [[classifying topos]] of the theory of "intervals", meaning [[totally ordered sets]] equipped with distinct top and bottom elements.

Specifically, if $E$ is a topos containing such an interval $I$, then we obtain a functor $\Delta \to E$ sending $[n]$ to the subobject
$$ \{ (x_1,x_2,\dots,x_n) \;|\; x_1 \le x_2 \le \dots \le x_n \} \hookrightarrow I^n $$
The corresponding [[geometric realization]]/[[nerve]] [[adjunction]] $E \leftrightarrows Set^{\Delta^{op}}$ is the [[geometric morphism]] which classifies $I$.  In particular, the generic such interval is the simplicial 1-simplex $\Delta^1$; see [[generic interval]] for more.

The usual geometric realization into [[topological spaces]] cannot be obtained in this way precisely, since [[Top]] is not a topos.  However, there are [[Top]]-like categories which are toposes, such as [[Johnstone's topological topos]].

#### Cosimplicial sets

Similarly, also the category $Set^{\Delta}$ of cosimplicial sets is a classifying topos: for inhabited [[linear orders]]. See at _[[classifying topos]]_ the section _[For (inhabited) linear orders](http://ncatlab.org/nlab/show/classifying+topos#ForLinearOrders)_.

### As models in homotopy theory

(...) [[homotopy theory]] (...) [[Kan complex]] (...) [[quasi-category]] (...)

### Relation to dendroidal sets

For the moment see at _[[dendroidal set]]_ the section [Relation to simplicial sets](#http://ncatlab.org/nlab/show/dendroidal+set#RelationToSimplicialSets)

## Variants

* A [[symmetric set]] is a simplicial set equipped with additional *transposition maps* $t^n_i: X_n \to X_n$ for $i=0,\ldots,n-1$.  These transition maps generate an [[action]] of the [[symmetric group]] on $X_n$ and satisfy certain commutation relations with the face and degeneracy maps.

## Related concepts

* [[simplicial map]]

* [[simplicial object]]

  * **simplicial set**

  * [[pointed simplicial set]]

  * [[simplicial object in an (∞,1)-category]]

* [[semi-simplicial object]]

  * [[semisimplicial set]]

* [[simplicial homotopy theory]]

* [[simplicial group]], [[reduced simplicial set]]

* [[bisimplicial set]]

* [[symmetric set]], [[cyclic set]], [[skew-simplicial set]]

* [[globular set]], [[cubical set]], [[cellular set]], [[dendroidal set]]

## References

The original definition:

* [[Samuel Eilenberg]], [[Joseph A. Zilber]], Section 8 in: _Semi-simplicial complexes and singular homology_, Annals of Mathematics 51:3 (1950), 499--513 ([jstor:1969364](https://www.jstor.org/stable/1969364))

Further early discussions (aimed at [[simplicial homotopy theory]]):

* [[Pierre Gabriel]], [[Michel Zisman]], Chapter II of *[[Calculus of fractions and homotopy theory]]*, Ergebnisse der Mathematik und ihrer Grenzgebiete **35**, Springer (1967) [[doi:10.1007/978-3-642-85844-4](https://link.springer.com/book/10.1007/978-3-642-85844-4), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf)]

* [[Peter May]], _Simplicial objects in algebraic topology_, University of Chicago Press 1967 ([ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]])

*  [[Edward B. Curtis]], _Simplicial homotopy theory_, Advances in Mathematics 6 (1971) 107–209 (doi:10.1016/0001-8708(71)90015-6, [MR279808](http://www.ams.org/mathscinet-getitem?mr=279808))

Exposition and introduction:

* {#Friedman08} [[Greg Friedman]], _An elementary illustrated introduction to simplicial sets_, Rocky Mountain J. Math. 42(2): 353-423 (2012) ([arXiv:0809.4221](http://arxiv.org/abs/0809.4221), [doi:10.1216/RMJ-2012-42-2-353](https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-42/issue-2/Survey-Article-An-elementary-illustrated-introduction-to-simplicial-sets/10.1216/RMJ-2012-42-2-353.full))

* [[Emily Riehl]], _A leisurely introduction to simplicial sets_, 2008, 14 pages ([pdf](http://www.math.jhu.edu/~eriehl/ssets.pdf)).

* [[Francis Sergeraert]], _Introduction to Combinatorial Homotopy Theory_, July 7, 2013, [pdf](https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Trieste-Lecture-Notes.pdf).

* Christian Rüschoff, *Simplicial Sets*, Lecture Notes 2017 ([pdf](https://www.mathi.uni-heidelberg.de/~rueschoff/ss17sset/sset.pdf), [[Rueschoff_SimplicialSets.pdf:file]])

Further discussion in the context of [[simplicial homotopy theory]]/[[algebraic topology]]:

* [[Klaus Lamotke]], _Semisimpliziale algebraische Topologie_, Grundlehren der mathematischen Wissenschaften 147 (1968).

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Chapter I of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/))

* {#Cisinski06} [[Denis-Charles Cisinski]], Section 2 of: _[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]_, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf))

* [[André Joyal]], [[Myles Tierney]], _Notes on simplicial homotopy theory_, March 7, 2008, [pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern47.pdf).

category: topology

category: simplicial object

[[!redirects simplicial sets]]

[[!redirects cosimplicial set]]
[[!redirects cosimplicial sets]]

[[!redirects boundary map]]
[[!redirects boundary maps]]
[[!redirects face map]]
[[!redirects face maps]]
[[!redirects degeneracy map]]
[[!redirects degeneracy maps]]
[[!redirects coface map]]
[[!redirects coface maps]]
[[!redirects codegeneracy map]]
[[!redirects codegeneracy maps]]

[[!redirects face and degeneracy maps]]


simplicial sheaf

#Contents#
* table of contents
{:toc}

## Definition

A _simplicial sheaf_ $A$ is equivalently

* a [[simplicial object]]

  $$
    A \in SSh(C) := [\Delta^{op}, Sh(C)]
  $$

  in a [[category of sheaves]] $Sh(C)$ for some [[site]] $C$;

* a [[simplicial presheaf]]

  $$
    A \in SPSh(C) := [\Delta^{op}, PSh(C)] \simeq [\Delta^{op}, [C^{op}, Set]]
  $$

  that satisfies degreewise the [[sheaf]] condition;

* an [[SSet]]-valued [[presheaf]]

  $$
    A \in PSh(C,SSet) := [C^{op}, SSet] \simeq [C^{op}, [\Delta^{op}, Set]]
  $$

  which, when regarded under the equivalence

  $$
    PSh(C,SSet) \simeq SPSh(C)
  $$

  is degreewise a [[sheaf]].

## Joyal model structure ##

The Jardine local [[model structure on simplicial presheaves]] restricts to the standard [[model structure on simplicial sheaves]].  This restriction is a [[Quillen equivalence]], so that equipped with this model structure $SSh(C)$ is a [[models for ∞-stack (∞,1)-toposes|model]] for the [[hypercomplete (infinity,1)-topos]] over the site $C$.

Historically, [[Joyal]] constructed his model structure on simplicial sheaves first and [[Jardine]] later constructed his model structure on [[simplicial presheaves]].

## Local projective model structure

Blandner proved that the category of simplicial sheaves
admits a model structure whose weak equivalences are the same as in the Joyal model structure and acyclic fibrations are the projective (i.e., degreewise) acyclic fibrations.

## Related concepts

* [[simplicial presheaf]]

## References

The original construction due to [[Joyal]] in 1984 is in

* [[André Joyal]], _Lettre d’André Joyal à Alexandre Grothendieck_, April 11, 1984, [PDF](https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf).

A discussion of the [[homotopy theory]] of [[simplicial objects]] in [[toposes]] using [[Cisinski model structures]] is in

* Garth Warner, _Homotopical topos theory_ ([pdf](http://www.math.washington.edu/~warner/HTT_Warner.pdf))

The last part of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

is announced to be about simplicial objects in toposes, but that part does not exist yet.

For more see at _[[simplicial presheaf]]_ and _[[model structure on simplicial presheaves]]_.

[[!redirects simplicial sheaves]]


simplicial skeleton

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

> This entry is about the notion of (co)skeleta of [[simplicial sets]]. For the notion of skeleton of a [[category]] see at _[[skeleton]]_.

#Contents#
* table of contents
{:toc}

## Definition

For $\Delta$ the [[simplex category]] write $\Delta_{\leq n}$ for its [[full subcategory]] on the objects $[0], [1], \cdots, [n]$. The inclusion $\Delta|_{\leq n} \hookrightarrow \Delta$ induces a truncation functor

$$
  tr_n : sSet = [\Delta^{op}, Set] \to [\Delta_{\leq n}^{op},Set] = sSet_{\leq n}
$$

that takes a [[simplicial set]] and restricts it to its degrees $\leq n$.

This functor has a fully faithful [[left adjoint]], given by left [[Kan extension]]

$$
  sk_n
  \;\colon\;
  sSet_{\leq n} \to sSet
$$

called the $n$-**skeleton**

and a fully faithful [[right adjoint]], given by right [[Kan extension]]

$$
  cosk_n
  \;\colon\;
  sSet_{\leq n} \to sSet
$$

called the $n$-**coskeleton**.

$$
  ( sk_n \dashv tr_n \dashv cosk_n) \;\;
   \colon
   \;\;
   sSet_{\leq n}
   \stackrel{\overset{sk_n}{\longrightarrow}}{\stackrel{\overset{tr_n}{\longleftarrow}}{\underset{cosk_n}{\longrightarrow}}}
   sSet
  \,.
$$

The $n$-skeleton produces a simplicial set that is freely filled with degenerate simplices above degree $n$. Conversely, the $n$-coskeleton produces a simplicial set having a simplex of degree $m \gt n$ whenever there is a compatible family of $m$-faces.

Write

$$
  \mathbf{sk}_n := sk_n \circ tr_n: sSet \to sSet
$$

and

$$
  \mathbf{cosk}_n := cosk_n \circ tr_n: sSet \to sSet
$$

for the composite functors. Often by slight abuse of notation we suppress the boldface and just write $sk_n : sSet \to sSet$ and $cosk_n : sSet \to sSet$.

these in turn form an [[adjunction]]

$$
  ( \mathbf{sk}_n \dashv \mathbf{cosk}_n)
  \;\;
  :
  \;\;
  sSet
  \stackrel{\leftarrow}{\to}
  sSet
  \,.
$$

So the $k$-coskeleton of a simplicial set $X$ is given by the formula

$$
  \mathbf{cosk}_k X : [n] \mapsto Hom_{sSet}(\mathbf{sk}_k \Delta[n], X)
  \,.
$$

Simplicial sets isomorphic to objects in the image of $cosk_n$ are called **$n$-coskeletal** simplicial sets.

## Properties {#Properties}

### General

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

For $X \in $ [[sSet]], the following are equivalent:

* $X$ is $n$-coskeletal;

* on $X$ the [[adjunction unit]] $X \to \mathbf{cosk}_n(X)$ is an [[isomorphism]];

* the map

  $$
    X_k
    \;=\;
    Hom(\Delta[k], X)
     \stackrel{tr_n}{\to}
    Hom(tr_n(\Delta[k]), tr_n(X))
  $$

  is a [[bijection]] for all $k \gt n$

* for $k \gt n$ and every morphism $\partial\Delta[k] \to X$ from the [[boundary of a simplex|boundary]] of the $k$-[[simplex]] there exists a _unique_ filler $\Delta[k] \to X$

  $$
    \array{
       \partial \Delta[k] &\to& X
       \\
       \downarrow & \nearrow
       \\
       \Delta[k]
    }
  $$

=--

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

So in particular if $X$ is an $n$-coskeletal [[Kan complex]], all its [[simplicial homotopy group]]s above degree $(n-1)$ are trivial.

=--

### Compatibility with Kan conditions

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

The coskeleton operations $\mathbf{cosk}_n$ preserve [[Kan complexes]].

More generally, $\mathbf{cosk}_n$ preserves those [[Kan fibrations]] between [[Kan complexes]] whose [[codomains]] have [[trivial group|trivial]] [[homotopy group]] $\pi_n$.

=--

([Dwyer & Kan 1984, p. 141 (4 of 9)](#DwyerKan84), proofs are spelled out by [Low 2013](#Low13), [Deflorin 2019, Lemma 10.12](#Deflorin19))

### Truncation and Postnikov towers {#Truncation}

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

For each $n \in \mathbb{N}$, the [[unit of an adjunction|unit of the adjunction]]

$$
  X \longrightarrow \mathbf{cosk}_n(X)
$$

induces an [[isomorphism]] on all [[simplicial homotopy groups]] in degree $\lt n$.

=--

It follows from the above that for $X$ a [[Kan complex]], the sequence

$$
  X = \underset{\leftarrow}{\lim}\, cosk_n X \to \cdots \to cosk_{n+1} X \to cosk_{n} X \to \cdots \to *
$$

is a [[Postnikov tower]] for $X$.

See also the discussion in [Dwyer & Kan 1984, p. 140, 141](#DwyerKan84).

For the interpretation of this in terms of  [[(n,1)-topos]]es inside the [[(∞,1)-topos]] [[∞Grpd]] see [[n-truncated object in an (∞,1)-category]], example In ∞Grpd and Top.

## Examples
 {#Examples}

\begin{example}
\label{CoskeletalityOfSimplicialNervesOfCategories}
**(coskeletality of simplicial nerves of categories)**
\linebreak
The [[simplicial nerve]] of a [[category]] (i.e. of a [[1-category]]) is a [[coskeleton|2-coskeletal]] [[simplicial set]] ([this Prop.](nerve#NerveOfACategoryIsTwoCoskeletal)): The unique filler of the [[boundary of a simplex|boundary of an $n \geq 3$-simplex]] encodes the [[associativity]]-condition on [[n-tuple|$n$-tuples]] of composable morphisms.

Of course there is more to a category than its associativity condition, and hence the converse fails: Not every [[coskeleton|2-coskeletal]] [[simplicial set]] is the [[nerve of a category]]. For example the [[boundary of a simplex|boundary of the 2-simplex]], $\partial \Delta^2$, is 2-coskeletal but not the nerve of a category, since it is missing a [[composition]] of the edges $0 \to 1 \to 2$, namely it is missing a filler of this [[inner horn]].

In fact, a [[simplicial set]] is the [[nerve]] of a [[category]] iff it has *unique* [[inner horn|inner $n$-horn]]-fillers for $n \geq 2$ (e.g. [this Prop.](nerve#sSetIsNerveOfCategoryIffAllInnerHornsHaveUniqueFillers)).
But [[coskeleton|2-coskeletality]] already implies that all $k \geq 4$-horns have unique filler (first uniquely fill the missing $k-1$-face then the interior $k$)-cell. Together this implies that:

A [[simplicial set]] is the [[nerve]] of a [[category]] iff

1. it is [[coskeleton|2-coskeletal]],

1. all *[[inner horn|inner]]* 2- and 3-[[horns]] have unique fillers (encoding [[composition]] and [[associativity]]).

Similarly for [[groupoids]] (by [this Prop.](nerve#sSetIsNerveOfGroupoidIffAllHornsHaveUniqueFillers)):

A [[simplicial set]] is the [[nerve]] of a [[groupoid]] iff

1. it is [[coskeleton|2-coskeletal]],

1. all 2- and 3-[[horns]] have unique fillers.

For better or worse, such a simplicial set has at times also been called a *[[1-hypergroupoid]]*, pointing to the fact that this is the first non-trivial stage in a pattern that recognizes $n+1$-coskeletal [[Kan complexes]] with unique horn fillers as models for [[n-groupoids|$n$-groupoids]]

Notice that a [[Kan complex]] which is [[coskeleton|2-coskeletal]] but with possibly non-unique 2-horn fillers is still a [[homotopy 1-type]] and may still be called a *[[1-groupoid]]* in the sense of [[homotopy theory]], but need not be the [[nerve]] of a [[groupoid]]. It may be thought of as a [[bigroupoid]] (2-[[hypergroupoid]]) which happens to be just a [[homotopy 1-type]].
\end{example}

Accordingly, essentially by definition:

* A [[Kan complex]] that is $(n+1)$-coskeletal is equivalent to (the [[nerve]] of) an [[n-groupoid|$n$-groupoid]].

Also:

* A 0-coskeletal simplicial set $X$ is (-1)-[[truncated]] and hence either empty or a [[contractible]] [[Kan complex]] , $X \stackrel{\simeq}{\to} *$ that is the [[nerve]] $X = N(C)$ of a [[groupoid]] $C$ that has a [[equivalence of categories]] $C \simeq *$.

## Related concepts

* [[Postnikov tower]], [[Whitehead tower]]

* [[Eilenberg subcomplex]]

* [[adjoint modality]]

* [[hypergroupoid]]

## References

* [[Peter May]], Section II.8 of: _Simplicial objects in algebraic topology_, The University of Chicago Press 1967 ([djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html))

* {#DwyerKan84} [[William Dwyer]], [[Dan Kan]], Section 1.2 (vi) of: *An obstruction theory for diagrams of simplicial sets*, Indagationes Mathematicae (Proceedings) **87** 2 (1984) 139-146 [doi:10.1016/1385-7258(84)90015-5]

* [[Paul Goerss]], [[J. F. Jardine]], Section VI.3 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999)  Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4))

Also:

* {#Low13} [[Zhen Lin]], [Math.SE:a/597990](http://math.stackexchange.com/a/597990/58526)

* {#Deflorin19} Gian Deflorin, Section 10.2 of: *The Homotopy Hypothesis*, Zurich 2019 ([2019](http://user.math.uzh.ch/cattaneo/deflorin.pdf), [[Deflorin_HomotopyHypothesis.pdf:file]])

The [[level of a topos]]-structure of simplicial (co-)skeleta is discussed in

* {#KRRZ10} C. Kennett, [[Emily Riehl|E. Riehl]], M. Roy, M. Zaks, _Levels in the toposes of simplicial sets and cubical sets_ , JPAA **215** no.5 (2011) pp.949-961. ([arXiv:1003.5944](http://arxiv.org/abs/1003.5944))

[[!redirects coskeletal]]
[[!redirects coskeleton]]
[[!redirects simplicial coskeleton]]

[[!redirects coskeleta]]
[[!redirects skeleta]]

[[!redirects coskeletons]]
[[!redirects skeletons]]

[[!redirects simplicial skeleta]]

[[!redirects n-skeleton]]
[[!redirects n-skeleta]]

[[!redirects cellular skeleton]]
[[!redirects cellular skeleta]]


simplicial space

The term **simplicial space** refers to either

* a [[simplicial simplicial set]], i.e. a [[simplicial object]] in the [[category]] of [[simplicial sets]];
* or a [[simplicial topological space]], i.e. a [[simplicial object]] in the [[category]] of [[topological spaces]].

[[!redirects simplicial spaces]]


simplicial Stein site

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The [[category]] of [[second countable space|second countable]]  [[complex manifold]]s and [[holomorphic map]]s is [[simplicially enriched category|simplicially enriched]]: $Map_\Delta(X,Y)$ consists of all  singular simplices $\Delta^{op}\to Map(X,Y)$ where $Map(X,Y)$ is the space of holomorphic maps from $X$ to $Y$ with [[compact-open topology]].

The full simplicially enriched subcategory spanned by [[Stein manifold]]s has a [[Grothendieck topology]] on the category of homotopy components: namely a [[cover]] of a Stein manifold $S$ is a family of holomorphic maps $\{X_\alpha\to S\}_\alpha$ such that we can deform each of them (by a [[homotopy]] within $Map(X_\alpha,S)$) to a biholomorphic map onto a Stein open subset in $S$, such that these Stein [[open subset]]s cover $S$.

This [[simplicial site]] is called the **simplicial Stein site** $Stein_\Delta$.

## References

* [[Finnur Larusson]], _Model structures and Oka principle_, [math.CV/0303355](http://arxiv.org/abs/math/0303355)


simplicial T-complex

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

#Contents#
* automatic table of contents goes here
{:toc}

##Idea

A **simplicial T-complex** is  a [[Kan complex]] equipped with a _choice_ of [[horn]] fillers, which satisfy certain 'equations'. It is thus a special case of an [[algebraic Kan complex]].

There is quite a difference between the [[Kan complex]] structure

1. of the [[nerve|nerve]] $N(\mathcal{G})$  of a [[groupoid]], $\mathcal{G}$,

1. and that of, say,  a [[path infinity-groupoid|singular complex]] $Sing X$ of some [[topological space]] $X$.

In the first, if we are given a $(n,i)$-[[horn]], then there is _exactly one_ $n$-simplex in  $Ner(G)$, since the $(n,i)$-horn has a chain of $n$-composable arrows of $G$ in it (at least unless $(n,i) = (2,0)$ or $(2,2)$, which cases are slightly different) and that chain gives the required $n$-simplex.  In other words, there is a 'canonical' filler for any horn.  In $Sing(X)$,  there will usually be many fillers; however the fact that this simplicial set is Kan is a property of retractions on standard simplices, and is not specifically a property of the space $X$ - that is the basic intuition.

Abstracting, in part, from this idea, Brown and Higgins developed the idea of a [[cubical T-complex]]. This was a [[cubical set|cubical set]] with in each dimension $n$,  a subset of the $n$-[[cube]]s being declared _'thin'_. The term was adopted to indicate that they, somehow, were of lower dimension than they looked to be.  The theory was initiated in a simplicial context in the 1977 Bangor thesis of Keith Dakin listed below, and used by Brown and Higgins who showed that cubical $T$-complexes were equivalent to  [[crossed complex|crossed complexes]].  The corresponding simplicial $T$-complex theory was further developed in the 1978  Bangor thesis of  Nick Ashley, (see below for publication).

##Simplicial $T$-complex: the definition

+--{: .un_defn}
######Definition######
A _simplicial $T$-complex_ is a pair $(K,T)$,
where $K$ is a [[simplicial set|simplicial set]] and $T = (T_n)_{n\geq 1}$ is a graded subset of
$K$ with $T_n\subseteq  K_n$.  Elements of $T$ are called _[[thin]]_.  The
thin structure satisfies the following axioms:

* Every degenerate element is thin.
* Every [[horn]] in $K$ has a unique thin filler.
* If all faces but one of a thin element are thin, then so also is the remaining face.
=--

+--{: .un_ex}
######Examples######
1. A closely related idea is that of  [[group T-complex]]. Group $T$-complexes form a category equivalent to reduced [[crossed complex]]es.  Any group $T$-complex has an underlying simplicial set, which is a simplicial $T$-complex.

1. The nerve of a [[crossed complex]] has a natural T-complex structure. In a bit more detail, if $\mathsf{C}$ is a crossed complex, its nerve is given by $Ner(\mathsf{C})_n = Crs(\pi(n),\mathsf{C})$, where $\pi(n)$ is the free crossed complex on the $n$-simplex, $\Delta[n]$. This singular complex description shows that if we have an $n$-simplex $f : \pi(n) \to \mathsf{C}$, and declare it to be \emph{thin} if the image $f(\iota_n)$ of the top dimensional generator in $\pi(n)$ is trivial, then the resulting collection of thin simplices determines  a $T$- complex structure on the nerve.
=--

### Results

* Simplical $T$-complexes together with maps between them which preserve 'thinness' form a category that is equivalent to that of [[crossed complexes]] and thus to strict $\infty$-groupoids. The uniqueness of the thin filler is exactly what gives a definite composition in the model.

## Related concepts

* [[T-complex]]

  * **simplicial $T$-complex**

  * [[cubical T-complex]]

  * [[group T-complex]]

* [[algebraic Kan complex]]

Together with very similar ideas of [[John Roberts]], adapted by
[[Ross Street]], the notion of $T$-complex is one of the precursors of [[Dominic Verity]]'s notion of [[complicial set|complicial set]].

They are also related to [[Jack Duskin]]'s notion of [[hypergroupoid]]. (The connection is explored in the papers by Nan Tie listed below.)

##References

Relevant references for simplicial T-complexes include:

* [[Ronnie Brown]], _An Introduction to simplicial T-complexes_,  [Esquisses Math. 32 (1983) Part 1](http://ehres.pagesperso-orange.fr/Cahiers/brownem32.pdf)

* M.K. Dakin, _Kan complexes and multiple groupoid structures_, Ph.D Thesis, University of Wales, Bangor, 1977. [Esquisses Math. (1983) 32 Part 2](http://ehres.pagesperso-orange.fr/Cahiers/dakinEM.pdf)

* N. Ashley, _Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan_, Ph.D Thesis University of Wales, Bangor, 1978; [Dissertationes Math., 165, (1989), 11 – 58](https://eudml.org/doc/268359).
[Esquisses Math. (1983)  32 Part 3](http://ehres.pagesperso-orange.fr/Cahiers/AshleyEM32.pdf)

* G. Nan Tie, _A Dold-Kan theorem for crossed complexes_, J. Pure Appl. Alg., 56, (1989.), 177 -– 194.

* G. Nan Tie, _Iterated W and T-groupoids_, J. Pure Appl. Alg., 56, (1989), 195 -– 209.

* [[Ronnie Brown]],  and [[P.J. Higgins]], _On the algebra of cubes_
J. Pure Appl. Algebra 21 (1981) 233--260.

* R. Brown,  and P.J. Higgins, _The equivalence of $\omega $-groupoids and cubical   $T$-complexes_ Cahiers Topologie G\'eom. Diff\'erentielle 22 (1981) 349--370.

[[!redirects simplical T-complex]]
[[!redirects simplicial T-complexes]]


simplicial T-complexes > history

< [[simplicial T-complexes]]
[[!redirects simplicial T-complexes -- history]]


simplicial topological group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A _simplicial topological group_ is a [[simplicial object]] in the [[category]] of [[topological groups]].

=--

For various applications the ambient category [[Top]] of [[topological space]]s is taken specifically to be

* the [[category]] of [[compactly generated space|compactly generated]] [[weakly Hausdorff space]]s, or

* or the category of [[k-space]]s.

We take [[Top]] to be the category of [[k-space]]s in the following.

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

A simplicial topological group $G$ is called **[[well-pointed simplicial topological group|well-pointed]]** if for $*$ the trivial simplicial topological group and $i : * \to G$ the unique [[homomorphism]], all components $i_n : * \to G_n$ are [[closed cofibrations]].

=--

For $B \in Top$ a fixed base object, it is often desirable to work in "$B$-parameterized families", hence in the [[over-category]] $Top/B$ (see [MaySigurdson](#MaySigurdson)). There is the
[[Strøm model structure|relative Strøm model structure]] on $Top/B$.

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

A simplicial group in $G$ in $Top/B$ is called **well-sectioned** if for $B$ the trivial simplicial topological group over $B$ and $i : B \to G$ the unique [[homomorphism]], all components $i_n : B \to G_n$ are $\bar f$-cofibrations.

=--

## Properties

Recall for a [[discrete group|discrete]] [[simplicial group]] $G$ the [[simplicial classifying space]] coprojection  $W G \to \overline{W} G$, being a [[Kan complex]] presentation of the [[universal principal infinity-bundle]] $\mathbf{E}G \to \mathbf{B}G$ from [[simplicial group]]. These constructions for discrete simplicial groups have immediate analogs for simplicial topological groups.

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

Let $G$ be a simplicial topological group. Write $\bar W G \in Top^{\Delta^{op}}$ for the [[simplicial topological space]] whose topological space of $n$-[[simplices]] is the [[product]]

$$
  \bar W G_n := G_{n-1} \times G_{n-2} \cdots \times G_{0}
$$

in [[Top]], equipped with the evident face and degeneracy maps (see at *[[simplicial classifying space]]*).

=--

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

We say a morphism $f : X \to Y$ of [[simplicial topological space]]s is a **global Kan fibration** if for all $n \in \mathbb{N}$ and $0 \leq k \leq n$ the canonical morphism

$$
  X_n \to Y_n \times_{sTop(\Lambda^n_k, Y)} sTop(\Lambda^n_k, X)
$$

in [[Top]] has a [[section]], where

* $\Lambda^n_k \in $ [[sSet]] $\hookrightarrow Top^{\Delta^{op}}$ is the $k$th $n$-[[horn]] regarded as a [[discrete space|discrete]] [[simplicial topological space]]:

* $sTop(-,-) : sTop^{op} \times sTop \to Top$ is the [[Top]]-[[hom object]].

We say a [[simplicial topological space]] $X_\bullet \in Top^{\Delta^{op}}$ is **(global) Kan simplicial space** if the unique morphism $X_\bullet \to *$ is a global Kan fibration, hence if for all $n \in \mathbb{N}$ and all $0 \leq i \leq n$ the canonical [[continuous function]]

$$
  X_n \to sTop(\Lambda^n_k, X)
$$

into the [[topological space]] of $k$th $n$-[[horn]]s admits a [[section]].

=--

This global notion of Kan simplicial spaces is considered for instance in ([Brown & Szczarba 1989](#BrownSzczarba89)) and ([May](#May)).

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

Let $G$ be a simplicial topological group. Then

1. $G$ is a globally Kan simplicial topological space;

1. $\bar W G$ is a globally Kan simplicial topological space;

1. $W G \to \bar W G$ is a global Kan fibration.

=--

+-- {: .proof}
###### Proof

The first statement appears as ([Brown & Szczarba 1989, theorem 3.8](#BrownSzczarba89)), the second is noted in ([Roberts & Stevenson 2012](#RobertsStevenson12)), the third as ([Brown & Szczarba 1989, lemma 6.7](#BrownSzczarba89)).

=--

\begin{proposition}\label{RealizationOfWellPointedIsWellPointed}
If $G$ is a [[well-pointed simplicial topological group]] (Def. \ref{WellPointedSimplicialTopologicalGroup}), then

1. its [[underlying]] [[simplicial topological space]] is [[good simplicial topological space|good]];

1. $\overline{W} G$ is a [[nice simplicial topological space|proper simplicial topological space]];

1. the [[geometric realization of simplicial topological spaces|geometric realization]] $|G|$ is well-pointed.

\end{proposition}

([Roberts & Stevenson 2012, Prop. 3](#RobertsStevenson12), for the last item see also [Baez & Stevenson 2008, Lem. 1](#BaezStevenson08))

## Related concepts

* [[simplicial topological space]], [[nice simplicial topological space]]

* [[simplicial group]], [[simplicial abelian group]]

* [[geometric realization of simplicial topological spaces]]

## References

Basic discussion of simplicial topological groups:

* {#BrownSzczarba89} [[Edgar H. Brown]], [[R. H. Szczarba]], *Continuous cohomology and real homotopy type* , Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 ([doi:10.1090/S0002-9947-1989-0929667-6](https://doi.org/10.1090/S0002-9947-1989-0929667-6),  [pdf](http://www.ams.org/journals/tran/1989-311-01/S0002-9947-1989-0929667-6/S0002-9947-1989-0929667-6.pdf))

* {#May} [[Peter May]], _Geometry of iterated loop spaces_ , SLNM 271, Springer-Verlag, 1972 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf))

Discussion of their [[geometric realization of simplicial topological spaces|geometric realization]] and [[principal infinity-bundles|principal $\infty$-bundles]]:

* {#BaezStevenson08} [[John Baez]], [[Danny Stevenson]], _The Classifying Space of a Topological 2-Group_,  In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) Algebraic Topology_. Abel Symposia, vol 4. Springer 2009 ([arXiv:0801.3843](https://arxiv.org/abs/0801.3843), [doi:10.1007/978-3-642-01200-6_1](https://doi.org/10.1007/978-3-642-01200-6_1))

* {#RobertsStevenson12} [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundles in parametrized spaces_, New York Journal of Mathematics Volume 22 (2016) 405-440 ([arXiv:1203.2460](http://arxiv.org/abs/1203.2460), [nyjm:22-19](http://nyjm.albany.edu/j/2016/22-19.html))

Discussion of [[homotopy theory]] over a base $B$ is in

* [[Peter May]], [[Johann Sigurdsson]], _[[Parametrized Homotopy Theory]]_ ([web](http://www.math.uiuc.edu/K-theory/0716/))

[[!redirects simplicial topological groups]]

[[!redirects topological simplicial group]]
[[!redirects topological simplicial groups]]


simplicial topological space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _simplicial topological space_ is a [[simplicial object]] in [[Top]].

Often this is called just a _simplicial space_ , if the context is clear.

A special case is that of [[simplicial manifold]]s.

Often one is interested in simplicial topological spaces with extra nice properties. See [[nice simplicial topological space]] for more on that.

## Applications

As with simplicial objects in general, simplicial spaces may serve to model [[internal ∞-groupoid]]s in [[Top]]. Notably there is a rich theory of [[simplicial topological group]]s.

## Related notions

* [[nice simplicial topological space]]

* [[simplicial topological group]]

* [[geometric realization of simplicial spaces]]

* [[semi-simplicial topological space]]

## References

A standard textbook reference is chapter 11 of

* {#May} [[Peter May]], _The geometry of iterated loop spaces_ ([pdf](http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf))

[[!redirects simplicial topological spaces]]