simplicial connection

Simplicial connection is a [[connection on a bundle|connection]] on a simplicial principal bundle.

* [[Peter May]], _Simplicial objects in algebraic topology_, [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu)

This has been used recently in

* [[Branislav Jurčo]], _Crossed module bundle gerbes; classification, string group and differential geometry_, [math.DG/0510078](http://arxiv.org/abs/math/0510078)

There is an unrelated notion of simplicial connection related to numerical methods at simplicial triangulations in

* I. A. Dynnikov, S. P. Novikov, _Laplace transforms and simplicial connections_, [pdf](http://www.mi.ras.ru/~snovikov/138.pdf)


simplicial de Rham complex

#Contents#
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## Idea

The _simplicial de Rham complex_ of a [[simplicial manifold]] $X_\bullet$ is the analog of the [[de Rham complex]] of [[differential form]]s of an ordinary [[manifold]]: it is the complex whose elements in degree $k$ may be thought of as $k$-[[differential form|form]]s on $X_\bullet$.

One useful conceptual way to think of this is to notice that a a [[simplicial object|simplicial]] [[manifold]] may be thought of as a degreewise [[representable functor|representable]] [[simplicial presheaf]] on [[Diff]]
and then to realize that by way of the standard [[model structure on simplicial presheaves]] such a simplicial presheaf presents an [[∞-stack]] on [[Diff]] which we may think of as a [[Lie ∞-groupoid]]. From that perspective we expect that

+-- {: .standout}

**Slogan**. The simplicial de Rham complex is the complex of differential forms on a geometric [[∞-Lie groupoid]].

=--

We shall discuss this in more detail below.

## Definition

There are several definitions that are [[quasi-isomorphism|quasi-isomorphic]]. The first one we give is the conceptually most straightforward one. The second one we give is sometimes more useful in computations.

Let $X_\bullet : \Delta^{op} \to Diff$ be a [[simplicial object|simplicial]] [[manifold]].

+-- {: .num_defn}
###### Definition
**(simplicial de Rahm complex, first version)**

Write

$$
  \mathcal{A}(X_\bullet)
  :=
  Tot \Omega^\bullet(X_\bullet)
$$

for the [[cochain complex]] that is the [[total complex]] of the [[double complex]] on the degreewise [[de Rham complex]] with one differential the simplicial-degree-wise de Rham differential and the other one the de Rham-degree-wise alternating sum of pullback along the face maps

$$
  \array{
   \Omega^p(X_q) &\stackrel{\sum_i (-1)^i \delta_i^*}{\to}
    & \Omega^{p}(X_{q+1})
    \\
    \downarrow^{d_{dR}} && \downarrow^{d_{dR}}
    \\
    \Omega^{p+1}(X_q)
     &\stackrel{\sum_i (-1)^i \delta_i^*}{\to}
    & \Omega^{p+1}(X_{q+1})
  }
  \,.
$$

=--

So an element $\omega \in \mathcal{A}(X_\bullet)$ in degree $n$ is a collection $(\omega^p_q \in \Omega^p(X_q))_{p+q = n}$ of ordinary differential forms.

+-- {: .num_defn}
###### Definition
**(simplicial de Rahm complex, second version)**

Write $\Delta^n_{Diff}$ for the standard $n$-[[simplex]] in its standard incarnations as a smooth [[manifold]] (with boundary). These arrange in the obvious way into the [[cosimplicial object]] $\Delta_{Diff} : \Delta \to Diff$.

Say that a differential form $\omega^p_q \in \Omega^\bullet(\Delta^p_{Diff}\times X_q)$ is _compatible_ if for each face map $\delta_i$ we have

> (some condition, need to look something up...)

There is a decomposition

$$
  \Omega^n(\Delta^p_{Diff} \times X_q)
  \simeq
  \oplus_{k+l=n} \Omega^k(\Delta^p_{Diff})
   \otimes_{C^\infty(\Delta^p_{Diff} \times X_q)}
   \Omega^l(X_q)
  \,.
$$

This defines a bidegree and $A(X_\bullet)$ is the obvious total complex of the obvious double complex here

> will polish this up later...

=--

The following proposition says that and how these two complexes are related.

+-- {: .num_prop}
###### Proposition
**(Dupont)**

Consider the map between the two [[double complex]]es involved above which integrates each element in degree $(p,q)$ over $\Delta^p_{Diff}$. This induces a map on the corresponding total complexes
$$
  \int_\Delta : A(X_\bullet) \to \mathcal{A}(X_\bullet)
  \,.
$$
This is a [[morphism]] of [[cochain complex]]es which is a [[quasi-isomorphism]].

=--

## Examples

### Differential forms on $\mathbf{B}G$

A main application of this technology is to the simplicial manifold $\mathbf{B}G = (\cdots G \times G \stackrel{\stackrel{\to}{\to}}{\to} G \stackrel{\to}{\to} {*})$ that represents the smooth groupoid which is the [[delooping]] of a [[Lie group]] $G$.

Dupont exhibits a [[Chern-Weil homomorphism]]

$$
  I(G) \to A(\mathbf{B}G)
$$

and constructs a canonical [[connection on a bundle|connection]] on the [[generalized universal bundle|universal G-principal-bundle]]

$$
  \mathbf{E}G \to \mathbf{B}G
$$

(which, in traditional [[simplicial group]] notation, reads $W G \to \bar W G$ -- but recall that these are "Lie" simplicial groups here).

## Reformulations in synthetic differential geometry

+-- {: .query}

[[Urs Schreiber]]: This section is supposed to provide a useful reformulation of the simplicial de Rham complex in the context described at [[schreiber:∞-Lie theory]].

In the language uses there, the statement we establish is the following:

=--

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

Let $C = $ [[CartSp]]${}_{th}$ and $\mathbf{H} = (sPSh(C)_{proj}^{loc})^\circ$ the [[(∞,1)-category of (∞,1)-sheaves]] on [[CartSp]], the [[(∞,1)-topos]] of [[Lie ∞-groupoid]]s. This is a [[locally contractible (∞,1)-topos]] (as discussed there). Accordinly we have its [[schreiber:path ∞-groupoid]] and [[schreiber:infinitesimal path ∞-groupoid]] $\mathbf{\Pi}_{inf}(-)$.

Then

* the [[schreiber:Chevalley-Eilenberg algebra]] of $\mathbf{\Pi}_{inf}(X)$ is [[quasi-isomorphism|quasi-isomorphic]] to the simplicial de Rham complex

  $$
    CE( \Pi^{inf}(X)) \simeq \mathcal{A}(X_\bullet)
    \,.
  $$

=--

The following discussion breaks this down and then describes the proof.

As a preparation, recall from the discussion at [[differential forms in synthetic differential geometry]] that if we pass from [[Diff]] to a [[smooth topos]] $(\mathcal{T},R)$ that models the axioms of [[synthetic differential geometry]], then for sufficiently well-behaved objects $X \in \mathcal{T}$ there is the [[infinitesimal singular simplicial complex]] $X^{(\Delta^\bullet_{inf})} : \Delta^{op} \to \mathcal{T}$, the [[simplicial object]] that in degree $k$ is the [[space]] of [[infinitesimal object|infinitesimal]] $k$-[[simplex|simplices]] in $X$.

As discussed there, this is such that under the [[Dold-Kan correspondence]] the [[cosimplicial algebra]] $Hom( X^{\Delta^\bullet_{inf}}, R )$ maps to the [[de Rham complex]] (and under the [[monoidal Dold-Kan correspondence]] to the full [[de Rham dg-algebra]]):

$$
  C_{Moore} :
  C^\infty( X^{(\Delta^\bullet_{inf}}))
  \mapsto
  \Omega^\bullet(X)
  \,.
$$

It would be nice to have an analog of this statement for simplicial objects and the simplicial de Rham complex. I am thinking that the answer should be the following:

Let $X_\bullet : \Delta^{op} \to \mathcal{T}$ be a [[simplicial object]] that is degreewise of the sort such that the [[infinitesimal singular simplicial complex]] $(X_n)^{\Delta^\bullet_{inf}}$ exists. Use that $\mathcal{T}$ is canonically [[copower|tensored]] over $Set$ to find that simplicial objects in $\mathcal{T}$ are canonically tensored over [[simplicial set]]s. Then consider the _realization_

$$
  \mathbf{\Pi}_{inf}(X_\bullet) := \int^{[n] \in \Delta}
    \Delta[n] \cdot X_n^{(\Delta^{\bullet}_{inf})}
$$

where

* $\Delta[n]$ is the standard [[simplicial set|simplicial]] $n$-[[simplex]]

* the integrand is the tensor operatoin of simplicial objects by simplicial sets

* the integral sign denotes the [[coend]].

By the lemma _expression in terms of simplicial realization_ at [[schreiber:infinitesimal path ∞-groupoid]] this is the same as $\mathbf{\Pi}_{inf}(X)$.

The above proposition now reads in pedestrian terms:

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

The [[Moore complex|Moore cochain complex]] of the [[cosimplicial algebra]] $C^\infty(\mathbf{\Pi}_{inf}(X_\bullet)) := Hom_{\mathcal{T}}(\mathbf{\Pi}_{inf}(X_\bullet),R)$ is [[quasi-isomorphism|quasi-isomorphic]] to the simplicial de Rham complex of $X_\bullet$.

=--

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

We use the cosimplicial and the simplicial version of the [[Eilenberg-Zilber theorem]] together with the fact that for a [[bisimplicial set]] the diagonal is given by the realization (as discussed there) $Diag F_{\bullet,\bullet} \simeq \int^{[n] \in \Delta} \Delta[n] \cdot F_{n,\bullet}$ to compute

$$
  \begin{aligned}
    C ( C^\infty(\mathbf{\Pi}_{inf}(X_\bullet)) )
    & :=
    C ( Hom( \int^{[n]} \Delta[n] \cdot X_n^{(\Delta^\bullet_{inf})}
    , R))
    \\
    & \simeq
    C ( Hom( Diag X_\bullet^{(\Delta^\bullet_{inf})} , R))
    \\
    & \simeq
    C Diag Hom( X_\bullet^{(\Delta^\bullet_{inf})} ,R)
    \\
    & \simeq_{q i}
    Tot C Hom( X_\bullet^{(\Delta^\bullet_{inf})} ,R)
    \\
    & \simeq
    Tot \Omega^\bullet(X_\bullet)
    \,.
  \end{aligned}
$$

Here in the last step we used the following reasoning on the [[bisimplicial object]] $([p],[q]) \mapsto (X_p)^{(\Delta^q_{inf})}$.
We know that

* for fixed $p$, the normalized [[Moore complex]] of the [[cosimplicial algebra]] $Hom((X_p)^{(\Delta^\bullet_{inf})},R)$ is the [[de Rham complex]] of $X_p$ -- this is the statement about combinatorial [[differential forms in synthetic differential geometry]].

* for fixed $q$ the [[Moore complex]] of the [[cosimplicial algebra]] $Hom((X_\bullet)^{((\Delta^q_{inf}))}, R)$ is the complex of functions on simplices whose differential is the alternating sum of pullbacks along face maps -- by the very definition of the [[Moore complex]].

This means that the simplicial de Rham complex is (quasi-isomorphic to) the total complex of the bi-cosimplicial algebra

$$
  \mathcal{A}(X_\bullet) \simeq
  Tot C(Hom((X_\bullet)^{(\Delta^\bullet_{inf})}, R))
  \,.
$$

So it remains to show that this total complex is also (quasi-isomorphic to) the [[Moore complex]] of $Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)$. For this use [exercise 1.6 here](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi) which says (transported from [[Set]] to $\mathcal{T}$) that this is the diagonal [[simplicial object]] of our [[bisimplicial object]]

$$
  d((X_\bullet)^{\Delta^\bullet_{inf}})
  \simeq
  \int^{[n]} \Delta^n \cdot (X_n)^{(\Delta^\bullet_{inf})}
  \,.
$$

This implies that the [[Moore complex]] of $Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)$ is the Moore complex of the diagonal of the bisimplicial algebra $Hom(X_\bullet^{(\Delta^\bullet_{inf})}),R)$.

This way the desired statement recudes to the quasi-isomorphism

$$
  diag (Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)
  \simeq
  Tot (Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)
  \,.
$$

But this (even their chain-homotopy equivalence) is the content of the generalized [[Eilenberg-Zilber theorem]].

=--

## References

Canonical references on simplicial de Rham cohomology are by [[Johan Louis Dupont]]. For instance

* [[Johan Louis Dupont]], _Simplicial de Rham Cohomology and characteristic classes of flat bundles_ Topology 15 (1976)

* [[Johan Louis Dupont]], _A dual simplicial de Rham complex_, Lecture notes in Mathematics 1318 (1988) ([journal](http://www.springerlink.com/content/65784420x146888h/))

* chaper 3 in _Curvature and Characteristic Classes_

>(I am still looking for the best survey reference...)

When restricted to low degree this is closely related to or synonymous to considerations of de Rham cohomology in the context of [[differentiable stack]]s.

> (need to dig out references)

A [[de Rham theorem]] for simplicial manifolds was proven in the classical

* {#BottShulmanStasheff76} [[Raoul Bott]], [[Herbert Shulman]], [[Jim Stasheff]], _On the de Rham theory of certain classifying spaces_, Advances in Mathematics, Volume 20, Issue 1, April 1976, Pages 43-56 (doi:10.1016/0001-8708(76)90169-9, [pdf](https://core.ac.uk/download/pdf/82496263.pdf))

In principle closely related is the discussion of a de Rham theorem for [[∞-stack]]s as discussed in

* [[nLab:Carlos Simpson|Carlos Simpson]], [[nLab:Constantin Teleman|Constantin Teleman]], _de Rham theorem for $\infty$-stacks_ ([pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf))

though a simplicial de Rham complex is only somewhat implicit in that article.

[[!redirects simplicial deRham complex]]


simplicial diagram

#Contents#
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{:toc}

## Definition

A [[diagram]] of shape the [[simplex category]]. Hence a _[[simplicial object]]_ in the [[codomain]] [[category]].

## Related concepts

* [[simplicial object]]

* [[simplicial model category]].

[[!redirects simplicial diagrams]]


simplicial functor

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###Context###
#### Enriched category theory
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[[!include enriched category theory contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The term _simplicial functor_ is usually used to refer to [[sSet]]-[[enriched functors]], hence to _simplicially enriched functors_, hence to [[morphisms]] between [[simplicially enriched categories]].

In principle, but less commonly so in practice, it may also refer to  a morphism of [[simplicial object in Cat]].
See the discussion at _[[simplicial category]]_.

Another meaning was introduced by [[Manos Lydakis]]:
simplicial functors are [[reduced excisive functors]] from
the category of pointed [[simplicial functors]] (in the first sense above)
from the [[simplicial category]] of [[compact]] [[simplicial sets]]
to the [[simplicial category]] of [[simplicial sets]].
This [[simplicial category]] is equipped with a stable [[model structure]]
that makes it [[Quillen equivalent]] to the Bousfield–Friedlander
model structure on [[simplicial spectra]].

See [[model structure on reduced excisive functors]] for more information.

## Related concepts

* [[simplicial category]], [[simplicial model category]]

* [[simplicial groupoid]]

[[!redirects simplicial functors]]


simplicial fundamental groupoid

# Contents
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{: toc}

## Definitions

### Definition using the category of simplices

The __fundamental groupoid__ of a [[simplicial set]] $X$
is the [[localization]] $C[C^{-1}]$
of its [[category of simplices]] $C$
(a special case of the [[category of elements]] of a [[presheaf]]).

### Definition using the fundamental category

The __fundamental groupoid__ of a [[simplicial set]] $X$
is the [[localization]] $C[C^{-1}]$
of its [[fundamental category]] $C$,
defined as the [[left adjoint functor]] to the [[nerve functor]]
$$N\colon Cat \to sSet.$$

### Definition using generators and relations

The __fundamental groupoid__ of a [[simplicial set]] $X$
is a [[groupoid]] defined via the following system
of [[generators and relations]] for a [[groupoid]]:

* objects are precisely the vertices of $X$;

* generating isomorphisms are precisely the edges of $X$;

* for every 2-simplex of $X$ we have a relation
$$d_1(\sigma)=d_0(\sigma)\circ d_2(\sigma).$$

## See also

* [[fundamental groupoid]]
* [[fundamental category]]


simplicial group

+-- {: .rightHandSide}
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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Recall that a [[simplicial set]] is a combinatorial [[model structure on simplicial sets|model]] for a [[topological space]]. This relation is most immediate when the simplicial set is in fact a [[Kan complex]] (an [[∞-groupoid]]).

A _simplicial group_ is a simplicial set with the structure of a [[group]] on it. It turns out that this necessarily means that it is also a [[Kan complex]]. Therefore a simplicial group is

* an [[∞-groupoid]] with an extra group structure on it;

* a model for a topological space with a group structure.

Accordingly (as discussed at [[group]]) a simplicial group $G$ gives rise to

* a one-object $\infty$-groupoid $\mathbf{B} G$ whose explicit standard realization as a [[simplicial set]] is denoted $\bar W G$

* an $\infty$-groupoid $\mathbf{E} G$ whose explicit standard realization as a [[simplicial set]] (even a simplicial group, again) is denoted $W G$

* such that there is a fibration
  $$
    \array{
       \mathbf{E} G
       &:=& W G
       \\
       \downarrow && \downarrow
       \\
       \mathbf{B} G
       &:=&
       \bar W G
    }
  $$
  which is the [[generalized universal bundle|universal G-bundle]].

Simplicial abelian groups are models for [[connective spectrum|connective]] [[module spectrum|modules]] over the [[Eilenberg-Mac Lane spectrum]] $H \mathbf{Z}$; see [[Dold-Kan correspondence]] and [[stable Dold-Kan correspondence]].

## Definition

A **simplicial group**, $G$, is a [[simplicial object]] in the [[category]] [[Grp]] of [[group]]s.

The [[category]] of simplicial groups is the category of functors from $\Delta^{op}$ to [[Grp]].  It will be denoted $\Simp\Grp$.

## Properties

### As Kan complexes
 {#AsKanComplexes}

\begin{theorem}\label{EverySimplicialGroupIsAKanComplex}
The [[simplicial set]] [[underlying]] any simplicial group
(by [[forgetful functor|forgetting]] the group [[structure]]) is a [[Kan complex]].
\end{theorem}

This is due to [Moore 1954, Theorem 3 on p. 18-04](#Moore54), review in [Quillen 67, II 3.8](#Quillen67), [May 67, Theorem 17.1](#MaySimplicialObjects),$\;$ [Curtis 1971, Sec. 3, Lem. 3.1](#Curtis71), $\;$ [Weibel 1994, Lem. 8.2.8](#Weibel94), $\;$ [Joyal & Tierney 2005, p. 14](#JoyalTierney05).

In fact, not only are the [[horn]] fillers guaranteed to exist, but there is an algorithm that provides explicit fillers.  This implies that constructions on a simplicial group that use fillers of horns can often be adjusted to be functorial by using the algorithmically defined fillers.  An argument that just uses 'existence' of fillers can be refined to give something more 'coherent'.

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

Let $G$ be a simplicial group.

Here is the explicit algorithm that computes the horn fillers:

Let $(y_0,\ldots, y_{k-1}, -,y_{k+1}, \ldots, y_n)$ give a [[horn]] in $G_{n-1}$, so the $y_i$s are $(n-1)$ simplices that fit together as if they were all but one, the $k^{th}$ one,  of the faces of an $n$-simplex. There are three cases:

1. if $k = 0$:

   * Let  $w_n = s_{n-1}y_n$ and then $w_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i$ for  $i = n-1, \ldots, 1$, then  $w_1 $ satisfies $d_i w_1 = y_i$, $i\neq  0$;

1. if $0\lt k \lt n$:

   * Let $w_0 = s_0 y_0$ and $w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i$ for $i = 1, \ldots, k-1$, then take $w_n = w_{k-1}(s_{n-1}d_n w_{k-1})^{-1}s_{n-1}y_n$, and finally a downwards induction given by $w_i = w_{i+1}(s_{i-1}d_{i}w_{i+1})^{-1}s_{i-1}y_i$, for $i = n-1, \ldots, k+1$, then $w_{k+1}$ gives $d_{i}w_{k+1} = y_i$ for $i \neq k$;

3. if $k=n$:

   * use $w_0 = s_0 y_0$ and $w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i$ for $i = 1, \ldots, n-1$, then $w_{n-1}$ satisfies $d_i w_{n-1} = y_i$, $i\neq n$.

=--

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

* The filler for any horn can be chosen to be a _product of degenerate elements_.

* The [[simplicial homotopy groups]] of a simplicial group, $G$, can be calculated as the homology groups of the [[Moore complex]] of $G$.  This is, in general,  a non-Abelian chain complex.

* A simplicial group can be considered as a [[simplicial groupoid]] having exactly one object. If $G$ is a simplicial group, the suggested notation for the corresponding simplicially enriched groupoid would be $\mathbf{B}G$ according to notational conventions suggested elsewhere in the nLab.

* There is a functor due to Dwyer and Kan, called the [[Dwyer-Kan loop groupoid]] that takes a  simplicial set to a simplicial groupoid. This has a left adjoint $\overline{W}$ (see below), called the _[[simplicial classifying space]]_ functor, and together they give an equivalence of categories between the homotopy category of simplical sets and that of simplicial groupoids. We thus have that all homotopy types are modelled by simplicial groupoids ... and for connected homotopy types by simplicial groups. One *important fact* to note in this equivalence is that it shifts dimension by 1, so if $G(K)$ is the simplicial group corresponding to the connected simplicial set $K$ then $\pi_k(K)$ is the same as $\pi_{k-1}(G(K))$.  This is important when considering algebraic models for a [[homotopy n-type]].

=--

### Fiber sequences
 {#FiberSequences}

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

Let $G$ be a simplicial group and $G_0$ its group of 0-cells, regarded as a simplicially constant simplicial group. Write $G/G_0$ for the evident [[quotient]] of simplicial groups.

The evident morphisms

$$
  G_0 \to G \to G/G_0 \simeq \mathbf{B} \Omega G/G_0
$$

form a [[fiber sequence]] in [[sSet]].

=--

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

One checks that for $X$ any simplicial set and $G$ a simplicial group acting freely on it, the [[quotient]] map

$$
  X \to X/G
$$

is a [[Kan fibration]]. This is for instance
([Weibel, exercise 8.2.6](#Weibel94)). By the disucssion at [[fiber sequence]] it is therefore sufficient to observe that

$$
  \array{
    G_0 &\to& *
    \\
    \downarrow && \downarrow
    \\
    G &\to& G/G_0
  }
$$

is an ordinary [[pullback]] of simplicial sets.
This is clear since the action of $G_0$ on $G$ is degreewise free (being the action of a subgroup).

=--

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

Let $(G_1 \stackrel{\delta}{\to} G_0)$ be a [[crossed module]] of groups, write

$$
  [G_1 \stackrel{\delta}{\to} G_0]
  =
  \left(
    G_0 \times G_1
    \stackrel{\overset{(\delta p_2)\cdot p_1}{\to}}{\underset{p_1}{\to}}
    G_0
  \right)
$$

for [[groupoid]] which is the corresponding [[strict 2-group]] and write $N[G_1 \to G_0]$ for the [[nerve]] being the corresponding simplicial group. Then the above says that

$$
  G_0 \to [G_1 \stackrel{\delta}{\to} G_0]
  \to
  \mathbf{B}G_1
$$

is a fiber sequence of [[groupoid]]s.

=--

###  Free simplicial groups

+-- {: .num_lemma}
###### Lemma

The [[stuff, structure, property|forgetful functor]]

$$
  U : AbSGrpg \to SSet
$$

from simplicial [[abelian group]]s to the underlying [[simplicial sets]] has a [[left adjoint]]

$$
  \mathbb{Z} : SSet \to AbSimpGrp
$$

from [[simplicial sets]] to abelian simplicial groups,
the **free simplicial abelian group** functor
that sends the set $X_n$ of $n$-simplices to the free abelian group $(\mathbb{Z}X)_n = \mathbb{Z} X_n$ over it.

This functor $\mathbb{Z}$ has the following properties:

* it preserves weak equivalences

* $\mathbb{Z} X$ is a cofibrant simplicial group

=--

(...)

### Looping and delooping

Let $sSet_0 \hookrightarrow$ [[sSet]] be the category of _reduced simplicial sets_ (simplicial sets with a single 0-cell).

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

For $X \in sSet_0$ define $\Omega X \in sGrpd$ by

$$
  \Omega X :  [n] \mapsto  (F X_{n+1})/ s_0 F(X_n)
$$

and

$$
  \Omega X : ([n] \to [k]) \mapsto ...
$$

=--

### As $\infty$-groups
 {#AsInfinityGroups}

Simplicial groups are models for [[∞-group]]s. This is exhibited by the [[model structure on simplicial groups]]. See also
models for group objects in ∞Grpd.

Another equivalent model is that of [[connected]] [[Kan complex]]es.

At the abstract level of [[(∞,1)-category theory]] this equivalence is induced by forming [[loop space object]]s and [[delooping]]

$$
  \Omega
  :
  \infty Grpd_{con} \stackrel{\leftarrow}{\to}
  \infty Grp
  :
  \mathbf{B}
  \,.
$$

This [[equivalence of quasi-categories|(∞,1)-equivalence]] is modeled by a [[Quillen equivalence]] of [[model categories]] whose [[right adjoint]] Quillen functor is the operation $\overline{W}$ discussed above.

$$
  \mathcal{G} : sSet_0 \stackrel{\stackrel{\simeq_{Quillen}}{\longleftarrow}}{\longrightarrow} sGrp
  : \overline{W}
  \,.
$$

This is for instance in [GoerssJardine, chapter 5](#GoerssJardine).

See also group object in an (∞,1)-category -- models for groups in ∞Grpd.

### Closed monoidal structure
 {#ClosedMonoidalStructure}

The [[category]] $sAb$ of simplicial abelian groups is naturally a [[monoidal category]], with the tensor product being degreewise that of abelian groups. This is indeed a [[closed monoidal category]]. For $A, B$ The [[internal hom]] $[A,B]$ is the simplicial abelian group whose underlying simplicial set is

$$
  [A,B] : [n] \mapsto Hom_{sAb}(A \otimes \mathbb{Z}[\Delta[n]], B)
  \,,
$$

where $\mathbf{Z}[-] : sSet \to sAb$ is degreewise the [[free construction|free]] abelian group functor.

## Delooping and simplicial principal bundles  {#DeloopingAndBundle}

For $G$ a simplicial group, we describe its [[delooping]] [[Kan complex]] $\mathbf{B}G  \in sSet$ and the corresponding [[generalized universal bundle]] $\mathbf{E}G \to \mathbf{B}G$ such that the ordinary [[pullback]]

$$
  \array{
    P_\bullet &\to& \mathbf{E}G
    \\
    \downarrow && \downarrow
    \\
    X_\bullet &\stackrel{g}{\to}& \mathbf{B}G
  }
$$

in [[sSet]] models the [[homotopy pullback]] in $sSet_{Quillen}$ / [[(∞,1)-pullback]] in [[∞Grpd]]

$$
  \array{
    P_\bullet &\to& *
    \\
    \downarrow &\swArrow& \downarrow
    \\
    X_\bullet &\to& \mathbf{B}G
  }
  \,.
$$

in the standard [[model structure on simplicial sets]] and hence produces the [[principal ∞-bundle]] $P_\bullet \to X_\bullet$ classified by $X_\bullet \to \mathbf{B}G$.

For all these constructions exist very explicit combinatorial formulas that go by the symbols

* $\overline{W}G$ for the [[delooping]] $\mathbf{B}G$

* $W G$ for the [[generalized universal bundle]] $\mathbf{E}G$

* $\tau : X_\bullet \to G_\bullet$ (called the _[[twisting function]]_) for the [[cocycle]] $X_\bullet \to \mathbf{B}G$;

* $X_\bullet \times_g W G$ for $P_\bullet$ (called _[[twisted Cartesian product]]_ ).

All of these constructions are functorial and hence lift from the context of [[simplicial set]]s to that of [[simplicial presheaves]] over some [[site]] $C$. There they provide models for strict [[group object in an (infinity,1)-category|group objects]], [[delooping]] and [[principal ∞-bundle]]s in the corresponding [[(∞,1)-topos]]es over $C$. In particular in the projective model structure on $[C^{op}, sSet]$ the pullback of the objectwise $W G \to \overline{W}G$ is still a homotopy pullback and models the corresponding principal $\infty$-bundles.

### Delooping {#Delooping}

A simplicial group $G$ is a [[group object in an (infinity,1)-category|group object]] [[internalization|internal]] to the category of [[Kan complex]]es. Accordingly, there should be a [[Kan complex]] $\mathbf{B}G$ which is the [[delooping]] of $G$, i.e. a Kan complex with an essentially unique object, such that the [[loop space object]] of that Kan complex reproduces $G$.

An explicit construction of $\mathbf{B}G$ from $G$ goes traditionally by the symbol $\bar W G \in KanCplx$. Another one by $d B G$.

#### Delooping modeled by $\bar W G$

It is immediate to deloop the simplicial group $G$ to the [[simplicial groupoid]] that in degree $k$ is the 1-[[groupoid]] with a single object and $G_k$ as its collection of morphisms.

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

For $\mathcal{G}$ a [[simplicial groupoid]] that on objects is a constant simplicial set, define a [[simplicial set]] $\bar W \mathcal{G}$ as follows.

* $(\overline{W}\mathcal{G})_0 := ob(\mathcal{G}_0)$, the set of objects of the groupoid of 0-simplices (and hence of the groupoid at each level);

* $(\overline{W}\mathcal{G})_1 = Mor(\mathcal{G}_0)$, the collection of [[morphism]]s of the groupoid $\mathcal{G}_0$:

and for $n \geq 2$,

*  $(\overline{W}\mathcal{G})_n = \{(h_{n-1}, \ldots ,h_0)| h_i \in Mor(\mathcal{G}_i)$  and $s(h_{i-1}) = t(h_i), 0\lt i\lt n\}$.

Here  $s$ and $t$ are generic symbols for the domain and codomain mappings of all the groupoids involved.  The face and degeneracy mappings between
$\overline{W}(\mathcal{G})_1$ and $\overline{W}(\mathcal{G})_0$ are the source and target maps and the identity maps of $\mathcal{G}_0$, respectively; whilst the face and degeneracy maps at higher levels are given as follows:

The face and degeneracy maps are given by

* $d_0(h_{n-1}, \ldots, h_0) = (h_{n-2}, \ldots, h_0)$;

*  for $0 \lt  i\lt  n$, $d_i(h_{n-1}, \ldots, h_0)  = (d_{i-1}h_{n-1}, d_{i-2}h_{n-2}, \ldots, d_0h_{n-i}h_{n-i-1},h_{n-i-2}, \ldots , h_0)$;

and

*  $d_n(h_{n-1}, \ldots, h_0) = (d_{n-1}h_{n-1}, d_{n-2}h_{n-2}, \ldots, d_1h_{1})$;

whilst

*  $s_0(h_{n-1}, \ldots, h_0) = (id_{dom(h_{n-1})},h_{n-1}, \ldots, h_0) $;

and,

* for $0\lt i \leq n$, $s_i(h_{n-1}, \ldots, h_0) = (s_{i-1}h_{n-1}, \ldots, s_0h_{n-i}, id_{cod(h_{n-i})},h_{n-i-1}, \ldots,  h_0) $.

=--

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

For $G$ a simplicial group and $\mathcal{G}$ the corresponding one-object [[simplicial groupoid]], one writes

$$
  \overline{W}G := \overline{W}\mathcal{G}
  \,.
$$

=--

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

The above construction has a straightforward [[internalization]] to contexts other than [[Set]]. For instance if $G$ is a [[simplicial object]] in [[topological group]]s or in [[Lie group]]s, then $\overline{W}G$ with

$$
  (\overline{W}G)_n :=
  G_{n-1} \times G_{n-2} \times \cdots \times G_0
$$

is a [[simplicial object]] in this context ([[topological spaces]], [[smooth manifold]]s, etc.)

In particular, if $C$ is a [[small category]] and $G : C^{op} \to sSet$ is a [[simplicial presheaf]] that is objectwise a simplicial group, then we have the simplicial presheaf

$$
  \overline{W}G : c \mapsto \overline{W}(G(c))
  \,.
$$

=--

#### Delooping modeled by $d B G$

For $G$ a [[simplicial group]], write $B G$ for the [[bisimplicial set]] obtained by taking degreewise the [[nerve]] of the [[delooping]] [[groupoid]]. Write $d B G \in $ [[sSet]] for its [[delooping]].

+-- {: .num_theorem}
###### Theorem

There is a [[weak homotopy equivalence]]

$$
  \bar W G \simeq d B G
  \,.
$$

=--

This is shown for instance in ([JardineLuo](#JardineLuo)) and in ([CegarraRemedios](#CegarraRemedios)).

#### Examples

If $G$ is an ordinary [[group]], regarded as a simplicially constant simplicial group, then $\overline{W}G$ is the usual bar complex of $G$:

$$
  \overline{W}G =
  \left(
    \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *
  \right)
  \,.
$$

### Cocycles {#UniversalBundle}

For $X_\bullet$ a [[simplicial set]] a morphism

$$
  g : X_\bullet \to \overline{W}G
$$

in [[sSet]] corresponds precisely to what is called a [[twisting function]], a family of maps

$$
  \{\phi(g)_n : X_n \to G_{n-1}\}
$$

satisfying the relations

$$
\array{
  d_0 \phi(x) = \phi(d_1 x)(\phi(d_0 x))^{-1}
  \\
  d_i \phi(x) = \phi(d_{i+1}x), i\gt 0,
  \\
  s_i\phi(x) = \phi(s_{i+1}x), i\geq 0,
  \\
  \phi(s_0 x) = 1_G.
}
$$

### Simplicial Principal bundles {#PrincipalBundles}

Simplicial groups model all [[∞-group]]s in [[∞Grpd]]. Accordingly all [[principal ∞-bundle]]s in [[∞Grpd]] should be modeled by [[simplicial principal bundle]]s.

+-- {: .num_defn}
###### Definition
**(principal action)**

Let $G$ be a simplicial group. For $P$ a [[Kan complex]], an
[[action]] of $G$ on $E$

$$
  \rho : E \times G \to E
$$

is called **principal** if it is degreewise principal, i.e. if for all $n \in \mathbb{N}$ the only elements $g \in G_n$ that have any fixed point $e \in E_n$ in that $\rho(e,g) = e$ are the neutral elements.

=--

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

The canonical action

$$
  G \times G \to G
$$

of any simplicial group on itself is principal.

=--

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

For $G$ a simplicial group, a morphism $P \to X$ of [[Kan complex]]es equipped with a $G$-[[action]] on $P$ is called a $G$-**simplicial principal bundle** if

* the action is principal;

* the base is isomorphic to the quotient $E/G := \lim_{\to}(E \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to} E})$ by the action:

  $$
    E/G \simeq X
    \,.
  $$

=--

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

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

=--

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

This appears as Lemma 18.2 in [MaySimpOb](#MaySimplicialObjects).

=--

#### Universal simplicial $G$-principal bundle {#UniversalSimplicialBundle}

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

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

$$
  W G := Dec \overline{W}G
  \,.
$$

=--

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 $g$.

=--

+-- {: .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 $i \gt 0$

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

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

=--

#### References {#SimplicialBundleReferences}

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)).

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]], _[[Tim Porter:crossed menagerie|The 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)

## Related concepts

* [[simplicial group action]]

* [[simplicial topological group]]

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

* [[group]]

* [[2-group]], [[crossed module]], [[differential crossed module]]

* [[3-group]], [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]

* [[n-group]]

* [[∞-group]], **simplicial group**, [[crossed complex]], [[hypercrossed complex]]

* [[model structure on simplicial groups]]

* [[Borel model structure]]

## References

* {#Quillen67} [[Daniel Quillen]], II §3.8 of: *[[Homotopical Algebra]]*, Lecture Notes in Mathematics **43**, Springer (1967) [[doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)]

* {#MaySimplicialObjects} [[Peter May]], Chapter IV, Section 17 of: _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]])

* {#Curtis71} [[Edward B. Curtis]], Section 3 of: _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))

* {#GoerssJardine} [[Paul Goerss]], [[J. F. Jardine]], Chapter V 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/))

* {#Weibel94} [[Charles Weibel]], Chapter 8 of: _An introduction to homological algebra_, Cambridge (1994)

The algorithm for finding the horn fillers in a simplicial group is given in the proof of theorem 17.1, page 67 there.

This proof that simplicial groups are Kan complexes is originally due to  Moore, see theorem 3.4 in

* [[John Moore]],  _Semi-Simplicial Complexes And Postnikov Systems_, in _Symposium International De Topologia Algebraica_ , 1956 conference, book published in 1958

which earlier appeared in more detail as Theorem 3 on p. 18-04 of

* {#Moore54} [[John Moore]], *Homotopie des complexes monoideaux, I*, Seminaire Henri Cartan (1954-55) [[numdam:SHC_1954-1955__7_2_A8_0](http://www.numdam.org/item?id=SHC_1954-1955__7_2_A8_0)]

and is often referenced to

* {#Moore} [[John Moore]], _Algebraic homotopy theory_, lecture notes, Princeton University, 1955--1956

In fact, it seems that this is the origin of the very notion of [[Kan complex]].

A proof is also on p. 14 of

* {#JoyalTierney05} [[André Joyal]], [[Myles Tierney]] chapter I of _An introduction to simplicial homotopy theory_, 2005  ([web](http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01), [[JoyalTierneySimplicialHomotopyTheory.pdf:file]])

Section 1.3.3 of

* [[Tim Porter]], _[[Tim Porter:crossed menagerie|The Crossed Menagerie]]_

discusses simplicial groups in the context of [[nonabelian algebraic topology]].

Additional useful references include

* {#JardineLuo} [[Rick Jardine]], Luo, _Higher order principal bundles_ ([web](http://www.math.uiuc.edu/K-theory/0681/))

* {#CegarraRemedios} [[Antonio Cegarra]], [[Josué Remedios]], _The relationship between the diagonal and the bar constructions on a bisimplicial set_ ([pdf](http://www.ugr.es/~acegarra/Paperspdfs/TRBDWC.pdf))

category: simplicial object

[[!redirects simplicial groups]]

[[!redirects sGrp]]
[[!redirects SimplicialGroups]]


simplicial group action

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

#Contents#
* table of contents
{:toc}

## Definition

Given a [[simplicial group]] $\mathcal{G} \in Groups(sSet)$, hence a [[group object]] [[internalization|internal]] to [[SimplicialSets]], a [[group action]] of $\mathcal{G}$ on a [[simplicial set]] $X$ is a $\mathcal{G}$-[[action object]] [[internalization|internal]] to [[SimplicialSets]], hence a morphism of simplicial sets of the form

$$
  \mathcal{G} \times X
  \xrightarrow{\;\; \rho \;\;}
  X
$$

satisfying the action property.

The [[category]] of simplicial $\mathcal{G}$-action may be understood as the [[sSet]]-[[enriched functor|enriched]] [[functor category]] from the one-object [[sSet-category]] $\mathbf{B}\mathcal{G}$ to [[sSet]]:

\[
  \label{CategoryOfSimplicialGroupActions}
  \mathcal{G}Actions(sSet)
  \;\;
  \simeq
  \;\;
  sSetCat
  \big(
    \mathbf{B}\mathcal{G},
    \,,
    sSet
  \big)
\]

## Properties

### Model category structure

\begin{proposition}
**([[model structure on simplicial group actions]])** \linebreak
  There is a [[model category]]-structure on simplicial group actions (eq:CategoryOfSimplicialGroupActions) whose [[weak equivalences]] and [[fibrations]] are those in the underlying [[classical model structure on simplicial sets]], hence are the [[simplicial weak equivalences]] and [[Kan fibrations]] of the underlying simplicial sets.
\end{proposition}

([DDK 80, Prop. 2.2. (ii)](#DDK80), $\;$ [Guillou, Prop. 5.3](#Guillou), $\;$ [Goerss & Jardine 09, V Lem. 2.4](#GoerssJardine09)).

## Related concepts

* [[Borel model structure]]

* [[G-set]], [[topological G-space]]

* [[∞-action]]

## References

* {#DDK80} [[Emmanuel Dror]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 ([jstor:2043448](http://www.jstor.org/stable/2043448))

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.2 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/))

* {#Guillou} [[Bert Guillou]], _A short note on models for equivariant homotopy theory_, 2006 ([pdf](http://www.math.uiuc.edu/~bertg/EquivModels.pdf), [[GuillouModelsForEquivariantHomotopyTheory.pdf:file]])

[[!redirects simplicial group actions]]


simplicial groupoid

+-- {: .rightHandSide}
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#### Enriched category theory
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[[!include enriched category theory contents]]
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#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The notion of *simplicial groupoids* pairs the concepts of *[[groupoids]]* and *[[simplicial sets]]*. Via the [[Dwyer-Kan loop groupoid]] functor [[Dwyer & Kan (1984)](#DwyerKan84)] their [[homotopy]] theory is equivalent to the [[classical model structure on simplicial sets|classical]] homotopy theory of [[simplicial sets]]/[[Kan complexes]] (both being models for [[infinity-groupoids|$\infty$-groupoids]]).

## Definition
 {#Definition}

A priori, the term *simplicial groupoid* may refer to [[simplicial objects]] in the ([[1-category|1-]])[[category]] [[Grpd]] of ([[small category|small]]) [[strict category|strict]] [[groupoids]].

However (cf. discussion at *[[simplicial category]]*), in applications (notably in [[homotopy theory]]) one is interested only in those simplicial objects in [[Grpd]] whose [[underlying]] simplicial [[sets]] of [[objects]] are *simplicially constant*. This more restrictive notion --- equivalent to *[[sSet]]-[[enriched groupoids]]* --- is traditionally still referred to as "simplicial groupoids" [e.g. [Dwyer & Kan (1984),  §1.2.(ii)](#DwyerKan84), [Goerss & Jardine (2009), V.7](GoerssJardine09)]; but for the moment let us call these instead "simplicial DK-groupoids", for clarity, and let us denote the [[full subcategory]] they form by

$$
  sSet\text{-}Grpd
  \;\simeq\;
  sGrpd_{DK}
    \overset{\phantom{---}}{\hookrightarrow}
  Func(\Delta^{op}, Grpd)
  \,.
$$

So given such a simplicial DK-groupoid $\mathcal{C}_\bullet \,\in\, sGrpd_{DK}$ we have:

1. a plain [[set]] of [[objects]] $Obj(\mathcal{C}_\bullet) \,\in\, Sets$;

1. for every [[pair]] $x, y \,\in\, Obj(\mathcal{C}_\bullet)$ of [[objects]] the degree-wise [[hom-sets]] form a [[simplicial set]]

   $$
     \mathcal{C}_\bullet(x,y)
     \;\coloneqq\;
     Hom_{\mathcal{C}_\bullet}(x,y)
     \;\in\;
     sSet
     \,;
   $$

1. where the degreewise [[composition]] operations constitutes an [[sSet]]-[[enriched category|enriched]] composition operation

   $$
     \mathcal{C}_\bullet(x,y)
     \,\times\,
     \mathcal{C}_\bullet(y,z)
     \overset{\circ}{\longrightarrow}
     \mathcal{C}_\bullet(x,z)
   $$

1. making $\mathcal{C}_\bullet$ an [[sSet-enriched category]].

Conversely, simplical DK-groupoids are therefore in [[bijection]] to those [[sSet-enriched categories]] whose degreewise [[categories]] formed by the $n$-cell [[morphisms]] for any $n \,\in\, \mathbb{N}$ are [[groupoids]].

This makes the category $sGrpd_{DK}$ be a [[full subcategory]] of the category of ([[small category|small]]) [[sSet-enriched categories]]:

$$
  sGrpd_{DK}
    \overset{\phantom{---}}{\hookrightarrow}
  sSet Cat
  \,.
$$

## Remarks

*  Simplicially enriched groupoids are related to [[simplicial set|simplicial sets]] via an [[adjoint functor|adjunction]] found independently by Dwyer & Kan and Joyal & Tierney; see *[[Dwyer-Kan loop groupoid]]*.  This adjunction gives an [[equivalence of categories|equivalence]] of [[homotopy category|homotopy categories]] so that simplicially enriched groupoids model all [[homotopy type|homotopy types]].

   *  This is a groupoidal version of the equivalence between [[quasi-category|quasi-categories]] and [[simplicially enriched category|simplicially enriched categories]] via the [[homotopy coherent nerve]].

*  A simplicially enriched groupoid having exactly one object is essentially the same as a [[simplicial group]]. Notationally however it is often important to distinguish a simplicial group form the corresponding single object simplicially enriched groupoid.

*  Many constructions on simplicial groups, such as that of its [[Moore complex]] carry over to simplicialy enriched groupoids without difficulty.

## Properties

### Relation to simplicial groups
 {#RelationToSimplicialGroups}

\begin{example}\label{SimplicialDeloopingGroupoids}
**([[simplicial groupoid|simplicial]] [[delooping groupoid]])**
\linebreak
For $\mathcal{G} \,\in\, Grp(sSet)$ a [[simplicial group]], we obtain its simplicial [[delooping groupoid]] $\mathbf{B}\mathcal{G}$ by setting

* $Obj(\mathbf{B}\mathcal{G}) \,\coloneqq\, \ast$ (the [[singleton set]]),

* $(\mathbf{B}\mathcal{G})(\ast,\ast) \,\coloneqq\, \mathcal{G}$

* with [[composition]] given by the group operation of $\mathcal{G}$.

Conversely, for $\mathcal{X} \,\in\, sSet\text{-}Grpd$ and $x \,\in\, Obj(\mathcal{X})$, the [[endomorphism]]-[[hom-object]] at $x$ canonically carries the [[structure]] of a [[simplicial group]]:
$$
  \mathcal{X}(x,x) \,\in\, Grp(sSet)
  \,.
$$
\end{example}

\begin{definition}
**(some notation)**
\linebreak
For $\mathcal{X} \,\in\, sSet\text{-}Grpd$ a DK-simplicial groupoid, write:

* $\mathcal{X}_0 \,\in\, Grpd$ for its [[underlying]] plain (i.e. [[Set]]-enriched) [[groupoid]] (its [[change of enrichment]] along the [[terminal category|terminal]] functor $sSet \to \ast$)

* $\pi_0(\mathcal{X}) \,\coloneqq\, \pi_0(\mathcal{X}_0) \,\in\, Set$ for the set of [[isomorphism classes]] of [[objects]], hence for the set of *[[connected components]]* of $\mathcal{X}$,

* $\mathcal{X}_{[x]} \hookrightarrow \mathcal{X}$ for the enriched [[full subcategory]] on all objects in the connected component $[x] \,\in\, \pi_0(\mathcal{X})$.

\end{definition}

\begin{proposition}
\label{SimplicialGroupoidsAdjointEquivalentToSkeleta}
**(simplicial groupoids adjoint equivalent to skeletons)**
\linebreak
  Assuming the [[axiom of choice]], every [[sSet]]-[[enriched groupoid]] is [[sSet]]-[[enriched adjoint equivalence|enriched adjoint equivalent]] to a [[skeletal groupoid|skeletal]] simplicial groupoid, namely to a [[disjoint union]] of simplicial delooping groupoids (Exp. \ref{SimplicialDeloopingGroupoids}), one for each [[connected components]], in that given a choice of representative [[objects]] $(i, x_i) \,\in\, \big(i \in \pi_0(\mathcal{X})\big) \times Obj(\mathcal{X}_i)$ of all the [[connected components]], the [[full subcategory|inclusion]] [[enriched functor]]
\[
  \label{FullInclusionOfConnectedComponents}
  \iota
  \,\colon\,
  \left(
  \underset{i \in \pi_0(\mathcal{X})}{\amalg}
  \mathbf{B}\big(\mathcal{X}(x_i,\,x_i)\big)
  \right)
  \xhookrightarrow{\phantom{---}}
  \mathcal{X}
\]
has a strict [[left inverse]] and a [[right inverse]] up to [[enriched natural transformation|enriched natural]] [[isomorphism]].

In other words, every [[sSet]]-[[enriched groupoid]] has a *[[deformation retraction]]* onto an [[sSet]]-[[skeletal groupoid]].
\end{proposition}
\begin{proof}
  This follows in direct [[enriched category theory|enriched]]-analogy to the corresponding statement for plain [[groupoids]] (as for instance spelled out [here](Introduction+to+Topology+--+2#EveryGroupoidIsEquivalentToDisjointUnionOfGroupDeloopings)) using (only) that the [[cosmos]] [[sSet]] is [[cartesian monoidal category|cartesian monoidal]]:

Choosing for each $x \in Obj(\mathcal{X})$ an element

$$
  \gamma_x
  \,\colon\,
  \ast \to \mathcal{X}(x_{[x]}, x)
$$

induces

1. an [[enriched functor]]

   \[
     \label{ContractionOntoConnectedComponents}
     \array{
       \mathcal{X}
         &\xrightarrow{
          \phantom{----}
          p
          \phantom{----}
         }&
       \underset{i \in \pi_0(\mathcal{X})}{\amalg}
       \mathbf{B}\big( \mathcal{X}(x_i,\,x_i) \big)
       \\
       x &\mapsto& x_{[x]}
       \\
       \mathcal{X}(x,y)
         &\xrightarrow{
            \phantom{----}
            p_{x,y}
            \phantom{----}
         }&
       \mathcal{X}(x_{[x]}, x_{[y]})
       \\
       \mathllap{\simeq}\Big\downarrow
       &&
       \Big\uparrow\mathrlap{\circ}
       \\
       \ast \times \mathcal{X}(x,y) \times \ast
       &
       \underset{
         \big((-)^{-1}\circ \gamma_y\big)
         \times
         id
         \times
         \gamma_x
       }{\longrightarrow}
       &
         \mathcal{X}(y,x_{[y]})
         \times
         \mathcal{X}(x,y)
         \times
         \mathcal{X}(x_{[x]},x)
     }
   \]

1. an [[enriched natural transformation]]

   \[
     \label{ContractingHomotopy}
     \gamma
       \,\colon\,
     \iota \circ p
       \longrightarrow
     id_{\mathcal{X}}
   \]

   with components $\gamma_x$ (which satisfies its "enriched [[naturality square]]-condition" [here](enriched+natural+transformation#eq:ExtranaturalitySquare) essentially by construction of $p$).

Similarly there is an enriched transformation $id_{\mathcal{X}} \longrightarrow  p \circ \iota$, and these make the [[adjunction unit|unit]] and [[adjunction counit|counit]] of an "enriched [[adjoint equivalence]]".
But if we choose $\gamma_{x_{[x]}} \coloneqq id_{x_{[x]}}$ --- as we may --- then there is already an [[equality]] $p \circ \iota = id$ (hence a [[retraction]] onto the skeleton).
\end{proof}

Some consequences:

\begin{remark}
\label{sSetPresheavesOnSimplicialGroupoidsAreProductsOfsSetGroupActions}
**([[simplicial presheaves]] on [[simplicial groupoids]] are [[equivalence of categories|equivalently]] [[products]] of [[simplicial group actions]])**
\linebreak
  For

  * $\mathcal{X}$ a DK-enriched groupoid regarded as an [[sSet-enriched category]]

  * $\mathbf{C}$ any [[sSet-enriched category]]

the [[enriched functor category]] between the two is [[equivalence of categories|equivalent]] (as a plain [[locally small category]]) to a [[product category|product]] of [[enriched functor categories]] on simplicial delooping groupoids (Exp. \ref{SimplicialDeloopingGroupoids}), one for each connected component of $\mathcal{X}$,
$$
  sFunc(\mathcal{X},\,\mathbf{C})
  \;\simeq\;
  \underset{
   i \in \pi_0(\mathcal{X})
  }{\prod}
  sFunc\big(
    \mathbf{B}\mathcal{X}(x_i,x_i)
    ,\,
    \mathbf{C}
  \big)
  \,.
$$

This follows by observing that the functor of precomposion with the inclusion (eq:FullInclusionOfConnectedComponents)
$$
  \iota^\ast
  \;\colon\;
  sFunc(\mathcal{X},\,\mathbf{C})
  \longrightarrow
  \underset{
   i \in \pi_0(\mathcal{X})
  }{\prod}
  sFunc\big(
    \mathbf{B}\mathcal{X}(x_i,x_i)
    ,\,
    \mathbf{C}
  \big)
$$
is inverse to $p^\ast$ (eq:ContractionOntoConnectedComponents)
up to a [[natural isomorphism]] whose component at any [[enriched functor]] $F \colon \mathcal{X} \longrightarrow \mathbf{C}$ is the [[enriched natural transformation]] $p^\ast \iota^\ast F\longrightarrow F$ obtained as the [[horizontal composition]] ("[[whiskering]]" of enriched transformations, see [there](enriched+natural+transformation#RightWhiskering)) by $F$ of the enriched transformation (eq:ContractingHomotopy). Notice that this is already an [[adjoint equivalence]].

If $\mathbf{C}$ is moreover [[tensoring|tensored]] over [[sSet]], then this, in turn, is [[equivalence of categories|equivalent]] to a product of categories of [[simplicial group|simplicial]]$\;$[[group actions]] on objects in $\mathbf{C}$:
$$
  sFunc(\mathcal{X},\,\mathbf{C})
  \;\simeq\;
  \underset{
   i \in \pi_0(\mathcal{X})
  }{\prod}
  \Big(
  \big(\mathcal{X}(x_i,x_i)\big)
  Act(\mathbf{C})
  \Big)
  \,.
$$
\end{remark}

\begin{example}
\label{BorelModelStructureOnSimplicialGroupActions}
**([[Borel model structure]] on [[simplicial group actions]] over simplicial groupoids)**
\linebreak
For $\mathcal{G} \,\in\, Grp(sSet)$ a [[simplicial group]] with [[sSet]]-[[enriched groupoid|enriched]] [[delooping groupoid]] denoted $\mathbf{B}\mathcal{G} \in sSet\text{-}Grpd$, an [[sSet]]-[[enriched functor]] $\mathbf{B}\mathcal{G} \longrightarrow sSet$ is equivalently a [[simplicial group action]] of $\mathcal{G}$.

Under this identification, the projective model structure on [[simplicial functors]] ([this Prop.](model+structure+on+functors#ExistenceOfProjectiveStructureOnEnrichedFunctors)) is equivalently the *[[Borel model structure]]* on [[simplicial group actions]], a context of Borel-[[equivariant homotopy theory]]:

$$
  sFunc\big(
    \mathbf{B}\mathcal{G}
    ,\,
    sSet
  \big)_{proj}
  \;\;
  =
  \;\;
  \mathcal{G}Act(sSet)_{Borel}
  \,.
$$

More generally, for $\mathcal{X} \in sSet\text{-}Grpd$ an [[sSet]]-[[enriched groupoid]] (Dwyer-Kan [[simplicial groupoid]]) with a single [[connected component]] $\pi_0(\mathcal{X}) \simeq \{[x]\}$, so that the inclusion
$$
  \iota
    \,\colon\,
  \mathbf{B}(\mathcal{X}(x,x))
  \xhookrightarrow{\phantom{---}}
  \mathcal{X}
$$
is an [[sSet]]-[[enriched adjoint equivalence]] (see discussion [there](simplicial+groupoid#RelationToSimplicialGroups)) the projective model structure on [[simplicial functors]] (from [that Prop.](model+structure+on+functors#ExistenceOfProjectiveStructureOnEnrichedFunctors)) is [[transferred model structure|transferred]] under the induced [[adjoint equivalence]] of [[sSet]]-[[enriched functor categories]]
$$
  sFunc\big(
    \mathcal{X}
    ,\,
    sSet
  \big)_{proj}
  \underoverset
    {\underset{\iota^\ast}{\longrightarrow}}
    {\overset{\iota_!}{\longleftarrow}}
    {\;\; \bot_{\simeq} \;\;}
  sFunc\Big(
    \mathbf{B}\big(\mathcal{X}(x,x)\big)
    ,\,
    sSet
  \Big)_{proj}
  \;=\;
  \big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}
$$
By [this example](transferred+model+structure#RightTransferAlongAdjointEquivalence) it follows that morphisms in all three classes $(\mathrm{W}, Fib, Cof)$ in $sFunc(\mathcal{X}, \, sSet)_{proj}$ are those which restrict on $x \in Obj(\mathcal{X})$ to the respective class in $\big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}$.

It follows that for $\mathcal{X} \,\in\, sSet\text{-}Grpd$ a [[simplicial groupoid]] with any set $\pi_0(\mathcal{X})$ of [[connected components]], the projective model structure of simplicial functors over it is the [product model structure](model+category#ProductModelStructure) of the [[Borel model structures]] of [[simplicial group actions]], one for each connected component:
$$
  sFunc\big(
    \mathcal{X}
    ,\,
    sSet
  \big)_{proj}
  \underoverset
    {\underset{\iota^\ast}{\longrightarrow}}
    {\overset{\iota_!}{\longleftarrow}}
    {\;\; \bot_{\simeq} \;\;}
  \underset{i \in \pi_0(\mathcal{X})}{\prod}
  sFunc\Big(
    \mathbf{B}\big(\mathcal{X}(x_i,x_i)\big)
    ,\,
    sSet
  \Big)_{proj}
  \;=\;
  \underset{i \in \pi_0(\mathcal{X})}{\prod}
  \big(\mathcal{X}(x_i,x_i)\big) Act(sSet)_{Borel}
  \,.
$$

The analogous statement holds (still by [that Prop.](model+structure+on+functors#ExistenceOfProjectiveStructureOnEnrichedFunctors)) for the [[codomain]] [[sSet]] replaced by any [[combinatorial simplicial model category]] $\mathbf{C}$:
$$
  sFunc\big(
    \mathcal{X}
    ,\,
    \mathbf{C}
  \big)_{proj}
  \underoverset
    {\underset{\iota^\ast}{\longrightarrow}}
    {\overset{\iota_!}{\longleftarrow}}
    {\;\; \bot_{\simeq} \;\;}
  \underset{i \in \pi_0(\mathcal{X})}{\prod}
  sFunc\Big(
    \mathbf{B}\big(\mathcal{X}(x_i,x_i)\big)
    ,\,
    \mathbf{C}
  \Big)_{proj}
  \;=\;
  \underset{i \in \pi_0(\mathcal{X})}{\prod}
  \big(\mathcal{X}(x_i,x_i)\big) Act(\mathbf{C})_{Borel}
  \,.
$$

Moreover, if $I_{\mathbf{C}}, J_{\mathbf{C}} \,\subset\, \mathbf{C}$ denote [[classes]] of (acyclic) [[generating cofibrations]] of $\mathbf{C}$, then [that Prop.](model+structure+on+functors#ExistenceOfProjectiveStructureOnEnrichedFunctors) gives [[generating cofibrations]] of $\mathcal{G}Act(\mathbf{C})_{Borel}$ to be
$$
  I^{\mathcal{G}}_{\mathbf{C}} \,\coloneqq\,
  \big\{
    \mathcal{G} \cdot i
    \;\vert\;
    i \in I_{\mathbf{C}}
  \big\}
  \,,\;\;\;\;
  J^{\mathcal{G}}_{\mathbf{C}} \,\coloneqq\,
  \big\{
    \mathcal{G} \cdot j
    \;\vert\;
    i \in J_{\mathbf{C}}
  \big\}
  \,.
$$
\end{example}

### Cartesian closure
 {#CartesianClosedStructure}

\begin{proposition}
  The category of [[sSet]]-[[enriched groupoids]] is [[cartesian closed category|cartesian closed]].
\end{proposition}
\begin{proof}
The ambient category of [[sSet-enriched categories]] is cartesian closed (as any [[category of enriched categories|category of $\mathcal{V}$-categories]] for [[cartesian closed category|cartesian closed]] [[complete category|complete]] [[cosmos]] $\mathcal{V}$), given by forming [[enriched product categories]] and [[enriched functor categories]].
Therefore we just need to check that for $\mathbf{D},\,\mathbf{C} \,\in\, sSet\text{-}Grpd$ also the $sSet$-[[enriched product category]] $\mathbf{C} \times \mathbf{D}$ and the $sSet$-[[enriched functor category]] $[\mathbf{D},\,\mathbf{C}]$ are in fact [[enriched groupoids]].

For the product this is immediate, the [[inverse morphism|inversion]]-operation is given factorwise.

To see the statement for the [[internal hom]], write $\mathbf{I}_S$ for the [[sSet-enriched category]] with set objects $\{0,1\}$ and [[hom objects]] given by:

* $\mathbf{I}_{S}(0,0) \,\coloneqq\, \ast$

* $\mathbf{I}_{S}(1,1) \,\coloneqq\, \ast$

* $\mathbf{I}_{S}(1,0) \,\coloneqq\, \varnothing$

* $\mathbf{I}_{S}(0,1) \,\coloneqq\, S$

(whence there is no non-trivial [[composition]] to be defined).

Now for $n \in \mathbb{N}$ the $sSet$-[[enriched functors]]

$$
  \mathbf{I}_{\Delta[n]}
    \longrightarrow
  [\mathbf{D},\,\mathbf{C}]
$$

are in [[bijection]] with the $(n+1)$-morphisms in $[\mathbf{D},\,\mathbf{C}]$, and we need to exhibit an inversion operation on these.

But by the [[cartesian closed monoidal category|cartesian closure]] of [[VCat|sSet Cat]] these are also in natural bijection to enriched functors of the form
$$
  \mathbf{D}
  \times
  \mathbf{I}_{\Delta[n]}
  \longrightarrow
  \mathbf{C}
$$
which are manifestly a simplicial diagram of [[natural transformations]] between the restrictions $\mathbf{D} \times \{0\} \to \mathbf{C}$ and $\mathbf{D} \times \{1\} \to \mathbf{C}$. The corresponding system of inverse natural transformations corresponds to the inverse $n+1$-morphism that we are after.
\end{proof}

## Related concepts

* [[model structure on simplicial groupoids]]

* [[simplicial category]]

  * [[simplicially enriched category]]

  * [[simplicial object in Cat]]

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

* [[model structure on presheaves of simplicial groupoids]]

## References

Original discussion in the context of [[simplicial localization]]

* {#DwyerKan80} [[William Dwyer]], [[Daniel Kan]], §5.5 of: *Simplicial localizations of categories*, J. Pure Appl. Algebra **17** 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3, [pdf](http://www.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf)]

and in the context of a [[model structure on simplicial groupoids]]:

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]],  _Homotopy theory and simplicial groupoids_, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

Textbook accounts

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.7 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/)]

* {#Jardine15} [[John F. Jardine]], §9.3 in: *[[Local homotopy theory]]*, Springer Monographs in Mathematics (2015) [[doi:10.1007/978-1-4939-2300-7](https://doi.org/10.1007/978-1-4939-2300-7)]

See also:

* [[Philip Ehlers]], _Simplicial groupoids as models for homotopy type_, Master's thesis (1991) [[pdf](https://ncatlab.org/nlab/files/Ehlers-MSc-thesis.pdf)]

* [[Philip J. Ehlers]], *Algebraic Homotopy in Simplicially Enriched Groupoids*, PhD thesis (1993) [[[Ehlers-thesis.pdf|pdf:file]]]

[[!redirects simplicial groupoids]]
[[!redirects simplicially enriched groupoids]]
[[!redirects simplicially enriched groupoid]]

[[!redirects sSet-enriched groupoid]]
[[!redirects sSet-enriched groupoids]]


simplicial homology

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

#Contents#
* table of contents
{:toc}

## Idea

For $S$ a [[simplicial set]] and $A$ an [[abelian group]], the _simplicial [[homology]]_ of $S$ is the [[chain homology]] of the [[chain complex]] corresponding under the [[Dold-Kan correspondence]] to the [[simplicial abelian group]] $S \cdot A$ of $A$-[[chains]] on $S$: [[formal linear combinations]] of [[simplices]] in $S$ with [[coefficients]] in $A$.

## Definition

Let $S$ be a [[simplicial set]] and $A$ an [[abelian group]].

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

For $n \in \mathbb{N}$ write

$$
  S_n \cdot A \coloneqq \mathbb{Z}[S_n] \otimes A
$$

for the [[free abelian group]] on the set $S_n$  of $n$-[[simplices]] [[tensor product of abelian groups|tensored]] with $A$: the group of [[formal linear combinations]] of $n$-simplices with [[coefficients]] in $A$.

These abelian groups arrange to a [[simplicial abelian group]]

$$
  X \cdot A \in Ab^{\Delta^{op}}
  \,.
$$

The [[alternating face map complex]] of these groups is called the **complex of simplicial chains** on $S$

$$
  C_\bullet(S \cdot A) = C_\bullet(S,A)
  \,.
$$

The **simplicial homology** of $S$ is the [[chain homology]] of the complex of simplicial chains:

$$
  H_\bullet(S, A)
  \coloneqq
  H_\bullet(C_\bullet(S,A))
  \,.
$$

=--

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

This means that the [[differentials]] in $C_\bullet(S,A)$ are given on [[basis]] elements $\sigma \in S_n$ by the [[formal linear combination]]

$$
  \partial \sigma
    =
  \sum_{k = 0}^{n} (-1)^k d_k \sigma
  \,,
$$

where $d_k : S_n \to S_{n-1}$ are the [[face maps]] of $S$.

=--

## Examples

### Explicit

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

Let $S = \partial \Delta^3$ be the [[boundary of a simplex|boundary]] of the simplicial  3-[[simplex]], the (hollow) simplicial [[tetrahedron]].

Since this has

* 4 non-degenerate vertices

* 6 non-degenerate edges

* 4 non-degenerate faces

the [[normalized chain complex]] of $\mathbb{Z}$ is of the form

$$
  \cdots \to 0 \to 0 \to \mathbb{Z}^4 \to \mathbb{Z}^6 \to \mathbb{Z}^4 \to 0
  \,.
$$

By writing out the two non-trivial differentials, one can deduce explicitly that

* $H_0(\partial \Delta^3) = \mathbb{Z}$ (reflecting the fact that the tetrahedron is [[connected topological space|connected]]);

* $H_1(\partial \Delta^3) = 0$ (reflecting the fact that it is [[simply-connected topological space|simply-connected]]);

* $H_2(\partial \Delta^3) = \mathbb{Z}$ (reflecting the fact, by the [[Hurewicz theorem]], that the second [[homotopy group]] of the 2-sphere is $\mathbb{Z}$ );

=--

### General

* For $X$ a [[topological space]] and $S = Sing X$ the [[singular simplicial complex]] of $X$, the simplicial homology of $Sing X$ is called the _[[singular homology]]_ of $X$, denoted $H_\bullet(X,A)$.

## Related concepts

* [[cochain on a simplicial set]]

* [[singular homology]]

## Terminology and a bit of history

The term simplicial homology is also used in the literature for the homology of polyhedral spaces, based on the theory of [[simplicial complex]]es.  That homology is defined by first looking at a chain complex of simplicial chains on, say, a triangulation of a space, and then passing to the corresponding homology.  The theory then proceeds by proving that the end result is independent of the triangulation used.  The resulting homology theory is isomorphic to singular homology, but historically was the earlier theory.

## References

A basic discussion is for instance around application 1.1.3 of

* [[Charles Weibel]]. _[[An Introduction to Homological Algebra]]_

Homology for spaces  is discussed in chapter 2 of

*  [[Alan Hatcher]], [Algebraic topology](http://www.math.cornell.edu/~hatcher/AT/AT.pdf), Cambridge Univ. Press 2002, starting p. 106

and this includes a discussion of the homology of simplicial complexes.

[[!redirects chain on a simplicial set]]
[[!redirects chains on a simplicial set]]

[[!redirects simplicial chain]]
[[!redirects simplicial chains]]


simplicial homotopy

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

#Contents#
* table of contents
{:toc}

## Idea

A **simplicial homotopy** is a [[homotopy]] in the classical [[model structure on simplicial sets]]. It can also be defined combinatorially; in that form one can define a homotopy 2-cell between morphisms of simplicial objects in any category $C$.

##Definition via cylinders##

[[SSet]] has a [[cylinder functor]] given by [[cartesian monoidal category|cartesian]] product with the standard 1-[[simplex]], $I := \Delta[1]$. (In fact, one can define simplicial  cylinders, $\Delta[1]\odot X$, more generally, for example for $X$ being a simplicial object in an [[cocomplete category]] $C$,(see below).)

Therefore for $f,g : X \to Y$ two morphisms of [[simplicial set]]s, a [[homotopy]] $\eta : f \Rightarrow g$ is a morphism $\eta : X \times \Delta[1] \to Y$ such that the diagram

$$
  \array{
    X \simeq X\times \Delta[0] &\stackrel{Id \times \delta^1}{\to}& X \times \Delta[1]
   &
   \stackrel{Id \times \delta^0}{\leftarrow}&
   X \times \Delta[0]
   \simeq X
   \\
   &
    {}_f\searrow &\downarrow^\eta& \swarrow_{g}
   \\
   &&
   Y
  }
$$

commutes.

Remark: Since in the standard [[model structure on simplicial sets]] every simplicial set is cofibrant, this indeed defines left homotopies.

##Combinatorial definition##

Given morphisms $f,g,:X\to Y$ of simplicial objects in any category $C$, a __simplicial homotopy__ is a family of morphisms, $h_i:X_n\to Y_{n+1}$, $i= 0,\ldots,n$ of $C$, such that $d_0 h_0 = f_n$, $d_{n+1} h_n = g_n$ and

$$ d_i h_j = \left\lbrace\array{
h_{j-1}d_i, & i\lt j \\
d_i h_{i-1}, &i=j\neq 0\\
h_j d_{i-1}, & i\gt j+1.
}\right.$$

$$ s_i h_j = \left\lbrace\array{
h_{j+1} s_i, & i\le j\\
h_j s_{i-1}, & i\gt j.
}\right.$$

Note that we did not explicitly say what $d_i h_j$ should be when $i = j + 1$, but if we look at the rule for $d_{j + 1} h_{j + 1}$, we see this requires $d_{j + 1} h_{j + 1} = d_{j + 1} h_j$, which is the relevant compatibility condition.

##Remark / warning on conventions

The above formulae give one of the, at least,  two  forms of the combinatorial specfication of a homotopy between $f$ and $g$. (When trying to construct a specific homotopy using a combinatorial form, check which convention is being used!) The two forms correspond to different conventions such as saying that this is a homotopy from $g$ to $f$, or reversing the labelling of the $h_i$.

##Commentary##
It is fairly easy to prove that the combinatorial definition of homotopy agrees with the one via the cylinder both for simplicial sets and for simplicial objects in any [[finitely cocomplete category]], $C$.  This uses the fact that the category of simplicial objects in a cocomplete category, $C$, has [[copower|copowers]] with finite simplicial sets and hence in particular with $\Delta[1]$. (As there are explicit formulae for the construction of [[copower|copowers]] ...)
+-- {: .query}
[[Tim Porter|Tim]]:  With only my own resources available, I was unable to find them so was hoping someone kind would come up with them. They derive from the coend formulae/Kan extension formulae using some combinatorics to discuss the indexing sets. I think Quillen gave some form of them, but have not got a copy of HA. I needed them recently and could not find them in any of the usual sources, and did not manage to work them out using the Kan extension idea either (Help please anyone). We could do with those formulae or with a reference to them at least.
=--

In the case of the category of (not necessarily abelian) groups, the combinatorial definition equals the one via cylinder only if the role of "cylinder" for a group $G$ is played by a simplicial object in the category of groups which in degree $n$ equals the free product of $(n+2)$ copies of $G$, indexed by the set $\Delta[1]_n$ (noted by Swan and quoted in exercise 8.3.5 of Weibel: _Homological algebra_).

+--{: .query}
[[Tim Porter]]:   Perhaps we need an explicit description of [[copower|copowers]] in simplicial objects also. I pointed out in an edit above that the combinatorial description is much more general than just for  simplicial objects in an abelian category.

Can specific references to Swan be given, anyone?

[[Zoran Škoda]]: I agree that one should talk about copowers etc.

Anonymous Coward: With exercise 8.3.5 of Weibel in mind, what is the notion of "cylinder" meant in the assertion "the combinatorial definition of homotopy agrees with the one via the cylinder both for simplicial sets and for simplicial objects in any finitely cocomplete category" for a general finitely cocomplete category?

[[Tim Porter]]:  I have transferred this question to the [nForum](https://nforum.ncatlab.org/discussion/5627/simplicial-homotopy/#Comment_69108) where it will be easier for others to reply.  In the meantime some indication is given in Kamps and Porter as referenced below. I myself do not quite understand your question as it is presently stated, but this may be that I am too near to the subject matter to see the difficulty.
=--

##Properties##

+-- {: .un_lemma}
###### Lemma

Precisely when $Y$ is a [[Kan complex]], the relation
$$
  (f \sim g) \Leftrightarrow
  (\exists simplicial homotopy f \Rightarrow g : X \to Y )
$$
is an [[equivalence relation]].

=--

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

Since [[Kan complex]]es are precisely the fibrant objects with respect to the standard [[model structure on simplicial sets]] this follows from general statements about [[homotopy]] in model categories.

The following is a direct proof.

We first show that the homotopy between points $x,y : \Delta[0] \to Y$ is an equivalence relation when $Y$ is a [[Kan complex]].

We identify in the following $x$ and $y$ with vertices in the image of these maps.

* -**reflexivity**- For every vertex $x \in Y_0$, the degenerate 1-simplex $s_0 x \in S_1$ has, by the [[simplicial identities]], 0-faces $d_0 s_0 x = x$ and $d_1 s_0 x = x$.
  $$
    (d_1 s_0 x) \stackrel{s_0 x}{\to} (d_0 s_0 x)
  $$
  Therefore the morphism $\eta : \Delta[0] \times \Delta[1] \to Y$ that takes the unique non-degenerate 1-simplex in $\Delta[1]$ to $s_0 x$ constitutes a homotopy from $x$ to itself.

* -**transitivity**- let $v_2 : x \to y$ and $v_0 : y \to z$ in $Y_1$ be 1-cells. Together they determine a map from the [[horn]] $\Lambda^2_1$ to $Y$,
  $$
    (v_2, v_2) : \Lambda^2_1 \to Y
    \,.
  $$
  By the [[Kan complex]] property there is an extension $\theta$ of this morphism through the 2-[[simplex]] $Delta^2$
  $$
    \array{
      \Lambda^2_1 &\stackrel{(v_0,v_2)}{\to}& Y
      \\
      \downarrow & \nearrow_{\theta}
      \\
      \Delta[2]
    }
    \,.
  $$
  If we again identify $\theta$ with its image (the image of its unique non-degenerate 2-cell) in $Y_2$, then
  using the [[simplicial identities]] we find
  $$
    \array{
       && (d_0 d_2 \theta) = (d_1 d_0 \theta)
       \\
       & {}^{d_2 \theta }\nearrow &
       \Downarrow \theta & \searrow^{d_0 \theta}
       \\
       (d_1 d_2 \theta) = (d_1 d_1 \theta) && \stackrel{d_1 \theta}{\to}  &&
       (d_0 d_1 \theta)
       =
       (d_0 d_1 \theta)
    }
  $$
 that the 1-cell boundary bit $d_1 \theta$ in turn has 0-cell boundaries
  $$
    d_0 d_1 \theta = d_0 d_0 \theta = z
  $$
  and
  $$
    d_1 d_1 \theta = d_1 d_2 \theta = x
    \,.
  $$
  This means that $d_1 \theta$ is a homotopy $x \to z$.

* -**symmetry**- In a similar manner, suppose that $v_2 : x \to y$ is a 1-cell in $Y_1$ that constitutes a homotopy from $x$ to $y$. Let $v_1 := s_0 x$ be the degenerate 1-cell on $x$. Since $d_1 v_1 = d_1 v_2$ together they define a map
$\Lambda^2_0 \stackrel{v_1, v_2}{\to} Y$ which by the Kan property of $Y$ we may extend to a map $\theta'$
  $$
   \array{
    \Lambda^2_0 &\stackrel{v_1, v_2}{\to}& Y
    \\
    \downarrow & \nearrow_{\theta'}
    \\
    \Delta[2]
   }
  $$
  on the full 2-simplex.

  Now the 1-cell boundary $d_0 \theta'$ has, using the [[simplicial identities]], 0-cell boundaries
  $$
    d_0 d_0 \theta' = d_0 d_1 \theta' = x
  $$
  and
  $$
    d_1 d_0 \theta' = d_0 d_2 \theta' = y
  $$
  and hence yields a homotopy $y \to x$. So being
  homotopic is a symmetric relation on vertices in a
  Kan complex.

Finally we use the fact that [[SSet]] is a [[cartesian closed category]] to deduce from this statements about vertices the corresponding statement for all map:

  a morphism $f : X \to Y$ is the Hom-[[adjunct]] of a morphism $\bar f : \Delta[0] \to [X,Y]$, and a homotopy $\eta : X \times \Delta[1] \to Y$ is the [[adjunct]] of a morphism $\bar \eta : \Delta[1] \to [X,Y]$. Therefore homotopies $\eta : f \Rightarrow g$ are in bijection with homotopies $\bar \eta : \bar f \to \bar g$.

=--

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

Let $A$ be an abelian category and $f,g : X\to Y$ homotopic morphisms of simplicial objects in $A$. Then the induced maps $f_*, g_* : N(X)\to N(Y)$ of their normalized chain complexes are chain homotopic.

=--
+-- {: .un_proposition}
###### Proposition

Let $A$ be an abelian category. The morphisms of simplicial objects (variant: of unbounded chain complexes) in $A$, which are homotopic to zero, form an ideal. More precisely

* being homotopic is an equivalence relation on the class of morphism,

* $f_1\sim 0$ and $f_2\sim 0$ implies $f_1+f_2 \sim 0$,

* if $f\circ h$ (resp. $h\circ f$) exists and if $f\sim 0$ then $f\circ h\sim 0$ (resp. $h\circ f\sim 0$).
=--

##References##

* Goerss, Jardine, _Simplicial homotopy theory_ ([pdf](http://dodo.pdmi.ras.ru/~topology/books/goerss-jardine.pdf))

Cylinder based homotopy is also discussed extensively in

* K. H. Kamps and T. Porter, _Abstract Homotopy and Simple Homotopy Theory_, World
Scientific Publishing Co. Inc., River Edge, NJ.


simplicial homotopy category

## Idea

A refinement of the notion of [[homotopy category]] of a [[model category]] (more generally, [[relative category]]) that keeps track of homotopies between morphisms.

Dwyer–Kan implement the notion of a simplicial homotopy category as a simplicial category that is Dwyer–Kan equivalent to the [[hammock localization]].

## Related concepts

* [[homotopy category]]

* [[category of fractions]]

* [[hammock localization]]

## References

See §5.1 in

* {#DwyerKan80FunctionComplexes} [[William G. Dwyer]], [[Daniel M. Kan]], _Function complexes in homotopical algebra_, Topology 19 (1980), 427-440 [doi:10.1016/0040-9383(80)90025-7, [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dwyer-kan-3.pdf)]


simplicial homotopy equivalence

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#Contents#
* table of contents
{:toc}

## Idea

A _simplicial homotopy equivalence_ is a [[homotopy equivalence]] between [[simplicial sets]].

## Definition

A [[morphism]] $f\colon A\to B$ of [[simplicial sets]]
is a __simplicial homotopy equivalence__ if there is a morphism $g\colon B\to A$ and [[homotopies]] $p\colon \Delta^1\times A\to A$ from $id_A$ to $g \circ f$ and $q\colon \Delta^1\times B\to B$ from $f \circ g$ to $id_B$ (where $id_X$ denotes the [[identity morphism]] on the simplicial set $X$).

## Properties

* If $A$ and $B$ are [[Kan complexes]],
then $f$ is a simplicial homotopy equivalence
if and only if $f$ has the [[right homotopy lifting property]] with respect to the [[boundary]] inclusion $\partial\Delta^n\to\Delta^n$ of [[simplices]].

* All simplicial homotopy equivalences are [[simplicial weak equivalences]]. The converse is true if $A$ and $B$ are [[Kan complexes]].

## Related notions

* [[simplicial weak equivalence]]

* [[simplicial Whitehead theorem]]

* [[classical model structure on simplicial sets]]


simplicial homotopy group

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#Contents#
* table of contents
{:toc}

## Idea

Simplicial homotopy groups are the basic invariants of [[simplicial sets]]/[[Kan complexes]] in [[simplicial homotopy theory]].

Given that a [[Kan complex]] is a special [[simplicial set]] that [[homotopy hypothesis|behaves like]] a combinatorial model for a [[topological space]], the _simplicial homotopy groups_ of a  Kan complex are accordingly the combinatorial analog of the [[homotopy groups]] of [[topological spaces]]: instead of being maps from topological [[spheres]] modulo maps from topological disks, they are maps from the [[boundary of a simplex]] modulo those from the [[simplex]] itself.

Accordingly, the definition of the discussion of simplicial homotopy groups is essentially literally the same as that of [[homotopy groups]] of topological spaces.  One technical difference is for instance that the definition of the group structure is slightly more non-immediate for simplicial homotopy groups than for topological homotopy groups (see below).

## Definition

Recall the classical [[model structure on simplicial sets]]. Let $X$ be a fibrant simplicial set, i.e. a [[Kan complex]].

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

For $X$ a [[Kan complex]], then its **0th homotopy group** (or **set of [[connected components]]**) is the set of [[equivalence classes]] of vertices modulo the [[equivalence relation]] $X_1 \stackrel{(d_1,d_0)}{\longrightarrow} X_0 \times X_0$

$$
  \pi_0(X) \colon X_0/X_1
  \,.
$$

For $x \in X_0$ a vertex and for $n \in \mathbb{N}$, $n \geq 1$, then the underlying [[set]]  of the **$n$th homotopy group** of $X$ at $x$ -- denoted $\pi_n(X,x)$ -- is, the set of [[equivalence classes]] $[\alpha]$ of morphisms

$$
  \alpha \colon \Delta^n \to X
$$

from the simplicial $n$-[[simplex]] $\Delta^n$ to $X$, such that these take the [[boundary of a simplex|boundary of the simplex]] to $x$, i.e. such that they fit into a [[commuting diagram]] in [[sSet]] of the form

$$
  \array{
     \partial \Delta[n] & \longrightarrow &  \Delta[0]
     \\
     \downarrow && \downarrow^{\mathrlap{x}}
     \\
     \Delta[n] &\stackrel{\alpha}{\longrightarrow}& X
   }
  \,,
$$

where two such maps $\alpha, \alpha'$ are taken to be equivalent is they are related by a [[simplicial homotopy]] $\eta$

$$
  \array{
     \Delta[n]
     \\
     \downarrow^{i_0} & \searrow^{\alpha}
     \\
     \Delta[n] \times \Delta[1]
     &\stackrel{\eta}{\longrightarrow}&
     X
     \\
     \uparrow^{i_1} & \nearrow_{\alpha'}
     \\
     \Delta[n]
  }
$$

that fixes the boundary in that it fits into a [[commuting diagram]] in [[sSet]] of the form

$$
  \array{
     \partial \Delta[n] \times \Delta[1]
     & \longrightarrow &
     \Delta[0]
     \\
     \downarrow && \downarrow^{\mathrlap{x}}
     \\
     \Delta[n]  \times \Delta[1]
     &\stackrel{\eta}{\longrightarrow}&
     X
  }
  \,.
$$

=--

These sets are taken to be equipped with the following group structure.

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

For $X$ a [[Kan complex]], for $x\in X_0$, for $n \geq 1$ and for  $f,g \colon \Delta[n] \to X$ two representatives of $\pi_n(X,x)$ as in def. \ref{UnderlyingSetsOfSimplicialHomotopyGroups}, consider the following $n$-simplices in $X_n$:

$$
  v_i
  \coloneqq
  \left\{
    \array{
      s_0 \circ s_0 \circ \cdots \circ s_0 (x) & for \; 0 \leq i \leq  n-2
      \\
      f & for \; i = n-1
      \\
      g & for \; i = n+1
    }
  \right.
$$

This corresponds to a morphism $\Lambda^{n+1}[n] \to X$ from a [[horn]] of the $(n+1)$-[[simplex]] into $X$. By the [[Kan complex]] property of $X$ this morphism has an [[extension]] $\theta$ through the $(n+1)$-[[simplex]] $\Delta[n]$

$$
  \array{
     \Lambda^{n+1}[n] & \longrightarrow & X
     \\
     \downarrow & \nearrow_{\mathrlap{\theta}}
     \\
     \Delta[n+1]
  }
$$

From the [[simplicial identities]] one finds that the boundary of the $n$-simplex arising as the $n$th boundary piece $d_n \theta$ of $\theta$ is constant on $x$

$$
  d_i d_{n} \theta = d_{n-1} d_i \theta = x
$$

So $d_n \theta$ represents an element in $\pi_n(X,x)$ and we define a product operation on $\pi_n(X,x)$ by

$$
  [f]\cdot [g] \coloneqq [d_n \theta]
  \,.
$$

=--

(e.g. [Goerss-Jardine 96, p. 26](#GoerssJardine96))

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

All the degenerate $n$-simplices $v_{0 \leq i \leq n-2}$ in def. \ref{ProductOnSimplicialHomotopyGroups} are just there so that the gluing of the two $n$-cells $f$ and $g$ to each other can be regarded as forming the boundary of an $(n+1)$-simplex except for one face. By the Kan extension property that missing face exists, namely $d_n \theta$.  This is a choice of gluing composite of $f$ with $g$.

=--

+-- {: .num_lemma}
###### Lemma

The product on homotopy group elements in def. \ref{ProductOnSimplicialHomotopyGroups} is
well defined, in that it is independent of the
choice of representatives $f$, $g$ and of the extension $\theta$.

=--

e.g. ([Goerss-Jardine 96, lemma 7.1](#GoerssJardine96))

+-- {: .num_lemma}
###### Lemma

The product operation in def. \ref{ProductOnSimplicialHomotopyGroups} yields a [[group]] structure on $\pi_n(X,x)$, which is [[abelian group|abelian]] for $n \geq 2$.

=--

e.g. ([Goerss-Jardine 96, theorem 7.2](#GoerssJardine96))

Finally:

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

The simplicial homotopy groups of any [[simplicial set]], not necessarily [[Kan complex|Kan]], are those of any of its [[Kan fibrant replacements]] according to def. \ref{UnderlyingSetsOfSimplicialHomotopyGroups}.

=--

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

The first homotopy group, $\pi_1(X,x)$, is also called the _[[fundamental group]]_ of $X$.

=--

## Properties

### Relation to topological homotopy groups

The simplicial homotopy groups of a Kan complex coincide with the homotopy groups of its [[geometric realization]], see e.g. ([Goerss-Jardine 96, page 60](#GoerssJardine96)).

### Relation to homotopy equivalence

A morphism of simplicial sets which induces an [[isomorphism]] on all simplicial homotopy groups is called a [[weak homotopy equivalence]]. If it goes between [[Kan complexes]] then it is actually a [[homotopy equivalence]].

### Relation to chain homology groups of associated Moore complexes

Another way to get the group structure on the homotopy groups of a Kan complex, $X$, is via its [[Dwyer-Kan loop groupoid]] and the [[Moore complex]]. This gives a [[simplicial groupoid|simplicially enriched groupoid]] $G(X)$, or if we restricted to the pointed case, and just look at the loops at the base vertex, a [[simplicial group]]. (We will assume for the sake of simplicity that $X$ is _reduced_, that is to say, $X_0$ is a singleton, and thus that $G(X)$ is a simplicial group.)

The construction of $G(X)$ is then given by  the free group functor on the various levels, shifted by 1, and with a twist in the zeroth face map (see [[Dwyer-Kan loop groupoid]] and simplify to the reduced case.)

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

There is an isomorphism between $\pi_n(X)$ as defined above and $H_{n-1}(N G(X))$, the $(n-1)$th homology group of the [[Moore complex]] of the simplicial group, $G(X)$.

=--

### Long exact sequences of a Kan fibration

For $f \colon X \longrightarrow Y$ a [[Kan fibration]], for $x\in X_0$ any vertex, for $y \coloneqq f(x) \in Y$ its image and $F_x \coloneqq f^{-1}(y)$ the [[fiber]] at that point, then the induced [[homomorphism]] of simplicial homotopy groups form a [[long exact sequence of homotopy groups]]

$$
  \cdots
  \to
  \pi_{n+1}(Y,y)
  \stackrel{}{\longrightarrow}
  \pi_n(F,x)
  \stackrel{}{\longrightarrow}
  \pi_n(X,x)
  \stackrel{}{\longrightarrow}
  \pi_n(Y,y)
  \stackrel{}{\longrightarrow}
  \pi_{n-1}(F,x)
  \to
  \cdots
$$

$$
  \cdots
  \to
  \pi_1(F,x) \longrightarrow
  \pi_0(F) \stackrel{}{\longrightarrow} \pi_0(X) \longrightarrow \pi_0(Y)
$$

i.e. a [[long exact sequence]] of [[groups]] ending in a long exact sequence of [[pointed sets]].

(e.g. [Goerss-Jardine 96, lemma 7.3](#GoerssJardine96))

## Examples

+-- {: .num_lemma }
###### Lemma

Let $C$ be a [[groupoid]] and $\mathcal{N}(C)$ its [[nerve]].

Then

* $\pi_0 \mathcal{N}(C,c)$ is the set of isomorphism classes of $C$ with the class of $c$ as base point

* $\pi_1 \mathcal{N}(C,c)$ is the [[automorphism group]] $Aut_C(c)$ of $c$

* $\pi_{n \geq 2} \mathcal{N}(C,c)$ is trivial

=--

In particular a [[functor]]
$f : C \to D$ of [[groupoids]] is a [[equivalence of categories]] if under the nerve it induces a weak equivalence $\mathcal{N}(f) : \mathcal{N}(C) \to \mathcal{N}(D)$ of [[Kan complexes]]:

* that $\pi_0 \mathcal{N}(f,c) : \pi_0(C,c) \to \pi_0(D,f(c))$ is an isomorphism implies that $f$ is an [[essentially surjective functor]] and is implied by $f$\'s being a [[full functor]];
* that $\pi_1 \mathcal{N}(f,c) : \pi_1(C,c) \to \pi_1(D,f(c))$ is an isomorphism is equivalent to $f$\'s being a [[full and faithful functor]].

## References

Textbook accounts include

* {#GoerssJardine96} [[Paul Goerss]], [[Rick Jardine]], chapter I, section 7 of _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1996)

Originally homotopy groups of simplicial sets had been defined in terms of the ordinary [[homotopy groups]] of the [[topological spaces]] [[geometric realization|realizing]] them. Apparently the first or one of the first discussions of the purely combinatorial definition is

* {#Kan58} [[Dan Kan]], _A combinatorial definition of homotopy groups_, Annals of Mathematics Second Series, Vol. 67, No. 2 (Mar., 1958), pp. 282-312 ([jstor](http://www.jstor.org/stable/1970006?seq=8))

[[!redirects simplicial homotopy groups]]


Simplicial homotopy theory

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=--
=--
=--

This page collects material related to the textbook

* [[Paul Goerss]], [[J. F. Jardine]]:

  **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/) with ps and dvi version - note that section and theorem etc numbers differ from the published version!)

on [[simplicial homotopy theory]] via the [[classical model structure on simplicial sets]].

There is also an article by [[E. B. Curtis]] with this name, from 1971.

We list the notions explained in the book for searchability.

Alexander-Whitney map, almost free morphism, anodyne extension, barycentric subdivision, bisimplex, bisimplicial abelian group, bisimplicial set, bonding map, Bousfield factorization, Bousfield-Kan R-completion, Bousfield-Kan R-resolution, G-bundle, category translation, resolution of a category, classifying object, classifying space, closed n-loop, model structure (Reedy, Moerdijk, f-local, Q-structure, cofibrantly generated, stable, strict, towers), cocycle, codegeneracy, coface, coglueing lemma, cohomotopy groups, pointed cone, continuous functor, coskeleton, covering system, equivariant Dold-Kan correspondence, Dwyer-Kan theorem, $Ex^{\infty}$ functor, Ext-p complete groups, Eilenberg subcomplex, Eilenberg-Zilber theorem, endpoint preserving, equivariant cohomology, expontial law, extension condition, external product, extra degeneracy, f-injective map, f-local space, f-localization, various kinds of fibration (Reedy, Serre, Q-fibration, acyclic, diagonal, global, minimal, pointwise, principal, stable, strict), Freudenthal suspension theorem, fringed spectral sequence, function complex, glueing lemma, Grothendieck construction, group completion, free groupoid, fundamental groupoid, homotopically full and faithful, homotopy cartesian, homotopy coherent diagram, homotopy colimit, homotopy fibre sequence, homotopy inverse limit, homotopy left Kan extension, homotopy pullback, homotopy spectral sequence (for a cosimplicial space, for a homotopy inverse limit, of a tower), horisontal normalization, horn, Hurewicz map, Hurewicz theorem, interchange law, join functor, k-invariant, Kan complex, Kan extension, Kan suspension, latching object, lax functor, left homotopy, local coefficient system, loop group, loop groupoid, loop spectrum, fake loop spectrum, loop space, matching object, matching space, Milnor FK-construction, Milnor exact sequence, Moore complex, Moore bicomplex, Moore-Postnikov section, Moore-Postnikov tower, nilpotent space, p-completion of a space, p-completion of an abelian group, Postnikov section, Postnikov tower, projective generator, pseudo cross-section, quasi-isomorphism, realization, n-reduced, relative homotopy group, retract, right lifting property, saturated, segment, Serre spectral sequence, shuffle map, simplical [abelian gruop, algebra, category, circle, coalgebra, functor, graded algebra, graded module, group, group action, groupoid, homotopy, map, model category, module, presheaf], simplicial set (n-fold, n-truncated, diagonal, reduced), singular set, skeleton, small object argument, spectrum, stable category, suspension ,Theorem B, total derived functor theorem, transgression, translation object, tree, trisimplicial set, universal cover, unstable algebra, van Kampen theorem, vertical path component, vertical simplicial set, weak equivalence, f-local equivalence, Q-weak equivalence, pointwise WE, Reedy WE, stable equivalence, strict WE of spectra, weak r-equivalence.

---

category: reference

[[!redirects Simplicial Homotopy Theory]]


simplicial homotopy theory

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=--
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#Contents#
* table of contents
{:toc}

## Idea

_Simplicial homotopy theory_ is the study of [[homotopy theory]] by means of [[simplicial sets]], but also the study of those properties of simplicial sets detectable by means of techniques adapted from  [[topological homotopy theory]].

A key tool is the _[[classical model structure on simplicial sets]]_.

## Related entries

* [[simplex category]], [[simplicial identities]],

* [[simplicial object]],

  * [[simplicial set]], [[simplicial group]], [[simplicial topological space]],

* [[nerve]]

* [[singular simplicial complex]]

* [[simplicial classifying space]], [[simplicial loop space]]

* [[Kan fibration]], [[Kan complex]],

  * [[anodyne morphism]], [[left fibration]], [[minimal fibration]]

* [[simplicial homotopy]]

* [[simplicial homotopy group]]

* [[geometric realization]]

  * [[geometric realization of categories]]

  * [[geometric realization of simplicial topological spaces]]

* [[weak homotopy equivalence]]

* [[model structure on simplicial sets]]

* [[homotopy hypothesis]]-theorem

## References

(See also the references at *[[homotopy theory]]*, *[[algebraic topology]]* and *[[(infinity,1)-category theory|$(\infty,1)$-category theory]]*.)

* [[Pierre Gabriel]], [[Michel Zisman]], _[[Calculus of fractions and homotopy theory]]_, _Ergebnisse der Mathematik und ihrer Grenzgebiete_, Band 35. Springer, New York (1967)

* [[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))

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.4 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/)]

* {#JoyalTierney05} [[André Joyal]], [[Myles Tierney]] _Notes on simplicial homotopy theory_, Lecture at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM 2008  ([[JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf:file]])

* {#JoyalTierney05} [[André Joyal]], [[Myles Tierney]], _An introduction to simplicial homotopy theory_, 2009  ([web](http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01), [[JoyalTierneySimplicialHomotopyTheory.pdf:file]])

* [[Paul Goerss]], [[Kirsten Schemmerhorn]], _Model categories and simplicial methods_, Notes from lectures given at the University of Chicago, August 2004, in: _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics 436, AMS 2007([arXiv:math.AT/0609537](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

* [[Francis Sergeraert]], _Introduction to Combinatorial Homotopy Theory_, 2008 ([pdf](https://www-fourier.ujf-grenoble.fr/~%20sergerar/Papers/Trieste-Lecture-Notes.pdf), [[SergeraertCombinatorialHomotopyTheory.pdf:file]])

* [[Zhen Lin Low]], _[[Notes on homotopical algebra]]_

* [[Emily Riehl]], _[[Categorical Homotopy Theory]]_

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

* [[Jacob Lurie]], _[[Kerodon]]_