homotopy category of chain complexes

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#### Homotopy theory
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[[!include homotopy - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
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## Definition

Given an [[abelian category]] $\mathcal{A}$, write $Ch_\bullet(\mathcal{A})$ for its [[category of chain complexes]].

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

The [[relation]] of [[chain homotopy]] is an [[equivalence relation]] on the [[hom sets]] of this category.

=--

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

Write $
  \mathcal{K}(\mathcal{A})$
for the [[category]] whose [[objects]] are [[chain complexes]], and whose [[morphisms]] between objects $C_\bullet, D_\bullet$ are chain homotopy-equivalence classes

$$
  \mathcal{K}(\mathcal{A})(C_\bullet, D_\bullet)
  \coloneqq
  Ch_\bullet(\mathcal{A})(C_\bullet, D_\bullet)_{\sim_{ch}}
  \,.
$$

=--

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

Analogous to the variants discussed at _[[category of chain complexes]]_ one considers also the [[full subcategories]]

$$
  K^{+,-,b}(\mathcal{A}) \hookrightarrow \mathcal{K}(\mathcal{A})
$$

on the chain complexes which are bounded above or bounded below or bounded, respectively.

=--

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

The category $\mathcal{K}(\mathcal{A})$ is sometimes called the "homotopy category of chain complexes". But beware that there is another category that also and more properly deserves to be called by this name:

the _[[homotopy category]]_ $Ho(Ch_\bullet(\mathcal{A}))$ in the sense of the result of [[localization]] at the [[chain maps]] that are [[quasi-isomorphisms]] is traditionally called the _[[derived category]]_ $\mathcal{D}(\mathcal{A})$ of $\mathcal{A}$:

$$
  \mathcal{D}(\mathcal{A}) \coloneqq Ho(Ch_\bullet(\mathcal{A}))
  \,.
$$

=--

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

By construction there is a canonical [[functor]]

$$
  Ch_\bullet(\mathcal{A}) \to \mathcal{K}(\mathcal{A})
$$

which is the [[identity]] on objects and the [[quotient]] [[projection]] on [[hom-sets]].

=--

## Properties

### Additive category structure

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

Under componentwise addition of [[chain maps]], $\mathcal{K}(\mathcal{A})$ is an [[additive category]].

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In fact using the [[abelian group]]-structure on chain maps, we can equivalently reformulate the quotient by chain homotopy as follows:

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

For $C_\bullet, D_\bullet \in Ch_\bullet(\mathcal{A})$, let

$$
  NHt(C_\bullet, D_\bullet) \hookrightarrow Ch_\bullet(C_\bullet, D_\bullet)
$$

be the [[abelian group|abelian]] [[subgroup]] on those [[chain maps]] which are [[null homotopy|null homotopic]], hence for which there is a [[chain homotopy]] to the [[zero morphism]].

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

For $C_\bullet, D_\bullet \in Ch_\bullet(\mathcal{A})$, there is an [[natural isomorphism]] of [[abelian groups]]

$$
  \mathcal{K}(\mathcal{A})(C_\bullet, D_\bullet)
  \simeq
  Ch_\bullet(\mathcal{A})/NHt(C_\bullet, D_\bullet)
$$

of the [[hom-objects]] in $\mathcal{K}(\mathcal{A})$ with the [[quotient group]] of all chain maps by those whose are null homotopic.
=--

### Triangulated category structure

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

A **[[distinguished triangle]]** in $\mathcal{K}(\mathcal{A})$ is a sequence of [[morphisms]] of the form

$$
  Y_\bullet \to (X\sslash Y)_\bullet \to X[1]_\bullet \stackrel{}{\to} Y[1]_\bullet
  \,,
$$

where $C[1]_\bullet$ denotes the [[suspension of a chain complex]], which is [[isomorphism|isomorphic]] to the [[image]] under the projection fuctor def. \ref{ProjectionFunctor} of a [[mapping cone]] sequence in $Ch_\bullet(\mathcal{A})$

$$
  Y_\bullet \stackrel{}{\hookrightarrow} cone(f)_\bullet
   \stackrel{}{\to}
  X[1]_\bullet
   \stackrel{f[1]_\bullet}{\to}
  Y[1]_\bullet
$$

of a [[chain map]] $f_\bullet : X_\bullet \to Y_\bullet$ in $Ch_\bullet(\mathcal{A})$.

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###### Theorem

With

* [[shift functor]] given by [[suspension of a chain complex|suspension of chain complexes]];

* [[distinguished triangles]] as in def. \ref{DistinguishedTriangles}

the category $\mathcal{K}(\mathcal{A})$ is a [[triangulated category]].

=--

### Relation to the derived category
 {#RelationToDerivedCategory}

See at _[[derived category]]_ in the section [derived category - Properties](derived+category#Properties).

## Examples

### Chain homotopies that ought to exist but do not
 {#ChainHomotopiesThatOughtToExistButDoNot}

We discuss some basic examples of [[chain maps]]
that ought to be identified in [[homotopy theory]], but which are not yet identified in $\mathcal{K}(\mathcal{A})$, but only in the [[derived category]] $\mathcal{D}(\mathcal{A})$.

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

In $Ch_\bullet(\mathcal{A})$ for $\mathcal{A} = $ [[Ab]] consider the [[chain map]]

$$
  \array{
    \cdots &\to& 0 &\to& 0 &\to& 0 &\to& \mathbb{Z}_2
    \\
    && \downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{id}}
    \\
    \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2
  }
  \,.
$$

The [[codomain]] of this map is an [[exact sequence]], hence is [[quasi-isomorphism|quasi-isomorphic]] to the 0-chain complex. Therefore in [[homotopy theory]] it should behave entirely as the 0-complex itself. In particular, every [[chain map]] to it should be [[chain homotopy|chain homotopic]] to the [[zero morphism]] (have a _[[nLab:null homotopy]]_).

But the above chain map is chain homotopic precisely only to itself. This is because the degree-0 component of any chain homotopy out of this has to be a homomorphism of abelian groups $\mathbb{Z}_2 \to \mathbb{Z}$, and this must be the 0-morphism, because $\mathbb{Z}$ is a [[free group]], but $\mathbb{Z}_2$ is not.

This points to the problem: the components of the domain chain complex are not _free enough_ to admit sufficiently many maps out of it.

Consider therefore a [[free resolution]] of the above domain complex by the [[quasi-isomorphism]]

$$
  \array{
     \cdots
    &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z}
    \\
    && \downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{mod\,2}}
    \\
    \cdots &\to& 0 &\to& 0 &\to& 0 &\to& \mathbb{Z}_2
  }
  \,,
$$

where now the domain complex consists entirely of [[free groups]].
The composite of this with the original chain map is now

$$
  \array{
     \cdots
    &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z}
    \\
    && \downarrow && \downarrow && \downarrow^{0} && \downarrow^{\mathrlap{mod\,2}}
    \\
    \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2
  }
  \,.
$$

This is the corresponding [[resolution]] of the original chain map. And _this_ indeed has a [[null homotopy]]:

$$
  \array{
     \cdots
    &\to& 0 &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z}
    \\
      &&
    \downarrow
     &\swarrow&
    \downarrow
      &\swarrow_{-id}&
    \downarrow^{0}
      &\swarrow_{\mathrlap{id}}&
    \downarrow^{\mathrlap{mod\,2}}
    \\
    \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\stackrel{mod\,2}{\to}& \mathbb{Z}_2
  }
  \,.
$$

Indeed, for this to happen it is sufficient that the [[resolution]] is by a degreewise [[projective object|projective]] complex. This is the statement of _[this lemma](projective+resolution#MapsProjectiveIntoExactAreNullHomotopic)_ at _[[projective resolution]]_.

=--

## Related pages

* [[Hurewicz model structure on chain complexes]]

## References

Lecture notes include

section 3.1 and 7.1 of

* [[Pierre Schapira]], _Categories and homological algebra_ (2011) ([pdf](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf))
 {#Schapira}

[[!redirects homotopy categories of chain complexes]]


homotopy class

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

## Definition

A a _homotopy class_ is an [[equivalence class]] under [[homotopy]]:

For $f \;\colon\; X \to Y$ a [[continuous function]] between [[topological spaces]] which admit the structure of [[CW-complexes]], its _homotopy class_ is the [[morphism]] in the [[classical homotopy category]] that is represented by $f$. The sets of such homotopy classes if often denoted $[X,Y]$ or similar.

Similarly, if $f$ is a base-point preserving function between [[pointed topological spaces]] admitting the structure of [[CW-complexes]], then its homotopy class ("pointed homomotopy class") represents a morphism in the [[homotopy category]] of [[pointed homotopy types]]. The sets of such pointed homotopy classes if often denoted $[X,Y]_\ast$ or similar.

## Examples

### Homotopy groups and Cohomotopy sets

If the [[domain]] $X \simeq S^n$ is an [[n-sphere]], then the homotopy classes of maps $f \colon S^n \to Y$ form the $n$th *[[homotopy group]]* of $Y$.

If the [[codomain]] is an [[n-sphere]] $Y \simeq S^n$, then the homotopy classes of maps $f \colon X \to S^n$ form the [[Cohomotopy set]] of $X$.

## Related concepts

* [[homotopy category]]

* [[cohomology]]

* [[isomorphism class]]

[[!redirects homotopy classes]]

[[!redirects pointed homotopy class]]
[[!redirects pointed homotopy classes]]


homotopy coend

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###Context###
#### Limits and colimits
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[[!include infinity-limits - contents]]
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A **homotopy end** or **coend** is a [[homotopy limit|homotopy (co)limit]] corresponding to an [[end]] or coend.

For now, see:

* [[coend in a derivator]],

* [derived coends over tensors](Quillen+bifunctor#CoendsOverTensors),

* [[homotopy weighted colimits]],

* [[fat geometric realization]]

## Related concepts

* [[homotopy weighted colimit]], [[homotopy weighted limit]],

* [[homotopy colimit]], [[homotopy limit]],

* [[homotopy Kan extension]]

## References

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

* {#ArkhipovOrsted18} [[Sergey Arkhipov]], [[Sebastian Ørsted]], *Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories* ([arXiv:1807.03266](https://arxiv.org/abs/1807.03266))

[[!redirects homotopy coends]]
[[!redirects homotopy end]]
[[!redirects homotopy ends]]


homotopy coequalizer

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#Contents#
* table of contents
{:toc}

## Idea

In generalization of the notion of *[[equalizers]]* from [[category theory]] to [[homotopy theory]],
*homotopy coequalizers* are the special case of [[homotopy colimits]],
where the indexing [[diagram]] is the [[walking parallel pair]], consisting of a [[pair]] of [[parallel morphisms]], i.e., two [[objects]], 0 and 1, and exactly two nonidentity [[morphisms]], both of the form $0\to 1$.

Homotopy coequalizers can be defined in any [[relative category]],
just like [[homotopy colimits]],
but practical computations are typically carried out
in presence of additional structures such as [[model structures]].

## Computation
 {#Computation}

In any [[model category]], the homotopy coequalizer
of a pair of arrows $f,g\colon A\to B$ can be computed as a [[homotopy pushout]] as follows (compare the ordinary expression of [[coequalizer]] as [[pushouts]], [here](coequalizer#CoequalizedAsAPushout)).

First, if $A$ is not [[cofibrant]] and the [[model category]] is not [[left proper]], construct a [[cofibrant replacement]] $q\colon QA\to A$
and replace $(f,g)$ with $(f q,g q)$.

Assume now that $A$ is [[cofibrant]] or the [[model category]] is [[left proper]].

In the special case when the map

$$
  (f,g)
  \,\colon\, A\sqcup A \longrightarrow B
$$
happens to be a [[cofibration]],
one can compute the ordinary [[coequalizer]] of $f$ and $g$,
which is a homotopy coequalizer.

In the general case,
factor the [[codiagonal map]] $\nabla \,\colon\, A\sqcup A \longrightarrow A$
as a [[cofibration]] $A\sqcup A \longrightarrow C A$ followed by a [[weak equivalence]] $CA \longrightarrow A$,
then compute the (ordinary) [[pushout]] of
$$
  CA
    \longleftarrow
  A\sqcup A
   \overset{(f,g)}{\longrightarrow}
  B
  \,.
$$
This is the homotopy coequalizer of $f$ and $g$.

## Examples

In [[simplicial sets]] with [[simplicial weak equivalences]],
the homotopy coequalizer of $f,g\colon A\to B$
can be computed as the pushout
$$
  (\Delta^1 \times A)
    \overset{A\sqcup A}{\amalg}
  B
  \,.
$$
The analogous formula works for [[topological spaces]] with [[weak homotopy equivalences]], then using the [[closed interval]] $\Delta=[0,1]$ and also known as the *[[double mapping cylinder]]*-construction, see for instance [Dwyer, Farjoun & Ravenel (1999), p. 1856](#DwyerFarjounRavenel99) (cf. *[[mapping cylinder]]*).

For [[chain complexes]] with [[quasi-isomorphisms]],
the homotopy coequalizer can be computed
(expanding the analogous formula with $C A =\mathrm{N} \mathbf{Z} [\Delta^1]\otimes A$)
as
$$
  B \oplus A[1]
  \,,
$$
where

$$
  d(b\oplus a)
   \;=\;
  d b+f(a)-g(a) \oplus d a
  \,.
$$

## Related notions

* [[mapping cylinder]]

* [[coproduct]]

* [[homotopy colimit]]

* [[homotopy pushout]]

* [[homotopy coproduct]]

* [[homotopy realization]]

* [[homotopy equalizer]]

## References

* {#DwyerFarjounRavenel99} [[William G. Dwyer]], [[Emmanuel Dror Farjoun]], [[Douglas C. Ravenel]], pp. 1856 in: *Bousfield Localizations of Classifying Spaces of Nilpotent Groups*, Proceedings of the American Mathematical Society **127** 6 (1999) [[jstor:119499](https://www.jstor.org/stable/119499)]

* {#Topospaces} Topospaces, *[Double mapping cylinder](https://topospaces.subwiki.org/wiki/Double_mapping_cylinder)*

[[!redirects homotopy coequalizers]]

[[!redirects double mapping cylinder]]
[[!redirects double mapping cylinders]]


homotopy coherent category theory

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[[!include category theory - contents]]
***
[[!include homotopy - contents]]
</div>

_Homotopy coherent category theory_ or [[enriched homotopy theory]] is the attempt to understand those situations that arise in [[homotopy theory]], [[homotopical algebra]], and non-Abelian [[homological algebra]], in which quite naturally occuring diagrams are not commutative, yet are commutative 'up-to-homotopy'.

If the diagram is just commutative in the [[homotopy category]], that is not much use and one can do little with it.  Surprisingly often however, the homotopies involved in such a diagram's 'almost commutative' nature can be specified, but then the question arises as to whether those homotopies form some sort of diagram, so homotopies between composite homotopies become involoved. This begins to look like parts of 2-category theory, or rather '[[higher category theory|higher weak category theory]]' and the development of homotopy coherent category theory was initiated in an attempt to merge homotopy theory with categorical tools for handling higher categories.

In particular:

* [[homotopy theory]] using [[model category|model categories]] and similar structures;
* [[enriched category theory]].

In the case that the category $V$ one is [[enriched category|enriching over]] is itself a [[model category]] or at least a [[category with weak equivalences]] one wishes to generalize [[limit]]s and in particular the [[weighted limit]]s such as as [[end]]s and coends in [[enriched category theory]] to constructions which satisfy the familiar universal properties only _up to coherent [[homotopy]]_.

## References

These articles deal with the theory of homotopy coherent diagrams:

*  R. Vogt, Homotopy limits and colimits , Math. Z., 134, (1973), 11 – 52.

* J.-M. Cordier and T. Porter, Vogt's theorems on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986), 65–90.

* J.-M. Cordier and T. Porter,  Maps between homotopy coherent diagrams, Top. and its Applications, 28 (1988) 255-275.

* J.-M. Cordier and T. Porter, Fibrant diagrams, rectifications and a construction of Loday, J. Pure Appl. Alg 67 (1990), 111–124.

A discussion of homotopy limits is in

* D. Bourn and J.-M. Cordier, A general formulation of homotopy limits , J. Pure Appl. Algebra, 29, (1983), 129–141,

* J.-M. Cordier, Sur les limites homotopiques de diagrammes homotopiquement  cohérents, Comp. Math. 62 (1987), 367–388.

In

* J-M Cordier and T. Porter, _Homotopy coherent category theory_, Trans. Amer. Math. Soc. 349 (1997) 1-54. ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.3273))

a main point is the definition and discussion of a [[homotopy coherent end]] for the case of enrichment over the [[model structure on simplicial sets|model category of simplicial sets]].

In

* Mike Shulman, _Homotopy limits and colimits and enriched homotopy theory_ ([arXiv](http://arxiv.org/abs/math.AT/0610194v1))

the general issue of [[enriched homotopy theory]] is addressed and  [[enriched homotopical category|enriched homotopical categories]] are introduced, which are a coherent combination of the notion of [[enriched category]] with that of [[homotopical category]]


homotopy coherent diagram

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#### Higher category theory
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
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{:toc}

## Idea

A _homotopy coherent diagram_ is a [[diagram]] of [[object]]s in a [[homotopical category]], where [[commutative diagram|commutativity]] is replaced by explicit [[homotopies]], those homotopies are to then be [[coherent]]ly linked by higher homotopies ... and so on.

It is a model for an [[(∞,1)-functor]].

The idea perhaps intuitively makes sense but the management of the interactions between the various levels of homotopy requires care. The ideas were handled in various ways , but we will concentrate on approaches linked to the initial work of [[Michael Boardman]] and [[Rainer Vogt]] and then developed further by [[Jean-Marc Cordier]] and [[Tim Porter]]. There was another more-or-less equivalent approach by [[Dwyer]] and [[Kan]], but this did not make the relationship with homotopy coherence quite so explicit.

We will often use h.c. as an abbreviation for ''homotopy coherent''.

## Definition

### In components

The original definition of [Vogt, 1973](#Vogt) is essentially the following.

Suppose now that we have the h.c. diagram $F : S(\mathbb{A}) \to \mathcal{B}$.  This is specified by assignments:

* to each [[object]] $a$ of $\mathbb{A}$, it assigns an object $F(a)$ of $\mathcal{B}$;

* for each string of composable [[morphism]]s in $\mathbb{A}$,

$$\sigma = (f_0, \ldots, f_n)$$

starting at $a$ and ending at $b$, a simplicial map

$$F(\sigma) : S(\mathbb{A})(0,n+1) \to \mathcal{B}(F(a), F(b)),$$

that is, a higher homotopy

$$F(\sigma) : \Delta[1]^n \to \mathcal{B}(F(a), F(b)),$$

such that

(i) if $f_0 = id$, $F(\sigma) = F(\partial_0\sigma)(proj \times \Delta[1]^{n-1})$

(ii) if $f_i = id$, $0\lt i \lt n$

$$F(\sigma) = F(\partial_i\sigma(.(I^i \times m \times I^{n-i}),$$

where $m : I^2 \to I$ is the multiplicative structure on $I = \Delta[1]$ by the 'max' function on $\{0,1\}$;

(iii) if $f_n = id$, $F(\sigma) = F(\partial_n \sigma)(I^{n-1} \times proj)$;

(iv)$_{i}$  $F(\sigma)|(I^{i-1}\times \{0\} \times I^{n-i}) = F(\partial_i\sigma), 1 \leq i \leq n-1$;

(v)$_{i}$  $F(\sigma)|( I^{i-1}\times \{1\} \times I^{n-i}) = F(\sigma^\prime_i) . F(\sigma_i)$, where $\sigma_i = (f_0, \ldots, f_{i-1})$
and $\sigma^\prime = (f_i, \ldots, f_n)$.  We have used $\partial_i$ for the face operators in the nerve of $\mathbb{A}$.

This original form can be very useful for checking (bare hands!) within an application that a diagram is h.c., although the $SSet$-functor approach is for many uses more compact and maniable and allows functorial constructions more easily. The link with the [[bar construction]] and [[comonadic resolution]] approaches give suggestive links to interpretation of cohomology classes.

### As algebras over an operad.

For $\mathcal{E}$ a [[symmetric monoidal category]], and $C$ a small $\mathcal{E}$-[[enriched category]],
there is an [[operad]] $Diag_C$ whose [[algebras over an operad]] are $\mathcal{E}$-[[enriched functor]]s

$$
  F : C \to \mathcal{E}
$$

hence $C$-[[diagram]]s in $\mathcal{E}$.

If $\mathcal{E}$ is also a [[monoidal model category]] with an [[interval object]] $H$ in a sufficiently nice way, then there exists the [[Boardman-Vogt resolution]]

$$
  HoCoDiag_C := W(H, Diag_C)
  \,.
$$

The algebras over this operad are then precisely homotopy coherent diagrams over $C$ in $\mathcal{E}$. For $\mathcal{E} = $ [[Top]] regarded with the standard [[model structure on topological spaces]] and $H = [0,1]$ the standard interval, this reproduces the ordinary notion of homotopy coherent diagrams ([BergerMoerdijk](#BergerMoerdijkAlgebras))

## Properties

### Equivalence

(i) If $X : \mathbf{A}\to $ [[Top]] is a commutative diagram and we
replace some of the $X(a)$ by homotopy equivalent $Y(a)$ with specified
homotopy equivalence data:

$$f(a) : X(a) \to Y(a), \quad g(a) : Y(a) \to X(a)$$

$$H(a) : g(a)f(a) \simeq Id, \quad K(a) : f(a)g(a) \simeq Id,$$

then we can combine these data into the construction of a h. c. diagram $Y$
based on the objects $Y(a)$ and homotopy coherent maps

$$f : X\to Y, \quad g :
Y \to X,  etc.,$$

making $X$ and $Y$ homotopy equivalent as h.c. diagrams.

(This applied to a $G$-space, $X$, shows that if we replace $X$ by
a homotopy equivalent $Y$, then $Y$ will be a h. c. version of a $G$-space, i.e. a
h. c. diagram of shape $BG$, the corresponding one object groupoid to $G$.)

### Cordier's Homotopy coherent nerve

+-- {: .un_theorem}
###### Theorem
**Cordier (1980)**

For each a [[small category]] $\mathbb{A}$, the [[sSet]]-[[enriched category]]
${S(\mathbb{A})}$ defined in [[homotopy coherent nerve#the_cosimplicial_category]] is such that a
h.c. diagram of shape ${\mathbb{A}}$ in [[Top]] is given precisely by an
[[sSet]]-[[enriched functor]]

$$
  F : {S(\mathbb{A})} \to Top
$$

=--

This suggested the extension of h.c. diagrams to other contexts such as a
general locally Kan $SSet$-category, $\mathcal{B}$ and further suggests the definition of homotopy coherent diagram in a $\mathcal{S}$-category and thus a [[homotopy coherent nerve]] of an $SSet$-category.  This was first done by Cordier and Porter in 1986, (see references). If the $SSet$-category is "locally Kan", this homotopy coherent nerve is a [[quasicategory]].

To understand simplical h.c. diagrams and thus the h.c. simplicial [[nerve]] $N(\mathcal{B})$, we  unpack the definition of homotopy coherence, for convenience, repeating some points made in [[homotopy coherent nerve]].

The first thing to note is that for any $n$ and $0\leq i\lt j\leq n$, $S[n](i,j) \cong \Delta[1]^{j-i-1}$, the $(j-i-1)$-cube given by the product of $j-i-1 $ copies of $\Delta[1]$.  Thus we can reduce the higher homotopy data to being just that, maps from higher dimensional cubes.

Next some notation:

Given simplicial maps

$$f_1: K_1 \to \mathcal{B}(x,y),$$

$$f_2: K_2 \to \mathcal{B}(y,z),$$

we will denote the composite

$$K_1 \times K_2 \to \mathcal{B}(x,y)\times \mathcal{B}(y,z) \stackrel{c}{\to} \mathcal{B}(x,z)$$

just by $f_2.f_1$ or $f_2f_1$.  (We will seen this in the h.c. diagram below for
$\mathbb{A} = [3]$.  $X(123)X(01)$ is actually $X(123)(I \times X(01) )$, whilst $X(23)X(012)$ is exactly what it states.)

### Rectification

Every homotopy coherent diagram is weakly equivalent to a strict diagram, a phenomenon known as _[[rectification]]_.

#### Vogt's theorem

+-- {: .un_theorem}
###### Theorem
**(Vogt)**

If $\mathbf{A}$ is a small category, there is a category $\mathbf{Coh(A,Top)}$
  of h.c. diagrams and homotopy classes of h. c. maps between them.  Moreover
  there is an equivalence of categories

$$\mathbf{Coh(A,Top)} \stackrel{\simeq}{\to} \mathbf{Ho(Top^A)}.$$

=--

#### Berger-Moerdijk theorem

If we think of hc diagrams as algebras over an operad, then this rectification is a special case of the general rectification theorem for such algebras. See [[model structure on algebras over an operad]] for details.

## Examples

**h.c. diagrams in a category with [[cylinder functor]], denoted $-\times I$**

1.  A diagram indexed by the small category, $[2]$.

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is h.c. if there is specified a homotopy

$$X(012) : X(0)\times I \to X(2),$$

$$X(012) : X(02) \simeq X(12)X(01).$$

2. For a diagram indexed by $[3]$: Draw a 3-simplex, marking the vertices
$X(0), \ldots, X(3)$, the edges $X(ij)$, etc., the faces $X(ijk)$, etc.  The
homotopies $X(ijk)$ fit together to make the sides of a square

$$
 \begin{matrix}
  X(1 3)X(0 1)&\xrightarrow{X(1 2 3)X(0 1)}&X(2 3)X(1 2)X(0 1)\\
  \mathllap{X(0 1 3)}\left\uparrow\space{30}{20}{0}\right.& &\left.\space{30}{20}{0}\right\uparrow\mathrlap{X(2 3) X(0 1 2)}\\
  X(0 3)&\xrightarrow[\qquad X(0 2 3)\qquad]{\quad}&X(2 3)X(0 2)
 \end{matrix}
$$
and the
diagram is made h.c. by specifying a second level homotopy

$$X(0123) : X(0)\times I^2\to X(3)$$

filling this square, in the sense that restricting to each side of the square in the double homotopy gives the correspondingly labelled homotopy from the diagram.

These can be continued for larger $[n]$, and the results glued together to
make larger h.c. diagrams.  Of course, this is not how it is actually done, but may
provide some help in understanding  the basic idea.

## References and Literature

For [[Rainer Vogt|Vogt]]'s theorem, the original reference is

* {#Vogt73} [[Rainer Vogt]], _Homotopy limits and colimits_, Math. Z., **134** (1973) 11-52 [[doi:10.1007/BF01219090](https://doi.org/10.1007/BF01219090), [eudml:171965](https://eudml.org/doc/171965)]

A generalisation of his theorem using simplicially enriched categories and the [[homotopy coherent nerve]] of such a thing, is to be found in

* [[J.-M. Cordier]] and [[T. Porter]], _Vogt's Theorem on Categories of Homotopy Coherent Diagrams_,  Math. Proc. Camb. Phil. Soc. 100 (1986) pp. 65-90.

A neat application to changing objects in diagrams within a homotopy type can be found in

* J.-M. Cordier and T. Porter, _Maps between homotopy coherent diagrams_, Top. and its Appls.,  28, (1988), 255 – 275.

See also

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

A summary of homotopy coherence can be found in Chapter 5 of

* [[Klaus Heiner Kamps]], [[Tim Porter]], _Abstract Homotopy and Simple Homotopy Theory_, World Scientific 1997 ([doi:10.1142/2215](https://doi.org/10.1142/2215), [GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1))

and in chapter 10 of

* [[Tim Porter]], _[[Crossed Menagerie]]_

The operad-theoretic description of homotopy-coherent diagrams is in

* {#BergerMoerdijkAlgebras}[[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_ ([arXiv:math/0512576](http://arxiv.org/abs/math/0512576))

See [[model structure on algebras over an operad]] for more on this.

[[!redirects homotopy coherent diagrams]]

[[!redirects Vogt's theorem]]

[[!redirects homotopy coherent functor]]
[[!redirects homotopy coherent functors]]

[[!redirects homotopy commutative diagram]]
[[!redirects homotopy commutative diagrams]]


homotopy coherent nerve

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

#Contents#
* table of contents
{:toc}

## Idea

The _homotopy coherent nerve_ (also called _simplicial nerve_) of a [[simplicially enriched category]] is a [[simplicial set]] which includes information about all the higher homotopies present in the hom-spaces.  It generalizes the ordinary [[nerve]] of an ordinary [[category]].

The homotopy coherent nerve operation

$$
  N \colon sSet\text{-}Cat \longrightarrow sSet
  \,.
$$

is induced, by the general machinery of [[nerve and realization]], by a  [[cosimplicial object|cosimplicial]] [[simplicially enriched category]], namely a [[functor]]

$$
  \Delta \longrightarrow sSet\text{-}Cat
$$

from the [[simplex category]] to the category of
[[simplicially enriched categories]] which regards each
$n$-[[simplex]] as a [[sSet]]-[[enriched category]]
with $n$ objects analogous to how the [[orientals]]
regard the $n$-simplex as an [[strict omega-category|strict $n$-category]].

## Definitions

### The cosimplicial $sSet$-category

We here describe the cosimplicial [[sSet]]-[[enriched category]]

$$
  S \colon \Delta \longrightarrow sSet\text{-}Cat
$$

that induces the homotopy coherent nerve.

### A categorical description

Recall that a [[graph]], $\Gamma$, is _[[reflexive graph|reflexive]]_ if for each [[vertex]] $v$ it is equipped with a (specified) [[edge]] $v \to v$. Similarly, a **reflexive [[directed graph]]** has a specified identity edge $i_X \colon X \to X$ on each object (vertex) $X$.

The [[free category]] on a reflexive directed graph has

* as [[objects]] the [[vertices]] of the graph,

* [[identity morphisms]] corresponding to the identity edges,

* non-identity morphisms consisting of sequences of non-identity edges.

The [[free category]]-construction extends to a [[functor]]

$$
  F
  \;\colon\;
  ReflxDGraph \longrightarrow Cat
$$
and as such
is [[left adjoint]] to the [[forgetful functor]]

$$
  F
  \;\colon\;
  Cat \longrightarrow ReflxDGraph
  \,.
$$

Hence the composite $G = F U \colon Cat\to Cat$ constitutes a [[comonad]] on $Cat$.

The [[simplicial resolution]] of this comonad gives an augmented simplicial [[endofunctor]] $S \colon \Delta \longrightarrow sSet\text{-}Cat$ with natural augmentation $S\to Id$, and which is a [[cofibrant replacement]]-construction in the [[Bergner model structure]] on $sSet-Cat$ ("model structure for simplicially enriched categories").

\begin{remark}
Some *words of caution*, as always with simplicial resolutions, there are two conventions which differ by being the opposite simplicial object of each other.  In the original paper, Cordier uses a different one of these conventions from some of the other sources mentioned here. A similar problem occurs in the following combinatorial description as some sources use 'reverse inclusion' where others just use 'inclusion' for the partial order on the poset. This corresponds more or less exactly to the distinction between 'op-lax' and 'lax' functors in the theory of 2-categories.

Because of this, it is always important to test the definition being used on a simple example, $[2]$ will do, before commiting to the use of any specific formulae. We will see this again in another Remark later on in this entry.
\end{remark}

### A combinatorial description

For $[n]$ the finite [[ordinal number|ordinal]] $[n] \coloneqq \{0 \lt 1 \lt  \cdots \lt n\}$ and for $\Delta[n]$ be  standard [[simplicial set|simplicial]] [[simplex|$n$-simplex]], define the [[sSet-enriched category]] $S[n]$ as follows:

* the [[objects]] of $S[n]$ are $\{0,1, \cdots, n\}$;

* the [[hom-objects]] $S[n]_{i,j} \in sSet$ for $i, j \in \{0,1,\cdots, n\}$ are the [[nerves]]

  $$
    S[n](i,j) \coloneqq N(P_{i,j})
  $$

  of the [[poset]] $P_{i,j}$ which is equivalently

  1. the [[poset]] of [[subsets]] of $[i,j]$ that contain both $i$ and $j$ (so in particular if $i \gt j$ then $P(i,j)$ is [[empty set|empty]] and hence so is its nerve) with the partial order is given by reverse inclusion.

   1. the poset of [[path category|paths]] in $[n]$ that start at $i$ and finish at $j$ (hence is empty if $i\gt j$), the order relation is given by 'subdivision', i.e. path $a$ is less than path $b$ in $P(i,j)$ if $a$ visits all the vertices that $b$ does ... and perhaps some others as well.

      Of course, the way you go between the two descriptions is that a path corresponds to the set of vertices it visits and _vice versa_.

Notice that the simplicial set $N(P_{i,j})$ is [[isomorphism|isomorphic]] to the $j-i-1$ cube in $sSet$:

$$
  N(P_{i,j}) = (\Delta[1])^{\times (j-i-1)}
  \,.
$$

Under this isomorphism for instance the vertex $(0,0,1,0,1) \in (\Delta[1])^{\times (j-i-1)}$ corresponds to the subset $\{i,i+3,i+5,j\} \subset [i,j]$ and to the path $i \to i+3 \to i+5 \to j=i+6$.

(We will look at an example after this definition.)

* the [[composition]] operation on hom-objects

  $$
    \circ_{i,j,k} : S[n]_{i,j} \times S[n]_{j,k} \to S[n]_{i,k}
  $$

  is induced by 'concatenation of the corresponding paths' and thus essentially by union of the sets involved.

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

The choice to order paths by reverse inclusion agrees with constructions such as the [[Duskin nerve]]. However, the other convention where they are ordered by inclusion also appears in the literature, such as [[Higher Topos Theory]] definition 1.1.5.1., and corresponds to the convention used in the original paper by Cordier (see earlier Remark). The resulting theory is more or less equivalent in as much as the results true when using one convention have analogous results when using the other.

=--

### The homotopy coherent nerve

The **homotopy coherent nerve** functor

$$
  N
    \coloneqq
  Hom_{sSet Cat}(S[\bullet],-)
  \colon sSet Cat \longrightarrow sSet
$$

is the [[nerve]] defined by the [[cosimplicial object|cosimplicial]] [[sSet-enriched category]] $S \colon \Delta \to sSet Cat$ defined above.

For $C \in sSet Cat$ any [[sSet-enriched category]], the homotopy coherent nerve  $N(C)$ is the [[simplicial set]] uniquely characterized as giving a [[natural isomorphism]] of the form

$$
  Hom_{SSet}\big(\Delta[n], N(C)\big)
   \;\simeq\;
  Hom_{SSet Cat}\big(S[n], C\big)
  \,.
$$

By the general logic of [[nerve and realization]], this functor has a [[left adjoint]]

$$
  S(-) \colon SSet \to SSet Cat
$$

the **realization** functor given by the [[coend]] formula

$$
  S(X)
  \coloneqq
  \int^{[n] \in \Delta} X_n \cdot S[n]
  \,,
$$

also known as the operation of *[[rigidification of quasi-categories]]*.

This functor does extend the functor $S : \Delta \to sSet Cat$  in that there is a canonical isomorphism

$$
  S(\Delta[n]) \cong S[n]
$$

and hence may consistently be named $S$.

## Examples and illustrations

### For the cosimplicial $sSet$-category {#IllustratOfCosimpSSetCat}

We illustrate here the nature of the cosimplicial $sSet$-category $S : [n] \mapsto S[n]$, viewed from the combinatoiral viewpoint above.

We will examine the lowest dimensional cases.

For $n = 0$ there is nothing of note.

For $n = 1$ we have that

$$
  P_{0,1} = \left\{ (0,1) \right\} = \Delta[0] = \Delta[1]^0
$$

is the poset with a single object.

For $n = 2$, there are unique paths in $[2]$ from $[0]$ to $[1]$, and $[1]$ to $[2]$, so the corresponding homs in $S[2]$ are copies of $\Delta[0]$ (or, if you prefer, of $\Delta[1]^0$!). Things are slightly more interesting for $S[2](0,2)$. Looking at this from the 'subsets' viewpoint, as above, there clearly are two subsets of $\{0,1,2\}$ containing both $0$ and $2$, one corresponds to the direct route in $[2]$ from $0$ to $2$, the other goes via $1$ so is $0\to 1\to 2$.

$$
  P_{0,2} =
  \left\{
    (0,1,2) \to (0,2)
  \right\}
  =
  \Delta[1]
  \,,
$$

so in $S[2](0,2)$, there is a 1-simplex $k$ starting at $\{0,1,2\}$ and ending at $\{0,2\}$.

\begin{center}\begin{xymatrix} &1\ar[dr]^{(1,2)}&\\
 0\ar[ur]^{(0,1)} \ar[rr]_{(0,2)}\ar@{}[rr]<3ex>^{\Large{\Downarrow} (0,1,2)}&&2\end{xymatrix}\end{center}

Since $(0,1,2)$ is a product of $(0,1)$ and $(1,2)$, this simplex can also be depicted as $(1,2) \cdot (0,1) \Rightarrow (0,2)$.

Everything else, in higher dimensions, is degenerate, so $S[2](0,2)\cong \Delta[1]$. Sometimes it is useful to think of this 1-simplex as 'rewriting' the direct path to that via 1, all this happening in the free category on the underlying graph of the poset $[2]$. (The construction of $S[n]$ in general has a nice interpretation in terms of higher dimensional [[rewriting]]. This can be given using the language of [[polygraph]]s or [[computad]]s.)

In this example there are no significant compositions. To see examples of those, you need to look at $n = 3$. In $S[3]$, the simplicial hom-sets $S[3](i,j)$ for $(i,j) \neq (0,3)$, can all be analysed by the same sort of argument to the above.  The new features occur in $S[3](0,3)$.  The vertices of this simplicial set are the subsets
corresponding to the direct path $0\to 3$ and then the three others. Rewriting the direct path can be done in two immediate ways, to go via the left or via the right route. Each of these can be 'rewritten' to give the longest path /  largest subset.  There is also, of course, an inclusion of the smallest to the largest of these, so that in total the poset here looks like:

$$
  P_{0,3}
  =
  \left\{
    \array{
       \{0,1,2,3\}&\rightarrow & \{0,1,3\}
       \\
       \downarrow & \searrow &\downarrow\\
       \{0,2,3\}&\rightarrow &\{0,3\}
     }
  \right\}
  =
  \Delta[1]^{\times 2}
  \,.
$$

In addition, there will be 2-simplexes filling the two triangles, coming from the chains $\{0,1,2,3\}\subset \{0,1,3\}\subset \{0,3\}$ and $\{0,1,2,3\}\subset \{0,2,3\}\subset \{0,3\}$ in the poset.

$$
  \array{
    & \nearrow && \searrow^{\mathrlap{\{0,3\}}}
    \\
    & & \Uparrow
    \\
    0  &&\stackrel{\{0,1,3\}}{\to}&& 3
    \\
    && \Uparrow
    \\
    & \searrow && \nearrow_{\mathrlap{\{0,1,2,3\}}}
  }
  \;\;\;\;\;
  \;\;\;\;\;
  \;\;\;\;\;
  \,,
  \;\;\;\;\;
  \;\;\;\;\;
  \;\;\;\;\;
  \array{
    & \nearrow && \searrow^{\mathrlap{\{0,3\}}}
    \\
    & & \Uparrow
    \\
    0  &&\stackrel{\{0,2,3\}}{\to}&& 3
    \\
    && \Uparrow
    \\
    & \searrow && \nearrow_{\mathrlap{\{0,1,2,3\}}}
  }
  \,.
$$

We thus get $S[3](0,3) \cong \Delta[1]^2$, a square.

The composition maps

$$S[3](1,3)\times S[3](0,1)\to S[3](0,3)$$

and similarly for the one with 1 replaced by 2, *are* now fairly obvious.

For $n = 4$, the corresponding diagram for $S[4](0,4)$ gives a cube
but here there is an interesting feature.

Five of the six faces of the cube $|N(P_{0,4})|$ correspond to the  associativity of composition of triples of composable morphisms in $[4]$. These correspond to the 5 faces of the 4-simplex $\Delta[4]$, as depicted for instance at [[oriental]] and at [[monoidal category]].

But the cube has one more face

$$
  \array{
    (0,1,2,3,4) &\to& (0,1,2,4)
    \\
    \downarrow &\searrow& \downarrow
    \\
    (0,2,3,4) &\to& (0,2,4)
  }
$$

which does not correspond to associativity: instead, this encodes the [[exchange law]]

$$
  \array{
    && 1 &&&& 3
    \\
    & \nearrow &\Downarrow& \searrow && \nearrow && \searrow
    \\
    0 &&\to&& 2 &&  && 4
    \\
    &&  &&&& 3
    \\
    & && && \nearrow &\Downarrow& \searrow
    \\
    0 &&\to&& 2 && \to && 4
  }
  \;\;\;\;\;\;
  \simeq
  \;\;\;\;\;\;
  \array{
    && 1 &&&& 3
    \\
    & \nearrow && \searrow && \nearrow &\Downarrow& \searrow
    \\
    0 &&&& 2 && \to && 4
    \\
    && 1 &&&&
    \\
    & \nearrow &\Downarrow& \searrow &&  &&
    \\
    0 &&\to && 2 && \to && 4
  }
$$

or, if preferred, to the fact that

$$S[4](2,4)\times S[4](0,2)\to S[4](0,4)$$

is to be a simplicial map.

A similar phenomenon occurs in higher dimensions.  There are two 'extra faces' in $S[5](0,5)$, and so on.

### For other simplicial sets

The description of $S[n]$ works more generally for the [[nerve]] of a [[poset]]. Explicitly, if $P$ is a poset, $S(NP)$ is isomorphic to the simplicially enriched category with the structure

   * The objects of $S(NP)$ are the elements of $P$
   * The [[hom-object]] $S(NP)(x,y)$ is the nerve of the poset of [[chains]] $S \subseteq P$ with $\min(S) = x$ and $\max(S) = y$, ordered by reverse inclusion
   * Composition is given by taking unions

The simplicially enriched categories constructed from spheres and [[inner horns]] also have simple descriptions.

For $(i,j) \neq (0,n)$, the hom-objects are the same as the ambient simplex: $S(\Lambda^n_k)(i,j) = S(\partial \Delta^n)(i,j) = S(\Delta^n)(i,j)$.

For the remaining hom-object, while $S(\Delta^n)(0,n) \cong \square^{n-1}$ is the $(n-1)$-cube, $S(\partial \Delta^n)(0,n) = \partial \square^{n-1}$ is the boundary, and $S(\Lambda^n_k)(0,n)$ further omits the corresponding face.

To see this, observe that the nondegenerate simplices of $S(\Delta^n)(0,n)$ are either decomposable (i.e. the product of two or more simplices from other hom-sets) or are the image of a top simplex of $S(\Delta^m)(0,m) \to S(\Delta^n)(0,n)$ for some monomorphism $\Delta^m \to \Delta^n$ preserving top and bottom elements, depending on whether the simplex contains the top vertex $(0,n)$.

Thus, the decomposable faces of $S(\Delta^n)(0,n)$ are the ones omitting the top vertex $(0,n)$, and the remaining faces come from inner faces $\Delta^{n-1} \to \Delta^n$.

### For the homotopy coherent nerve

*  Any [[2-category]] gives a simplicially enriched category using the embedding of [[Cat]] into [[sSet]] via the usual [[nerve]] functor. The homotopy coherent nerve of a 2-category considered in this way is, sometimes, called the [[geometric nerve]] of the 2-category. The [[Duskin nerve]] of a [[bicategory]] is an extension of this construction.

   A particular case of this nerve is the nerve of a [[2-group]] [[delooping|considered as]] a 2-category.

## Properties {#Properties}

* If $C \in sSet Cat$ is such that all [[hom-object]]s $C(x,y) \in sSet$ are [[Kan complex]]es, then the homotopy coherent nerve $N(C)$ is a [[weak Kan complex]]/[[quasi-category]].

  If $f : C \to D$ is a morphism of such Kan-complex enriched categories which is a weak equivalence (in the [[model structure on sSet-categories]]) in that

  * the induced functor

    $$
      Ho(f) : Ho(C) \to H(D)
    $$

    on the ordinary underlying [[homotopy category of an (infinity,1)-category|homotopy categories]] (obtained by taking hom-wise connected component sets) is [[essentially surjective functor|essentially surjective]]

  * its component on each hom-object

    $$
      f_{x,y} : C(x,y) \to D(f(x),f(y))
    $$

    is a [[homotopy equivalence]],

  then its homotopy coherent nerve

  $$
    N(f) : N(C) \to N(C)
  $$

  is a [[model structure for quasi-categories|weak equivalence of]] [[quasi-categories]].

  We may think of [[category]] $\Delta[n]$ trivially as a simplicially enriched category. In the [[model structure on sSet-categories]] the object $S[n]$ is a cofibrant replacement of $\Delta[n]$. And Kan-complex enriched categories are fibrant. So on these the homotopy coherent nerve is given by the [[derived hom-space]] functor

  $$
    N(C) = \mathbb{R}Hom(\Delta[\bullet], C)
    \,.
  $$

## Related concepts

* [[rigidification of quasi-categories]]

* [[relation between quasi-categories and sSet-enriched categories]]

### Comonadic resolution
(To be edited)

The use of $S[A]$, above, extends that given at the start of this page. Here $S$ is related  to the [[left adjoint]] of the homotopy coherent nerve, but is defined using a [[comonadic resolution]].  The [[comonad]] comes from the adjunction between small categories and directed graphs with distinguished 'unit' loops. The 'forgetful' part of the adjunction forgets the composition in the category, but remembers that the identity arrows are special. The left adjoint /  'free' part of the adjunction takes a directed graph (with distinguished 'identity' loops, and forms the free category on the non-identity arrows. As usual, we can form a [[comonad]] from this  and hence form a functorial [[simplicial resolution]] of any small category, $A$.

This can also be seen to be a case of a [[bar resolution]] construction, related to the [[bar construction]]. Here the adjoint pair also give a [[monad]] on   directed graphs with distinguished 'unit' loops and the small category $A$ is an algebra for this monad.

Since the functors involved preserve the identities on the objects of $A$, the resulting simplicial category  is a simplicially enriched category, and this is $S[A]$.  The $n$-dimensional arrows between objects, $a$ and $b$ in $S[A]$ correspond to a  path from $a$ to $b$ in $A$ containing no identity arrows, together with a bracketting of the
resulting string having depth $n$.

### $W$-Construction of topological operads {#WConstruction}

By hom-wise precomposition with the singular complex functor

$$
  Sing : Top \to sSet
  \,,
$$

which is a [[monoidal functor]], the homotopy coherent nerve extends to a nerve of [[Top]]-[[enriched category|categories]]

$$
  N : Top Cat \to sSet
  \,.
$$

As such, it is a special case of the [[Michael Boardman|Boardman]]-[[Rainer Vogt|Vogt]] [[W-construction]] for [[model structure on operads|cofibrant replacement of]] topological [[operads]]. See also _[[dendroidal homotopy coherent nerve]]_.

In this construction, roughly, for $T$ a [[tree]] in an operad $O$, the tree is replaced with the topological space $e(T) \to [0,1]$ of maps from the set of edges of $T$ to the topological unit interval.

We may restrict this construction to the $n$-[[simplex]] $\Delta[n]$, regarded as a [[category]] and then trivially regarded as a $Top$-category. Then a tree in $\Delta[n]$ is necessarily a linear tree $\to \to \cdots \to$ of some length $k$ and is hence mapped to the topological space of functions $k \to [0,1]$, i.e. to the space $[0,1]^k$. This is the [[geometric realization]] of the simplicial cubes $(\Delta[1])^k$ that we saw above.

### Relation to quasi-categories

As mentioned above, the simplicial or h.c. nerve, together with its [[adjoint functor|left adjoint]], serves to relate the two models of [[(∞,1)-category|(∞,1)-categories]] given by [[quasi-category|quasi-categories]] and [[simplicially enriched category|simplicially enriched categories]].

The homotopy coherent nerve extends to a [[Quillen equivalence]] between the [[Joyal model structure]] $SSet_{Joyal}$ that models [[quasi-categories]] and the [[model structure on SSet-categories]].

See

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

for details.

### Models for $(\infty,1)$-categories

[[!include table - models for (infinity,1)-operads]]

### Other kinds of nerves

* [[nerve]]

* [[Duskin nerve]]

* [[∞-nerve]]

* **homotopy coherent nerve**

* [[dg-nerve]]

## History

The original motivation for the introduction of the homotopy coherent nerve is that it provides a neat simplicial formulation of idea of [[homotopy coherent diagram|homotopy coherent diagrams]]. Homotopy coherent algebraic structures were studied in the 1970s by [[Michael Boardman|Boardman]] and [[Rainer Vogt|Vogt]] in joint work,  and then Vogt individually looked at homotopy coherent diagrams.  The  homotopy coherent nerve was first explicitly defined by [[Jean-Marc Cordier|Cordier]] (reference below).
He  realised that, with a slight modification in the definition, Vogt's definition of homotopy coherent diagram, indexed by a small category $A$, say, corresponded exactly to a simplicially enriched functor from the $SSet$-category $S[A]$ to the $SSet$-category $Top$. They thus also corresponded to simplicial maps from the [[nerve]] of $A$ to $N(Top)$, (although that latter object was 'too large' to be a simplicial 'set'). This allowed a good definition of homotopy coherent diagrams in arbitrary simplicially enriched categories to be given.

This definition works best when the simplicially enriched category is 'locally Kan', in other words it is enriched in the category of [[Kan complex|Kan complexes]]. These locally Kan $SSet$-categories are the fibrant ones in the [[model structure on sSet-categories]].

Cordier and [[Tim Porter|Porter]] (1986) proved that if $C$ is a locally Kan simplicially enriched category then $N(C)$ is a '[[weak Kan complex]]', in other words, a [[quasi-category]].  Some of the main ideas behind this result can be traced to [[Rainer Vogt|Vogt]]'s paper of 1973.

In more modern terminology as [[Kan complex]]es can be considered as [[∞-groupoid]]s, these locally Kan simplicially enriched categories are one particularly nice model for an [[(infinity,1)-category]], and so this result is one of the earliest giving the transition from one model for [[(infinity,1)-categories]] to another, the 'weak Kan complexes' or [[quasi-categories]].

## References and Literature.

The homotopy coherent nerve operation was introduced, explicitly, in

* {#Cordier82} [[Jean-Marc Cordier]], _Sur la notion  de diagramme homotopiquement cohérent_, Cahier Top. et Geom. Diff. XXIII 1 (1982) 93-112 [[numdam:CTGDC_1982__23_1](http://www.numdam.org/numdam-bin/feuilleter?id=CTGDC_1982__23_1)]

making a link with earlier work in

* R. D. Leitch, _The homotopy commutative cube_, J. London Math. Soc. **2** 9 (1974) 23-29 [[doi:10.1112/jlms/s2-9.1.23](https://doi.org/10.1112/jlms/s2-9.1.23)]

as well as with [Vogt (1973)](#Vogt73) and with

* [[Michael Boardman]], [[Rainer Vogt]], *Homotopy invariant algebraic structures on topological spaces*, Lec. Notes in Math. __347__ Springer (1973) [[doi:10.1007/BFb0068547](https://doi.org/10.1007/BFb0068547)]

Poof of an [[equivalence of categories|equivalence]] between the category obtained by inverting the 'levelwise' homotopy equivalence in a category of diagrams, and the [[homotopy category]] of [[homotopy coherent diagrams]]:

* {#Vogt73} [[Rainer Vogt]], *Homotopy limits and colimits*, Math. Z. __134__ (1973) 11-52 [[eudml:171965](https://eudml.org/doc/171965), [doi:10.1007/BF01219090](https://doi.org/10.1007/BF01219090)]

and explicit proof that the homotopy coherent nerve of a locally Kan simplicially enriched category is a [[quasicategory]], as well as the harder result on when [[outer horns]] in this quasicategory can be filled:

* {#CordierPorter86} [[Jean-Marc Cordier]], [[Tim Porter]], *Vogt's theorem on categories of homotopy coherent diagrams*, Math. Proc. Cambridge Philos. Soc. __100__ (1986) 65-90 [[doi:10.1017/S0305004100065877](https://doi.org/10.1017/S0305004100065877), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/Cordier-Porter.pdf)]

Two other papers are relevant to this:

*  [[Jean-Marc Cordier]], [[Tim Porter]], _Maps between homotopy coherent diagrams_, Top. and its Appl. __28__, (1988), 255 – 275.

* [[Jean-Marc Cordier]], [[Tim Porter]], _Fibrant diagrams, rectifications and a construction of Loday_, J. Pure. Applied Alg. __67__, (1990), 111 – 124.

An elementary discussion of the concept of homotopy coherence forms Chapter V of

*  [[K. H. Kamps]], [[Tim Porter]], _Abstract homotopy and simple homotopy theory_, World Scientific 1997.

For the role played by Cordier's simplicial nerve in the context of relating quasi-categories to simplicially enriched categories as models for $(\infty,1)$-categories see

* [[André Joyal]], *[[The Theory of Quasi-Categories and its Applications]]*, lectures at *[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)*, CRM (2008) [[pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]]]

* [[André Joyal]], *Notes on quasi-categories* (2008) [[pdf](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf), [[JoyalNotesOnQuasiCategories.pdf:file]]]

* [[Jacob Lurie]], section 1.1.5 in: *[[Higher Topos Theory]]* (2009)

This emphasizes the [[adjoint functor|adjunction]] corresponding to the homotopy coherent ("simplicial") nerve construction.

A review of this latter aspect is also in

* {#Dhand} Vivek Dhand, _The simplicial nerve of a simplicial category_ ([pdf](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=374dc56d9a4352b8f9f71cc12dd7331e9285bb0c), [[Dhand-SimplicialNerve.pdf:file]])

* Mitya Boyarchenko, _Notes and exercise on $\infty$-categories_ ([pdf](http://math.uchicago.edu/~mitya/langlands/quasicategories.pdf))

* [[Vladimir Hinich]], _Simplicial nerve in Deformation theory_ ([arXiv:0704.2503](http://arxiv.org/abs/0704.2503))

* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets and simplicial operads_, [arxiv/1109.1004](http://arxiv.org/abs/1109.1004) (a Quillen equivalence for Segal vs. simplicial operads using coherent nerve)

* [[Emily Riehl]], _On the structure of simplicial categories associated to quasi-categories, Math. Proc. Camb. Phil. Soc. 150 (2011), 489 - 504.

For more references see [[relation between quasi-categories and simplicial categories]].

Two query-discussions on terminology and concrete description of the coherent/"simplicial" nerve are archived at nForum [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=754&Focus=20607#Comment_20607).  For an overview of the 2009 paper by Dugger and Spivak, see also:

* [[Emily Riehl]], [Understanding the homotopy coherent nerve](https://golem.ph.utexas.edu/category/2010/04/understanding_the_homotopy_coh.html), n-Category Café, 29 April 2010.

Relating the [[Dwyer-Kan loop groupoid]]-construction to the homotopy coherent nerve-construction:

* {#MinichielloRiveraZeinalian23} [[Emilio Minichiello]], [[Manuel Rivera]], [[Mahmoud Zeinalian]], *Categorical models for path spaces*, Advances in Mathematics **415** (2023) 108898 [[arXiv:2201.03046](https://arxiv.org/abs/2201.03046), [doi:10.1016/j.aim.2023.108898](https://doi.org/10.1016/j.aim.2023.108898)]

[[!redirects homotopy coherent nerves]]
[[!redirects homotopy coherent nerve functor]]
[[!redirects homotopy coherent nerve functors]]
[[!redirects homotopy-coherent nerve]]
[[!redirects homotopy-coherent nerves]]
[[!redirects coherent nerve]]
[[!redirects coherent nerves]]
[[!redirects nerve of a simplicial category]]
[[!redirects simplicial nerve of simplicial categories]]


homotopy coinvariant

[[!redirects homotopy coinvariants]]

(...)

[[homotopy quotient]], homotopy [[coinvariant]]

(...)

[[!include homotopy type representation theory -- table]]


homotopy coinvariants functor

#Contents#
* table of contents
{:toc}

## Idea

Given a [[monoidal category]] $(\mathbf{M},\otimes,I)$ and a [[comonoid]] $C$ in $\mathbf{M}$ with coaugmentation $\eta:I\to C$, one can define the following pair $Triv \dashv Coinv$ of [[adjoint functor]]s:

$$ Triv: \mathbf{M}\to \mathbf{Comod}_C, \,\,\,\,X\mapsto X\otimes\eta $$

$$Coinv:\mathbf{Comod}_C\to \mathbf{M},\,\,\,\,(M,\rho)\mapsto M^{co C}:=M\Box_C I $$

where $\Box_C$ denotes the [[cotensor product]] bifunctor and $\rho:M\to M\otimes C$ is a right $C$-[[coaction]]. $Triv$ is called the __(co)free__ or __trivial comodule functor__ and $Coinv$ the __functor of coinvariants__.

If $(\mathbf{M},\otimes,I)$ is in fact a [[monoidal model category]], then we can ask whether this pair of functors is a Quillen pair. If so then the the **homotopy coinvariants functor** is the total right derived functor

$$ \mathbb{R}Coinv: Ho(\mathbf{Comod}_C)\to Ho(\mathbf{M}).$$

Given a $C$-[[comodule]] $(M,\rho)$, any representative of $\mathbb{R}Coinv(M,\rho)$ is called a *model* of the homotopy coinvariants of $M$.

## Related concepts

* [[coinvariant]]

## References

* K. Hess, *Homotopic Hopf-Galois extensions: foundations and examples*, [arxiv/0902.3393](http://arxiv.org/abs/0902.3393)


homotopy coproduct

[[!redirects homotopy coproducts]]
## Idea

__Homotopy coproducts__ are a special case of [[homotopy colimits]],
when the indexing diagram is a [[discrete category]].

Homotopy coproducts can be defined in any [[relative category]],
just like [[homotopy colimits]],
but practical computations are typically carried out
in presence of additional structures such as [[model structures]].

## Computation

In any [[model category]], the homotopy coproduct
of a family of objects $\{A_i\}_{i\in I}$
can be computed by [[cofibrantly replacing]] each $A_i$
and computing the (ordinary) [[coproduct]]
of the resulting family $\{QA_i\}_{i\in I}$ of [[cofibrant replacements]].

If [[weak equivalences]] are closed under small [[coproducts]],
then homotopy coproducts can be computed as ordinary coproducts,
because the map $\coprod_{i\in I}QA_i\to \coprod_{i\in I}A_i$
is a [[weak equivalence]].

## Examples

In [[simplicial sets]] with [[simplicial weak equivalences]],
[[topological spaces]] with [[weak homotopy equivalences]],
and [[chain complexes]] with [[quasi-isomorphisms]],
homotopy coproducts can be computed as ordinary coproducts
because [[weak equivalences]] are closed under small [[coproducts]].

In [[commutative differential graded algebras]] with [[quasi-isomorphisms]] homotopy coproducts
can be computed by resolving each object by a [[cofibrant commutative differential graded algebra]], and then computing their ordinary [[tensor product]].

## Related notions

* [[coproduct]]

* [[homotopy colimit]]

* [[homotopy pushout]]

* [[homotopy coequalizer]]

* [[homotopy realization]]

[[!redirects homotopy coproducts]]


homotopy dimension

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

#Contents#
* table of contents
{:toc}

## Definition

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

An [[(∞,1)-topos]] $\mathcal{X}$ has **homotopy dimension** $\leq n \in \mathbb{N}$ if every [[n-connected|(n-1)-connected]] [[object]] $A$ has a [[global element]], a [[morphism]]  $* \to A$ from the
[[terminal object in an (∞,1)-category|terminal object]] into it.

=--

This appears as [[Higher Topos Theory|HTT, def. 7.2.1.1]].

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

An [[(∞,1)-topos]] $\mathcal{X}$ is **locally of homotopy dimension** $\leq n \in \mathbb{N}$ if there exists a collection $\{U_i \in \mathcal{X}\}$ of [[object]]s such that

* the $\{U_i\}$ generate $\mathcal{X}$ under [[(∞,1)-colimit]]s;

* each [[over-(∞,1)-topos]] $\mathcal{X}/U_i$ has homotopy dimension $\leq n$.

=--

This appears as [[Higher Topos Theory|HTT, def. 7.2.1.8]].

## Properties

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

If an [[(∞,1)-topos]] $\mathcal{X}$ is locally of homotopy dimension $\leq n$ for some $n \in \mathbb{N}$ then it is a [[hypercomplete (∞,1)-topos]].

=--

This appears as [[Higher Topos Theory|HTT, cor. 7.2.1.12]].

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

If $\mathcal{X}$ has homotopy dimension $\leq n$ then it also has [[cohomological dimension]] $\leq n$.

The converse holds if $\mathcal{X}$ has finite homotopy dimension and $n \geq 2$.

=--

This appears as [[Higher Topos Theory|HTT, cor. 7.2.2.30]].

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

An [[(∞,1)-topos]] $\mathcal{X}$ has homotopy dimension $\leq n$ precisely if the [[global section]] [[(∞,1)-geometric morphism]]

$$
  (\Delta \dashv \Gamma) : \mathcal{X}
    \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
$$

has the property that $\Gamma$ sends $(k\geq n)$-[[n-connective|connective]] morphisms to $(k-n)$-[[n-connective|connective]] morphisms.

=--

This is [[Higher Topos Theory|HTT, lemma 7.2.1.7]]

## Examples

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

Up to [[equivalence in an (∞,1)-category|equivalence]], the unique [[(∞,1)-topos]] of homotopy dimension $\leq -1$ is the the [[terminal category]] $ * \simeq Sh_{(\infty,1)}(\emptyset)$.

=--

This is [[Higher Topos Theory|HTT, example. 7.2.1.2]].

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

An object $X \in \mathcal{X}$ is $(-1)$-[[connected]] if the morphism $X \to *$to the [[terminal object in an (∞,1)-category]] is. This is the case if it is an [[effective epimorphism]].

Since the [[global section]] [[(∞,1)-functor]] is corepresented by the terminal object, $X$ is 0-connective precisely if $\Gamma(X) \to \Gamma(*) = *$ is an epimorphism on connected components. By the discussion at [[effective epimorphism]], this is the case precisely if $\Gamma(X) \to *$ is an effective epimorphism in [[∞Grpd]].

So $\mathcal{X}$ has homotopy dimension $\leq 0$ if $\Gamma$ preserves effective epimorphisms. This is the case if it preserves [[finite limit|finit]] [[(∞,1)-limit]]s (the [[(∞,1)-pullback]]s defining a [[Cech nerve]]) and all [[(∞,1)-colimit]]s (over the resulting Cech nerve). being a [[right adjoint|right]] [[adjoint (∞,1)-functor]] $\Gamma$ always preserves [[(∞,1)-limit]]s.  If $\mathcal{X}$ is [[local (∞,1)-topos|local]] then $\Gamma$ is by definition also a [[left adjoint]] and hence also preserves [[(∞,1)-colimit]]s.

=--

+-- {: .num_prop #LocalToposesHaveHomotopDimension0}
###### Proposition

Every [[local (∞,1)-topos]] has homotopy dimension $\leq 0$.

=--

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

Let

$$
  (\Delta \dashv \Gamma \dashv \nabla)
  :
  \mathbf{H} \to \infty Grpd
$$

be [[generalized the|the]] [[global section geometric morphism|terminal geometric morphism]] of the local $(\infty,1)$-topos, with $\nabla$ being the extra [[right adjoint]] to the [[global section]] [[(∞,1)-geometric morphism]] functor that characterizes locality.

By prop \ref{RecognitionByGlobalSections} it is sufficient to show that $\Gamma$ send [[(-1)-connected]] morphisms to (-1)-connected morphisms, hence [[effective epimorphism in an (∞,1)-category|effective epimorphisms]] to effective epimorphisms.

By the existence of $\nabla$ we have that $\Gamma$ preserves not only [[(∞,1)-limit]]s but also [[(∞,1)-colimit]]s. Since effective epimorphisms are defined as certain colimits over diagrams of certain limits, $\Gamma$ preserves effective epimorphisms.

=--

So in particular for $C$ any [[(∞,1)-category]] with a [[terminal object in an (∞,1)-category|terminal object]], the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ is an [[(∞,1)-topos]] of homotopy dimension $\leq 0$. Notably [[Top]] $\simeq$ [[∞Grpd]] $\simeq PSh_{(\infty,1)}(*)$ has homotopy dimension $\leq 0$.

This is [[Higher Topos Theory|HTT, example. 7.2.1.3]].

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

Every [[(∞,1)-category of (∞,1)-presheaves]] is an [[(∞,1)-topos]] of local homotopy dimension $\leq 0$.

=--

This appears as [[Higher Topos Theory|HTT, example. 7.2.1.9]].

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

If a [[paracompact topological space]] $X$ has [[covering dimension]] $\leq n$, then the [[(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(X) := Sh_{(\infty,11)}(Op(X))$ is an [[(∞,1)-topos]] of homotopy dimension $\leq n$.

=--

This is [[Higher Topos Theory|HTT, theorem 7.2.3.6]].

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

For $X \in $ [[∞Grpd]] $\simeq$ [[Top]] an object, the [[over-(∞,1)-topos]] $\infty Grpd/X$ has homotopy dimension $\leq n$ precisely if $X \in Top$ a [[retract]] in the [[homotopy category of an (∞,1)-category|homotopy category]] $Ho(Top)$ of a [[CW-complex]] of [[dimension]] $\leq n$.

=--

This is [[Higher Topos Theory|HTT, example 7.2.1.4]].

## Related concepts

[[!include notions of dimension -- contents]]

## References

The [[(∞,1)-topos theory|(∞,1)-topos theoretic]] notion is discuss in section  7.2.1 of

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


homotopy equalizer

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

#Contents#
* table of contents
{:toc}

## Idea

__Homotopy equalizers__ are a special case of [[homotopy limits]],
when the indexing [[diagram]] is the [[walking parallel pair]],
which consists of a pair of [[parallel morphisms]],
i.e., two [[objects]], 0 and 1, and exactly two nonidentity [[morphisms]], both of the form $0\to 1$.

Homotopy equalizers can be defined in any [[relative category]],
just like [[homotopy colimits]],
but practical computations are typically carried out
in presence of additional structures such as [[model structures]].

In the absence of nontrivial [[homotopies]] (in a bare [[1-category]]), homotopy equalizers reduce to ordinary [[equalizers]].

## Properties

### Relation to homotopy pullbacks
 {#RelationToHomotopyPullbacks}

In any [[model category]], the homotopy equalizer
of a pair of arrows $f,g\colon A\to B$ can be computed as follows.

First, if $B$ is not [[fibrant]] and the [[model category]] is not [[right proper]], construct a [[fibrant replacement]] $r\colon B\to R B$
and replace $(f,g)$ with $(r f,r g)$.

Assume now that $B$ is [[fibrant]] or the [[model category]] is [[right proper]].

In the special case when the map
$$
  (f,g)
  \;\colon\;
  A \longrightarrow B\times B
$$
happens to be a [[fibration]], we can compute the ordinary [[equalizer]] of $f$ and $g$, which is a homotopy equalizer.

In the general case,
factor the [[diagonal map]] $\Delta \;\colon\; B\to B\times B$
as a [[weak equivalence]] $B \to P B$ followed by a [[fibration]] $P B \longrightarrow B\times B$, then compute the (ordinary) [[pullback]] of
$$
  A \longrightarrow B\times B \longleftarrow P B
  \,.
$$
This is the homotopy equalizer of $f$ and $g$.

See also

* [this Prop.](homotopy+pullback#HomotopyPullbackByFactorizationLemma) at *[[homotopy pullback]]*

and

* [this Prop](https://ncatlab.org/nlab/show/Mayer-Vietoris+sequence#SequenceFromDiagonal) at *[[Mayer-Vietoris sequence]]*.

### Relation to homotopy fibers

For [[pointed model categories]], the homotopy equalizer
of $f\colon A\to B$ and the [[zero morphism]] $0\colon A\to B$ is known as the __[[homotopy fiber]]__ of $f$.
See there for more information.

## Examples

In [[simplicial sets]] with [[simplicial weak equivalences]],
the homotopy equalizer of $f,g\colon A\to B$
can be computed as the pullback
$$A\times_{B\times B}B^{\Delta^1}$$
if $B$ is a [[Kan complex]].
(Otherwise, compose with a [[fibrant replacement]]
like $B\to Ex^\infty B$ first.)
The same formula works for [[topological spaces]] with [[weak homotopy equivalences]], using $\Delta=[0,1]$.

For [[chain complexes]] with [[quasi-isomorphisms]],
the homotopy equalizer can be computed
(expanding the analogous formula with $P B =B^{\mathrm{N} \mathbf{Z} [\Delta^1]}$)
as
$$A\oplus B[-1],$$
where
$$d(a\oplus b)=d a\oplus d b+f(a)-g(a).$$

## Related notions

* [[homotopy coequalizer]]

* [[product]]

* [[homotopy limit]]

* [[homotopy pullback]]

* [[homotopy product]]

* [[homotopy totalization]]

[[!redirects homotopy equalizers]]


homotopy equivalence

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

#Contents#
* table of contents
{:toc}

## Definition

A **homotopy equivalence** between [[topological spaces]] (or other [[objects]] in a [[category]] equipped with a notion of [[homotopy]]) $X$ and $Y$ is a [[morphism]] $f:X\to Y$  which has a [[homotopy inverse]], hence such that there exists a morphism $g:Y\to X$ and [[homotopy|homotopies]] $g \circ f \sim 1_X$ and $f \circ g \sim 1_Y$.

(Note the similarity with the strict notion of [[equivalence of categories]].)

If there exists a homotopy equivalence between $X$ and $Y$ we say that $X$ and $Y$ are **homotopy equivalent** or that they have the same **[[homotopy type]]**.  Being homotopy equivalent is evidently an [[equivalence relation]].

For many purposes, one wants instead [[weak homotopy equivalences]].  Often when speaking of a homotopy equivalence or a _[[homotopy type]]_, one actually means the weak version.  (This is especially likely on this wiki, due to the relationship to $\infty$-[[infinity-groupoids]] suggested by the [[homotopy hypothesis]].)

The [[homotopy category]] of [[Top]] with respect to homotopy equivalences is [[Ho(Top)]]${}_{he}$.

## Examples

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

The following three [[graphs]]

(i.e. the evident [[topological subspaces]] of the [[plane]] $\mathbb{R}^2$ that these pictures indicate) are not [[homeomorphism|homeomorphic]]. But they are [[homotopy equivalence|homotopy equivalent]], in fact they are each homotopy equivalent to the [[disk]] with two points removed, by the homotopies indicated by the following pictures:

> graphics grabbed from [Hatcher](#Hatcher)

=--

More generally, in any [[(∞,1)-category]] $C$ other than [[Top]], a homotopy equivalence in $C$ is just an equivalence in $C$.

## Properties

Any homotopy equivalence is also a [[weak homotopy equivalence]].  Conversely, in the context of [[topological spaces]], any weak homotopy equivalence between [[m-cofibrant spaces]] (spaces that are homotopy equivalent to [[CW complexes]]) is a homotopy equivalence. This is the _[[Whitehead theorem]]_.

The homotopy equivalences are the [[weak equivalences]] in the Strøm [[model structure on topological spaces]].  The [[homotopy category]] resulting from inverting all homotopy equivalences in [[Top]] is the same as that resulting from identifying homotopic maps.

## Strong homotopy equivalences

Sometimes an apparently stronger form of homotopy equivalence is needed.  These seem to have been first noticed by [[Richard Lashof]] in 1970 in work on triangulation and smoothing. The point is one of the interaction between the two homotopies, which are not made explicit in the definition above. Let us set this up slightly differently:

In a homotopy equivalence there are the two maps $f: X\to Y$ and $g: Y\to X$, then two [[homotopy|homotopies]]

$$H : X\times I \to X,   H:g \circ f \sim 1_X$$

and

$$K: Y\times I \to Y,   K : f \circ g \sim 1_Y.$$

The [[whiskering]] actions of maps on homotopies (and more generally in any [[(∞,1)-category]] or category with a [[cylinder functor]]) gives
two homotopies $f\circ g \circ f \sim f$, namely $f_*H = f\circ H : X\times I \to Y$ and $f^*K = K\circ (f\times I)$.  Similarly, of course, there are two homotopies $g\circ f \circ g \sim g$, namely $g_*K$ and $g^*H$.  In the usual definition of homotopy equivalence, there is no coherence required between these. That is handled precisely by the notion of **strong homotopy equivalence**. More precisely Lashof defined

+-- {: .un_defn}
###### Definition
A **strong homotopy equivalence** between spaces $X$ and $Y$ is a quadruple $(f,g,H,K)$,
as above, such that $f_*H \sim f^*K$ and $g_*K\sim g^*H$.
=--

Thus this imposes a minimal coherence condition on the data making up the homotopy equivalence.  The definition clearly can be generalised to any reasonable setting with a notion of homotopy.

For example, in a [[2-category]] where a homotopy is a [[natural isomorphism]], and a homotopy between homotopies is just an [[equality]], a homotopy equivalence is simply an [[equivalence]], while a strong homotopy equivalence is the same as an [[adjoint equivalence]].

The question naturally arises as to whether all homotopy equivalences are strong. [[Rainer Vogt]](1972) proved

+-- {: .un_lemma}
###### Vogt's Lemma
If $f: X\to Y$ be a morphism that is a homotopy equivalence in $Top$, let $g: Y\to X$ be a homotopy inverse and $H:g \circ f \sim 1_X$ a homotopy.  Then there is a homotopy $K: f \circ g \sim 1_Y$ such that $(f,g,H,K)$ is a strong homotopy equivalence.
=--

Various versions of this are known in other settings, e.g. $SSet$-enriched categories.  In a 2-category the corresponding fact is that any equivalence can be improved to an adjoint equivalence, by changing at most one of the 2-cell isomorphisms involved.

Note, though, that the coherence supplied by Vogt's lemma is not required to continue to higher levels. There is no condition of compatibilty between the two 'homotopies between the homotopies'.  In some situations, at least, some higher compatibility is known to be derivable; for instance, any biequivalence in a [[3-category]] can be improved to an adjoint biequivalence.

Vogt's Lemma plays a key role in the proof of Vogt's theorem (cf. [[homotopy coherent diagram]]s).

## Related concepts

* [[simplicial homotopy equivalence]]

* [[equality]]

* [[isomorphism]]

* [[equivalence]]

* [[weak equivalence]]

* [[homeomorphism]]

* **homotopy equivalence**, [[weak homotopy equivalence]]

  * [[equivalence in homotopy type theory]]

  * [[n-equivalence]]

  * [[stable weak homotopy equivalence]]

* [[homotopy equivalence of toposes]]

* [[equivalence in a 2-category]]

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

* [[equivalence of (∞,1)-categories]]

## References

* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

[[!redirects homotopy equivalences]]
[[!redirects homotopy equivalent]]


homotopy equivalence of toposes

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

#Contents#
* table of contents
{:toc}

## Definitions

There are several things that one might mean by a "(possibly [[weak homotopy equivalence|weak]]) [[homotopy equivalence]] of [[toposes]] (or [[(∞,1)-toposes]])."   Some candidates are:

1. A [[geometric morphism]] that induces an [[equivalence]] of the [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]], or more generally of [[shape of an (∞,1)-topos|shapes]].

1. A geometric morphism which induces an isomorphism on all [[nonabelian cohomology]] with coefficients in [[constant ∞-stacks]].

1. A geometric morphism which induces an [[isomorphism]] on all [[abelian sheaf cohomology]] with coefficients in [[locally constant sheaves|locally constant]] sheaves (of complexes) of [[abelian groups]], as well as [[nonabelian cohomology]] in dimension one (the most classical notion).

1. A geometric morphism which has an inverse up to "homotopy" in the classical sense of a map $E\times [0,1]\to F$, where since $E$ and $F$ are toposes, we have to incarnate $[0,1]$ as the topos $Sh([0,1])$.

Of these, the first three are arguably a notion of *weak* homotopy equivalence.

## Relationship between definitions.

The first two definitions are equivalent, since (viewing a topos $E$ equivalently as the $(\infty,1)$-topos of sheaves on the same [[site]], if necessary), a constant ∞-stack on $E$ is one of the form $LConst A$ for an ∞-groupoid $A$, and the cohomology of $E$ with coefficients in $LConst A$ is just $\pi_0(Hom_E(*,LConst A))$.  But when $E$ is locally $\infty$-connected, $\Pi_E$ is left adjoint to $LConst$, so this is the same as $\pi_0 Hom_{\infty Gpd}(\Pi_E(*),A) = Hom_{Ho(\infty Gpd)}(\Pi_\infty(E),A)$ by definition of $\Pi_\infty(E)$.  Thus, by the Yoneda lemma, a map of toposes induces an equivalence of fundamental ∞-groupoids, i.e. an isomorphism in $Ho(\infty Gpd)$, iff it induces an isomorphism on all such cohomology.  Since in the non--locally-$\infty$-connected case, the [[shape of an (infinity,1)-topos|shape]] is just the functor that would be represented by $\Pi_\infty(E)$ if it existed, the equivalence is even easier in that case.

These first two definitions also imply the third, since the abelian cohomology $H^n(E,A)$ with coefficients in an abelian sheaf $A$ can be identified with nonabelian cohomology in the locally constant stack $B^n A$ which is the $n$-fold [[delooping]] of $A$, and locally constant stacks on $E$ are represented by maps out of $\Pi_\infty(E)$.

+--{: .query}
[[Mike Shulman]]: There's something a little funny here about "constant" versus "locally constant," but I don't have time to figure it out now.  Maybe someone else can explain it.
=--

Conversely, it is proven in Artin-Mazur that (in our language) a geometric morphism of 1-topoi satisfying the third definition necessarily induces isomorphisms on all homotopy progroups.  (The idea is essentially that nonabelian cohomology is controlled by abelian cohomology with local coefficients, via [[Postnikov tower in an (infinity,1)-category|Postnikov decompositions]].)   In the locally $\infty$-connected case, this is equivalent to inducing an equivalence of fundamental $\infty$-groupoids---the first definition.  Thus, at least for locally $\infty$-connected 1-topoi, the first three definitions are all equivalent.

+--{: .query}
[[Mike Shulman]]: Possibly in the non--locally-$\infty$-connected case, the third definition is weaker, since a map of pro-$\infty$-groupoids can induce an isomorphism on all homotopy progroups without being an equivalence?  It's also not immediately obvious whether the result is also true as stated for all $(\infty,1)$-toposes.

I also seem to recall that the last notion of a "paths homotopy equivalence" implies the first few, but I don't remember why.
=--

## Adjunctions induce homotopy equivalences

One important thing to note is that any [[adjunction]] in the [[2-category]] [[Topos]] induces a homotopy equivalence in any of the above senses.  For the first (hence also the second and third) sense, we just observe that $\Pi_\infty$ is 2-functorial, whereas it lands in the (∞,1)-category ?Gpd where all 2-morphisms are invertible; hence adjunctions get sent to adjunctions, which are necessarily equivalences.

For the final definition, we note that has [[weighted limit|tensors]] with the [[interval category]], that are given by [[cartesian product]] with sheaves on the [[Sierpinski space]] $\Sigma$.  Thus, a natural transformation between two geometric morphisms $E\to F$ is the same as a single geometric morphism
$E\times Sh(\Sigma)\to F$.  Now just pick a map $[0,1]\to \Sigma$ that doesn't identify the endpoints, like the classifying map of $[0,\frac1 2]$, to get an actual "homotopy" between the same two geometric morphisms.  Hence, any 2-morphism in $Topos$ gives a homotopy, so any adjunction gives a homotopy equivalence.

In particular, this means that any [[local geometric morphism]] or [[totally connected geometric morphism]] is a homotopy equivalence.

## Related concepts

* [[equality]]

* [[isomorphism]]

* [[equivalence]]

* [[weak equivalence]]

* [[homotopy equivalence]], [[weak homotopy equivalence]]

* **homotopy equivalence of toposes**

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

* [[equivalence of (∞,1)-categories]]

## References

* M. Artin and B. Mazur, 1969, _Etale homotopy_, No. 100, Lecture Notes in Maths.,
Springer-Verlag, Berlin.

[[!redirects homotopy equivalence of toposes]]
[[!redirects homotopy equivalence of topoi]]
[[!redirects homotopy equivalences of toposes]]
[[!redirects homotopy equivalences of topoi]]
[[!redirects topos homotopy equivalence]]
[[!redirects topos homotopy equivalences]]
[[!redirects weak homotopy equivalence of toposes]]
[[!redirects weak homotopy equivalence of topoi]]
[[!redirects weak homotopy equivalences of toposes]]
[[!redirects weak homotopy equivalences of topoi]]
[[!redirects topos weak homotopy equivalence]]
[[!redirects topos weak homotopy equivalences]]


homotopy exact completion

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

\tableofcontents

## Idea

A generalization of [[exact completion]] from [[finitely complete categories]] to [[categories with path objects]], which turns a category with path objects into an [[exact category]].

## Examples

* Let $\mathcal{C}$ be the [[syntactic category]] of a [[dependent type theory]]. Then the category of [[setoids]] in $\mathcal{C}$ is the homotopy exact completion of $\mathcal{C}$.

## See also

* [[categories with path objects]]

* [[exact completion]]

* [[setoid]]

## References

* {#vdBergMoerdijk18} [[Benno van den Berg]], [[Ieke Moerdijk]], *Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories*, Journal of Pure and Applied Algebra **222** 10 (2018) 3137-3181 [[doi:10.1016/j.jpaa.2017.11.017](https://doi.org/10.1016/j.jpaa.2017.11.017), [arXiv:1603.02456](https://arxiv.org/abs/1603.02456)]


homotopy exact square

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

# Homotopy exact squares
* automatic table of contents
{:toc}

## Idea

A *homotopy exact square* is the analogue of an [[exact square]] which applies to [[homotopy Kan extensions]], or equivalently to [[(∞,1)-Kan extensions]].  It is especially important in the theory of [[derivators]], which provide a calculus for computing with homotopy Kan extensions whose primary tool is the use of homotopy exact squares.

## Definition

Let $A$, $B$, $C$, and $D$ be small [[categories]], and consider a square of [[functors]]
$$\array{A & \overset{f}{\to} & B\\
  ^u\downarrow & \swArrow & \downarrow^v\\
  C& \underset{g}{\to} & D}$$
which is inhabited by a [[natural transformation]] (which might be an [[identity]]).  Let $M$ be either

* a [[model category]],
* a [[simplicially enriched category]] (or a topologically enriched category, or a [[dg-category]]),
* an [[(∞,1)-category]] (of any sort), or
* a [[derivator]].

We write $M^A$, $M^B$, etc. for the model categories, simplicial categories, $(\infty,1)$-categories, or homotopy categories of diagrams in $M$ of whatever shape.  We write $f^*\colon M^B\to B^A$, $g^*\colon M^D\to M^C$, and so on for precomposition functors, which are always homotopically meaningful, and we write $u_!\colon M^A\to M^C$, $v_!\colon M^B\to M^D$ and so on for the homotopically meaningful notions of pointwise [[left Kan extension]].  Specifically:

* If $M$ is a model category, then $u_!$ denotes the left [[derived functor]] of pointwise Kan extension along $u$.
* If $M$ is a simplicially enriched category, then $u_!$ is the coherent pointwise [[homotopy left Kan extension]] along $u$, which may be defined explicitly in various ways, such as using a [[bar construction]].
* If $M$ is an $(\infty,1)$-category, then $u_!$ denotes the pointwise [[(∞,1)-Kan extensions]] along $u$.
* If $M$ is a derivator, then $u_!$ simply denotes the left adjoint of $u^*$ (which is assumed to exist and to "be pointwise" by the derivator axioms).

Assume that $M$ is such that the relevant extensions $u_!$ and $v_!$ exist.  Then there is a canonical [[Beck-Chevalley transformation]]
$$ u_! f^* \to g^* v_! $$
defined as the composite
$$ u_! f^* \to u_! f^* v^* v_! \to u_! u^* g^* v_! \to g^* v_!. $$
and we say that the given square is **$M$-exact** if this transformation is an [[equivalence]].  If the square is $M$-exact for all $M$, we say it is **homotopy exact**.  Note that by the general calculus of [[mates]], this is equivalent to requiring that the dual transformation
$$ v^* g_* \to f_* u^* $$
is an equivalence, where $f_*$ and $g_*$ denote the analogous sort of *right* Kan extension.

Of course when we say "for all $M$" we need to specify what sorts of $M$ we consider.  However, we actually get the *same* definition regardless of whether we mean "for all model categories $M$" or "for all simplicially enriched categories $M$" or "for all $(\infty,1)$-categories $M$" or "for all derivators $M$".  This is a nontrivial theorem, especially in the case of derivators.

## Remarks

Since any 1-category is a degenerate sort of $(\infty,1)$-category, any homotopy exact square is [[exact square|exact]] in the usual 1-categorical sense, but the converse is not true.  This also implies that a square can be $M$-exact for some particular $M$ without being homotopy exact.  However, there exists a "universal" $M$ such that $M$-exactness is equivalent to homotopy exactness, namely $M=\infty Gpd$.

## Characterization

Of course, the above definition is "functional", while in practice we want some more combinatorial characterization which is easier to check.  This can be done completely analogously to the characterization of ordinary [[exact squares]] using comma objects, except that at the last step we need to consider a more restricted notion of "equivalence" (i.e. a more restricted [[basic localizer]]).

The characterization is the following.  Given $b\in B$ and $c\in C$ and $\varphi\colon v(b) \to g(c)$, let $(b/A/c)_\varphi$ denote the category whose objects are triples $(a,\alpha,\beta)$ with $\alpha\colon u(a)\to c$ and $\beta\colon b\to f(a)$ such that $g(\alpha) \circ v(\beta) = \varphi$, and whose morphisms are morphisms $a\to a'$ making two triangles commute.

+-- {: .un_theorem}
###### Theorem
A square is homotopy exact if and only if each category $(b/A/c)_\varphi$ has a [[contractible space|contractible]] [[nerve]].
=--

* See [exact square](/nlab/show/exact+square#CommaObj) for the proof that is being generalized.
* See [derivator](/nlab/show/derivator#ExactSquares) for the proof in the case of derivators, which relies on Cisinski's characterization of [[basic localizers]].
* The proof in the (∞,1)-categorical case generalizes the characterization of [final (∞,1)-functors](/nlab/show/final+(∞,1%29-functor#Properties).

## Examples

### Comma squares

Any [[comma square]] is homotopy exact.  In other words, if $A=(v/g)$ is the [[comma category]] with $f$ and $u$ the canonical projections, then the square is homotopy exact.  If in addition $C=*$ is the [[terminal category]], then $g$ just picks out an object $d\in D$ and $A$ is the comma category $(v/d)$; thus this says that (pointwise) homotopy Kan extensions can be computed pointwise as homotopy limits over such comma categories.

### Fully faithful functors

If $u\colon A\to B$ is a [[fully faithful functor]], then the square
$$\array{A & \overset{id}{\to} & A\\
  ^{id}\downarrow && \downarrow^u\\
  A & \underset{u}{\to} & B}$$
is homotopy exact.  This just says that the unit $Id_A \to u^* u_!$ is an isomorphism, i.e. that left (and equivalently right) homotopy Kan extensions along $u$ are "honest" extensions.

### Final functors

A functor $u\colon A\to B$ is a [[homotopy final functor]] if and only if the square
$$\array{A & \overset{u}{\to} & B\\
  \downarrow && \downarrow\\
  *& \underset{}{\to} & *}$$
is homotopy exact.  Homotopy exactness of this square says that for $F\colon B\to M$, the canonical map $hocolim_A u^*F \to hocolim_B F$ is an isomorphism, which is one equivalent definition of when $u$ is homotopy final.  In this case, the characterization theorem reduces to saying that $u$ is homotopy final if and only if each comma category $b/u$ has a contractible nerve, which is a known characterization of homotopy final functors.

### Pullbacks of fibrations {#PullbacksOfFibrations}

This example is due to [Moritz Groth](#Groth).  Let $p\colon C\to D$ be a [[Grothendieck opfibration]], and suppose that
$$\array{ A & \xrightarrow{u} & C \\
  ^v\downarrow && \downarrow^p\\
  B & \xrightarrow{q} & D } $$
is a [[pullback]] in [[Cat]].  Then we claim that this square is homotopy exact.

For a proof, suppose given $b\in B$ and $c\in C$ and a morphism $\phi\colon p(c) \to q(b)$, and let $X$ be the category whose contractibility we must check.  By definition, since $A = B\times_D C$, an object of $X$ consists of objects $c'\in C$ and $b'\in B$ such that $p(c') = q(b')$, and morphisms $\alpha\colon c\to c'$ and $\beta\colon b' \to b$ such that $q(\beta) . p(\alpha) = \phi$.

Let $Y$ be the full subcategory of $X$ consisting of those objects with $b'=b$ and $\beta$ the identity, so that an object of $Y$ is an object $c'\in C$ such that $p(c') = q(b)$ together with a morphism $\alpha\colon c\to c'$ such that $p(\alpha) = \phi$.  Then the inclusion $Y\hookrightarrow X$ has a left adjoint, which sends $(b',c',\alpha,\beta)$ to $(b, q(\beta)_!(c'), \overline{q(\beta)}.\alpha, 1_b)$, where $\overline{q(\beta)}\colon c' \to q(\beta)_!(c')$ is an opcartesian arrow over $q(\beta)$.  Therefore, the nerves of $Y$ and $X$ are homotopy equivalent.

But $Y$ has an initial object, namely $(\phi_!(c), \overline{\phi})$, where $\overline{\phi}\colon c \to \phi_!(c)$ is an opcartesian arrow over $\phi$.  Thus the nerve of $Y$ is contractible, and thus so is the nerve of $X$.

Groth has also proved that homotopy-exactness of such squares can be used to replace that of comma squares in the definition of a [[derivator]].

## References

* An [nCafé post](http://golem.ph.utexas.edu/category/2010/06/exact_squares.html) about (mostly ordinary) [[exact squares]]

* [[Moritz Groth]], "Derivators, pointed derivators, and stable derivators" [PDF](http://www.math.uni-bonn.de/~mgroth/groth_derivators.pdf)
{#Groth}

* [[Georges Maltsiniotis]], "Carrés exacts homotopiques, et dérivateurs", Cahiers de Topologie et Géométrie Différentielle Catégoriques 53 (2012), 3-63
[PDF](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/carhex.pdf)
{#Maltsiniotis}

[[!redirects homotopy exact squares]]
[[!redirects homotopically exact square]]
[[!redirects homotopically exact squares]]