simplicial identities

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###Context###
#### Combinatorics
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[[!include combinatorics - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The *simplicial identities* encode the relationships between the face and degeneracy maps in a [[simplicial object]], in particular, in a [[simplicial set]].

## Definition

The **simplicial identities** are the duals to the _simplicial relations_  of coface and codegeneracy maps described at [[simplex category]]:

Let $S \in $ [[sSet]] with

* face maps  $\partial_i : S_n \to S_{n-1}$ obtained by omitting the $i$th vertex;

* degeneracy maps $s_i : S_n \to S_{n+1}$ obtained by repeating the $i$th vertex.

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

The **simplicial identities** satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):

\[
\array{
  \partial_i
    \circ
  \partial_j
  \;=\;
  \partial_{j-1}
    \circ
  \partial_i
  &
  \text{if}
  &
  i \lt j
  \\
  s_i
    \circ
  s_j
  \;=\;
  s_j
    \circ
  s_{i-1}
  &
  \text{if}
  &
  i \gt j
}
\]

\[
  \label{MixedSimplicialIdentities}
  {\phantom{AAAAA}}
  \partial_i
    \circ
  s_j
  \;=\;
  \left\{
    \array{
      s_{j-1}
        \circ
      \partial_i
      &
      \text{if}
      &
      i \lt j
      \\
      id
      &
      \text{if}
      &
      i \in \{j, j+1\}
      \\
      s_j
        \circ
      \partial_{i-1}
      &
      \text{if}
      &
      i \gt j+1
    }
  \right.
\]

=--

(e.g. [Goerss & Jardine 1999/2009, I.1. (1.3)](#GoerssJardine09))

\begin{remark}\label{DegeneracyMapsAreSplitMono}
  The middle case in (eq:MixedSimplicialIdentities) implies in particular that degeneracy maps are ([[split monomorphisms|split]]) [[monomorphisms]]: One may think of them as including degenerate simplices into all simplices of a given dimension.
\end{remark}

## Properties

### Relation to nilpotency of differentials
 {#RelationToNilpotencyOfDifferentials}

The simplicial identities of def. \ref{SimplicialIdentities} can be understood as a non-abelian or "unstable" generalization of the identity

$$
  \partial \circ \partial = 0
$$

satisfied by [[differentials]] in [[chain complexes]] (in [[homological algebra]]).

Write $\mathbb{Z}[S]$ be the [[simplicial abelian group]] obtained form $S$ by forming degreewise the [[free abelian group]] on the set of $n$-simplices, as discussed at _[[chains on a simplicial set]]_.

Then using these [[formal linear combinations]] we can sum up all the $(n+1)$ face maps $\partial_i : S_n \to S_{n-1}$ into a single map:

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

The **alternating face map differential** in degree $n$ of the simplicial set $S$
is the linear map

$$
  \partial : \mathbb{Z}[S_n] \to \mathbb{Z}[S_{n-1}]
$$

defined on [[basis]] elements $\sigma \in S_n$ to be the alternating sum of the simplicial face maps:

\[
  \label{AlternatingFaceMapDifferential}
  \partial \sigma \coloneqq \sum_{k = 0}^n (-1)^k \partial_k \sigma
  \,.
\]

=--

This is the [[differential]] of the _[[alternating face map complex]]_ of $S$:

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

The simplicial identity def. \ref{SimplicialIdentities} (1) implies that def. \ref{TheAlternatingFaceMapDifferential} indeed defines a [[differential]] in that $\partial \circ \partial = 0$.

=--

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

By linearity, it is sufficient to check this on a basis element $\sigma \in S_n$. There we compute as follows:

$$
  \begin{aligned}
    \partial \partial \sigma
    & =
    \partial \left(
      \sum_{j = 0}^n (-1)^j \partial_j \sigma
    \right)
    \\
    & =
    \sum_{j=0}^n \sum_{i = 0}^{n-1} (-1)^{i+j} \partial_i \partial_j \sigma
    \\
     & =
     \sum_{0 \leq i \lt j \leq n} (-1)^{i+j} \partial_i \partial_j \sigma
     +
     \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} \partial_i \partial_j \sigma
     \\
     & =
     \sum_{0 \leq i \lt j \leq n} (-1)^{i+j} \partial_{j-1} \partial_i \sigma
     +
     \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} \partial_i \partial_j \sigma
     \\
     & =
     -
     \sum_{0 \leq i \leq j \lt n} (-1)^{i+j} \partial_{j} \partial_i \sigma
     +
     \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} \partial_i \partial_j \sigma
     \\
     & = 0
  \end{aligned}
  \,.
$$

Here

1. the first equality is (eq:AlternatingFaceMapDifferential);

1. the second is (eq:AlternatingFaceMapDifferential) together with the linearity of $d$;

1. the third is obtained by decomposing the sum into two summands;

1. the fourth finally uses the simplicial identity def. \ref{SimplicialIdentities} (1) in the first summand;

1. the fifth relabels the summation index $j$ by $j +1$;

1. the last one observes that the resulting two summands are negatives of each other.

=--

## Related concepts

* [[simplicial map]]

## References

For original references see at *[[simplicial set]]*.

Review includes:

* {#May67} [[Peter May]], Def. 1.1 in: _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]])

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Eq. (1.3) in Section I.1 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/))

[[!redirects simplicial identity]]


simplicial Lawvere theory

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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The definition of (the [[syntactic category]] of) a _[[Lawvere theory]]_ as a [[category]] with certain properties has an immediate generalization to [[simplicial categories]].

## Details

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

Let $\Gamma = (Skel(FinSet^{\ast/}))^{\mathrm{op}}$ be [[Segal's category]], the [[opposite category]] of a [[skeleton]] of [[finite set|finite]] [[pointed sets]].

A _[[simplicial Lawvere theory]]_ is a a pointed [[simplicial category]] $T$ equipped with a [[functor]] $i \;\colon\;\Gamma \to T$ such that

1. $T$ has the same [[set]] of [[objects]] as $\Gamma$;

2. $i$ is the identity on objects

3. $i$ preserves [[finite products]]

Given a simplicial theory $T$, then a _simplicial $T$-algebra_ is a product preserving [[simplicial functor]] $X$ to the [[simplicial category]] of [[pointed simplicial sets]]. The simplicial set

$$
  X(1_+) \in sSet
$$

(the value on the pointed 2-element set) is called the _underlying simplicial set_ of the $T$-algebra.

A [[homomorphism]] of $T$-algebras is a simplicial [[natural transformation]] between such functors. Write

$$
  T Alg \in sSet Cat
$$

for the resulting [[simplicial category]].

A homomorphism is called a _weak equivalence_ or a _fibration_ if on underlying simplicial sets it is a weak equivalence or fibration, respectively, in the [[classical model structure on simplicial sets]].
Write

$$
  (T Alg)_{proj}
$$

for the category equipped with these [[classes]] of morphisms.

=--

([Schwede 01, def. 2.1 and def. 2.2 and beginning of section 3](#Schwede01))

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

For $T$ a [[simplicial Lawvere theory]] (def. \ref{SimplicialLawvereTheory}) the category $(T Alg)_{poj}$ from def. \ref{SimplicialLawvereTheory} is a [[simplicial model category]].

=--

This is due to ([Reedy 74, theorem I](#Reedy74)), reviewed in ([Schwede 01](#Schwede01)). For more see at _[[model structure on simplicial algebras]]_.

The analogous statement with the [[classical model structure on simplicial sets]] replaced by the [[classical model structure on topological spaces]] is due to ([Schwänzl-Vogt 91](#SchwänzlVogt91)9

## Related concepts

* [[(infinity,1)-algebraic theory]]

## References

* {#Reedy74} [[Christopher Reedy]], _Homology of algebraic theories_, Ph.D. Thesis, University of California, San Diego, 1974

* {#SchwänzlVogt91} [[Roland Schwänzl]], [[Rainer Vogt]], _The categories of $A_\infty$- and $E_\infty$-monoids and ring spaces as closed simplicial and topological model categories_, Archives of Mathematics 56 (1991) 405-411 ([doi:10.1007/BF01198229](https://doi.org/10.1007/BF01198229))

  > (the [[model structure on simplicial algebras]])

* {#Schwede01} [[Stefan Schwede]], _Stable homotopy of algebraic theories_, Topology 40 (2001) 1-41 ([pdf](http://www.math.uni-bonn.de/people/schwede/stable.pdf))

[[!redirects simplicial Lawvere theories]]


simplicial Lie algebra

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###Context###
#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

A _simplicial Lie algebra_ is a [[simplicial object]] in the [[category]] of [[Lie algebra]]s.

## Definition

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

Let $k$ be a [[field]]. Write $LieAlg_k$ for the [[category]] of [[Lie algebras]] over $k$. Then the category of _simplicial Lie algebras_ is the category $(LieAlg_k)^{\Delta^{op}}$ of [[simplicial objects]] in Lie algebras.

=--

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

Let $(\mathfrak{g}, [-,-]$ be a simplicial Lie algebra according to def. \ref{CategoryOfSimplicialLieAlgebras}. Then the [[normalized chains complex]] $N \mmathfrak{g}$ of the underlying [[simplicial abelian group]] becomes a [[dg-Lie algebra]] by equipping it with the [[Lie bracket]] given by the following [[composition|composite]] morphisms

$$
  [-,-]_{N \mathfrak{g}}
  \;\;
  (N \mathfrak{g}) \otimes_k (N \mathfrak{g})
    \overset{\nabla}{\longrightarrow}
  N (\mathfrak{g} \otimes_k \mathfrak{g})
    \overset{N([-,-])}{\longrightarrow}
  N (\mathfrak{g})
$$

where the first morphism is the [[Eilenberg-Zilber map]].

This construction extends to a [[functor]]

$$
  N
    \;\colon\;
  LieAlg_k^{\Delta^{op}}
    \longrightarrow
  dgLieAlg_k
$$

from simplical Lie algebras to [[dg-Lie algebras]].

=--

([Quillen 69, (4.3)](#Quillen69))

## Properties
 {#Properties}

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

The functor $N$ from simplicial Lie algebras to dg-Lie algebras from def. \ref{dgLieAlgebraOfASimplicialLieAlgebra} has a [[left adjoint]]

$$
  (N^* \dashv N)
    \;\colon\;
  LieAlg_k^{\Delta^{op}}
    \underoverset
      {\underset{N}{\longrightarrow}}
      {\overset{N^*}{\longleftarrow}}
      {\bot}
  dgLieAlg_k
  \,.
$$

=--

This is ([Quillen 69, prop. 4.4](#Quillen69)).

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

There is a standard structure of a [[category with weak equivalences]] on both these categories, hence there are corresponding [[homotopy categories]]. (See also at _[[model structure on simplicial Lie algebras]]_ and _[[model structure on dg-Lie algebras]]_.) The following asserts that the above adjunction is compatible with this structure.

=--

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

For $k$ a [[field]] of [[characteristic zero]] the corresponding [[derived functors]] constitute an [[equivalence of categories]] between the corresponding [[homotopy categories]]

$$
  (L N^* \dashv \tilde N)
   \;\colon\;
  Ho(LieAlg^\Delta)_1
   \stackrel{\overset{L N^*}{\leftarrow}}{\underset{\tilde N}{\to}} Ho(dgLieAlg)_1
$$

of [[1-connected]] objects on both sides.

=--

This is in the proof of ([Quillen, theorem. 4.4](#Quillen)).

## Related concepts

* [[model structure on simplicial Lie algebras]]

* [[simplicial group]]

## References

An early account is in

* {#Quillen69} [[Dan Quillen]], part I, section 4 of _Rational homotopy theory_, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([JSTOR](http://www.jstor.org/stable/1970725))

See also

* Christian Rüschoff, section 8.3 of _relative algebraic $K$-theory and algebraic cyclic homology_ ([pdf](http://archiv.ub.uni-heidelberg.de/volltextserver/22305/1/Dissertation%20-%20Rueschoff.pdf))

* İ. Akça and Z. Arvasi, _Simplicial and crossed Lie algebras_ Homology Homotopy Appl. Volume 4, Number 1 (2002), 43-57.

On the [[homotopy theory]] of simplicial Lie algebras see also

* [[Stewart Priddy]], _On the homotopy theory of simplicial Lie algebras_, ([pdf](http://www.ams.org/journals/proc/1970-025-03/S0002-9939-1970-0270371-9/S0002-9939-1970-0270371-9.pdf))

* [[Graham Ellis]], _Homotopical aspects of Lie algebras_ Austral. Math. Soc. (Series A) 54 (1993), 393-419 ([web](http://anziamj.austms.org.au/JAMSA/V54/Part3/Ellis.html))

[[!redirects simplicial Lie algebras]]


simplicial local system

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### Context
#### Differential Graded Objects
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[[!include differential graded objects - contents]]
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=--

# Simplicial local system

## Definition

A __simplicial local system__
on a [[simplicial set]] $X$
with coefficients in a category $C$
(like [[abelian groups]], for example)
is a [[functor]]
$$\pi_{\le1}(X)\to C,$$
where $\pi_{\le1}(X)$
is (one of the models for) the [[simplicial fundamental groupoid]] of $X$.

## References

* NB. There is an entry at [[local system|local systems]] together with a blog link to David Speyer: _Three ways of looking at a local system_

  * [Introduction and connection to cohomology theories](http://sbseminar.wordpress.com/2009/04/20/three-ways-of-looking-at-a-local-system-introduction-and-connection-to-cohomology-theories/)

Here we will concentrate on the combinatorial and simplicial version of local systems.

##Local Systems in a simplicial context

By the category of **$n$-graded spaces**, we mean the category whose objects are the $n$-graded vector spaces

$$V = \sum_{p_1,\ldots,p_n\geq0}V^{p_1,\ldots,p_n}$$

and whose morphisms are the linear maps, homogeneous of multidegree zero.

The category of $n$-graded differential vector spaces has  for objects pairs $(V,d)$, where $V$ is an $n$-graded vector space, $d$ is a linear map of _total_ degree 1, and $d^2 = 0$.  The morphisms are the linear maps, homogeneous of multidegree zero, which commute with $d$.

We will denote by $\mathcal{C}$ one of the following categories:

* $n$-graded vector spaces.

* The category of $n$-graded algebras,

*  The subcategory of commutative $n$-graded algebras,

* $n$-graded differential vector spaces,

*  The subcategory of $n$-graded differential algebras,

*  The subcategory of  commutative $n$-graded differential algebras.

+-- {: .query}

[[Urs Schreiber|Urs]]: How does the $n$-grading affect the nature of the following definition? It seems that chain homotopies are not used in the following, just the 1-categorical structure?

=--

In the 'differential' examples, the differential will usually be denoted $d$.  Almost always we will be restricting ourselves to the case $n = 1$.  Extensions of any results or definitions to the general case are usually routine.

Let $K$ be a simplicial set.  A **local system** $F$ on $K$ with values in $\mathcal{C}$ is:

1. a family of objects $F_\sigma =\sum_{p\geq 0} F^p_\sigma$ in $\mathcal{C}$ indexed by the simplices $\sigma$ of $K$;

1. a family of morphisms (called the _face_ and _degeneracy_ operators)

$$d_i :F_\sigma \to F_{d_i\sigma} \quad and\quad s_i : F_\sigma \to F_{s_i\sigma}$$

satisfying the simplicial identities.

###Remarks###

* Here we will often just refer to 'local system' rather than the fuller 'simplicial local system', if no confusion will be likely to result.

*  There is an obvious way of assigning a small category to a simplicial set in which the simplices are the objects and the face and degeneracy maps generate the morphisms:

regarding the [[simplicial set]] as a [[functor]]

$$
  K : \Delta^{op} \to Set
$$

on the [[simplex category]], its category of cells is the [[comma category]]

$$
  (Y, const_K) =
  \left\{
    \array{
       Y(\Delta^n) &&\stackrel{}{\to}&& Y(\Delta^{n'})
       \\
       & {}_c\searrow && \swarrow_{c'}
       \\
       && K
    }
  \right\}
$$

where $Y : \Delta \to [\Delta^{op}, Set]$ is the [[Yoneda embedding]] for which $Y(\Delta^n)$ is the standard simplicial $n$-[[simplex]], so that $c : Y(\Delta^n) \to K$ is an $n$-simplex $c \in K_n$ of the simplicial set $n$.

A simplicial local system is then just a functor

$$
  F : (Y,const_K) \to \mathcal{C}
$$

from that category to $\mathcal{C}$.

+-- {: .query}

[[Urs Schreiber|Urs]]: Here it says "a local system". I suppose "simplicial local system" is meant? We should have a discussion about how this notion of simplicial local system relates to the functors from fundamental groupoids discussed at [[local system]].

[[Tim Porter|Tim]]: That has been amended! Halperin just calls them 'local systems', so in the notes that were the basis for this so did I. I copied and pasted from them, so this slip may occur elsewhere as well.

=--

###Back to discussion###

Let $\varphi : L \to K$ be a simplicial map and $F$ a local system over $K$.  The _pullback_ of $F$ to $L$ (or along $\varphi$) is the local system $\varphi^*F$ over $L$ given by

$$(\varphi^*F)_\sigma = F_{\varphi\sigma} ; \quad d_i = d_i ; \quad s_i = s_i.$$

If $\varphi$ is an inclusion of a simplicial subset then we may say that $\varphi^*F$ is the _restriction_ of $F$ to $L$.

Now let $F$ be a local system on $K$ with values in $\mathcal{C}$.  Define a graded space $F(K)$ as follows : an element $\Phi$ of  $F^p(K)$ is a function which assigns to each simplex $\sigma$ of $K$ an element $\Phi_\sigma \in F^p_\sigma$ such that for all $\sigma$

$$\Phi_{d_i\sigma} = d_i(\Phi_\sigma) \quad  and  \quad \Phi_{s_i\sigma}  = s_i(\Phi_\sigma).$$

+-- {: .query}

[[Urs Schreiber|Urs]]: Do I understand correctly that when the simplicial local system is expressed as a functor, then $F(K)$ is the space of natural transformations from the simplicial local system constant on the [[generator]] (if any) of $\mathcal{C}$ (for instance the tensor unit if $\mathcal{C}$ is graded vector spaces).

For ordinary [[local system]]s this gives the flat sections.

[[Tim Porter|Tim]]: I'm not sure.

=--

The linear structure is the obvious one, defined 'componentwise' and if $\mathcal{C}$ is one of the algebra (resp. differential) variants of the generic receiving category then the multiplication (resp. the differential) is defined componentwise as well. In this way $F(K)$ becomes an object of $\mathcal{C}$, called the **object of global sections** of $F$.

+-- {: .query}
[[Tim Porter|Tim]]: This construction also has (I think) a neat categorical description, that will be worth investigating.  It would seem to be the analogue of the Grothendieck construction / homotopy colimit (at least partially) in this context. (enlightenment sought!!!)
=--

If $\varphi : L \to K$ is a simplicial map, it determines a morphism $F(\varphi) : (\varphi^*F)(L)\to F(K)$ given by

$$(F(\varphi)\Phi)_\sigma = \Phi_{\varphi\sigma}.$$

If $\varphi$ is an inclusion of $L$ into $K$, then we denote $(\varphi^*F)(L)$ simply by $F(L)$ and call the morphism $F(K)\to F(L)$ _restriction_.

Now suppose $F$ is a local system over $K$.  Assume $M_n \subset K_n$ are subsets ($n \geq 0$) such that $d_i : M_n \to M_{n-1}$  This family $\{M_n\}$ generates a subsimplicial set $L\subset K$ and if $s_i\sigma \in M_{n+1}$ then $\sigma = d_i s_i\sigma \in M_n$.

+-- {: .query}

[[Urs Schreiber|Urs]]: So what are simplicial local systems used for? Is there a good motivating example? Relating it to the other definition of [[local system]], maybe?

[[Tim Porter|Tim]]: Aha! All will be revealed in the next entry 'Differential forms on a simplicial set' ... when I get to putting it in!  There is some more to go here as well, describing special properties, but it was getting late last night.

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

Suppose $\Phi_\sigma \in F^p_\sigma$ ( $\sigma \in M_n$), $n \geq 0$, satisfy $\Phi_{d_i\sigma} = d_i\Phi_\sigma$ and $\Phi_{s_i\sigma} = s_i\Phi_\sigma$ (this is with $s_i\sigma\in M_n$, and $n\geq 0$). Then there is a unique element $\Phi\in F^p(L)$ extending $\Phi_\sigma$.
=--

The proof is by induction and can be found in Halperin's notes if required.

For any simplicial set $K$, any $n$-simplex $\sigma \in K_n$ determines a unique simplicial map, which we will also write as $\sigma$ from $\Delta[n]$ to $K$ that sends the unique non-degenerate $n$-simplex of the standard $n$-simplex $\Delta[n]$ to the element $\sigma$. In particular, if $F$ is a local system over $K$, then we can form $\sigma^*F$ over $\Delta[n]$. We will say that $F$ is _extendable_ if for each $\sigma$ the restriction

$$\sigma^*(F)(\Delta[n]) \to \sigma^*(F)(\partial\Delta[n])$$

is surjective, where $\partial\Delta[n]$ is the boundary of the $n$-simplex.

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

Suppose $\varphi :L \to K$ is a simplicial map and $F$ is an extendable system over $K$, then $\varphi^*F$ is an extendable local system over $L$.

=--
The proof is easy.

+-- {: .un_prop }
######Proposition
Suppose that $L\subset K$ is a subsimplicial set and $F$ is an extendable local system over $K$.  Then the restriction morphism $F(K)\to F(L)$ is surjective.
=--
The proof is again by induction up the skeleta of $K$ and $L$, for details see Halperin, p.XII 10.

If $F$ is an extendable local system over $K$ and $L\subset K$, we denote the kernel of $F(K)\to F(L)$ by $F(K,L)$ and call it the *space of relative global sections*. (A description of $F(K,L)$ is given in detail in Halperin, p.XII-12.)

It may be useful to have some more of the terminology of local systems available.  A local system $F$ over $K$ is **constant**  if for some $F_0 \in \mathcal{C}$, each $F_\sigma = F_0$ and each $d_i$ and $s_j$ is the identity map on $F_0$. We say $F$ is **constant by dimension** if for some sequence $F_n\in \mathcal{C}$ ($n \geq 0$), $F_\sigma = F_n$, for $\sigma \in K_n$ and $d_i$, $s_j$ depend only on $\dim \sigma$.

 A local system $F$ over $K$ is a **local system of coefficients** if for each $\sigma$ and each $i$,

$$d_i : F_\sigma \to F_{d_i\sigma} \quad  and s_i : F_\sigma \to F_{s_i \sigma}$$

are isomorphisms.  Finally $F$ is a **local system of differential coefficients** if $\mathcal{C}$ is one of the categories with differentials above, and for each $\sigma$, and $i$

$$d_i^* : H(F_\sigma) \to H(F_{d_i\sigma}) \quad  and  s_i^* : H(F_\sigma) \to H(F_{s_i\sigma})$$

 are isomorphisms, in other words if the corresponding cohomology is a local system of coefficients.

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

Let $F$ and $G$ be extendable local systems of differential coefficients over $K$.  Assume we are given morphisms

$$\varphi_\sigma : F_\sigma \to G_\sigma, \quad \sigma \in K,$$

compatible with the face and degeneracy operators.  Then  a morphism $\varphi : F(K)\to G(K)$ is given by $(\varphi\Phi)\sigma = \varphi_\sigma (\Phi_\sigma)$, and

$$\varphi^* : H(F(K))\to H(G(K))$$

is an isomorphism.
=--

## See also

* [[local system]]

* [[twisted cohomology]]

## References

  * S. Halperin, _Lectures on minimal models_, Mémoires de la S. M. F. 2e série, tome 9-10 (1983), p. 1-261 ([eduml](http://eudml.org/doc/94833), [pdf](http://archive.numdam.org/article/MSMF_1983_2_9-10__1_0.pdf))


simplicial localization

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###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

> See also [[derived hom space]]

#Simplicial localization#
* table of contents
{:toc}

## Idea

A [[category with weak equivalences]] or  [[homotopical category]] is a [[category]] $C$ equipped with the information that some of its [[morphisms]], specifically, a subcategory $W \supset Core(C)$, are to be regarded as "weakly invertible".  One way to make this notion precise is through the concept of __simplicial localization__, one form of which is known as __Dwyer–Kan localization__.

The _simplicial localization_ $L C$ of a category $C$ is an [[(∞,1)-category]] realized concretely as a [[simplicially enriched category]] which is such that the original category injects into it,  $C \hookrightarrow L C$, such that every morphism in $C$ that is labeled as a weak equivalence becomes an actual equivalence in the sense of morphisms in [[(∞,1)-categories]] in $L C$. And $L C$ is in some sense universal with this property.

Passing to the [[homotopy category of an (∞,1)-category]] of $L C$ then reproduces the [[homotopy category]] that can also directly be obtained from $C$:

$$
  Ho_C(a,b) \simeq \Pi_0 (L C(a,b))
$$

(where $\Pi_0$ gives the  [[simplicial homotopy group|0th simplicial homotopy groupoid]]).

If the homotopical structure on $C$ extends to that of a (combinatorial) [[model category]], then there is another procedure to obtain a simplicially enriched category from $C$, the [[presentable (infinity,1)-category|(∞,1)-category presented by a combinatorial model category]]. This $(\infty,1)$-category is equivalent to the one obtained by simplicial localization but typically more explicit and more tractable.

See also [[localization of a simplicial model category]].

## Construction

See [[simplicial localization of a homotopical category]].

## Definition

### "Standard" simplicial localization

Let $U : \mathbf{Cat} \to \mathbf{Grph}$ be the [[forgetful functor]] that sends a (small) category to its underlying _[[reflexive graph|reflexive]]_ [[graph]], and let $F : \mathbf{Grph} \to \mathbf{Cat}$ be its [[left adjoint]]. We then get a [[comonad]] $\mathbb{G} = (G, \epsilon, \delta)$ on $\mathbf{Cat}$, and as usual this defines a functor $G_\bullet : \mathbf{Cat} \to [\mathbf{\Delta}^{op}, \mathbf{Cat}]$ from [[Cat]] to [[simplicial objects in Cat]] equipped with a canonical augmentation, where $G_n C = G^{n+1} C$.

+-- {: .num_defn}
###### Definition
The **standard resolution** of a small category $C$ is defined to be the simplicial category $G_\bullet C$.
=--

Note that this is also a [[simplicial category]] in the strong sense, i.e. $ob G_\bullet C$ is discrete! Thus we may also regard $G_\bullet C$ as an [[sSet]]-[[simplicially enriched category|category]]. This is a [[resolution]] in the sense that the augmentation $\epsilon : G_\bullet C \to C$ is a [[Dwyer-Kan equivalence]]. (In fact, for objects $X$ and $Y$ in $C$, the morphism $G_\bullet C (X, Y) \to C (X, Y)$ admits an extra degeneracy and hence a contracting homotopy.)

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

The **standard simplicial localization** of a [[relative category]] $(C, W)$ is the simplicial category $L_\bullet (C, W)$ where $L_n (C, W) = G_n C [{G_n W}^{-1}]$.

=--

This appears as ([DwyerKanLocalizations, def. 4.1](#DwyerKanLocalizations)).
Again, $L_\bullet C$ is a simplicial category in the strong sense, because $G_\bullet C$ is.

### Hammock localization

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

Let $(C,W)$ be a [[category with weak equivalences]].
For $X,Y \in C$ any two objects, write

$$
  L^H C(X,Y) \in sSet
$$

for the simplicial set defined as follows.  For each natural number $n$ there is a [[category]] defined as follows:

* its [[objects]] are length-$n$ zig-zags of morphisms in $C$

  $$
    X \stackrel{\simeq}{\leftarrow}
    K_1 \to K_2 \stackrel{\simeq}{\leftarrow} K_3 \to
    \cdots \to Y
    \,,
  $$

  where the left-pointing morphisms are to be in $W$;

* its [[morphisms]] are "natural transformations" between such objects, fixing the endpoints:

  $$
    \array{
       && K_1 &\to& K_2 &\stackrel{\simeq}{\leftarrow}&
       \cdots
       \\
       & {}^{\mathllap{\simeq}}\swarrow
       &&&&& && \searrow^{}
       \\
      X
      && \downarrow^{\simeq}&& \downarrow^{\simeq}& \cdots&&  && Y
      \\
       & {}_{\mathllap{\simeq}}\nwarrow
       &&&&& && \nearrow^{}
      \\
       && L_1 &\to& L_2 &\stackrel{\simeq}{\leftarrow}&
       \cdots
    }
    \;
  $$

$L^H C(X,Y)$ is obtained by

* taking the [[coproduct]] of the [[nerves]] of these categories over all $n$, and

* quotienting by the [[equivalence relation]] generated by inserting or removing identity morphisms and composing composable morphisms.

For $X,Y,Z$ three objects, there is an evident compositing morphism

$$
  L^H C(X,Y) \times L^H C(Y,Z) \to L^H C(X,Z)
$$

given by horizontally concatenating hammock diagrams as above.

The [[simplicially enriched category]] $L^H C$ obtained this way is the **hammock localization** of $(C,W)$.

=--

This appears as ([DwyerKanCalculating, def. 2.1](#DwyerKanCalculating)).

## Properties

### Basic properties

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

For $(C,W)$ a [[category with weak equivalences]], write $L^H C \in sSet Cat$ for its hammock localization and $C[W^{-1}] \in Cat$ for its ordinary [[localization]].
Write $Ho(L^H C) \in Cat$ for the category with the same objects as $C$ and morphisms between $X$ and $Y$ being the [[connected components]] $\pi_0 L^H C(X,Y)$.

There is an [[equivalence of categories]]

$$
  Ho L^H C
   \simeq
  C[W^{-1}]
  \,.
$$

=--

This appears as ([DwyerKanCalculating, prop. 3.1](#DwyerKanCalculating)).

\begin{remark}
  Hammock localization is clearly a [[functor]] from the category of [[relative categories]] to [[sSet-enriched categories]]:

$$
  RelativeCat \overset{L^H_W}{\longrightarrow} SimplicialCategories
$$

\end{remark}

See also [Barwick & Kan 12, Sec. 1.5](#BarwickKan12), [Spitzweck 10, p. 3](#Spitzweck10).

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

Let $(C,W)$ be a [[category with weak equivalences]], and let

$$
  (X \overset{f}{\to} Y) \in W \subset Mor(C)
$$

be a [[weak equivalence]]. Then for all objects $U \in C$ we have that the  concatenation operation on hammocks induce [[weak homotopy equivalences]]

$$
  f_*
  \;\colon\;
  L^H C(U,X)
    \stackrel{\simeq}{\to}
  L^H C(U,Y)
$$

and

$$
  f^*
    \;\colon\;
  L^H C(Y,U)
    \stackrel{\simeq}{\to}
  L^H C(X, U)
  \,.
$$

=--

This appears as ([DwyerKanCalculating, prop. 3.3](#DwyerKanCalculating)).

### Simplical localization gives all $(\infty,1)$-categories

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

Up to Dwyer-Kan equivalence --the [[weak equivalences]] in the [[model structure on sSet-categories]] -- every [[simplicial category]] is the simplicial localization of a [[category with weak equivalences]].

This is ([DwyerKan 87, 2.5](#DwyerKanEquivalences)).

=--

If the [[category with weak equivalences]] in question happens to carry even the structure of a [[model category]] there exist more refined tools for computing the [[SSet]]-[[hom object]] of the simplicial localization. These are described at [[(∞,1)-categorical hom-space]].

### Equivalences between simplicial localizations

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

Let $(C,W)$ and $(C', W')$ be [[categories with weak equivalences]]. Write $L^H C, L^H C' \in sSet Cat$ for the corresponding [[hammock localizations]].

Then for $F_1, F_2 :  C \to C'$ two [[homotopical functors]] (functors respecting the weak equivalences, i.e. $F_i(W) \subset W'$) with

$$
  \eta : F_1 \Rightarrow F_2
$$

a [[natural transformation]] with components in the $W'$,
we have that for all objects $X,Y \in C$, there is induced a [[natural transformation|natural]] [[homotopy]] between morphisms of [[simplicial sets]]

$$
  \array{
    && L^H C'(F_1(X), F_1(Y) )
    \\
    &
    {}^{\mathllap{L^H F_1}}\nearrow
    &&
    \searrow^{\mathrlap{\eta(Y)_*}}
    \\
    L^H C(X,Y)
     && \Downarrow &&
    L^H C'(F_1(X), F_2(Y))
    \\
    &
     {}_{\mathllap{L^H F_2}}\searrow
    &&
     \nearrow_{\mathrlap{\eta(X)^*}}
    \\
    && L^H C' (F_2(X), F_2(Y))
  }
  \,.
$$

=--

This is ([DwyerKanComputations, prop. 3.5](#DwyerKanComputations)).

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

Let $i : (C_1, W_1) \hookrightarrow (C_2, W_2)$
be a [[full subcategory]] such that

1. $i$ is homotopy-essentially surjective: for every object $c_2 \in C_2$ there is an object $c_1 \in C_1$ and a weak equivalence $c_2 \stackrel{\simeq}{\to} i(c_1)$;

1. there is a [[functor]] $Q : (C_2,W_2) \to (C_1, W_1)$ and a [[natural transformation]]

   $$
     i \circ Q  \Rightarrow Id_{C_2}
     \,.
   $$

Then we have an [[equivalence of (∞,1)-categories]]

$$
  L^H C_1 \simeq L^H C_2
  \,.
$$

=--

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

We have to check that $i$ is an [[essentially surjective (∞,1)-functor]] and a [[full and faithful (∞,1)-functor]].

The first condition is immediate from the first assumption. The second follows with prop. \ref{SimplicialLocalizationOfNaturalTransformation} (using prop. \ref{PrePostCompositionWithWeakEquivalences}) from the second assumption.

=--

### Simplicial localization of model categories

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

Let $C$ be a [[simplicial model category]]. Write $C^\circ$ for the full $sSet$-subcategory on the fibrant and cofibrant objects.

Then $C^\circ$ and $L^H C$ are connected by an [[equivalence of (∞,1)-categories]].

=--

This is one of the central statements in ([Dwyer & Kan 80 FuncComp](#DwyerKan80FunctionComplexes)). The weak homotopy equivalence between $C^\circ(X,Y)$ and $L^H C(X,Y)$ is in corollary 4.7. The equivalence of $\infty$-categories stated above follows with this and the diagram of morphisms of simplicial categories in prop. 4.8.

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

A [[Quillen equivalence]] $D \leftrightarrows C$ between [[model categories]] induces a Dwyer-Kan-equivalence $L C \leftrightarrow L D$ between their [[simplicial localizations]].

=--

This is made explicit in [Mazel-Gee 15, p. 17](#MazelGee15) to follow from [Dwyer & Kan 80 FuncComp, Prop. 4.4 with 5.4](#DwyerKan80FunctionComplexes).

## Related concepts

* [[localizer]]

* [[localizing subcategory]]

## References
 {#References}

The original articles:

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

* {#DwyerKanCalculating} [[William Dwyer]], [[Daniel Kan]], _Calculating simplicial localizations_, J. Pure Appl. Algebra 18 (1980), 17-35 [doi:10.1016/0022-4049(80)90113-9]

* {#DwyerKan80FunctionComplexes} [[William Dwyer]], [[Daniel 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)]

* {#DwyerKanEquivalences} [[William Dwyer]], [[Daniel Kan]], *Equivalences between homotopy theories of diagrams*, in: *Algebraic topology and algebraic K-theory*, Ann. of Math. Stud. **113**, Princeton University Press (1988) [[doi:10.1515/9781400882113-009](https://doi.org/10.1515/9781400882113-009)]

and in modernized form:

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]],  [[Jeff Smith]], around 35.6 of : *[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]*, Mathematical Surveys and Monographs **113**, AMS (2004) [[ISBN: 978-1-4704-1340-8](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf)]

Survey:

* [[Tim Porter]], Section 4.1 of: _$S$-Categories, $S$-groupoids, Segal categories and quasicategories_ [[arXiv:math/0401274](http://arxiv.org/abs/math/0401274)]

* [[Clark Barwick]], Section 2 of: *On (enriched) Bousfield localization of model categories* [[arXiv:0708.2067](http://arxiv.org/abs/0708.2067)]

Further development:

* {#BarwickKan12} [[Clark Barwick]], [[Daniel Kan]], *A characterization of simplicial localization functors and a discussion of DK equivalences*, Indagationes Mathematicae Volume 23, Issues 1–2, March 2012, Pages 69-79 ([doi:10.1016/j.indag.2011.10.001](https://doi.org/10.1016/j.indag.2011.10.001))

* {#Hinich13} [[Vladimir Hinich]], *Dwyer-Kan Localization Revisited*, Homology, Homotopy and Applications Volume 18 (2016) Number 1 ([arXiv:1311.4128](https://arxiv.org/abs/1311.4128), [doi:10.4310/HHA.2016.v18.n1.a3](https://dx.doi.org/10.4310/HHA.2016.v18.n1.a3))

See also:

* {#Lurie09} [[Jacob Lurie]], Section A.3.2 of: _[[Higher Topos Theory]]_

* {#Spitzweck10} [[Markus Spitzweck]], *Homotopy limits of model categories over inverse index categories*, Journal of Pure and Applied Algebra Volume 214, Issue 6, June 2010, Pages 769-777 ([doi:10.1016/j.jpaa.2009.08.001](https://doi.org/10.1016/j.jpaa.2009.08.001))

* {#MazelGee15} [[Aaron Mazel-Gee]], _Quillen adjunctions induce adjunctions of quasicategories_, New York Journal of Mathematics Volume 22 (2016) 57-93 ([arXiv:1501.03146](https://arxiv.org/abs/1501.03146), [publisher](http://nyjm.albany.edu/j/2016/22-4.html))

[[!redirects simplicial localizations]]

[[!redirects simplicial localisation]]
[[!redirects simplicial localisations]]

[[!redirects Dwyer-Kan localization]]
[[!redirects Dwyer-Kan localisation]]
[[!redirects Dwyer?Kan localization]]
[[!redirects Dwyer?Kan localisation]]
[[!redirects Dwyer--Kan localization]]
[[!redirects Dwyer--Kan localisation]]
[[!redirects Hammock-Localization]]
[[!redirects Hammock localization]]
[[!redirects Hammock localisation]]

[[!redirects hammock localization]]
[[!redirects hammock localizations]]

[[!redirects hammock localisation]]
[[!redirects hammock localisations]]

[[!redirects (infinity,1)-categorical localization]]
[[!redirects (infinity,1)-categorical localizations]]
[[!redirects (∞,1)-categorical localization]]
[[!redirects (∞,1)-categorical localizations]]


simplicial localization of a homotopical category

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

#Contents#
* table of contents
{:toc}

## Idea

We give two constructions of the [[simplicial localization]] of a [[homotopical category]].

## Preparations

We first need the following preparations:

Let $\mathbf{T}$ be the [[zigzag category]].  We define the _categorical realization_ of a zigzag $t=(n,t_+,t_-)$ to be the category $[t]$ generated freely by the following [[quiver]] data:

* $[t]$ has $n+1$ vertices $t_0,...,t_n$
* If $i\in t_+$, there is an edge $g_i:t_{i-1}\to t_i$
* If $i\in t_-$, there is an edge $g_i:t_i\to t_{i-1}$

Let $C$ be a [[homotopical category]] with class of weak equivalences $W$.  Consider the functor category $Hom_{Cat}([t],C)$.  We define the category of _restricted_ $t$-zigzags $C^{t}$ to be the subcategory whose objects are zigzags $F$ such that all maps $F(t_i)\to F(t_{i-1})\in W$, and where morphisms are given by object-wise weak equivalences (natural weak equivalences) such that the maps on the initial and terminal vertices are identities.  Finally, we define $C^t(X,Y)$ to be the full subcategory of $C^t$ spanned by those objects $F$ such that $F(t_0)=X$ and $F(t_n=Y)$.

Let $f:t\to t'$ be a morphism of $\mathbf{T}$.  Then we define the functor $f_*:C^t(X,Y)\to C^{t'}(X,Y)$ to be the functor sending a zigzag $F:[t]\to C$ with connecting maps $g_i$ lying between $t_{i-1}$ and $t_i$ (not necessarily $t_{i-1}\to t_i$) to the zigzag $f_*F:[t']\to C$ with connecting maps $h_j$ defined to be the ordered composite of the maps $g_i$ such that $i\in f^{-1}(j)$ (Note: There should be a better combinatorial way to precisely specify this, but I'm having trouble doing so).  Notice that if $f^{-1}(j)$ is empty, this procedure yields the identity map.  On morphisms, everything remains well-defined by the naturality condition (implying that the diagrams do in fact commute).   This proves that the assignment $C^{(t)}(X,Y):\mathbf{T} \to Cat$ is functorial in $t$.

Notice that this is not actually functorial in $X$ and $Y$, but for a map $Z\to X$, we can give a functor $C^t(X,Y)\to C^{1\vee t}(Z,Y)$.  Similarly, for a map $Y\to Z$, we can give a map $C^{t}(X,Y)\to C^{t \vee 1}(X,Z)$.  This is a special case of concatenation, which we will describe as follows:

We may concatenate two combinatorial zigzags $t=(n,t_+,t_-)$ and $t'=(m,t'_+,t'_-)$ in the obvious way by taking $t\vee t'$ to be the combinatorial zigzag $(n+m, t_+ \cup n(t'_+), t_- \cup n(t'_-))$ where $n(S):=\{n+i: i\in S\}$ is the map adding $n$ to each index.  This is clearly functorial in both coordinates and defines a monoidal product $\vee:\mathbf{T}\times \mathbf{T}\to \mathbf{T}$.

We will lift this to a functor $\vee^{t,t'}_{XYZ}:C^t(X,Y)\times C^{t'}(Y,Z)\to C^{t\vee t'}(X,Z)$.  We send a pair of zigzags to their concatenation in the obvious way.  On morphisms, we concatenate natural transformation diagrams, the result of which remains natural by the hammock-shape of our natural transformation diagrams (see below), so $\vee^{t,t'}_{XYZ}$ is a functor in both coordinates.

$$\begin{matrix}
&&A_1&\to&A_2&\leftarrow&A_3&\to&\ldots&\to &A_{n-3}&\leftarrow &A_{n-2}&\to& A_{n-1}&&\\
&\nearrow&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&\searrow&\\
X&&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&&Y\\
&\searrow&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&\nearrow&\\
&&B_1&\to&B_2&\leftarrow&B_3&\to&\ldots&\to &B_{n-3}&\leftarrow &B_{n-2}&\to& B_{n-1}&&\end{matrix}$$

(Note that this picture is technically hiding the identity morphisms $X\to X$ and $Y\to Y$.  We omit them to show the hammock-shape, as noted below).

We note that we must define everything pointwise for the moment, but it's reasonable expect all of this data to lift to an actual composition functor in the colimit over $\mathbf{T}$, perhaps after some effort.

With this data in hand, we may proceed to actual constructions of the simplicial categories that we want.

Before that, a short note on terminology:  Traditionally, only the second construction is called the hammock localization, but this name comes from the shape of a natural transformation between two $t$-zigzags with the same initial and terminal vertices (notice how much the figure above looks like a hammock!), but as we will see, this features in both constructions.  We could say that both constructions are "hammock localizations", and indeed, they produce weakly equivalent results.

##Simplicial localization via the Grothendieck Construction

By the [[Grothendieck construction]], we may present the functor $C^t(X,Y):\mathbf{T}\to Cat$ as a [[fibered category]] $GrC^\mathbf{T}(X,Y)$ over $\mathbf{T}$.

We describe the resulting category $GrC^\mathbf{T}(X,Y)$ explicitly as follows:

* The objects are pairs $(t,F)$ with $t\in \mathbf{T}$ and $F\in C^t(X,Y)$.
* The morphisms are pairs $(f,g):(t,F)\to (t',G)$ where $f:t\to t'$ and $g:f_*F\to G$.
* Composition is given by the rule $(f',g')\circ (f,g)=(f'\circ f,g'\circ f'_* g)$.

In the sequel, we will construct the strict 2-category $GrC$, called the Grothendieck enrichment with the same objects as $C$ and with morphism categories $GrC^\mathbf{T}(X,Y)$.

The first order of business is to show that the $\vee^{t,t'}_{XYZ}$ assemble first to a functor $\vee_{XYZ}:GrC^\mathbf{T}(X,Y)\times GrC^\mathbf{T}(Y,Z)\to GrC^\mathbf{T}(X,Z)$.  After that, we must show that all $\vee_{XYZ}$ assemble to an actual law of composition.

We define the functor $\vee_{XYZ}:GrC^\mathbf{T}(X,Y)\times GrC^\mathbf{T}(Y,Z)\to GrC^\mathbf{T}(X,Z)$ as follows: On objects (suppressing subscripts), we have $(t,F)\vee (t',F'):= (t\vee t', F\vee^{t,t'} F')$.  This is clearly functorial by way of the results from our earlier preparations.

We now define the strict $2$-category GrC by specifying the following data:

* The objects are simply the objects of $C$.
* The hom-categories are given by $GrC^\mathbf{T}(X,Y)$.
* The identity morphism $id_X\in GrC^\mathbf{T}(X,X)$ is the unique empty zigzag.
* We define the law of composition to be $\vee$.

It's clear from the definitions that $\vee$ is associative and that $id_X\vee F=F$ whenever the composition makes sense.  Then $GrC$ is a $2$-category.

We define the categorical nerve: $N:2\text{-}Cat\to Set_\Delta\text{-}Cat$ to be the canonical functor arising from the ordinary nerve.  That is, $N$ is the identity on objects, and sends $GrC^\mathbf{T}(X,Y)\to \nu(GrC^\mathbf{T}(X,Y))$ where $\nu$ denotes the ordinary nerve.  This is well defined since $\nu$ is a right adjoint and therefore commutes with products.

We call the simplicial category $N(GrC)$ the simplicial localization of $C$.

**Theorem**: The categorical nerve $N(GrC)$ sending objects to objects and hom-categories to simplicial sets $\nu(GrC^\mathbf{T}(X,Y))$ equips $N(GrC)$ with the structure of a simplicial category.  Further, the morphisms of the homotopy category of this simplicial category coincide with those of the homotopy category arising from the ordinary construction using equivalence relations.  That is, $Ho_C(X,Y)\cong \pi_0 (N(GrC(X,Y)))$ naturally in $X$ and $Y$, where $\pi_0(-):=Hom_{Ho(SSet)}(Ho(\Delta^0), Ho(-))$.

## Hammock Localization

We define $L_H C^\mathbf{T}(X,Y)$ to be $\varinjlim_\mathbf{T} \nu(C^\mathbf{T}(X,Y))$.  We define the _Hammock Localization_ of $C$ to be the simplicial category $L_H C$ defined by the following data:

* The objects of $L_H C$ are precisely the objects of $C$.
* The hom-spaces are precisely the $L_H C^\mathbf{T}(X,Y)$
* Composition is given by $\nu(\vee)$
* The identities $Id_X$ are those maps $\Delta^0\to C^\mathbf{T}(X,X)$ classifying the unique empty zigzag

It remains to show that the $\nu(\vee^{t,t'}_{XYZ})$ assemble to a morphism of simplicial sets $\vee_{XYZ}:L_H C^\mathbf{T}(X,Y)\times L_H C^\mathbf{T}(Y,Z)\to L_H C^\mathbf{T}(X,Z)$ "in the limit", as it were.  By the universality  of colimits in $SSet$ and the fact that $\nu$ is a right adjoint, we have that $L_H C^\mathbf{T}(X,Y)\times L_H C^\mathbf{T}(Y,Z)\cong \varinjlim_\mathbf{T} \nu(C^\mathbf{T}(X,Y)\times C^\mathbf{T}(Y,Z)$.

Then to prove that everything assembles correctly, we may need to use some sort of [[coend]], but I'm not too sure.

(...)


simplicial loop space

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

#Contents#
* table of contents
{:toc}

## Idea

In [[simplicial homotopy theory]] there is a standard construction [[Kan (1958a)](#Kan58a), [Kan (1958b)](#Kan58b)] -- traditionally denoted "$G$" -- for the simplicial analog of the [[homotopy type]] of the [[loop space]] $\Omega X$ of a [[connected topological space]], now incarnated as a [[simplicial group]]: It is the [[left adjoint]] in an [[adjunction]]
$$
  (G \dashv \overline{W})
    \;\colon\;
  sSet_*
    \underoverset
      {\underset{\overline{W}}{\longleftarrow}}
      {\overset{G}{\longrightarrow}}
      {\bot}
  sGrp
$$

between [[pointed simplicial sets]] and [[simplicial groups]], whose [[right adjoint]] is the [[simplicial classifying space]] construction.

This adjunction is a [[Quillen adjunction]] for the [[Kan–Quillen model structure]] on [[pointed simplicial sets]] and its [[transferred model structure]] on [[simplicial groups]] (more on which at *[[model structure on simplicial groups]]*).

When restricted to [[reduced simplicial sets]], this [[Quillen adjunction]] becomes a [[Quillen equivalence]] which exhibits the [[homotopy theory]] of [[simplicial groups]] as [[equivalence of (infinity,1)-categories|equivalent]] to the [[classical homotopy theory]] of [[pointed homotopy type|pointed]] [[connected homotopy type]] [[homotopy type]] (cf. at *[[looping and delooping]]*).

The generalization of the Kan loop group construction to non-reduced simplicial sets -- and then taking values in [[simplicial groupoids]] -- is the *[[Dwyer-Kan loop groupoid]]* functor.

## Properties

### Quillen equivalence between simplicial groups and reduced simplicial sets

\begin{proposition}\label{QuillenEquivalenceBetweenSimplicialGroupsAndReducedSimplicialSets}
**([[Quillen equivalence between simplicial groups and reduced simplicial sets]])** \linebreak
  The [[simplicial classifying space]]-construction $\overline{W}(-)$ is the [[right adjoint]] in a [[Quillen equivalence]] between the projective [[model structure on simplicial groups]] and the injective [[model structure on reduced simplicial sets]].

$$
  Groups(sSet)_{proj}
  \underoverset
    {\underset{\overline{W}}{\longrightarrow}}
    {\overset{ \Omega }{\longleftarrow}}
    {\;\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;\;}
  (sSet_{\geq 0})_{inj}
  \,.
$$

The [[left adjoint]] $\Omega$ is the [[simplicial loop space]]-construction.
\end{proposition}

(e.g. [Goerss & Jardine 09, V Prop. 6.3](#GoerssJardine09))

## Related concepts

* [[simplicial delooping functor]]

* [[Dwyer-Kan loop groupoid]]

## References

The original articles:

* {#Kan58a} [[Daniel M. Kan]], Section 9 of: *A combinatorial definition of homotopy groups*, Annals of Mathematics **67** 2 (1958) 282–312 [[doi:10.2307/1970006](https://doi.org/10.2307/1970006)]

* {#Kan58b} [[Daniel M. Kan]], Sections 10–11 in: _On homotopy theory and c.s.s. groups_, Ann. of Math. 68 (1958) 38-53 [[jstor:1970042](https://www.jstor.org/stable/1970042)]

Early review:

* [[Peter May]], chapter VI of _Simplicial objects in algebraic topology_, Van Nostrand, Princeton (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]]]

A modern monograph account:

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.5 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/))

More details and relation to [[décalage]]:

* [[Danny Stevenson]], _Décalage and Kan's simplicial loop group functor_, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 [[arXiv:1112.0474](https://arxiv.org/abs/1112.0474), [tac:26-28](http://www.tac.mta.ca/tac/volumes/26/28/26-28abs.html)]

[[!redirects simplicial loop spaces]]
[[!redirects simplicial loop group]]
[[!redirects simplicial loop groups]]
[[!redirects simplicial loop functor]]
[[!redirects simplicial loop functors]]
[[!redirects simplicial loop space functor]]
[[!redirects simplicial loop space functors]]

[[!redirects Kan loop group]]
[[!redirects Kan loop groups]]


simplicial manifold

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

## Definition

A (smooth) _simplicial manifold_ is a [[simplicial object]] in the [[category]] [[Diff]] of [[smooth manifolds]].

Under the canonical inclusion [[Diff]] $\hookrightarrow$ [[Top]] this is in particular a [[simplicial topological space]].

## Constructions on simplicial manifolds

* The [[simplicial de Rham complex]] generalizes the [[de Rham complex]] from ordinary manifolds to simplicial manifolds.

## Interpretation in terms of higher category theory

The role of simplicial manifolds in most of the literature is best understood from the [[nPOV]], by thinking of $[\Delta^{op}, Diff]$ as a [[subcategory]] $[\Delta^{op}, Diff] \hookrightarrow [\Delta^{op}, PSh(Diff)]$ of that of [[simplicial presheaf|simplicial presheaves]] on [[Diff]]. Using the [[local model structure on simplicial presheaves|local]] [[model structure on simplicial presheaves]] which [[presentable (∞,1)-category|presents]] the [[(∞,1)-topos]] of [[∞-stack]]s, one is entitled to think of any simplicial manifold as representing a [[Lie ∞-groupoid]].

In fact, more is true: as discussed there in detail, in the projective local [[model structure on simplicial presheaves]] on [[Diff]] there is a cofibrant resolution functor with values in simplicial manifolds. (Here we allow [[Diff]] to be closed under coproducts indexed by sets; that is, we do not require our manifolds to be connected or second-countable, although we may require their components to be second-countable). Therefore every [[∞-stack]] on [[Diff]] may be presented by a simplicial manifold.

## Kan simplicial manifolds

For more information, see the article [[Kan simplicial manifold]].

## Related concepts

* [[Lie groupoid]]

* [[∞-stack]]

* [[Kan simplicial manifold]]

[[!redirects simplicial manifolds]]


simplicial map

[[!redirects simplicial maps]]

#Contents#
* table of contents
{:toc}

## Idea

A [[homomorphism]] between [[simplicial sets]] $X \overset{f}{\longrightarrow} Y$ is a sequence of [[functions]] between [[sets]] $X_n \overset{f_n}{\longrightarrow} Y_n$ for all $n \in \mathbb{N}$, commuting with the [[face]] and [[degeneracy]] maps.

These are the [[morphisms]] in the [[category of simplicial sets]].

## Related concepts

* [[simplicial set]]

* [[simplicial objects]]


simplicial mapping complex > history

See at *[[function complex]]*


simplicial model category

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

#Contents#
* table of contents
{:toc}

## Idea

The term _simplicial model category_ is short for _[[model structure on simplicial sets|sSet]]${}_{Quillen}$-[[enriched model category]]_.

A simplicial model category is a model or presentation for an [[(∞,1)-category]] that is half way in between a bare [[model category]] and a [[Kan complex]]-[[enriched category]].

Specifically, a simplicial model category is an [[sSet]]-[[enriched category]] $C$ together with the structure of a [[model category]] on its underlying [[category]] $C_0$ such that both structures are compatible in a reasonable way.

One important use of simplicial model categories comes from the fact that the full [[sSet]]-subcategory $C^\circ \hookrightarrow C$ on the fibrant-cofibrant objects -- which is not just [[sSet]]-enriched but actually [[Kan complex]]-enriched -- is the [[(∞,1)-category]]-enhancement of the [[homotopy category]] of the [[model category]] $C_0$.

For generalizations of this construction with [[sSet]] replaced by another [[monoidal model category]] see [[enriched homotopical category]].

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

The term _simplicial model category_ for the notion described here is entirely standard, but in itself a bit suboptimal. More properly one would speak of [[simplicially enriched category]], which is a proper special case of a [[simplicial object]] in [[Cat]] (that for which the simplicial set of objects is discrete).

The other caveat is that there are different model category structures on [[sSet]] and hence even the term $sSet$-[[enriched model category]] is ambiguous.

For instance the [[model structure for quasi-categories]] is an $sSet$-[[enriched model category]], but not for the standard Quillen model structure on the enriching category: since $sSet_{Joyal}$ is a [[closed monoidal category|closed]] [[monoidal model category]] it is enriched over itself, hence is a $sSet_{Joyal}$-enriched model category, not an $sSet_{Quillen}$-enriched one. So in the standard terminology, $sSet_{Joyal}$ is _not_ a "simplicial model category".

=--

## Definition

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

A **simplicial model category** is an [[enriched model category]] which is enriched over $sSet_{Quillen}$: the category [[sSet]] equipped with its standard [[model structure on simplicial sets]].

=--

Spelled out, this means that a simplicial model category is

* an [[sSet]]-[[enriched category]]

  * which is [[power|powered]] and [[tensoring|tensored]] over [[sSet]]

* with the structure of a [[model category]] on the underlying category $C_0$

* such that for every cofibration $i : A \to B$ and every fibration $p : X \to Y$ in $C_0$ the [[pullback powering]] of [[simplicial sets]] $C(B,X) \stackrel{i^* \times p_*}{\to} C(A,X) \times_{C(A,Y)} C(B,Y)$ is a [[Kan fibration]];

  * and such that this fibration is an acyclic fibration whenever either $i$ or $p$ are acyclic.

## Properties

### Enrichment, tensoring, and cotensoring
 {#EnrichmentTensoringCotensoring}

Let $\mathcal{C}$ be a category equipped with the structure of a [[model category]] and with that of an [[sSet]]-[[enriched category]], which is [[tensoring|tensored]] and [[cotensoring|cotensored]] over [[sSet]].

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

The following conditions -- that each make $\mathcal{C}$ into a _simplicial model category_ -- are equivalent:

1. the [[tensoring]] $\otimes : \mathcal{C} \times sSet \to \mathcal{C}$ is a left [[Quillen bifunctor]];

1. [pullback-power axiom](enriched+model+category#PullbackPowerAxiom):

   for any cofibration $X \to Y$ and fibration $A \to B$ in $\mathcal{C}$, the induced morphism

   $$
     \mathcal{C}(Y, A) \to \mathcal{C}(X, A) \times_{\mathcal{C}(X,B)} \mathcal{C}(Y,B)
   $$

   is a fibration, and is in addition a weak equivalence if either of the two morphisms is;

1. for any cofibration $X \to Y$ in $sSet$ and fibration $A \to B$ in $\mathcal{C}$, the induced morphism

   $$
     A^Y \to A^X \times_{B^X} B^Y
   $$

   is a fibration, and is in addition a weak equivalence if either of the two morphisms is.

=--

This follows directly, by [[Joyal-Tierney calculus]],  from the defining properties of [[tensoring]] and [[cotensoring]] (as in [this Prop.](enriched+model+category#MonoidalModelCategoryIsEnrichedModelOverItself)).

We list in the following some implications of these equivalent conditions.

Let $\mathcal{C}$ now be a simplicial model category.

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

If $A \in \mathcal{C}$ is fibrant, and $X \hookrightarrow Y$ is a cofibration in [[sSet]], then

$$
  A^{Y} \to A^{X}
$$

is a fibration in $\mathcal{C}$.

=--

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

Apply prop. \ref{EquivalenceOfConditions} to the case
of the cofibration $X \to Y$ and the fibration $A \to *$, where "$*$" denotes the [[terminal object]]. This yields that

$$
  A^Y \to A^X \times_{{*}^X} {*}^Y
$$

is a fibration. But ${*}^Y = {*}^X =  {*}$ and hence the claim follows.

=--

Similarly we have

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

If $X \in \mathcal{C}$ is cofibrant and $A \in \mathcal{C}$ is fibrant, then $\mathcal{C}(X,A)$ is fibrant in [[sSet]], hence is a [[Kan complex]].

=--

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

Apply prop. \ref{EquivalenceOfConditions} to the cofibration $\emptyset \to X$, where "$\emptyset$" denotes the [[initial object]], and to the fibration $A \to *$ to find that

$$
  \mathcal{C}(X, A) \to \mathcal{C}(\emptyset, A) \times_{\mathcal{C}(\emptyset,*)} \mathcal{C}(X,*)
$$

is a fibration. But since $\emptyset$ is initial and $*$ is terminal, all three simplicial sets in the fiber product on the right are the point, hence this is a fibration

$$
  \mathcal{C}(X,A) \to *
  \,.
$$

=--

### Derived hom-spaces

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

For $X$ and $A$ any two objects and $Q X $ and $P A$ a cofibrant and fibrant replacement, respectively, $\mathcal{C}(Q X, P A)$ is the correct [[derived hom-space]] between $X$ and $A$ (see the discussion there). In particular the full $sSet$-enriched [[subcategory]] on cofibrant fibrant objects is therefore an [[simplicially enriched category|sSet-enriched category]] which is fibrant in the [[model structure on simplicially enriched categories]]. Its [[homotopy coherent nerve]] is a [[quasi-category]]. All this are intrinsic incarnatons of the [[(∞,1)-category]] that is [[presentable (∞,1)-category|presented]] by $C$.

=--

Although $\mathcal{C}(X,A)$ need not have the correct homotopy type for the mapping space if $X$ is not cofibrant or $A$ is not fibrant, simplicial notions of homotopy are still *sufficient* to detect the model-categorical ones.

+-- {: .num_prop #SimplicialHomotopyImpliesHomotopy}
###### Proposition
If $f,g:X\to A$ are simplicially homotopic (i.e. in the same connected component of the simplicial set $\mathcal{C}(X,A)$), then they represent the same map in the [[homotopy category]] of $\mathcal{C}$.  Therefore, any [[simplicial homotopy equivalence]] in $\mathcal{C}$ is a weak equivalence.
=--
+-- {: .proof}
###### Proof
This is Lemma 9.5.15 and Proposition 9.5.16 of [Hirschhorn](#Hirschhorn).
=--

## Examples

### General

* The  [[classical model structure on simplicial sets]] $sSet_{Quillen}$ is a [[closed monoidal category|closed]] [[monoidal model category]] and is hence naturally enriched, as a model category, over itself. This is the archetypical simplicial model category.

* The [[classical model structure on topological spaces]] for [[compactly generated topological spaces]] ([here](classical+model+structure+on+topological+spaces#ModelStructureOnCompactlyGeneratedTopologicalSpaces)) is similarly enriched over itself. Under [[geometric realization]] this also makes it a simplicial model category.

  (See also for instance [Goerss-Schemmerhorn 06, p. 26](#GoerssSchemmerhorn06))

* For $C$ any small [[sSet]]-[[enriched category]] and $A$ [[simplicial combinatorial model category]], the [[global model structure on functors]] $[C^{op}, A]_{proj}$ and $[C^{op},A]_{inj}$ are themselved simplicial combinatorial model categories. See _[[model structure on simplicial presheaves]]_.

* The [[Bousfield localization of model categories|left Bousfield localization]] of a [[combinatorial simplicial model category]] at any set of morphisms is again a combinatorial simplicial model category. Large classes of examples arise this way.

### Simplicial Quillen equivalent models
  {#SimpEquivMods}

While many model categories do not admit an [[classical model structure on simplicial sets|$sSet_{Qu}$]]-[[enriched model category|enrichment]], for large classes of model categories one can find a [[Quillen equivalence]] to a model category that does carry an $sSet_{Qu}$-enrichment.

These are constructed as [[Bousfield localization of model categories|Bousfield localization]] of [[Reedy model structures]] on the [[category of simplicial objects]] in the given model category.

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

Let $C$ be a

* [[left proper model category|left proper]]

* [[combinatorial model category]].

By the discussion at *[[cofibrantly generated model category]]* in the section *[Presentation and generation](cofibrantly+generated+model+category#PresentationAndGeneration)*, there exists a [[small set]] $E \subset Obj(C)$ of [[objects]] that detect [[weak equivalences]]. For some such choice of $E$, let

$$
  S
   \;\coloneqq\;
   \Big\{
    \;
    e
      \cdot
    \big(
      \Delta[k]
        \longrightarrow
      \Delta[l]
    \big)
      \;
    \Big\}_{
      {e \in E,}
      \atop
      {([k] \to [l]) \in \Delta}
    }
   \;\subset\;
   Mor\big(
     [\Delta^{op}, C]
   \big)
$$

where $\cdot$ denotes the [[tensoring]] of $C$ over [[Set]], hence $e \cdot \Delta[n]$ denotes the [[simplicial object]] consisting of [[coproducts]] of $e$ as
$$
  e \cdot \Delta[k]
    \;\;\colon\;\;
  [n]
    \;\mapsto\;
  \underset
   {\Delta([n],[k])}
   {\amalg}
  e
  \,.
$$

Write

$$
  [\Delta^{op}, C]_{proj, S}
  \underoverset
    {\underset{id}{\longrightarrow}}
    {\overset{id}{\longleftarrow}}
    {\;\;\bot\;\;}
  [\Delta^{op}, C]_{proj}
$$

for the [[Bousfield localization of model categories|left Bousfield localization]] of the projective [[model structure on functors]] at this set $S$ of morphisms.

Similarly, write

$$
  [\Delta^{op}, C]_{Reedy, S}
  \underoverset
    {\underset{id}{\longrightarrow}}
    {\overset{id}{\longleftarrow}}
    {\;\;\bot\;\;}
  [\Delta^{op}, C]_{Reedy}
$$

for the left Bousfield localization of the [[Reedy model structure]] at $S$.

=--

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

Let $C$ be a [[cofibrantly generated model category]].

If $X \in [\Delta^{op}, C]$ is degreewise [[cofibrant object|cofibrant]] and has all structure maps being [[weak equivalences]], then all [[coprojections]] $X_i \to hocolim X$ into the [[homotopy colimit]] are [[weak equivalences]]
and hence $X \to const\,hocolim X$ is a weak equivalence.

=--

This appears as [Dugger, prop. 5.4 corollary 5.5](#Dugger).

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

The model structures from def. \ref{LocalizationOfStructureOnSimplicalObjects} have the following properties.

1. The [[weak equivalences]] in both are precisely those morphisms which become weak equivalences under [[homotopy colimit]] over $\Delta^{op}$.

1. The [[fibrant objects]] in both are precisely those objects that are fibrant in the corresponding unlocalized structures, and such that all the [[face and degeneracy maps]] are weak equivalences in $C$.

1. The [[colimit]]/[[constant functor|constant]] [[adjoint functors]]

   $$
     (\lim_{\to} \dashv const)
     \colon
     C
     \underoverset
       {\underset{\;const\;}{\longrightarrow}}
       {\overset{\underset{\to}{\lim}}{\longleftarrow}}
       {\;\;\bot\;\;}
       [\Delta^{op}, C]_{proj, S}
   $$

   constitute a [[Quillen equivalence]], the [[identity functors]] constitute a [[Quillen equivalence]]

   $$
     [\Delta^{op}, C]_{Reedy, S}
     \underoverset
       {\underset{id}{\longrightarrow}}
       {\overset{id}{\longleftarrow}}
       {\bot}
     [\Delta^{op}, C]_{proj, S}
     \,,
   $$

   and the [[constant functor|constant]]/[[limit]] [[adjoint functors]] (notice that the limit is [[evaluation]] in degree 0, since $\Delta^{op}$ has an [[initial object]]) constitute a Quillen equivalence

   $$
     (const \dashv ev_0)
     \;\colon\;
     [\Delta^{op}, C]_{Reedy,S}
     \underoverset
       {\underset{\;ev_0\;}{\longrightarrow}}
       {\overset{const}{\longleftarrow}}
       {\bot}
     C
     \,;
   $$

1. The canonical [[sSet]]-[[enriched category|enrichment]]/[[tensoring]]/[[powering]] of the [[category of simplicial objects]] $[\Delta^{op}, C]$ makes $[\Delta^{op}, C]_{Reedy,S}$ (but _not_ in general $[\Delta^{op}, C]_{proj,S}$) into a [[simplicial model category]].

=--

This is [Dugger, theorem 5.2, theorem 5.7, theorem 6.1](#Dugger).

So in particular every [[left proper model category|left proper]] [[combinatorial model category]] is [[Quillen equivalence|Quillen equivalent]] to a [[simplicial model category]].

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

We first show that the [[fibrant objects]] in $[\Delta^{op}, C]_{proj,S}$ are the objectwise fibrant objects all whose structure maps are weak equivalences in $C$. The argument for the fibrant objects in $[\Delta^{op}, C]_{Reedy,S}$ is directly analogous.

By general properties of [[Bousfield localization of model categories|left Bousfield localization]], the
fibrant objects in $[\Delta^{op}, C]_{proj,S}$ are the projective fibrant objects $X$ for which all induced morphisms on [[derived hom spaces]]

$$
  \mathbb{R}Hom_{[\Delta^{op}, C]_{proj}}(s \cdot(\Delta[k] \to \Delta[l]), X)
$$

are weak equivalences. Since $s$ is cofibrant in $C$ by definition, also  $s \cdot \Delta[k]$ is cofibrant in
$[\Delta^{op}, C]_{proj}$.

So for $X \in [\Delta^{op}, C]_{proj}$ fibrant,
let $X_\bullet \in [\Delta^{op}, [\Delta^{op}, C]]$
be a [simplicial framing](derived+hom+space#Framings) for it.
Notice that this means that for all $n \in \mathbb{N}$
also $X_\bullet([n])$ is a simplicial framing for $X([n])$.
This is because

1. $const X \to X_\bullet$ being a weak equivalence means that for all $n$ the morphism $X \to X_n$ is a weak equivalence, which means that for all $k$ the morphism $X([k]) \to X_n([k])$ is a weak equivalence.

1. $X_\bullet$ being fibrant in $[\Delta^{op}, [\Delta^{op}, C]_{proj}]_{Reedy}$ means that for all $n\in \mathbb{N}$ the morphism $X_{\Delta[n]} \to X_{\partial \Delta[n]}$ is a fibration in $[\Delta^{op}, C]_{proj}$, hence that for all $k \in \mathbb{N}$ the morphism $X_{\Delta[n]}([k]) \to X_{\partial \Delta[n]}([k])$ is a fibration in $C$, hence that $X([k])$ is Reedy fibrant.

Then we find

$$
  \begin{aligned}
    \mathbb{R}Hom_{[\Delta^{op}, C]_{proj}}( s \cdot(\Delta[k] \to \Delta[l]), X)
	& \simeq
	Hom_{[\Delta^{op}, C]}(s \cdot(\Delta[k] \to \Delta[l]), X_\bullet)
	\\
	& \simeq
	Hom_{C}(s , X_\bullet([l]) \to X_\bullet([k]))
	\\
	& \simeq
	\mathbb{R}Hom_C(s, X([l]) \to X([k]))
	\,.
  \end{aligned}
$$

By assumption on the set $S$, this implies the claim.

\linebreak

Now we show that the weak equivalences in $[\Delta^{op}, C]_{proj,S}$ are precisely
those morphisms that become weak equivalences under
the [[homotopy colimit]].

By functorial [[cofibrant resolution]] and [[two-out-of-three]], it is sufficient to  show that this holds for morphisms between [[cofibrant objects]].

By lemma \ref{HocolimOverHomotopyConstantSimplicialDiagram}, we have weak equivalences

$$
  \mathbb{R}Hom_{[\Delta^{op}, C]}(A,X)
  \stackrel{\simeq}{\to}
  \mathbb{R}Hom_{[\Delta^{op}, C]}(A, const Z)
  \stackrel{\simeq}{\to}
  \mathbb{R}Hom_{C}(\lim_\to A , Z)
$$

seen by computing the derived homs by [simplicial framings](derived+hom+space#Framings).

Now, by properties of left [[Bousfield localization of model categories|Bousfield localization]],
$A \to B$ is a weak equivalence if for all $S$-[[local objects]] $X$ the morphism of [[derived hom-spaces]] $\mathbb{R}Hom(A \to B, X)$ is a weak equivalence. Looking at the diagram

$$
  \array{
    \mathbb{R}Hom_{[\Delta^{op}, C]}(A,X)
    &\stackrel{\simeq}{\to}&
    \mathbb{R}Hom_{[\Delta^{op}, C]}(A, const Z)
    &\stackrel{\simeq}{\to}&
    \mathbb{R}Hom_{C}(\lim_\to A , Z)
   \\
   \uparrow && \uparrow && \uparrow
   \\
    \mathbb{R}Hom_{[\Delta^{op}, C]}(B,X)
    &\stackrel{\simeq}{\to}&
    \mathbb{R}Hom_{[\Delta^{op}, C]}(B, const Z)
    &\stackrel{\simeq}{\to}&
    \mathbb{R}Hom_{C}(\lim_\to B , Z)
  }
$$

we see that this is the case precisely if the vertical morphism on the right is a weak equivalence for all fibrant $Z \in C$, which is the case if $\lim_\to A \to \lim_\to B$ is a weak equivalence. Since $A$ and $B$ here are cofibrant in $[\Delta^{op}, C]_{proj}$, the colimits here are indeed [[homotopy colimits]] (as discussed there).

\linebreak

Now we discuss that $(\lim_\to \dashv const) \colon C \to [\Delta^{op}, C]_{proj,S}$
is a [[Quillen equivalence]].
First observe that on the global model structure $const \colon C \to [\Delta^{op}, C]_{proj}$ is clearly a right Quillen functor, hence we have a Quillen adjunction on the unlocalized structure. Moreover, by definition and by the above discussion, the [[derived functor]] of the [[left adjoint]] $\lim_\to$, namely the [[homotopy colimit]], takes the localizing set $S$ to weak equivalences in $C$. Therefore the assumptions of the discussion at _[Quillen equivalence - Behaviour under localization](http://ncatlab.org/nlab/show/Quillen+adjunction#BehaviourUnderLocalization)_ are met, and hence it follows that $(\lim_\to \dashv const)$ descends as a Quillen adjunction also to the localization.

To see that this is a Quillen equivalence,
it is sufficient to show that for $A \in [\Delta^{op}, C]_{proj}$ cofibrant and $Z \in C$ fibrant, a morphism $\lim_\to A \to Z$ is a weak equivalence in $C$ precisely if the [[adjunct]] $A \to const Z$ becomes a weak equivalence under the homotopy colimit.

For this notice that we have a [[commuting diagram]]

$$
  \array{
     hocolim A &\longrightarrow& hocolim(const Z)
     \\
     \big\downarrow
       &&
     \big\downarrow
     \\
     colim A &\longrightarrow& colim (const Z) & \simeq Z
  }
$$

and so our statement follows (by [[2-out-of-3]]) once we know that the vertical morphisms here are weak equivalences. The left one is because $A$ is cofibrant, by assumption, as before.  To see that the right one is, too, consider the factorization

$$
  Z \to (const Z)_i \to hocolim (const Z) \to Z
$$

of the [[identity morphism]] on $Z$, for any $i \in \mathbb{N}$. By lemma \ref{HocolimOverHomotopyConstantSimplicialDiagram} the first morphism is a weak equivalence, and hence so is the morphism in question.

\linebreak

Now we show that the weak equivalences in $[\Delta^{op}, C]_{Reedy,S}$ are the hocolim-equivalences.

By a general result on [functoriality of localization](Bousfield+localization+of+model+categories#FunctorialityOfLocalization), we have that the $(id \dashv id ) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\leftarrow}{\to} [\Delta^{op}, C]_{proj,S}$ is at least a [[Quillen adjunction]].

Let then $A \to B$ be a morphism in $[\Delta^{op}, C]$ and consider two fibrant replacements

$$
  \array{
    A &\to& \bar A &\to & \hat A
    \\
    \downarrow && \downarrow && \downarrow
    \\
    B &\to& \bar B &\to& \hat B
  }
  \,,
$$

where the first one ($\bar A \to \bar B$) is taken in $[\Delta^{op}, C]_{proj,S}$ and the second ($\hat A \to \hat B$) in $[\Delta^{op}, C]_{Reedy}$.

Assume first that $A \to B$ is a hocolim-equivalence. Then so is $\hat A \to \hat B$, because the horizontal morphisms are all objectwise weak equivalences. But $\hat A$ and $\hat B$ are fibrant in $[\Delta^{op}, C]_{Reedy}$, hence in $[\Delta^{op}, C]_{proj}$ by construction and at the same time all their structure maps are weak equivalences (use 2-out-of-3), so that they are in fact fibrant in $[\Delta^{op}, C]_{proj,S}$. By general properties of left Bousfield localization, weak equivalences between local fibrant objects are already weak equivalences in the unlocalized structure -- so $\hat A \to \hat B$ is indeed even an objectwise weak equivalence. It follows then that so is $\bar A \to \bar B$, which is therefore in partiular a weak equivalence in $[\Delta^{op}, C]_{Reedy, S}$. Finally the left horizontal morphisms are also weak equivalences in $[\Delta^{op}, C]_{Reedy,S}$, by the above Quillen adjunction. So finally by 2-out-of-3 in $[\Delta^{op}, C]_{Reedy,S}$ it follows that also $A \to B$ is a weak equivalence there.

By an analogous diagram chase, one shows the converse implication holds, that $A \to B$ being a weak equivalence in $[\Delta^{op}, C]_{Reedy,S}$ implies that it is a hocolim-equivalence.

With this now it is clear that the identity adjunction above is in fact a Quillen equivalence.

\linebreak

Finally we show that $(const \dashv ev_0) : [\Delta^{op}, C]_{Reedy,S}
\stackrel{\overset{const}{\leftarrow}}{\underset{ev_0}{\to}} C$ is a [[Quillen equivalence]].

First, it is immediate to check that $const \colon C \to [\Delta^{op}, C]_{Reedy}$ is left Quillen, and since $id : [\Delta^{op}, C]_{Reedy} \to [\Delta^{op}, C]_{Reedy,S}$ is left Quillen by definition of Bousfield localization, the above is at least a Quillen adjunction.

To see that it is a Quillen equivalence, let $A \in C$ be cofibrant and $X \in [\Delta^{op}, C]_{Reedy,S}$ be fibrant -- which by the above means that it is a simplicial resolution -- and consider a morphism $const A \to X$. We need to show that this is a weak equivalence, hence, by the above, that its hocolim is a weak equivalence, precisely if $A \to X_0$ is a weak equivalence in $C$.

To that end, find a cofibrant resolution $const \tilde A \to \tilde X$ of $const A \to X$ in $[\Delta^{op}, C]_{proj}$ and consider the diagram

$$
  \array{
    A &\stackrel{\simeq}{\leftarrow}&  \tilde A &\stackrel{\simeq}{\to}&
    colim(const \tilde A)
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X_0 &\stackrel{\simeq}{\leftarrow}& \tilde X_0
    &\stackrel{\simeq}{\to}& colim \tilde X
  }
  \,.
$$

The colimits on the right compute the homotopy colimit. By 2-out-of-3 it  follows that the right vertical morphism is a weak equivalence precisely if the left vertical morphisms is.

\linebreak

Finally it remains to show that $[\Delta^{op}, C]_{Reedy,S}$ is a simplicially enriched model category. (...)

=--

+-- {: .num_theorem #AUniquenessTheorem}
###### Remark
**(uniqueness)**

Let $C$ be a [[model category]]. Then there is a unique model category  structure on $s C \coloneqq [\Delta^{op}, C]$ such that

* every morphism that is degreewise a weak equivalence in $C$ is a weak equivalence in $s C$;

* the cofibrations are those of the [[Reedy model structure]];

* the fibrant objects are the Reedy-fibrant objects whose face and degeneracy maps are weak equivalences in $C$.

=--

This is [Rezk, Schwede & Shipley, theorem 3.1](#RezkSchwedeShipley).

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

By theorem \ref{LocalizationOfReedyOnSimplicialObjects}
at least one such model structure exists.
By the discussion at _[model category -- Redundancy of the axioms](model+category#RedundancyInTheAxioms)_, the classes of cofibrations and fibrant objects already determine a model category structure.

=--

+-- {: .num_cor #DerivedHomBySimplicialReplacement}
###### Corollary

For $C$ any [[proper model category|left proper]]
[[combinatorial model category]], the [[derived hom-space]]
between two objects $X, A$ may be computed by

* choosing a cofibrant replacement $\hat X$ of $X$ in $C$;

* choosing a Reedy fibrant replacement $\hat A$ of $const A$ in $[\Delta^{op}, C]$ such that all face and degeneracy maps are weak equivalences,

setting

$$
  Maps(X,A)
  \;\colon\;
  [n] \mapsto Hom_C(\hat X, \hat A_n)
  \,.
$$

=--

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

By theorem \ref{LocalizationOfReedyOnSimplicialObjects} we may compute the [[derived hom space]] in $[\Delta^{op}, C]_{Reedy,S}$ after the inclusion $const \colon C \to [\Delta^{op}, C]$. Since by that theorem $[\Delta^{op}, C]_{Reedy,S}$ is a simplicial model category, by prop. \ref{DerivedHomSpaceBySimplicialFunctionComplex}
the derived hom space is given by the simplicial function complex between a cofibrant replacement of $const X$ and a fibrant replacement of $const A$. If $\hat X$ is cofibrant, then $const \hat X$ is already Reedy cofibrant, and by the theorem $\hat A$ as stated is a a fibrant resolution of $const A$. Finally, the theorem says that the simplicial function complex is given by

$$
  \begin{aligned}
    [\Delta^{op}, C](const \hat X, \hat A)_n
    & =
    Hom_{[\Delta^{op}, C]}((const \hat X) \cdot \Delta[n], \hat A)
    \\
    &  \simeq
    Hom_{[\Delta^{op}, C]}((const \hat X) , \hat A^{\Delta[n]})
    \\
    & \simeq
    Hom_C(\hat X, \hat A_n)
  \end{aligned}
  \,.
$$

=--

There is also a version for [[stable model categories]]:

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

Every [[proper model category|proper]] [[cofibrantly generated model category|cofibrantly generated]] [[stable model category]] is [[Quillen equivalence|Quillen equivalent]] to a simplicial model category

=--

This is [Rezk, Schwede & Shipley, prop 1.3](#RezkSchwedeShipley).

### Combinatorial simplicial model categories

A particularly important type of simplicial model categories are those that are also [[combinatorial model category|combinatorial model categories]].

A [[combinatorial simplicial model category]] is precisely a _presentation_ for a [[locally presentable (∞,1)-category]]. See there for more details.

## Related concepts

* [[Dugger's theorem]]

* [[combinatorial model category]]

* [[Ho(CombModCat)]]

* [[simplicial monoidel model category]]

## References

The definition originates in

* {#Quillen67} [[Daniel Quillen]], chapter II, section 2 of: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics __43__, Springer (1967) [[doi:10.1007/BFb0097438](https://link.springer.com/book/10.1007/BFb0097438)]

Textbook accounts:

* {#Hirschhorn} [[Philip Hirschhorn]], Chapter 9 in: _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

* [[Jacob Lurie]], section A.3 in _[[Higher Topos Theory]]_

* [[Emily Riehl]], §3.8 and §11.4 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

Further review:

* {#GoerssSchemmerhorn06} [[Paul Goerss]], [[Kirsten Schemmerhorn]], _Model categories and simplicial methods_, Notes from lectures given at the University of Chicago in August 2004, published as: _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)]

On Quillen-equivalent simplicial enhancements of model categories:

* {#RezkSchwedeShipley} [[Charles Rezk]], [[Stefan Schwede]], [[Brooke Shipley]], _Simplicial structures on model categories and functors_, American Journal of Mathematics, Vol. 123, No. 3 (Jun., 2001), pp. 551-575  ([arXiv:0101162](http://arxiv.org/abs/math/0101162), [jstor](http://www.jstor.org/stable/pdfplus/25099071.pdf))

* {#Dugger} [[Dan Dugger]], *Replacing model categories with simplicial ones*, Trans. Amer. Math. Soc. **353** 12 (2001) 5003-5027 [[ams:S0002-9947-01-02661-7](https://www.ams.org/journals/tran/2001-353-12/S0002-9947-01-02661-7), [pdf](https://www.ams.org/journals/tran/2001-353-12/S0002-9947-01-02661-7/S0002-9947-01-02661-7.pdf)]

* {#Dugger01} [[Daniel Dugger]], _[[Combinatorial model categories have presentations]]_, Adv. Math. **164** (2001) 1 177-201 [[arXiv:math/0007068](http://arxiv.org/abs/math/0007068), [doi:10.1006/aima.2001.2015](https://doi.org/10.1006/aima.2001.2015)]

  > (cf. *[[Dugger's theorem]]*)

[[!redirects simplicial model categories]]
[[!redirects sSet-model category]]
[[!redirects sSet-model categories]]

[[!redirects simplicial model structure]]
[[!redirects simplicial model structures]]


simplicial model for weak omega-categories

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

#Contents#
* table of contents
{:toc}

## Idea

While an [[algebraic definition of higher category|algebraic]] definition of [[strict ∞-categories]] is comparatively straightforward, algebraic definitions of fully weak $\omega$-categories (aka $\infty$-[[infinity-category|categories]]) are difficult to work with (although some definitions exist, such as those of [[Batanin ∞-category|Batanin]], [[Trimble n-category|Trimble]], [[Leinster n-category|Leinster]], and [[n-category|Penon]]).

However, just like strict $\omega$-categories have a [[simplicial set|simplicial]] [[nerve]] -- a [[complicial set]] -- induced by the [[orientals]], and just like the category of strict $\omega$-categories is equivalent to the category of complicial sets, one expects that every weak $\omega$-category naturally has a simplicial nerve and that the theory of algebraically defined weak $\omega$-categories is equivalent to the theory of the simplicial sets that arise as their nerves.  In fact, one can hope that the theory is _simpler_ in the weak case: complicial sets are simplicial sets equipped with extra [[stuff, structure, property|structure]] (a [[stratified simplicial set|stratification]] representing the chosen 'strict' composites), while in the nerve of a weak $\omega$-category this structure might be reducible to a property.

In the context of _simplicial models for weak $\omega$-categories_ the goal is to characterize those [[simplicial sets]] (or [[stratified simplicial sets]]) which should arise as [[nerves]] of algebraically defined weak $\omega$-categories and thus provide a [[geometric definition of higher category]] generalizing the familiar simplicial description of $\omega$-[[infinity-groupoid|groupoids]] as [[Kan complexes]] and of $(\omega,1)$-[[(infinity,1)-category|categories]] as [[quasi-category|quasi-categories]] to general higher categories.

In effect, the goal is to _define_ a weak $\omega$-category to be a certain sort of ([[stratified simplicial set|stratified]]) simplicial set.  One could then hope to prove that these are precisely the [[nerve|nerves]] of weak $\omega$-categories defined in some other way.

To distinguish the study of weak $\omega$-categories in terms of their presumed nerves from the study of their would-be algebraic descriptions people speak of [[simplicial weak omega-category|simplicial weak ∞-categories]], see in particular the articles by Dominic Verity referenced below.  One should just beware that in this context [[simplicial weak ∞-category]] is not meant as [[simplicial object]] in the category of weak $\omega$-categories.

This program was originally begun by Ross Street and has been carried forward by Dominic Verity with the theory of [[weak complicial sets]].  It is expected that the (nerves of) weak $\omega$-categories will be weak complicial sets satisfying an extra "saturation" condition ensuring that "every [[equivalence]] is [[thin element|thin]]."

#References#

* [[Dominic Verity]], _Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory_ ([arXiv](http://arxiv.org/abs/math.CT/0604414))

* [[Dominic Verity]], _Weak complicial sets, a simplicial weak omega-category theory. Part II: nerves of complicial Gray-categories_ ([pdf](http://arxiv.org/PS_cache/math/pdf/0604/0604416v1.pdf))

[[!redirects simplicial model for weak ∞-categories]]
[[!redirects simplicial models for weak ∞-categories]]
[[!redirects simplicial models for weak omega-categories]]


simplicial monoidal model category

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

#Contents#
* table of contents
{:toc}

## Idea

A *simplicial monoidal model category* is a [[classical model structure on simplicial sets|$sSet_{Qu}$]]-[[enriched monoidal model category]], namely a [[model category]] equipped with

1. the [[structure]] of a [[monoidal model category]],

1. the [[structure]] of a [[simplicial model category]],

1. the compatibility structure that makes the [[underlying]] [[monoidal category|monoidal]] and [[sSet-enriched category|sSet-enriched]] structure an

   [[sSet]]-[[enriched monoidal category]].

## Examples

\begin{example}
  It ought to be true that the [[Bousfield localization of model categories|Bousfield localization]] at the "realization equivalences" (see [this Prop.](model+structure+on+chain+complexes#MonoidalSimplicialEnhancementOfUnboundedChainModelStructure)) of the [[Reedy model structure]] on [[simplicial objects]] in [[chain complexes]] is simplicial monoidal under the objectwise [[tensor product of chain complexes]].
\end{example}

## References

The notion may be implicit in many discussions of [[monoidal model category]] such as used, notably, to build [[symmetric monoidal smash products of spectra]]. One reference making more explicit the need to specify compatibility structure is:

* [[Jacob Lurie]], Def. 4.1.7.8 in *[[Higher Algebra]]* [[pdf](https://www.math.ias.edu/~lurie/papers/HA.pdf)]

with a precursor in

* [[Jacob Lurie]], Def. 1.6.4 in: *Derived Algebraic Geometry II: Noncommutative Algebra* [[arXiv:math/0702299](https://arxiv.org/abs/math/0702299)]

but the actual compatibility is maybe not made very explicit there either, for this see the references at *[[enriched monoidal category]]*.

[[!redirects simplicial monoidal model categories]]

[[!redirects monoidal simplicial model category]]

[[!redirects monoidal simplicial model structure]]
[[!redirects monoidal simplicial model structures]]

[[!redirects simplicial monoidal model structure]]
[[!redirects simplicial monoidal model structures]]


simplicial nerve > history

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


simplicial nerve of simplicial categories > history

< [[simplicial nerve of simplicial categories]]
[[!redirects simplicial nerve of simplicial categories -- history]]


simplicial object

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

#Contents#
* table of contents
{:toc}

## Idea

A _simplicial object_ $X$ in a [[category]] $C$ is a _[[simplicial set]]_ [[internalization|internal]] to $C$: a collection $\{X_n\}_{n \in \mathbb{N}}$ of [[objects]] in $C$ that behave as if $X_n$ were an object of $n$-dimensional [[simplices]] [[internalization|internal to]] $C$ equipped with maps between these spaces that assign faces and degenerate simplices.

For instance, and there is a longer list further down this page, a simplicial object in $Grps$ is a collection $\{G_n\}_{n\in \mathbb{N}}$ of groups, together with face and degeneracy _homomorphisms_ between them. This is just a [[simplicial group]]. We equally well have other important instances of the same idea, when we replace $Grps$ by other categories, or higher categories.

## Definition

A **simplicial object** in a [[category]] $C$ is a [[functor]] $\Delta^{op} \to C$, where $\Delta$ is the [[simplex category|simplicial indexing category]].

More generally, a simplicial object in an [[(∞,1)-category]] is an [[(∞,1)-functor]] $\Delta^{op} \to C$.

A **[[cosimplicial object]]** in $C$ is similarly a functor out of the [[opposite category]], $\Delta \to C$.

Accordingly, simplicial and cosimplicial objects in $C$ themselves form a [[category]] in an obvious way, namely the [[functor category]] $[\Delta^{op},C]$ and $[\Delta,C]$, respectively.

**Remark**

A **simplicial object** $X$ in $C$ is often specified by
the objects, $X_n$, which are the images under $X$, of the objects $[n]$ of $\Delta$, together with a description of the face and degeneracy morphisms, $d_i$ and $s_j$, which must satisfy the [[simplicial identities]].

## Examples
 {#Examples}

* A simplicial object in [[Sets]] is a [[simplicial set]].

* A simplicial object in [[Presheaves]] is a [[simplicial presheaf]].

* A simplicial object in [[TopologicalSpaces]] is a [[simplicial topological space]].

* A simplicial object in [[SmoothManifolds|Manifolds]] is a [[simplicial manifold]].

* A simplicial object in [[Groups]] is a [[simplicial group]].

* A simplicial object in [[AbelianGroups]] is a [[simplicial abelian group]].

* A simplicial object in [[TopologicalGroups]] is a [[simplicial topological group]].

* A simplicial object in [[Lie algebras]] is a [[simplicial Lie algebra]].

* A simplicial object in [[Rings]] is a [[simplicial ring]].

* A cosimplicial object in the category of [[rings]] ([[algebras]]) is a [[cosimplicial ring]] ([[cosimplicial algebra]]).

* A simplicial object in a category of simplicial objects is a [[bisimplicial object]].

* A cosimplicial object in [[sSet]] is a [[cosimplicial simplicial set]] (equivalently a simplicial object in cosimplicial sets).

* The [[bar construction]] produces a simplicial object from a [[monad]] and an algebra over that monad.

## Category of simplicial objects

For $D$ a [[category]], we write $D^{\Delta^{op}}$ for the [[functor category]] from $\Delta^{op}$ to $D$: its category of simplicial objects.

### Simplicial enrichment

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

Let $D$ be a [[nLab:category]] with all [[nLab:limit]]s and [[nLab:colimit]]s. This implies that it is [[nLab:copower|tensored]] over [[nLab:Set]]

$$
  \cdot : D \times Set \to D
  \,.
$$

This induces a functor

$$
  \cdot^{\Delta^{op}} : D^{\Delta^{op}} \times sSet \to D^{\Delta^{op}}
$$

which we shall also write just "$\cdot$".

For $X,Y \in D^{\Delta^{op}}$ write

$$
  D^{\Delta^{op}}(X,Y) := Hom_{D^{\Delta^{op}}}(X \cdot \Delta[\bullet], Y)
  \in sSet
$$

and for $X,Y,Z \in D^{\Delta^{op}}$ let

$$
  D^{\Delta^{op}}(X,Y)
    \times
  D^{\Delta^{op}}(Y,Z)
  \to
  D^{\Delta^{op}}(X,Z)
$$

be given in degree $n$ by

$$
  (X \cdot \Delta[n] \to Y, Y \cdot \Delta[n] \to Z)
  \mapsto
  ( X \cdot \Delta[n] \to X \cdot \Delta[n]\times \Delta[n] \to Y \cdot \Delta[n]  \to Z)
  \,.
$$

=--

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

With the above definitions $D^{\Delta^{op}}$ becomes an [[nLab:sSet]]-[[nLab:enriched category]] which is both [[nLab:copower|tensored]] as well as [[nLab:power|cotensored]] over $sSet$.

=--

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

We may regard the category of cosimplicial objects $D^{\Delta}$ as an $sSet$-enriched category using the above enrichment by identifying

$$
  D^{\Delta} \simeq ({D^{op}}^{\Delta^{op}})^{op}
  \,.
$$

=--

### Geometric realization

If $D$ is already a [[simplicially enriched category]] in its own right, with [[powers]] and [[copowers]], we can define the [[geometric realization]] of a simplicial object $X\in D^{\Delta^{op}}$ as a [[coend]]:

$$ |X| = \int^{[n]\in\Delta} \Delta[n] \odot X_n $$

where $\odot$ denotes the [[copower]] for the simplicial enrichment of $D$.  This is [[left adjoint]] to the "total singular object" functor $D \to D^{\Delta^{op}}$ sending $Y$ to the simplicial object $n\mapsto Y^{\Delta[n]}$, the [[power]] for the simplicial enrichment.

Perhaps surprisingly, this adjunction is even a *simplicially enriched* adjunction when $D^{\Delta^{op}}$ has its simplicial structure from Definition \ref{SimplicialEnrichment}, even though the latter makes no reference to the given simplicial enrichment of $D$.  A proof can be found in [RSS01, Proposition 5.4](#RSS01).

## Related concepts

* [[Reedy model structure]]

* [[simplex]], [[simplex category]]

* **simplicial object**

  * [[simplicial set]]

  * [[simplicial diagram]]

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

* [[semi-simplicial object]]

  * [[semisimplicial set]]

* [[globular set]], [[cubical set]]

* [[nerve]], [[nerve and realization]]

* [[Segal condition]]

## References

The original definition of simplicial objects, maps between them, and homotopies of such maps is due to [[Daniel M. Kan]]:

* [[Daniel M. Kan]], _On the homotopy relation for c.s.s. maps_, Boletín de la Sociedad Matemática Mexicana 2 (1957), 75–81.  [PDF](https://dmitripavlov.org/scans/kan-on-the-homotopy-relation-for-css-maps.pdf).

An early discussion of simplicial objects is in:

* [[Alexander Grothendieck]], p. 108 (11 of 21) in: [[FGA]] _Techniques de construction et théorèmes d'existence en géométrie algébrique III : préschémas quotients_, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf), English translation: [pdf](https://labs.thosgood.com/builds/fga-3-III.pdf), [web version](https://thosgood.com/fga/book/FGA-3-III)).

See also:

* [[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]]]

* [[Dai Tamaki]], [[Akira Kono]], Appendix A.1 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#RSS01} [[Charles Rezk]], [[Stefan Schwede]], and [[Brooke Shipley]], *Simplicial structures on model categories and functors*, [arxiv](https://arxiv.org/abs/math/0101162)

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Contractible simplicial objects*, Comm. Math. Univ. Carol. **60** 4 (2019) 473–495 [[pdf](https://math.mcgill.ca/barr/papers/cso.pdf) [eudml:295068](https://eudml.org/doc/295068)]

[[!redirects simplicial objects]]

[[!redirects category of simplicial objects]]

[[!redirects categories of simplicial objects]]

category: simplicial object