homotopically injective object

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## Definition

Let $\mathcal{A}$ be an [[abelian category]] [[category with translation|with translation]].

An [[object]] $I$ in the [[category of chain complexes]] modulo [[chain homotopy]], $K(\mathcal{A})$,  is **homotopically injective** if for every $X \in K(\mathcal{A})$ that is
[[quasi-isomorphism|quasi-isomorphic]] to $0$
we have
$$
  Hom_{K(\mathcal{A})}(X,I) \simeq 0
  \,.
$$

Let

$$
  QuasiIsoMono = \{f \in Mor(A_c) | f mono and quasiio\}
$$

be the set of morphisms in the [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$ which are both [[quasi-isomorphism]]s as well as [[monomorphism]]s.

Then

A [[complex]] $I$ is an [[injective object]] with respect to monomorphic quasi-isomorphisms precisely if

* it is homotopically injective in the sense of complexes in $\mathcal{A}$; and

* it is injective as an object of $\mathcal{A}$ (with respect to morphisms $f : X \to Y$ such that $0 \to X \stackrel{f}{\to} Y$ is exact).

## Properties

### In complexes in a Grothendieck category

**Proposition**
For $\mathcal{A}$ a [[Grothendieck category]] [[category with translation|with translation]] $T : \mathcal{A}  \to \mathcal{A}$, every [[complex]] $X$ in $Ch_\bullet(\mathcal{A})$ is quasi-isomorphic to a complex $I$ which is injective and homotopically injective (i.e. QuasiIsoMono-injective).

### Relation to derived categories

For $\mathcal{A}$ an [[abelian category|abelian]] [[Grothendieck category|Grothendieck category]] [[category with translation|with translation]] the [[full subcategory]] $K_{hi}(\mathcal{A}) \subset K(\mathcal{A})$ of _homotopically injective_ complexes realizes the [[derived category]] $D(\mathcal{A})$ of $\mathcal{A}$:

$$
  Q|_{K_{hi}(A)} : K_{hi}(A) \stackrel{\simeq}{\to} D(A)
  \,,
$$

where $Q : K(A) \to D(A)$ and $Q|_{K_{hi}(A)}$ has a [[adjoint functor|right adjoint]].

It follows that for $D$ any other [[triangulated category]], every [[triangulated functor]] $F : K(\mathcal{A}) \to D$ has a right [[derived functor]] $R F : D(\mathcal{A}) \to D$ which is computed by evaluating $F$ on _injective replacements_: for
$R : D(\mathcal{A}) \stackrel{\simeq}{\to} K_{hi}(\mathcal{A})$ a weak inverse to $Q$, we have

$$
  R F \simeq D(A) \stackrel{R}{\to} K_{hi}(A) \hookrightarrow K(A) \stackrel{F}{\to} D
  \,.
$$

## References#

Much of this discussion can be found in

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_

The general notion of [[injective objects]] is in section 9.5, the case of injective complexes in section 14.1.

[[!redirects homotopically injective objects]]


homotopy

> This page is about homotopy as a transformation.  For homotopy sets in [[homotopy categories]], see [[homotopy (as an operation)]].

***

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## Idea

In many [[category|categories]] $C$ in which one does [[homotopy theory]], there is a notion of _homotopy_ between [[morphisms]], which is closely related to the [[2-morphisms]] in [[higher category theory]]: a homotopy between two morphisms is a way in which they are equivalent.

If we regard such a category as a presentation of an $(\infty,1)$-[[(infinity,1)-category|category]], then homotopies $f\sim g$ present the 2-cells $f\Rightarrow g$ in the resulting $(\infty,1)$-category.

## Definition

### In topological spaces

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

For $f,g\colon X \longrightarrow Y$ two [[continuous functions]] between [[topological spaces]] $X,Y$, then a **left homotopy**

$$
  \eta \colon f \,\Rightarrow_L\, g
$$

is a [[continuous function]]

$$
  \eta \;\colon\; X \times I \longrightarrow Y
$$

out of the standard [[cylinder object]] over $X$: the [[product space]] of $X$ with the [[Euclidean space|Euclidean]] [[closed interval]] $I \coloneqq [0,1]$, such that this fits into a [[commuting diagram]] of the form

$$
  \array{
     X
     \\
     {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}}
     \\
     X \times I &\stackrel{\eta}{\longrightarrow}& Y
     \\
     {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}}
     \\
     X
  }
  \,.
$$

(graphics grabbed from J. Tauber [here](http://jtauber.com/blog/2005/07/01/path_homotopy/))

=--

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

Let $X$ be a [[topological space]] and let $x,y \in X$ be two of its points, regarded as functions $x,y \colon \ast \longrightarrow X$ from the point to $X$. Then a left homotopy, def. \ref{LeftHomotopy}, between these two functions is a commuting diagram of the form

$$
  \array{
     \ast
     \\
     {}^{\mathllap{\delta_0}}\downarrow & \searrow^{\mathrlap{x}}
     \\
     I &\stackrel{\eta}{\longrightarrow}& X
     \\
     {}^{\mathllap{\delta_1}}\uparrow & \nearrow_{\mathrlap{y}}
     \\
     \ast
  }
  \,.
$$

This is simply a continuous path in $X$ whose endpoints are $x$ and $y$.

=--

### In enriched categories

If $C$ is [[enriched category|enriched]] over [[Top]], then a **homotopy** in $C$ between maps $f,g:X\,\rightrightarrows \,Y$ is a map $H:[0,1] \to C(X,Y)$ in $Top$ such that $H(0)=f$ and $H(1)=g$.  In $Top$ itself this is the classical notion.

If $C$ has [[copower|copowers]], then an equivalent definition is a map $[0,1]\odot X\to Y$, while if it has [[power|powers]], an equivalent definition is a map $X\to \pitchfork([0,1],Y)$.

There is a similar definition in a [[simplicially enriched category]], replacing $[0,1]$ with the 1-simplex $\Delta^1$, with the caveat that in this case not all _simplicial homotopies_ need be composable even if they match correctly. (This depends on whether or not all (2,1)-[[horn]]s  in the simplicial set, $C(X,Y)$, have fillers.)   Likewise in a [[dg-category]] we can use the "chain complex interval" to get a notion of _chain homotopy_.

### In model categories

If $\mathcal{C}$ is a [[model category]], it has an intrinsic notion of homotopy determined by its factorizations. For more on the following see at _[[homotopy in a model category]]_.

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

Let $\mathcal{C}$ be a [[model category]] and $X \in \mathcal{C}$ an [[object]].

* A **[[path object]]** $Path(X)$ for $X$ is a factorization of the [[diagonal]] $\nabla_X \colon X \to X \times X$ as

$$
  \nabla_X
  \;\colon\;
   X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X
  \,.
$$

where $X\to Path(X)$ is a weak equivalence. This is called a **good path object** if in addition $Path(X) \to X \times X$ is a fibration.

* A **[[cylinder object]]** $Cyl(X)$ for $X$ is a factorization of the [[codiagonal]] (or "fold map") $\Delta_X: X \sqcup X \to X$ as

$$
  \Delta_X
  \;\colon\;
  X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X
  \,.
$$

where $Cyl(X) \to X$ is a weak equivalence. This is called a **good cylinder object** if in addition $X \sqcup X \to Cyl(X)$ is a cofibration.

=--

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

By the factorization axioms every object in a model category has both a good path object and as well as a good cylinder object according to def. \ref{PathAndCylinderObjectsInAModelCategory}. But in some situations one is genuinely interested in using non-good such objects.

For instance in the [[classical model structure on topological spaces]], the obvious object $X\times [0,1]$ is a cylinder object, but not a good cylinder unless $X$ itself is cofibrant (a [[cell complex]] in this case).

More generally, the path object $Path(X)$ of def. \ref{PathAndCylinderObjectsInAModelCategory} is analogous to the [[powering]] $\pitchfork(I,X)$ with an [[interval object]] and the cylinder object $Cyl(X)$ is analogous to the [[tensoring]] $I\odot X$ with an interval object.  In fact, if $\mathcal{C}$ is a $V$-[[enriched model category]] and $X$ is fibrant/cofibrant, then these powers and copowers  are in fact examples of (good) path and cylinder objects if the [[interval object]] is sufficiently good.

=--

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

Let $f,g \colon X \longrightarrow Y$ be two [[parallel morphisms]] in a [[model category]].

* A **left homotopy** $\eta \colon f \Rightarrow_L g$ is a morphism $\eta \colon Cyl(X) \longrightarrow Y$ from a [[cylinder object]] of $X$,  def. \ref{PathAndCylinderObjectsInAModelCategory}, such that it makes this [[commuting diagram|diagram commute]]:

$$
  \array{
    X &\longrightarrow& Cyl(X) &\longleftarrow& X
    \\
    & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}}
    \\
    &&
    Y
  }
  \,.
$$

* A **right homotopy** $\eta \colon f \Rightarrow_R g$ is a morphism $\eta \colon X \to Path(Y)$ to some [[path object]] of $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, such that this [[commuting diagram|diagram commutes]]:

$$
  \array{
    && X
    \\
    & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}}
    \\
    Y &\longleftarrow& Path(Y) &\longrightarrow& Y
  }
  \,.
$$

=--

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

By remark \ref{RemarkOnChoicesOfNonGoodPathAndCylinderObjects} it follows that in a $Top$-[[enriched model category]], any enriched homotopy between maps $X\to Y$ is a left homotopy if $X$ is cofibrant and a right homotopy if $Y$ is fibrant.  Similar remarks hold for other enrichments.

=--

For more see at _[[homotopy in a model category]]_.

### In (co-)fibration categories

Clearly the concept of left homotopy in def. \ref{PathAndCylinderObjectsInAModelCategory} only needs part of the model category axioms and thus makes sense more generally in suitable [[cofibration categories]]. Dually, the concept of path objects in def. \ref{PathAndCylinderObjectsInAModelCategory} makes sense more generally in suitable [[fibration categories]] such as [[categories of fibrant objects]] in the sense of Brown.

Likewise if there is a [[cylinder functor]], one gets functorially defined [[cylinder objects]], etc.

### In dependent type theory

In [[dependent type theory]], let $A$ be a type and let $P$ be a [[type family]] indexed by $A$, and let $f,g:\prod_{x:A} P(x)$ be two elements of a [[dependent product type]] of a [[type family]] $P$. The **type of homotopies** between $f$ and $g$ is the type

$$f \sim g \equiv \prod_{x:A} (f(x) =_{B(x)} g(x))$$

A **homotopy** between $f$ and $g$ is simply an element $H:f \sim g$.

Note that a homotopy is not the same as an [[identification]] $f = g$. However this can be made so if one assumes [[function extensionality]].

## Related concepts

[[!include homotopy-homology-cohomology]]

* [[homotopy theory]]

* [[left homotopy]], [[right homotopy]]

* [[homotopy relative boundary]]

* [[smooth homotopy]]

* [[isotopy]], [[smooth isotopy]]

* [[higher homotopy]]

* [[homotopy class]]

* [[transfor]]

  * [[natural transformation]]

  * [[pseudonatural transformation]]

  * [[lax natural transformation]]

* [[function extensionality]]

* [[whiskering]]

* [[homotopy analysis method]]

## References

See the references at _[[homotopy theory]]_ and at _[[model category]]_.

Discussion in [[computational topology]]:

* [[Marek Filakovský]], [[Lukáš Vokřínek]], _Are two given maps homotopic? An algorithmic viewpoint_, Found Comput Math (2019) ([arXiv:1312.2337](https://arxiv.org/abs/1312.2337), [doi:10.1007/s10208-019-09419-x](https://doi.org/10.1007/s10208-019-09419-x))

For homotopies in [[Martin-Löf dependent type theory]]:

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

[[!redirects left homotopy]]
[[!redirects right homotopy]]

[[!redirects left homotopies]]
[[!redirects right homotopies]]

[[!redirects homotopy (as a transformation)]]

[[!redirects homotopies]]

[[!redirects homotopic]]

[[!redirects type of homotopies]]
[[!redirects type of left homotopies]]
[[!redirects type of right homotopies]]


homotopy (as an operation)

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* This page is about homotopy as an operation.  For homotopy as a transformation, see [[homotopy]].

***

# Homotopy sets in homotopy categories
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{: toc}

## Idea

 Ordinary _homotopy_ is a way to probe objects in an [[(∞,1)-topos]] $\mathbf{H}$ by mapping [[spheres]] into them:

the ordinary [[homotopy group]] $\pi_n(X,x)$ of an object $X \in \mathbf{H}$ is the fiber over $x \in X$ of the morphism

$$
  [S^n, X]_{\mathbf{H}} \to \tau_0 X \simeq \pi_0(X)
$$

In this sense _homotopy_ is the notion that is [[Eckmann-Hilton duality|Eckmann-Hilton dual]] to [[cohomology]].

For a detailed discussion see

* [[homotopy groups in an (∞,1)-topos]]

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

This duality suggests that more generally we may be entitled to speak for $B$ and $X$ objects in $\mathbf{H}$ of
$$
  H(B,X) := \pi_0 \mathbf{H}(B,X)_*
$$
as the **homotopy of $X$ with co-coefficients in $B$**.

Examples of such constructions exist, but are rarely thought of (or even recognized as) generalizations of the notion of homotopy. Rather, by the above duality, the same situation is usually regarded in the context of [[cohomology]], which, still by the above duality, is just as well.

=--

## Examples

* [[group homotopy]]

* [[generalized (Eilenberg-Steenrod) homotopy]]

* abelian cosheaf homotopy

* nonabelian homotopy

* [[?ech homotopy]]

* differential homotopy

* twisted homotopy

## Related concepts

[[!include homotopy-homology-cohomology]]

* [[homotopy]], [[homotopy group]],

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

* [[homotopy groups in an (∞,1)-topos]] .

[[!redirects homotopy as an operation]]
[[!redirects homotopy (as an operation)]]


homotopy - contents

**[[homotopy theory]], [[(∞,1)-category theory]], [[homotopy type theory]]**

flavors: [[stable homotopy theory|stable]], [[equivariant homotopy theory|equivariant]], [[rational homotopy theory|rational]], [[p-adic homotopy theory|p-adic]], [[proper homotopy theory|proper]], [[geometric homotopy theory|geometric]], [[cohesive homotopy theory|cohesive]], [[directed homotopy theory|directed]]...

models:  [[topological homotopy theory|topological]], [[simplicial homotopy theory|simplicial]], [[localic homotopy theory|localic]], ...

see also **[[algebraic topology]]**

**Introductions**

* [[Introduction to Topology -- 2|Introduction to Basic Homotopy Theory]]

* [[Introduction to Homotopy Theory|Introduction to Abstract Homotopy Theory]]

* [[geometry of physics -- homotopy types]]

**Definitions**

* [[homotopy]], [[higher homotopy]]

* [[homotopy type]]

* [[Pi-algebra]], [[spherical object and Pi(A)-algebra]]

* [[homotopy coherent category theory]]

  * [[homotopical category]]

    * [[model category]]

    * [[category of fibrant objects]], [[cofibration category]]

    * [[Waldhausen category]]

  * [[homotopy category]]

    * [[Ho(Top)]]

* [[(∞,1)-category]]

  * [[homotopy category of an (∞,1)-category]]

**Paths and cylinders**

* [[left homotopy]]

  * [[cylinder object]]

  * [[mapping cone]]

* [[right homotopy]]

  * [[path object]]

  * [[mapping cocone]]

  * [[generalized universal bundle|universal bundle]]

* [[interval object]]

  * [[homotopy localization]]

  * [[infinitesimal interval object]]

**Homotopy groups**

* [[homotopy group]]

  * [[fundamental group]]

    * [[fundamental group of a topos]]

  * [[Brown-Grossman homotopy group]]

  * [[categorical homotopy groups in an (∞,1)-topos]]

  * [[geometric homotopy groups in an (∞,1)-topos]]

* [[fundamental ∞-groupoid]]

  * [[fundamental groupoid]]

    * [[path groupoid]]

  * [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]

  * [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]]

* [[fundamental (∞,1)-category]]

  * [[fundamental category]]

**Basic facts**

* [[fundamental group of the circle is the integers]]

**Theorems**

* [[fundamental theorem of covering spaces]]

* [[Freudenthal suspension theorem]]

* [[Blakers-Massey theorem]]

* [[higher homotopy van Kampen theorem]]

* [[nerve theorem]]

* [[Whitehead's theorem]]

* [[Hurewicz theorem]]

* [[Galois theory]]

* [[homotopy hypothesis]]-theorem


homotopy 0-type

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## Idea

A _homotopy 0-type_ is a _[[homotopy n-type]]_ for $n = 0$.

In [[homotopy type theory]] also called, an _[[h-set]]_.

## Related concepts

[[!include homotopy n-types - table]]

[[!redirects homotopy 0-types]]


homotopy 1-type

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##  Idea

A homotopy $1$-type is a space where we consider its properties with regard to the [[fundamental group]]s $\pi_1$ of its components.

## Definition

A continuous map $X \to Y$ is a **homotopy $1$-equivalence** if it induces isomorphisms on $\pi_0$ and $\pi_1$ at each basepoint.  Two spaces share the same **homotopy $1$-type** if they are linked by a zig-zag chain of homotopy $1$-equivalences.

For any space $X$, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space $Y$ so that the inclusion of $X$ into $Y$ is a homotopy $1$-equivalence; up to (weak) [[homotopy equivalence]], the result is the same for any space with the same homotopy $1$-type.  Accordingly, a **homotopy $1$-type** may alternatively be defined as a space with trivial $\pi_i$ for $i \gt 1$, or as the unique (weak) [[homotopy type]] of such a space, or as its fundamental $\infty$-[[fundamental infinity-groupoid|groupoid]] (which will be a $1$-[[1-groupoid|groupoid]]).

See the general discussion in [[homotopy n-type]].

## Classification

A connected pointed homotopy 1-type is completely determined, up to (weak) homotopy equivalence, by the one [[group]] $\pi_1$.  A connected homotopy 1-type with $\pi_1 = G$ is  an [[Eilenberg-Mac Lane space]] and is often written $K(G,1)$.  A general homotopy 1-type can then be written as a disjoint union of such $K(G,1)$s, and is completely determined by its [[fundamental groupoid]].

In the other direction, for any (discrete) group $G$ one can construct its [[classifying space]] $\mathcal{B}G$, which is a $K(G,1)$.  In fact, many versions of this construction (such as the geometric realization of the simplicial nerve $nerve(G)$; see [[Dold-Kan correspondence]]) apply just as well to any [[groupoid]].  We can obtain any 1-type in this way,  since a groupoid is up to homotopy type (of groupoids!) a disjoint union of groups. However this description is not natural in the category of groupoids, and is analogous to choosing a basis for a vector space.

Moreover, every continuous map between $K(G,1)$s is induced by a group homomorphism, every map between 1-types is induced by a [[functor]] between groupoids, and every homotopy is induced by a conjugation (aka a [[natural transformation]] between groupoids).  In fact, one can show that the $(\infty,1)$-[[(infinity,1)-category|category]] of homotopy 1-types is equivalent to the 2-category [[Grpd]] of [[groupoid]]s, via the above-described correspondence..

There are further aspects to this relationship; for instance, the [[van Kampen theorem]] for the fundamental groupoid shows how the algebra of groupoids models the gluing of spaces.  The general result for non-connected spaces is possible because groupoids model homotopy 1-types, having structure in dimensions 0 and 1.  For the search for algebraic structures that play an analogous role to groupoids for $n$-types with $n\gt 1$, see the pages [[homotopy hypothesis]], [[fundamental infinity-groupoid]], [[cat-n-group]], [[classifying space]], [[crossed complex]], and probably others.

## Related concepts

[[!include homotopy n-types - table]]

[[!redirects 1-type]]
[[!redirects 1-types]]
[[!redirects homotopy 1-types]]


homotopy 2-category

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{:toc}

## Idea

The _homotopy 2-category_ of an [[(∞,n)-category]] $\mathcal{C}$ is the [[2-category]] $Ho_2(\mathcal{C})$ with the same [[objects]] and [[1-morphisms]] as $\mathcal{C}$ and with the [[2-morphisms]] being the [[equivalence classes]] of 2-morphisms of $\mathcal{C}$.

In other words, for every [[pair]] $X,Y$ of [[objects]] in $\mathcal{C}$, the [[hom-category]] $Ho_2(\mathcal{C})(X,Y)$ is the ordinary [[homotopy category]] of the $(\infty,n-1)$-category $\mathcal{C}(X,Y)$.

## Examples

* [[homotopy 2-category of (∞,1)-categories]]

* [[homotopy category]], [[homotopy category of a model category]]

## Related concepts

* [[homotopy category]]

* [[localization of a 2-category]]

* [[model 2-category]]

* [[Ho(CombModCat)]]

## References

The homotopy 2-category of [[topological spaces]], [[continuous functions]] and [[homotopies]], regarded as a [[strict (2,1)-category]], hence a [[Grpd]]-[[enriched category]]:

* [[Peter H. H. Fantham]], [[Eric J. Moore]], *Groupoid Enriched Categories and Homotopy Theory*, Canadian Journal of Mathematics **35** 3 (1983) 385-416 ([doi:10.4153/CJM-1983-022-8](https://doi.org/10.4153/CJM-1983-022-8))

On the [[Toda bracket]] understood as [[homotopy coherent diagram|homotopy-coherent]] [[pasting diagrams]] in a [[pointed (infinity,1)-category|pointed]] [[homotopy 2-category]]:

* {#HardieKampsKieboom99} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Rudger  Kieboom]], _Higher homotopy groupoids and Toda brackets_, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 ([euclid:hha/1139840198](https://projecteuclid.org/euclid.hha/1139840198))

* [[Keith Hardie]], [[Howard Marcum]], [[Nobuyuki Oda]], _Bracket operations in the homotopy theory of a 2-category_, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) ([rendiconti:33/02](https://rendiconti.dmi.units.it/volumi/33/02.pdf))

* {#HardieKampsMarcum02} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Howard Marcum]], _The Toda bracket in the homotopy category of a track bicategory_, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

* [[Howard Marcum]], [[Nobuyuki Oda]], _Long Box Bracket Operations in Homotopy Theory_, Appl Categor Struct 19, 137–173 ([doi:10.1007/s10485-009-9186-3](https://doi.org/10.1007/s10485-009-9186-3))

In the context of the [[homotopy 2-category of (∞,1)-categories]] and [[formal (infinity,1)-category theory|formal $(\infty,1)$-category theory]]:

* {#Joyal08} [[André Joyal]], p. 158 (10 of 348) in: _The theory of quasicategories and its applications_, lectures at: _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, Quadern **45** 2, Centre de Recerca Matemàtica, Barcelona 2008 ([[JoyalTheoryOfQuasiCategories.pdf:file]])

* {#Riehl14} [[Emily Riehl]], Chapter 18 of: _[[Categorical Homotopy Theory]]_, Cambridge University Press, 2014 ([pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf), [doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457))

* {#RiehlVerity15} [[Emily Riehl]], [[Dominic Verity]], Section 3 of: _Fibrations and Yoneda's lemma in an $\infty$-cosmos_, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 ([arXiv:1506.05500](https://arxiv.org/abs/1506.05500), [doi:10.1016/j.jpaa.2016.07.003](https://doi.org/10.1016/j.jpaa.2016.07.003))

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], Section 1.3 in: _Infinity category theory from scratch_, Higher Structures Vol 4, No 1 (2020) ([arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf))

* {#RiehlVerity21} [[Emily Riehl]], [[Dominic Verity]], Chapter 1 of: *Elements of $\infty$-Category Theory*, 2021- ([pdf](https://math.jhu.edu/~eriehl/elements.pdf))

[[!redirects homotopy 2-categories]]


homotopy 2-category of (∞,1)-categories

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#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[homotopy 2-category]] $Ho_2\big((\infty,1)Cat\big)$ of the [[(∞,2)-category]] [[(∞,1)Cat]] of [[(∞,1)-categories]] has been argued ([Riehl & Verity 13](#RiehlVerity13), following [Joyal 08](#Joyal08) [p. 158](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=10)) to provide a useful context for [[(∞,1)-category theory]] (in the spirit of [[John Gray]]'s "[[Adjointness for 2-Categories|formal category theory]]" in the [[2-category]] [[Cat]] of plain [[categories]]).

For example, the notion of [[adjoint (∞,1)-functors]] turns out to equivalently reduce to plain [[adjunctions]] in this homotopy 2-category ([Joyal 08](#Joyal08) [p. 159](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=11), [Riehl & Verity 13, Sec 4](#RiehlVerity13), see [below](#AdjunctionsAndAdjointInfinityFunctors).

## Properties

### Adjunctions and adjoint $(\infty,1)$-functors
 {#AdjunctionsAndAdjointInfinityFunctors}

\begin{proposition}
\label{AdjointInfinityFunctorsInHomotopy2Category}
  An adjunction in the [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category of $(\infty,1)$-categories]] is equivalently a pair of [[adjoint (infinity,1)-functors|adjoint $(\infty,1)$-functors]].
\end{proposition}
([Riehl & Verity 2015, Rem. 4.4.5](adjoint+infinity1-functor#RiehlVerity15); [Riehl & Verity 2022, Prop. F.5.6](adjoint+infinity1-functor#RVElements); see [there](adjoint+infinity1-functor#InTheHomotopy2Category) for more).

\begin{remark}
  In view of its classical analog ([this Prop.](adjunction#AdjunctionsInCatAreAdjointFunctors)),
  the remarkable aspect of Prop. \ref{AdjointInfinityFunctorsInHomotopy2Category} is that the [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category of $\infty$-categories]] is sufficient to detect adjointness of [[(infinity,1)-functors|$\infty$-functors]], which would, *a priori*, be defined as a kind of homotopy-coherent adjointness in the full [[(infinity,2)-category|$(\infty,2)$-category]] [[(infinity,1)Cat|$Cat_{(\infty,1)}$]]. For more on this reduction of homotopy-coherent adjunctions to plain adjunctions see [Riehl & Verity 2016, Thm. 4.3.11, 4.4.11](adjoint+infinity1-functor#RiehlVerity16).
\end{remark}

### Relation to homotopy 2-category of model categories?
 {#RelationToHomotopy2CategoryOfCombinatorialModelCategories}

One might expect (by the discussion [there](locally+presentable+infinity-category#PresentedByCombinatorialSimplicialModelCategories)) that the [[homotopy 2-category]] of [[locally presentable (∞,1)-categories]]
([Joyal 08](#Joyal08) [p. 348](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=40))
is [[equivalence of 2-categories|equivalent]] to the [[localization of a 2-category|localization of the 2-category]] of [[combinatorial model categories]] at the [[Quillen equivalences]], [[Ho(CombModCat)]].

At least for the homotopy $(2,1)$-categories this is proven in [Pavlov 2021](HoCombModCat#Pavlov21)
and for presentable [[derivators]] it is proven in [Renaudin 2006](HoCombModCat#Renaudin06).

## Related concepts

* [[adjoint (∞,1)-functor]]

* [[Ho(CombModCat)]]

## References

With [[(∞,1)-categories]] modeled as [[quasi-categories]], their homotopy 2-category was considered first in

* {#Joyal08} [[André Joyal]], p. 158 (10 of 348) in: _The theory of quasicategories and its applications_, lectures at: _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, Quadern **45** 2, Centre de Recerca Matemàtica, Barcelona 2008 ([[JoyalTheoryOfQuasiCategories.pdf:file]])

and then developed further (in terms of [[∞-cosmoi]]) in:

* {#RiehlVerity13} [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 ([arXiv:1306.5144](http://arxiv.org/abs/1306.5144), [doi:10.1016/j.aim.2015.04.021](https://doi.org/10.1016/j.aim.2015.04.021))

Review and further discussion in:

* {#Riehl14} [[Emily Riehl]], Chapter 18 of: _[[Categorical Homotopy Theory]]_, Cambridge University Press, 2014 ([pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf), [doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457))

* {#RiehlVerity15} [[Emily Riehl]], [[Dominic Verity]], Section 3 of: _Fibrations and Yoneda's lemma in an $\infty$-cosmos_, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 ([arXiv:1506.05500](https://arxiv.org/abs/1506.05500), [doi:10.1016/j.jpaa.2016.07.003](https://doi.org/10.1016/j.jpaa.2016.07.003))

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], Section 1.3 in: _Infinity category theory from scratch_, Higher Structures Vol 4, No 1 (2020) ([arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf))

* {#RiehlVerity22} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) $[$[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)$]$

[[!redirects homotopy 2-category of (infinity,1)-categories]]

[[!redirects homotopy 2-category of infinity-categories]]
[[!redirects homotopy 2-category of infinity1-categories]]

[[!redirects homotopy 2-category of quasi-categories]]
[[!redirects homotopy 2-category of quasicategories]]


homotopy 2-type

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

## Idea

A homotopy $2$-type is a view of a space where we consider its properties only up to the $2$nd [[homotopy group]] $\pi_2$.  To make this precise, we look at maps that 'see' invariants in dimensions 0,1, and 2. These are the 2-equivalences:

## Definition

A continuous map $X \to Y$ is a **homotopy $2$-equivalence** if it induces isomorphisms on $\pi_i$ for $i = 0, 1, 2$ at each basepoint.  Two spaces share the same **homotopy $2$-type** if they are linked by a zig-zag chain of homotopy $2$-equivalences.

For any 'nice' space $X$, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space $Y$ so that the inclusion of $X$ into $Y$ is a homotopy $2$-equivalence; up to (weak) [[homotopy equivalence]], the result is the same for any space with the same homotopy $2$-type.  Accordingly, a **homotopy $2$-type** may alternatively be defined as a space with trivial $\pi_i$ for $i \gt 2$, or as the unique (weak) [[homotopy type]] of such a space, or as its fundamental $\infty$-[[fundamental infinity-groupoid|groupoid]] (which will be a $2$-[[2-groupoid|groupoid]]).

See the general discussion in [[homotopy n-type]].

## Classification

Homotopy $2$-types can be classified by various different types of algebraic data.

### Homotopy $2$-types as crossed modules

Homotopy $2$-types can be classified up to weak homotopy type by [[crossed module]]s of groupoids.  These are the $2$-truncated versions of [[crossed complex]]es.  Such a $C$ consists of a morphism
$$\delta: C_2 \to C_1$$
of [[groupoid]]s with object set $C_0$ such that $C_2$ is totally disconnected, i.e. is a family of groups $C_2(x), x \in C_0$. Further the groupoid $C_1$ operates on this family of groups so that (using right operations) if $a: x \to y$ in $C_1$ and $u \in C_2(x)$ then $u^a \in C(y)$; and the usual rules for an operation are satisfied, namely $(uv)^a=u^a v^a$, $u^1=u$, $(u^a)^b=u^{a b}$ when these are defined. Further the two basic crossed module rules hold:

CM1) $\delta(u^a)= a^{-1} (\delta u) a $;

CM2) $v^{-1} u v = u ^{\delta v} $;

for all $a \in C_1, u,v \in C_2$ when the rules make sense.

Such a crossed module may be extended to a crossed complex $sk^2 C$ by adding trivial elements in dimensions higher than 2. Hence there is a simplicial [[nerve]] $N^\Delta C$ which in dimension $n$ is
$$ Crs(\Pi (\Delta^n_*), sk^2 C).  $$

The geometric realisation of this is the [[classifying space]] $BC$. Its first and second homotopy groups at $x \in C_0$ are the cokernel and kernel of $\delta: C_2(x) \to C_1(x,x)$. Its components are those of the groupoid $C_1$.  All other homotopy groups are trivial.

If $X$ is a CW-complex then there is a bijection of homotopy classes
$$ [X,BC] \cong [\Pi X_*, sk^2 C],$$
and hence there is a map $X \to B(cotr^2 \Pi X_*)$ inducing isomorphisms of homotopy groups in dimensions 1 and 2.

Here the cotruncation $cotr^n D$ of a general crossed complex $D$ agree with $D$ up to dimension $(n-1)$, is $Cok \delta_{n+1}$ in dimension $n$, and is trivial in higher dimensions.

It is in this sense that crossed modules of groupoids classify weak homotopy $2$-types.

The category $Crs^2$ of such crossed modules of groupoids is equivalent to that of strict [[2-groupoid]]s. Further, $Crs^2$ is [[closed monoidal category|monoidal closed]]:
$$Crs^2(C \otimes D, E) \cong Crs^2(C, CRS^2(D,E))$$
and with a unit interval object $I$ so that (left) homotopies are determined as morphisms $Crs^2(I \otimes D,E)$ or as elements of $CRS^2(D,E)_1$.

### Homotopy $2$-types as simplicial group(oid)s

As a crossed module give rise to an internal groupoid in the category of groups (or groupoids), we can take the nerve of that structure and get a simplicial group (or [[simplicial groupoid|simplicially enriched groupoid]]). From a simplicial group(oid), $G$, one can define a simplicial set called the classifying space $\overline{W}G$ of the simplicial group, $G$, for which construction see [[simplicial group]]. We thus can start with a crossed module $C$ form a simplicial group and then take $\overline{W}$ of that to get another model of $\mathcal{B}C$.

### Homotopy $2$-types as $2$-groupoids

With respect to the standard [[homotopy theory]]-structure on [[2-groupoids]] ([[2-truncated]] [[infinity-groupoids]]) these are equivalent to homotopy 2-types. See at _[[homotopy hypothesis]]_ for more on this.

### Homotopy 2-types as double groupoids

see

* [[Antonio Martínez Cegarra]]; Benjamın A. Heredia ; Josué Remedios,
_Double groupoids and homotopy 2-types_
Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also [arXiv:1003.3820](http://arxiv.org/abs/1003.3820).

## Related concepts

* [[fundamental 2-groupoid]]

[[!include homotopy n-types - table]]

[[!redirects 2-type]]
[[!redirects 2-types]]

[[!redirects homotopy 2-types]]


homotopy 3-type

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

## Idea

A homotopy $3$-type is a [[homotopy type]] where we consider its properties only up to the $3$nd [[homotopy group]] $\pi_3$, a [[homotopy n-type]] for $n = 3$

## Definition

A continuous map $X \to Y$ is a **homotopy $3$-equivalence** if it induces isomorphisms on $\pi_i$ for $0 \leq i \leq 3$ at each basepoint.  Two spaces share the same **homotopy $3$-type** if they are linked by a zig-zag chain of homotopy $3$-equivalences.

For any nice space $X$, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space $Y$ so that the inclusion of $X$ into $Y$ is a homotopy $3$-equivalence; up to (weak) [[homotopy equivalence]], the result is the same for any space with the same homotopy $3$-type.  Accordingly, a **homotopy $3$-type** may alternatively be defined as a space with trivial $\pi_i$ for $i \gt 3$, or as the unique (weak) [[homotopy type]] of such a space, or as its fundamental $\infty$-[[fundamental infinity-groupoid|groupoid]] (which should be a $3$-[[3-groupoid|groupoid]]).

See the general discussion in [[homotopy n-type]].

## Models

There are many useful algebraic models for a homotopy $3$-type. (Assume the homotopy type is connected for simplicity.)

1. [[2-crossed module]]s
1. [[crossed square]]s
1. [[cat-2-group]]s
1. [[Gray-group]]s: one-object groupoidal [[Gray-category|Gray-categories]]

One measure of the usefulness of a given model may be its ease of calculation (e.g., with a generalised [[van Kampen theorem]]) and of extraction of topologically significant invariants. In the above a lot more is known, from this viewpoint, about the second and third model than for the first.

Of course, any sufficient weak notion of $3$-[[3-groupoid|groupoid]] ought to qualify, by the [[homotopy hypothesis]].

## Related concepts

[[!include homotopy n-types - table]]

[[!redirects homotopy 3-types]]


homotopy 3-types > history

< [[homotopy 3-types]]
[[!redirects homotopy 3-types -- history]]


homotopy > history

< [[homotopy]]<br />
< [[homotopy (as an operation)]]


homotopy algebra

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

## Idea

For each of various classical types of [[algebras]], like [[associative algebra|associative]], [[Lie algebra|Lie]], [[Leibniz algebra|Leibniz]], [[commutative algebra|commutative]] and so on, one can form a version where the required identities hold only up to [[homotopy]]. The most important modification is that the required identities hold up to *coherent* homotopy; such were traditionally called _strong homotopy algebras_. If $P$ is an [[operad]] in some homotopical context then to obtain the homotopy generalization of $P$-[[algebra of an operad|algebras]], one needs to [[resolution|resolve]] the operad $P$ to an appropropriate replacement $P_\infty$ which is minimal or cofibrant in certain sense.

## Terminology in the case of dg-operads

The classical case is in the setup of a [[dg-operad]].
Thus one had __strong homotopy associative__ algebras which are the same as $A_\infty$-[[A-infinity-algebra|algebras]], __strong homotopy commutative__ (associative) algebras which also called $C_\infty$-[[C-infinity-algebra|algebras]], __strong homotopy bialgebras__ or $B_\infty$-[[B-infinity algebra|algebras]], __strong homotopy Gerstenhaber__ or $G_\infty$-[[G-infinity-algebra|algebras]] and __strong homotopy Lie algebras__ or $L_\infty$-[[L-infinity-algebra|algebras]] and (strong) [[homotopy BV-algebra]]s or $BV_\infty$-[[BV-infinity-algebra|algebras]].

## Related concepts

* [[algebra over a monad]]

  [[∞-algebra over an (∞,1)-monad]]

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

* [[algebra over an operad]]

   * [[∞-algebra over an (∞,1)-operad]]

   * [[model structure on algebras over an operad]]

## References

The term _homotopy algebra_ appears explicitly for instance in the following references. But see the above pages and [[higher algebra]] for more general lists of references.

These two articles discuss the [[homotopy theory]] of [[dg-algebra]]s over a [[dg-operad]].

* [[Vladimir Hinich]], V. V. Schechtman, _On homotopy limit of homotopy algebras_, $K$-theory, arithmetic and geometry (Moscow, 1984--1986), 240--264, Lecture Notes in Math., 1289, Springer, Berlin, 1987.

* [[Vladimir Hinich]], _Homological algebra of homotopy algebras_, Comm. Algebra 25 (1997), no. 10, 3291--3323.

See also [[dg-geometry]].

This article discusses something like a model for [[∞-algebras over an (∞,1)-operad]]

* [[Tom Leinster]], _Homotopy algebras for operads_, [math.QA/0002180](http://arxiv.org/abs/math.QA/0002180), 101 pages, 2000

[[!redirects homotopy algebra]]
[[!redirects homotopy algebras]]
[[!redirects strong homotopy algebra]]
[[!redirects strong homotopy algebras]]


homotopy analysis method

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[ODE]]/[[PDE]]-theory, what is known as the *homtopy analysis method* ([Liao 92](#Liao92)) is a method for solving a *non-linear* [[differential equation]] $N_x u(x) = 0$ by considering its [[convex combination]] with a suitably chosen [[linear differential equation]] $L_x \big(u(x) - u_0(x)\big) = 0$ and then solving the combined equation

\[
 \label{CombinationOfDifferentialEquations}
  q \cdot N_x u(x,q)
  \;+\;
  (1-q) \cdot L_x\big(u(x,q) - u_0(x,q)\big)
  \;=\;
  0
  \,,
  {\phantom{AAA}}
  for\; q \in [0,1]
\]

in terms of a [[Taylor series]]-expansion of the desired solution $u$ in the new parameter $q$ ("homotopy Maclaurin series"). This transforms the original non-linear problem into an infinite sequence of [[linear differential equations]], that can (apparently) often be solved more readily.

The terminology "homotopy analysis method" alludes to regarding the  [[convex combination]] (eq:CombinationOfDifferentialEquations) as a "[[homotopy]] of [[differential equations]]" from the linear $L$ (at $q = 0$) to the non-linear $N$ (at $q = 1$). But it seems that there is no actual [[homotopy theory]] being used in discussion of the "homotopy analysis method".

## References

Original articles:

* {#Liao92} Shijun Liao, *The proposed homotopy analysis technique for the solution of nonlinear problems*, PhD thesis, Shanghai Jiao Tong University (1992)

* Shijun Liao, *Notes on the homotopy analysis method: Some definitions and theorems*, Communications in Nonlinear Science and Numerical Simulation, Volume 14, Issue 4, April 2009, Pages 983-997 ([doi:10.1016/j.cnsns.2008.04.013](https://doi.org/10.1016/j.cnsns.2008.04.013))

* Binfeng Pan, Yangyang Ma, Yang Ni, *A new fractional homotopy method for solving nonlinear optimal control problems*, Acta Astronautica, Volume 161, August 2019, Pages 12-23 ([doi:10.1016/j.actaastro.2019.05.005](https://doi.org/10.1016/j.actaastro.2019.05.005))

But see:

* Cheng-shi Liu, *The essence of the homotopy analysis method*, Applied Mathematics and Computation Volume 216, Issue 4, 15 April 2010, Pages 1299-1303 ([doi:10.1016/j.amc.2010.02.022](https://doi.org/10.1016/j.amc.2010.02.022))

Introduction and review:

* Shijun Liao, *Beyond Perturbation -- Introduction to the Homotopy Analysis Method*, CRC Press 2003 ([ISBN:9781584884071](https://www.routledge.com/Beyond-Perturbation-Introduction-to-the-Homotopy-Analysis-Method/Liao/p/book/9781584884071))

* Shijun Liao, *Homotopy Analysis Method in Nonlinear Differential Equations* Springer 2012 ([pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-642-25132-0.pdf))

* V.G. Gupta, Sumit Gupta, *Application of the Homotopy Analysis Method for solving nonlinear Cauchy Problem*, Surveys in Mathematics and its Applications, Volume 7 (2012), 105 – 116 ([ISSN:1842-6298](https://www.emis.de/journals/SMA/v07/a08.html), [pdf](https://www.emis.de/journals/SMA/v07/p08.pdf))

See also

* Wikipedia, *[Homotopy analysis method](https://en.wikipedia.org/wiki/Homotopy_analysis_method)*

Further development:

* M. Turkyilmazoglu, *A note on the homotopy analysis method*, Applied Mathematics Letters Volume 23, Issue 10, October 2010, Pages 1226-1230 ([doi:10.1016/j.aml.2010.06.003](https://doi.org/10.1016/j.aml.2010.06.003))


homotopy BV-algebra

+-- {: .standout}
rough notes from a talk by [[Bruno Vallette]] -- **raw material** to be polished

see also [[framed little 2-disk operad]]

=--

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

## Homotopy theory for $A_\infty$-algebras

**[[homotopy theory]] for $A_\infty$-[[A-infinity algebra|algebras]]**

for $V$ a complex with the structure of an $A_\infty$-[[A-infinity algebra|algebra]] and for $V \to W$ a morphism of chain cmoplexes, we get an induced $A_\infty$-structure on $W$.

Application: for $V$ a [[differential graded algebra]] its [[chain homology and cohomology|chain cohomology]] inherits the structure of an $A_\infty$-[[A-infinity algebra|algebra]]. The product operations are the [[Massey product]]s.

+-- {: .un_theorem}
###### Theorem (Getzler--Jones, Hinich, Berger--Moerdijk, Spitzweck)
There is a [[cofibrantly generated model category|cofibrantly generated]] [[model category]] structure on the [[category]] of differential graded [[operad]]s (operads in the monoidal cateory of [[chain complex]]es or [[cochain complex]]es).
=--

+-- {: .un_prop}
###### Proposition
If $P_\infty$ is a cofibrant dg-operad, then under some assumptions $P_\infty$-[[algebra for an operad]] structures are preserved by weak equivalences.
=--

+-- {: .un_defn}
###### Definition
A quasi-free dg-operad is one that is free after forgetting the differential
=--

+-- {: .un_prop}
###### Proposition
Cofibrant operads are the retracts of quasi-free operads.
=--

In particular quasi-free operads are cofibrant.

So we look for quasi-free resolutions.

+-- {: .un_defn}
###### Definition
Gerstenhaber algebra: essentially a [[Poisson algebra]] in the dg context.
=--

+-- {: .un_remark}
###### Question
How to define a Gerstenhaber algebra up to homotopy?
=--

Extend the operations dfined by Gerstenhaber on $CH(A,A)$ whjich induce the Gerstenhaber algebra on $HH(A,A)$ to a **Gerstenhaber algebra up to homotopy**.

So we have a strict structure on Hochschild homology $HH(A,A)$ and are asking for from which homotopy structure it may come on Hochschild chains in $CH(A,A)$.

So we define the [[Gerstenhaber algebra operad]] whose [[algebra for an operad]] are [[Gertsenhaber algebra]]s.

It is generated of course from the product operation and the bracket operation modulo the associativity constraint for the product, the Jacobi identity for the bracket and the Leibnitz property for their interaction.

These relations are always encoded in _quadratic_ expressions.

### Koszul duality theory for operads

the dual notion of [[operad]] is that of [[cooperad]] (reverse all arrows)

+-- {: .un_theorem}
###### Theorem (Ginzburg--Kapranov, Getzler--Jones)

There exist [[adjoint functors]]

$$
  B : \{dg operads\} \leftrightarrow \{dg cooperads\} : \Omega
$$

given by [[bar and cobar construction]]

under this equivalence

$$
  quadratic operad P \stackrel{Koszul duality}{\to}
  cooperad P'
$$

with $P_\infty := \Omega P' \to P$ morphism of [[dg-operad]]s.
=--

+-- {: .un_defn}
###### Definition (Koszul operad)

$P$ is Koszul if $P_\infty = \Omega P' \stackrel{\simeq}{\to} P$ is a quasi-isomorphism, i.e. a cofibrant replacement.
=--

+-- {: .un_prop}
###### Proposition (Ginzburg--Kapranov, Getzler--Jones)

If $P = F(V)/(R)$ be a quadratic operad with $dim(V) \lt + \infty$ The _linear dual_ $P'^*$ is a quadratic operad aabd the suspension of $P'^* \simeq ' := F(V^* \otimes sgn)/(R^{\perp})$.
=--

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

Compute $P'$ and its operadic structure with this formula, then dualize everything to get this formula.
=--

+-- {: .un_prop}
###### Proposition (Getzler--Jones, Markl)
The Gerstenhaber operad $G$ is Koszul.
=--

### Homotopy Gerstenhaber or $G_\infty$-algebras

$$
  G = G' = Com \circ Lie^1
$$

+-- {: .un_defn}
###### Definition (Batalin--Vilkovisky algebra)

This is a [[Gerstenhaber algebra]] $A$ with a unary operator $\Delta : A \to A$ of degree +1 such that

* $\Delta^2 = 0 $

* $[-,-]$ measures failure of $\Delta$ to be a derivation with respect to the product $\cdot$.
=--

This also is the algebra over an operad. But this operad is no longer a [[quadratic operad]].

So we define:

+-- {: .un_defn}
###### Definition (quadratic BV-algebra)

As above but now demand $\Delta$ a derivation of both the product $\cdot$ and the bracket $[-,-]$.
=--

This is a first step in the resolution process.

Now the corresponding operad is Koszul. So we get

$$
  \Rightarrow qBV_\infty := \Omega q BV' \stackrel{\simeq}{\to}
$$

as a quasi-free resolution.

+-- {: .un_defn}
###### Definition-proposition** A homotopy BV-algebra is an [[algebra over an operad]] for that.

In terms of this we have a homotopy BV-algebra structure on $CH(A,A)$.
=--

+-- {: .un_remark}
###### Idea
Add a suitable differential $d_1 : qBV' \to qBV'$.
=--

### Koszul duality theory

Let $P = F(V)/(R)$ be a quadratic and linear presentation of a dg-operad

let $q : F(V) \to F(V)^{(2)}$ the quadratic projection

and $qP := F(V)/(qR)$ the quadratic analogue of $P$

+-- {: .un_defn}
###### Definition (quadratic-linear Koszul operad)

$P = F(V)/(R)$ is a Koszul operad if

* $R \cap V = \{0\} \Leftrightarrow V is minimal$

* (more ...)

* $q P$ is Koszul
=--

This now yields a good machinery for cofibrant resolutions for Koszul operads.

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

When $P$ is a quadratic linear Koszul operad then

$$
  P_\infty := \Omega P' \stackrel{\simeq}{\to} P
$$
=--

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

This applies to the BV operad.
=--

+-- {: .un_defn}
###### Definition
A **$BV_\infty$-algebra** is an algebra over this cofibrant replacement for the BV operad.
=--

### Applications

+-- {: .un_theorem}
###### Theorem (Ginzburg, Tradler, Menichi)

* $A$ a [[Frobenius algebra]].

* Its [[Hochschild cohomology]] $HH(A,A)$ is a [[BV-algebra]].
=--

+-- {: .un_theorem}
###### Theorem (Cyclic Deligne conjecture)

...
=--

### Comparison to other definitions

Another definition of homotopy BV-algebra by Kravchenko -- this turns out to be a special case of the definition here by setting some operations to $0$ (her algebra is not an algebra over a cofibrant operad).

Another definition by Tamarkin--Tsygan: this is more general than the one here. TT have many more operations, namely operations with sevearl outputs.

The notion here also difers from that in Beilinson--Drinfeld.

### PBW isomorphism

The free operad $F(V)$ is filtered $\Rightarrow P = F(V)/(R)$ is filtered. There is then a morphism of operads $q P \to gr P$.

+-- {: .un_theorem}
###### Theorem
For $P$ Koszul we have

$$
  q P \simeq gr P
$$

For instance

$$
  q BV \simeq gr P
$$
=--

### Relation with framed little disks

Let $dD$ be the [[framed little disk operad]]; then:

Getzler: $H_\bullet(fD) \simeq BV$.

+-- {: .un_theorem}
###### Theorem (Gianciracusa--Salvatore--Severa)

$fD$ is formal, i.e. quasi-isomorphic by zig-zags to its homology
=--

### Relation with TCFT

A [[topological conformal field theory]] is an algebra over the [[PROP]] $C_\bullet(\mathcal{R})$ of [[Riemann surface]]s.

+-- {: .un_theorem}
###### Theorem
Any [[TCFT]] carries a homotopy BV-algebra structure which lifts the [[BV-algebra]] structure of Getzler in its [[homology]].
=--

### Homotopy theory for $P_\infty$ algebras

### Deformation theory

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

In some sense homotopy BV is a formal extension of homotopy Gerstenhaber.
=--

+-- {: .un_theorem}
###### Theorem (generalized Lian--Zuckermann conjecture)

For any [[topological vertex algebra]] $A$ with $\mathbb{N}$-graded conformal weight there exists an expliciit $BV_\infty$-algebra structure on $A$ which extends Lian--Zuckermann operations on $A$ and which lifts the [[BV-algebra]] structure on $H(A)$.
=--

## Related entries

* [[quantum L-infinity algebra]]

## References

The above material probably roughly follows the talk slides

* [[Bruno Vallette]], _Homotopy Batalin-Vilkovisky algebras_ , [slides](https://www.math.univ-paris13.fr/~vallette/download/Homotopy%20BV.pdf))

The corresponding article is

* [[Imma Galvez-Carrillo]], [[Andy Tonks]], [[Bruno Vallette]], _Homotopy Batalin-Vilkovisky algebras_ ([arXiv:0907.2246](http://arxiv.org/abs/0907.2246))

See also

* Imma Gálvez, Vassily Gorbounov, Andrew Tonks, _Homotopy Gerstenhaber structures and vertex algebras_, [math/0611231.QA](http://arxiv.org/abs/math/0611231)

[[!redirects homotopy BV algebra]]


homotopy canonicity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A version of [[canonicity]] where the [[definitional equalities]] are replaced with [[identity types]].

In the case for the [[natural numbers type]] homotopy canonicity states that every natural number $n:\mathbb{N}$ comes with a natural number $m:\mathbb{N}$ and an [[identification]] to the [[canonical form]] $p:n =_\mathbb{N} \mathrm{succ}^m(0)$.

## Examples

* [[Chris Kapulkin]] and [[Christian Sattler]] have an unpublished proof that [[homotopy type theory]] has homotopy canonicity, even though HoTT does not have [[canonicity]]

* [[cubical type theory]] has homotopy canonicity

## References

* [[Michael Shulman]], _Univalence for inverse diagrams, oplax limits, and gluing, and homotopy canonicity_ ([arXiv:1203.3253](http://arxiv.org/abs/1203.3253))

* [[Thierry Coquand]], [[Simon Huber]], [[Christian Sattler]], *Canonicity and homotopy canonicity for cubical type theory*, Logical Methods in Computer Science, Volume 18, Issue 1 (February 3, 2022) lmcs:9043, ([doi:10.46298/lmcs-18%281%3A28%292022](https://doi.org/10.46298/lmcs-18%281%3A28%292022), [arXiv:1902.06572](https://arxiv.org/abs/1902.06572))

* [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))


homotopy category

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Given a [[category with weak equivalences]] $(\mathcal{C},W)$, then its _homotopy category_ $Ho(\mathcal{C})$ is, if it exists, the result of universally forcing the [[weak equivalences]] to become actual [[isomorphisms]], also called the _[[localization]]_ at the weak equivalences

$$
  \mathcal{C} \longrightarrow Ho(\mathcal{C})= \mathcal{C}[W^{-1}]
  \,.
$$

The classical example is the category of [[topological spaces]] with weak equivalences those [[continuous functions]] which are _[[homotopy equivalences]]_ or _[[weak homotopy equivalences]]_. The corresponding homotopy category is often referred to as "the homotopy category", by default, or the "[[classical homotopy category]]" for emphasis. This turns out to be equivalent to the category of topological spaces or (for weak homotopy equivalences) of just those [[homeomorphism|homeomorphic]] to [[CW-complexes]] with [[left homotopy]]-[[homotopy class|classes]] of continuous functions between them, whence the name "homotopy category".

The existence of a homotopy category, as well as tractable presentations of it typically require extra [[properties]] of the class of weak equivalences (such as that they admit a [[calculus of fractions]]) or even extra [[structure]] (such as [[fibration category]]/[[cofibration category]] structure, or full [[model category]] structure, or further enhancements of that to [[simplicial model category]] structures, etc). See at _[[homotopy category of a model category]]_ for more on this.

More generally, to every [[(∞,1)-category]] is associated a homotopy category, whose morphisms are literally the [[homotopy classes]] of the original morphisms. See at _[[homotopy category of an (∞,1)-category]]_ for more on this.

These two concepts of "homotopy category" are compatible: to a [[category with weak equivalences]] is associated, if it exists, an [[(∞,1)-category]] obtained by universally forcing the [[weak equivalences]] to become actual [[homotopy equivalences]], also called the _[[simplicial localization]]_ $L_W \mathcal{C}$ at the weak equivalences. The homotopy categories of $(\mathcal{C},W)$ and of $L_W \mathcal{C}$ coincide, which justifies the terminology "homotopy category" generally.

## Definition

### For simplicially enriched categories

Given a simplicially enriched category $C$, we can form for each pair of objects, $x,y$, of objects of $C$, the set, $\pi_0C(x,y)$, of connected components of the 'function space' $C(x,y)$.  As $\pi_0$ preserves finite limits, this gives a category, denoted $\pi_0(C)$.  As 1-simplices in $C(x,y)$ can be often interpreted as being homotopies, this category $\pi_0(C)$ is often called the _homotopy category of $C$_, and then the notation $Ho(C)$ may be used.

This notions is closely related to the next, by using, say the [[hammock localisation]] of Dwyer and Kan, as then $\pi_0$ of that simplicially enriched category, coincides with the following.

### For categories with weak equivalences

Given a [[category with weak equivalences]] (such as a [[model category]]), its __homotopy category__ $Ho(C)$ is -- if it exists -- the [[category]] which is universal with the property that there is a [[functor]]

$$
 Q :  C \to Ho(C)
$$

that sends every weak equivalence in $C$ to an [[isomorphism]] in $Ho(C)$.

One also writes $Ho(C) := W^{-1}C$ or $C[W^{-1}]$ and calls it the [[localization]] of $C$ at the collection $W$ of weak equivalences.

More in detail, the universality of $Ho(C)$ means the following:

* for any (possibly [[large category|large]]) category $A$ and functor $F : C \to A$ such that $F$ sends all $w \in W$ to isomorphisms in $A$, there exists a functor $F_Q : Ho(C) \to A$ and a natural isomorphism

$$
  \array{
     C &&\stackrel{F}{\to}& A
     \\
     \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q}
     \\
     Ho(C)
  }
$$

* the functor $Q^* : Func(Ho(C),A) \to Func(C,A)$ is a [[full and faithful functor]].

The second condition implies that the functor $F_Q$ in the first condition is unique up to unique isomorphism.

## Properties

* If it exists, the homotopy category $Ho(C)$ is unique up to [[equivalence of categories]].

* As described at [[localization]], in general, the morphisms of $Ho(C)$ must be constructed using zigzags of morphisms in $C$ in which the backwards-pointing arrows are weak equivalences.  This means that in general, $Ho(C)$ need not be [[locally small category|locally small]] even if $C$ is.  However, in many cases (such as any [[model category]]) there is a more direct description of the morphisms in $Ho(C)$ as [[homotopy]] classes of maps in $C$ between suitably "good" (fibrant and cofibrant) objects.

* In [[2-limit|2-categorical terms]], the homotopy category $Ho(C)$ is the _coinverter_ of the canonical 2-cell
$$\array{& \to \\ W & \Downarrow & C\\ & \to}$$
where $W$ is the category whose objects are morphisms in $C$ and whose morphisms are commutative squares in $C$.

## Examples

* In classical [[homotopy theory]], _the_ homotopy category refers to the homotopy category [[Ho(Top)]] of [[Top]] with weak equivalences taken to be [[weak homotopy equivalences]].

* [[Ho(Top)]] is often restricted to the [[full subcategory]] of spaces of the [[homotopy type]] of a [[CW-complex]] (the full subcategory of CW-complexes in $Ho(Top)$). This is equivalent to $Ho(sSet_{Quillen})$, the homotopy category of the standard Quillen-[[model structure on simplicial sets]]. This equivalence is one aspect of the [[homotopy hypothesis]].

* In [[homological algebra]] the localization of the [[category of chain complexes]] at the [[quasi-isomorphisms]] is called the _[[derived category]]_. But see also at _[[homotopy category of chain complexes]]_.

* In [[stable homotopy theory]] one considers the [[stable homotopy category]] of [[spectra]].

* In [[equivariant stable homotopy theory]] one considers the [[equivariant stable homotopy category]] of spectra.

* For the homotopy category of [[Cat]], see [[Ho(Cat)]].

* For the homotopy category of that of [[combinatorial model categories]] see [[Ho(CombModCat)]].

## Related concepts

* [[homotopy category of a model category]]

* [[homotopy category of an (infinity,1)-category]]

* [[homotopy 2-category]]

* [[simplicial homotopy category]]

## References

Early discussion:

* {#Brown65} [[Edgar Brown]], _Abstract homotopy theory_, Trans. AMS 119 no. 1 (1965) $[$[doi:10.1090/S0002-9947-1965-0182970-6](https://doi.org/10.1090/S0002-9947-1965-0182970-6)$]$

For more see the references at *[[model category]]* or *[[classical homotopy category]]*.

[[!redirects homotopy categories]]


homotopy category of a dg-category

Let $A$ be a [[dg-category]], so that one has a mapping complex of morphisms $Map_A(x,y)$ between any two objects $x$ and $y$.  In analogy with [[simplicially enriched category|simplicially enriched categories]], the **homotopy category** of $A$ is the [[preadditive category]] $ho(A)$ whose objects are the objects of $A$, morphism groups are given by
  \[ \Hom_{ho(A)}(x, y) = H^0(Map_A(x,y)) \]
for any two objects $x$ and $y$, and composition is induced from the composition law of $A$.  Here $H^0(-)$ denotes the zeroth [[cohomology]] group of the complex (which is by convention graded cohomologically).

### Related entries

* [[homotopy category of an (infinity,1)-category]]


homotopy category of a model category

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

#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] structure $(\mathcal{C}(W,Fib,Cof))$ on some [[category]] $\mathcal{C}$ is a means to guarantee the [[locally small category|local smallness]] and to improve the tractability of the [[homotopy category]] of the underlying [[category with weak equivalences]] $(\mathcal{C},W)$. In particular, if a [[category with weak equivalences]] admits a [[model category]] structure, then its [[homotopy category]] (in the sense of [[localization]] $\mathcal{C}[W^{-1}]$ at the class of [[weak equivalences]]) is [[equivalence of categories|equivalent]] to the category of [[bifibrant objects]] whose morphisms are the actual [[homotopy classes]] of morphisms between them ([[left homotopy]] or [[right homotopy]] [[equivalence classes]] in the sense of [[homotopy in a model category]]) .

## Definition

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

Let $\mathcal{C}$ be a [[model category]]. Write $Ho(\mathcal{C})$ for the [[category]] whose

* [[objects]] are the [[bifibrant objects]] of $\mathcal{C}$;

* [[morphisms]] are the [[homotopy classes]] of morphisms of $\mathcal{C}$, hence the [[equivalence classes]] of morphism under [[left homotopy]].

=--

This is, up to [[equivalence of categories]], the _homotopy category of the model category_ $\mathcal{C}$.

## Universal property

We spell out that def. \ref{HomotopyCategoryOfAModelCategory} indeed satisfies the [[universal property]] that defines the [[homotopy category]] of a [[category with weak equivalences]].

+-- {: .num_lemma #WhiteheadTheoremInModelCategories}
###### Lemma
**([[Whitehead theorem]] in model categories)**

Let $\mathcal{C}$ be a [[model category]]. A [[weak equivalence]] between [[bifibrant objects]] is a [[homotopy equivalence]].

=--

(e.g. [Goerss-Jardine 99, part I, theorem 1.10](#GoerssJardine99))

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

By the factorization axioms in $\mathcal{C}$, every weak equivalence $f\colon X \longrightarrow Y$ factors through an object $Z$ as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with $X$ and $Y$ both [[bifibrant object|bifibrant]], so is $Z$, and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.

So let $f \colon X \longrightarrow Y$ be an acyclic fibration between fibrant-cofibrant objects, the case of acyclic cofibrations is [[formal dual|formally dual]]. Then in fact it has a genuine [[right inverse]] given by a lift $f^{-1}$ in the diagram

$$
  \array{
    \emptyset &\rightarrow& X
    \\
    {}^{\mathllap{\in cof}}\downarrow
     &{}^{{f^{-1}}}\nearrow&
   \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib \cap W}}
    \\
    Y &=& Y
  }
  \,.
$$

To see that $f^{-1}$ is also a [[left inverse]] up to [[left homotopy]], let $Cyl(X)$ be any [[cylinder object]] on $X$, hence a factorization of the [[codiagonal]] on $X$ as a cofibration followed by a an acyclic fibration

$$
  X \sqcup X \stackrel{\iota_X}{\longrightarrow} Cyl(X) \stackrel{\sigma}{\longrightarrow} X
$$

and consider the square

$$
  \array{
    X \sqcup X
    &\stackrel{(f^{-1}\circ f, id)}{\longrightarrow}& X
    \\
    {}^{\mathllap{\iota_X}}{}_{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W \cap Fib}}
    \\
    Cyl(X) &\underset{f\circ \sigma}{\longrightarrow}& Y
  }
  \,,
$$

which [[commuting square|commutes]] due to $f^{-1}$ being a genuine right inverse of $f$. By construction, this [[commuting square]] now admits a [[lift]] $\eta$, and that constitutes a [[left homotopy]] $\eta \colon f^{-1}\circ f \Rightarrow_L id$.

=--

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

Given a [[model category]] $\mathcal{C}$, consider a _choice_ for each object $X \in \mathcal{C}$ of

1. a factorization $\emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_x}{\longrightarrow} X$ of the initial morphism, such that when $X$ is already cofibrant then $p_X = id_X$;

1. a factorization $X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} R X \underoverset{\in Fib}{q_x}{\longrightarrow} \ast$ of the terminal morphism, such that when $X$ is already fibrant then $j_X = id_X$.

Write then

$$
  \gamma_{R,Q}
  \;\colon\;
  \mathcal{C}
   \longrightarrow
  Ho(\mathcal{C})
$$

for the [[functor]] to the homotopy category, def. \ref{HomotopyCategoryOfAModelCategory}, which sends an object $X$ to the object $R Q X$ and sends a morphism $f \colon X \longrightarrow Y$ to the [[homotopy class]] of the result of first lifting in

$$
  \array{
    \emptyset &\longrightarrow& Q Y
    \\
    {}^{\mathllap{i_X}}\downarrow &{}^{Q f}\nearrow& \downarrow^{\mathrlap{p_Y}}
    \\
    Q X &\underset{f\circ p_X}{\longrightarrow}& Y
  }
$$

and then lifting (here: [[extension|extending]]) in

$$
  \array{
    Q X &\overset{j_{Q Y} \circ Q f}{\longrightarrow}& R Q Y
    \\
    {}^{\mathllap{j_{Q X}}}\downarrow &{}^{R Q f}\nearrow& \downarrow^{\mathrlap{q_{Q Y}}}
    \\
    R Q X
    &\longrightarrow&
    \ast
  }
  \,.
$$

=--

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

The construction in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory} is indeed well defined.

=--

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

First of all, the object $R Q X$ is indeed both fibrant and cofibrant (as well as related by a [[zig-zag]] of weak equivalences to $X$):

$$
  \array{
     \emptyset
     \\
     {}^{\mathllap{\in Cof}}\downarrow & \searrow^{\mathrlap{\in Cof}}
     \\
     Q X &\underset{\in W \cap Cof}{\longrightarrow}& R Q X &\underset{\in Fib}{\longrightarrow}& \ast
     \\
     {}^{\mathllap{\in W}}\downarrow
     \\
     X
  }
  \,.
$$

Now to see that the image on morphisms is well defined. First observe that any two choices $(Q f)_{i}$ of the first lift in the definition are left homotopic to each other, exhibited by lifting in

$$
  \array{
    Q X \sqcup Q X &\stackrel{((Q f)_1, (Q f)_2 )}{\longrightarrow}& Q Y
    \\
    {}^{\mathllap{\in W \cap Cof}}\downarrow
     &&
    \downarrow^{\mathrlap{p_{Y}}}_{\mathrlap{\in Fib}}
    \\
    Cyl(Q X)
    &\underset{f \circ p_{X} \circ \sigma_{Q X}}{\longrightarrow}&
    Y
  }
  \,.
$$

Hence also the composites $j_{Q Y}\circ (Q_f)_i $ are [[left homotopy|left homotopic]] to each other, and since their domain is cofibrant, they are also [[right homotopy|right homotopic]] (via [this](homotopy+in+a+model+category#LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually) lemma) by a right homotopy $\kappa$. This implies finally, by lifting in

$$
  \array{
    Q X &\overset{\kappa}{\longrightarrow}& Path(R Q Y)
    \\
    {}^{\mathllap{\in W \cap Cof}}\downarrow
     &&
    \downarrow^{\mathrlap{\in Fib}}
    \\
    R Q X &\underset{(R (Q f)_1, R (Q f)_2)}{\longrightarrow}&
    R Q Y \times R Q Y
  }
$$

that also $R (Q f)_1$ and $R (Q f)_2$ are right homotopic, hence that indeed $R Q f$ represents a well-defined [[homotopy class]].

Finally to see that the assignment is indeed [[functor|functorial]], observe that the commutativity of the lifting diagrams for $Q f$ and $R Q f$ imply that also the following diagram commutes

$$
  \array{
    X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& R Q X
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}}
    \\
    Y &\underset{p_y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& R Q Y
  }
  \,.
$$

Now from the [[pasting]] composite

$$
  \array{
    X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& R Q X
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}}
    \\
    Y &\underset{p_Y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& R Q Y
    \\
    {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{Q g}} && \downarrow^{\mathrlap{R Q g}}
    \\
    Z &\underset{p_Z}{\longleftarrow}& Q Z &\underset{j_{Q Z}}{\longrightarrow}& R Q Z
  }
$$

one sees that $(R Q g)\circ (R Q f)$ is a lift of $g \circ f$ and hence the same argument as above gives that it is homotopic to the chosen $ R Q(g \circ f)$.

=--

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

For $\mathcal{C}$ a [[category with weak equivalences]], its _[[homotopy category]]_ (or: _[[localization]]_ at the weak equivalences) is, if it exists, a [[category]] $Ho(\mathcal{C})$ equipped with a [[functor]]

$$
  \gamma \colon \mathcal{C} \longrightarrow Ho(C)
$$

which sends weak equivalences to [[isomorphisms]], and which is [[universal property|universal with this property]]:

for $F \colon \mathcal{C} \longrightarrow D$ any [[functor]] out of $\mathcal{C}$ into any [[category]], such that $F$ takes weak equivalences to [[isomorphisms]], it factors through $\gamma$ up to a [[natural isomorphism]]

$$
  \array{
    \mathcal{C} && \overset{F}{\longrightarrow} && D
    \\
    & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}}
    \\
    && Ho(\mathcal{C})
  }
$$

and this factorization is unique up to unique isomorphism, in that for $(\tilde F_1, \rho_1)$ and $(\tilde F_2, \rho_2)$ two such factorizations, then there is a unique [[natural isomorphism]] $\kappa \colon \tilde F_1 \Rightarrow \tilde F_2$ making the evident diagram of natural isomorphisms commute.

=--

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

For $\mathcal{C}$ a [[model category]], the functor $\gamma_{R,Q}$ in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory} (for any choice of $R$ and $Q$) exhibits $Ho(\mathcal{C})$ as indeed being the [[homotopy category]] of the underlying [[category with weak equivalences]], in the sense of def. \ref{HomotopyCategoryOfACategoryWithWeakEquivalences}.

=--

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

First, to see that that $\gamma$ indeed takes weak equivalences to isomorphisms: By [[two-out-of-three]] applied to the [[commuting diagrams]] shown in the proof of lemma \ref{ConstructionOfLocalizationFunctorForModelCategoryIsWellDefined} the morphism $R Q f$ is a weak equivalence if $f$ is:

$$
  \array{
    X &\underoverset{\simeq}{p_X}{\longleftarrow}& Q X &\underoverset{\simeq}{j_{Q X}}{\longrightarrow}& R Q X
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{R Q f}}
    \\
    Y &\underoverset{p_y}{\simeq}{\longleftarrow}& Q Y &\underoverset{j_{Q Y}}{\simeq}{\longrightarrow}& R Q Y
  }
$$

With this the "Whitehead theorem for model categories", lemma \ref{WhiteheadTheoremInModelCategories}, implies that $R Q f$ represents an isomorphism in $Ho(\mathcal{C})$.

Now let $F \colon \mathcal{C}\longrightarrow D$ be any functor that sends weak equivalences to isomorphisms. We need to show that it factors as

$$
  \array{
    \mathcal{C} && \overset{F}{\longrightarrow} && D
    \\
    & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}}
    \\
    && Ho(\mathcal{C})
  }
$$

uniquely up to unique [[natural isomorphism]]. Now by construction of $R$ and $Q$ in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory}, $\gamma_{R,Q}$ is the identity on the [[full subcategory]] of fibrant-cofibrant objects. It follows that if $\tilde F$ exists at all, it must satisfy for all $X \stackrel{f}{\to} Y$ with $X$ and $Y$ both fibrant and cofibrant that

$$
  \tilde F([f]) \simeq F(f)
  \,.
$$

But by def. \ref{HomotopyCategoryOfAModelCategory} that already fixes $\tilde F$ on all of $Ho(\mathcal{C})$, up to unique [[natural isomorphism]]. Hence it only remains to check that with this definition of $\tilde F$ there exists any [[natural isomorphism]] $\rho$ filling the diagram above.

To that end, apply $F$ to the above [[commuting diagram]] to obtain

$$
  \array{
    F(X) &\underoverset{iso}{F(p_X)}{\longleftarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(R Q X)
    \\
    {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{F(Q f)}} && \downarrow^{\mathrlap{F(R Q f)}}
    \\
    F(Y) &\underoverset{F(p_y)}{iso}{\longleftarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(R Q Y)
  }
  \,.
$$

Here now all horizontal morphisms are [[isomorphisms]], by assumption on $F$. It follows that defining $\rho_X \coloneqq F(j_{Q X}) \circ F(p_X)^{-1}$ makes the required natural isomorphism:

$$
  \array{
    \rho_X \colon & F(X) &\underoverset{iso}{F(p_X)^{-1}}{\longrightarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(R Q X)
   &=&
   \tilde F(\gamma_{R,Q}(X))
    \\
    & {}^{\mathllap{F(f)}}\downarrow
     &&
     && \downarrow^{\mathrlap{F(R Q f)}}
     && \downarrow^{\tilde F(\gamma_{R,Q}(X))}
    \\
    \rho_Y\colon& F(Y) &\underoverset{F(p_y)^{-1}}{iso}{\longrightarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(R Q Y)
   &=&
   \tilde F(\gamma_{R,Q}(X))
  }
  \,.
$$

=--

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

Due to theorem \ref{UniversalPropertyOfHomotopyCategoryOfAModelCategory} we may suppress the choices of cofibrant $Q$ and fibrant replacement $R$ in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory} and just speak of [[generalized the|the]] [[localization functor]]

$$
  \gamma \;\colon\;
  \mathcal{C}
  \longrightarrow
  Ho(\mathcal{C})
$$

up to [[natural isomorphism]].

=--

While the construction of the homotopy category in def. \ref{HomotopyCategoryOfAModelCategory} combines the restriction to good (fibrant/cofibrant) objects with the passage to [[homotopy classes]] of morphisms, it is often useful to consider intermediate stages:

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

Given a [[model category]] $\mathcal{C}$, write

$$
  \array{
   && \mathcal{C}_{fc}
   \\
   & \swarrow && \searrow
   \\
   \mathcal{C}_c && && \mathcal{C}_f
   \\
   & \searrow && \swarrow
   \\
   && \mathcal{C}
  }
$$

for the system of [[full subcategory]] inclusions on the cofibrant objects ($\mathcal{C}_c$), the fibrant objects ($\mathcal{C}_f$) and the objects which are both fibrant and cofibrant ($\mathcal{C}_{fc}$), all regarded a [[categories with weak equivalences]], via the weak equivalences inherited from $\mathcal{C}$.

=--

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

Of course the subcategories in def. \ref{FullSubcategoriesOfFibrantCofibrantObjects} inherit more structure than just that of [[categories with weak equivalences]] from $\mathcal{C}$. $\mathcal{C}_f$ and $\mathcal{C}_c$ each inherit "half" of the factorization axioms. One says that $\mathcal{C}_f$ has the structure of a "[[fibration category]]" called a "[[category of fibrant objects]]", while $\mathcal{C}_c$ has the structure of a "[[cofibration category]]".

=--

The proof of theorem \ref{UniversalPropertyOfHomotopyCategoryOfAModelCategory} immediately implies the following:

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

For $\mathcal{C}$ a [[model category]], the restriction of the localization functor $\gamma\;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})$ from def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory} (using remark \ref{EssentialUniquenessOfLocalizationFunctorOfModelCategory}) to any of the sub-[[categories with weak equivalences]] of def. \ref{FullSubcategoriesOfFibrantCofibrantObjects}

$$
  \array{
   && \mathcal{C}_{fc}
   \\
   & \swarrow && \searrow
   \\
   \mathcal{C}_c && && \mathcal{C}_f
   \\
   & \searrow && \swarrow
   \\
   && \mathcal{C}
   \\
   && \downarrow^{\mathrlap{\gamma}}
   \\
   && Ho(\mathcal{C})
  }
$$

exhibits $Ho(\mathcal{C})$ equivalently as the [[homotopy category]] also of these subcategories. In particular there are [[equivalences of categories]]

$$
  Ho(\mathcal{C})
   \simeq
  Ho(\mathcal{C}_{f})
   \simeq
  Ho(\mathcal{C}_{c})
   \simeq
  Ho(\mathcal{C}_{fc})
  \,.
$$

=--

In fact, for computing [[hom-sets]] in the homotopy category, it is sufficient that the [[domain]] is [[cofibrant object|cofibrant]] and the [[codomain]] is [[fibrant object|fibrant]]:

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

For $X, Y \in \mathcal{C}$ with $X$ cofibrant and $Y$ fibrant, and for $P, Q$ fibrant/cofibrant replacement functors as in def. \ref{FibrantCofibrantReplacementFunctorToHomotopyCategory}, the morphism

$$
 Hom_{Ho(\mathcal{C})}(P X,Q Y)
  =
 Hom_{\mathcal{C}}(P X, Q Y)/_{\sim}
   \overset{Hom_{\mathcal{C}}(j_X, p_Y)}{\longrightarrow}
 Hom_{\mathcal{C}}(X,Y)/_{\sim}
$$

(on [[homotopy classes]] of morphisms, well defined by prop. \ref{BetweenCofibFibLeftAndRightHomotopyAreEquivalentEquivalenceRelations}) is a [[natural bijection]].

=--

([Quillen 67, I.1 corollary 1](#Quillen67))

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

We may factor the morphism in question as the composite

$$
  Hom_{\mathcal{C}}(P X, Q Y)/_{\sim}
    \overset{Hom_{\mathcal{C}}(id_{P X}, p_Y)/_\sim }{\longrightarrow}
  Hom_{\mathcal{C}}(P X, Y)/_{\sim}
    \overset{Hom_{\mathcal{C}}(j_X, id_Y)/_\sim}{\longrightarrow}
  Hom_{\mathcal{C}}(X,Y)/_{\sim}
  \,.
$$

This shows that it is sufficient to see that for $X$ cofibrant and $Y$ fibrant, then

$$
  Hom_{\mathcal{C}}(id_X, p_Y)/_\sim
   \;\colon\;
  Hom_{\mathcal{C}}(X, Q Y)/_\sim \to Hom_{\mathcal{C}}(X,Y)/_\sim
$$

is an isomorphism, and dually that

$$
  Hom_{\mathcal{C}}(j_X, id_Y)/_\sim
  \;\colon\;
  Hom_{\mathcal{C}}(P X, Y)/_\sim \to Hom_{\mathcal{C}}(X,Y)/_\sim
$$

is an isomorphism. We discuss this for the former; the second is [[formal dual|formally dual]]:

First, that $Hom_{\mathcal{C}}(id_X, p_Y)$ is surjective is the [[lifting property]] in

$$
  \array{
    \emptyset &\longrightarrow& Q Y
    \\
    {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_Y}}_{\mathrlap{\in W \cap Fib}}
    \\
    X &\overset{f}{\longrightarrow}& Y
  }
  \,,
$$

which says that any morphism $f \colon X \to Y$ comes from a morphism $\hat f \colon X \to Q Y$ under postcomposition with $Q Y \overset{p_Y}{\to} Y$.

Second, that $Hom_{\mathcal{C}}(id_X, p_Y)$ is injective is the lifting property in

$$
  \array{
    X \sqcup X &\overset{(f,g)}{\longrightarrow}& Q Y
    \\
    {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_Y}}_{\mathrlap{\in W \cap Fib}}
    \\
    Cyl(X) &\underset{\eta}{\longrightarrow}& Y
  }
  \,,
$$

which says that if two morphisms $f, g \colon X \to Q Y$ become homotopic after postcomposition with $p_Y \colon Q X \to Y$, then they were already homotopic before.

=--

## Derived functors
 {#DerivedFunctors}

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

For $\mathcal{C}$ and $\mathcal{D}$ two [[categories with weak equivalences]], then a [[functor]] $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ is called _[[homotopical functor]]_ if it sends weak equivalences to weak equivalences.

=--

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

Given a [[homotopical functor]] $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ (def. \ref{HomotopicalFunctor}) between [[categories with weak equivalences]] whose [[homotopy categories]] $Ho(\mathcal{C})$ and $Ho(\mathcal{D})$ exist (def. \ref{HomotopyCategoryOfACategoryWithWeakEquivalences}), then its _[[derived functor]]_ is the functor $Ho(F)$ between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. \ref{HomotopyCategoryOfACategoryWithWeakEquivalences}):

$$
  \array{
    \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D}
    \\
    {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow
    &\swArrow_{\simeq}&
    \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
    \\
    Ho(\mathcal{C})
      &\underset{\exists \; Ho(F)}{\longrightarrow}&
    Ho(\mathcal{D})
  }
  \,.
$$

=--

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

While many functors of interest between [[model categories]] are not homotopical in the sense of def. \ref{HomotopicalFunctor}, many become homotopical after restriction to the [[full subcategories]] of fibrant object or of cofibrant objects, def. \ref{FullSubcategoriesOfFibrantCofibrantObjects}. By corollary \ref{HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects} this is just as good for the purpose of [[homotopy theory]].

=--

Therefore one considers the following generalization of def. \ref{DerivedFunctorOfAHomotopicalFunctor}:

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

Consider a functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ out of a [[model category]] $\mathcal{C}$ into a [[category with weak equivalences]] $\mathcal{D}$.

1. If the restriction of $F$ to the [[full subcategory]] $\mathcal{C}_f$ of fibrant object becomes a [[homotopical functor]] (def. \ref{HomotopicalFunctor}), then the [[derived functor]] of that restriction, according to def. \ref{DerivedFunctorOfAHomotopicalFunctor}, is called the _[[right derived functor]]_ of $F$ and denoted by $\mathbb{R}F$:

   $$
     \array{
       & \mathcal{C}_f &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D}
       \\
       & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow
       &&
       \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
       &\swArrow_{\simeq}&
       \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
       \\
       \mathbb{R} F \colon & Ho(\mathcal{C}_f) &\simeq& Ho(\mathcal{C})
       &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D})
     }
     \,.
   $$

   Here the commuting square on the left is from corollary \ref{HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects}, the square on the right is that of def. \ref{DerivedFunctorOfAHomotopicalFunctor}.

1. If the restriction of $F$ to the [[full subcategory]] $\mathcal{C}_c$ of cofibrant object becomes a homotopical functor (def. \ref{HomotopicalFunctor}), then the [[derived functor]] of that restriction, according to def. \ref{DerivedFunctorOfAHomotopicalFunctor}, is called the _[[left derived functor]]_ of $F$ and denoted by $\mathbb{L}F$:

   $$
     \array{
       & \mathcal{C}_c &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D}
       \\
       & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow
       &&
       \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}}
       &\swArrow_{\simeq}&
       \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
       \\
       \mathbb{L} F \colon & Ho(\mathcal{C}_c) &\simeq& Ho(\mathcal{C})
       &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D})
     }
     \,.
   $$

   Here the commuting square on the left is from corollary \ref{HomotopyCategoryOfSubcategoriesOfModelCategoriesOnGoodObjects}, the square on the right is that of def. \ref{DerivedFunctorOfAHomotopicalFunctor}.

=--

The key fact that makes def. \ref{LeftAndRightDerivedFunctorsOnModelCategories} practically useful is the following

+-- {: .num_prop #KenBrownLemma}
###### Proposition
**([[Ken Brown's lemma]])**

Let $\mathcal{C}$ be a [[model category]] with [[full subcategories]] $\mathcal{C}_f, \mathcal{C}_c$ [[category of fibrant objects|of fibrant objects]] and [[cofibration category|of cofibrant objects]] respectively (def. \ref{FullSubcategoriesOfFibrantCofibrantObjects}). Let $\mathcal{D}$ be a [[category with weak equivalences]].

1. A [[functor]]

   $$
     F \;\colon\; \mathcal{C}_f \longrightarrow \mathcal{D}
   $$

  is a [[homotopical functor]], def. \ref{HomotopicalFunctor}, already if it sends acylic fibrations to weak equivalences.

1. A [[functor]]

   $$
     F \;\colon\; \mathcal{C}_c \longrightarrow \mathcal{D}
   $$

  is a [[homotopical functor]], def. \ref{HomotopicalFunctor}, already if it sends acylic cofibrations to weak equivalences.

=--

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

Let $\mathcal{C}, \mathcal{D}$ be [[model categories]] and consider $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ a [[functor]]. Then:

1. If $F$ preserves cofibrant objects and acyclic cofibrations between these, then its [[left derived functor]] (def. \ref{LeftAndRightDerivedFunctorsOnModelCategories}) $\mathbb{L}F$ exists, fitting into a [[diagram]]

   $$
     \array{
       \mathcal{C}_{c} &\overset{F}{\longrightarrow}& \mathcal{D}_{c}
       \\
       {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow
       &\swArrow_{\simeq}&
       \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
       \\
       Ho(\mathcal{C}) &\overset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D})
     }
   $$

1. If $F$ preserves fibrant objects and acyclic fibrants between these, then its [[right derived functor]] (def. \ref{LeftAndRightDerivedFunctorsOnModelCategories}) $\mathbb{R}F$ exists, fitting into a [[diagram]]

   $$
     \array{
       \mathcal{C}_{f} &\overset{F}{\longrightarrow}& \mathcal{D}_{f}
       \\
       {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow
       &\swArrow_{\simeq}&
       \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}}
       \\
       Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D})
     }
     \,.
   $$

=--

In practice it turns out to be useful to arrange for the assumptions in corollary \ref{LeftAndRightDerivedFunctors} to be satisfied in the following neat way:

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

Let $\mathcal{C}, \mathcal{D}$ be [[model categories]]. A pair of [[adjoint functors]] between them

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

is called a _[[Quillen adjunction]]_ (and $L$,$R$ are called left/right _Quillen functors_, respectively) if the following equivalent conditions are satisfied

1. $L$ preserves cofibrations and $R$ preserves fibrations;

1. $L$ preserves acyclic cofibrations and $R$ preserves acyclic fibrations;

1. $L$ preserves cofibrations and acylic cofibrations;

1. $R$ preserves fibrations and acyclic fibrations.

=--

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

The conditions in def. \ref{QuillenAdjunction} are indeed all equivalent.

=--

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

Observe that

* (i) _A [[left adjoint]] $L$ between [[model categories]] preserves acyclic cofibrations precisely if its [[right adjoint]] $R$ preserves fibrations.

* (ii) _A [[left adjoint]] $L$ between [[model categories]] preserves cofibrations precisely if its [[right adjoint]] $R$ preserves acyclic fibrations.

We discuss statement (i), statement (ii) is [[formal dual|formally dual]].  So let $f\colon A \to B$ be an acyclic cofibration in $\mathcal{D}$ and $g \colon X \to Y$ a fibration in $\mathcal{C}$. Then for every [[commuting diagram]] as on the left of the following, its $(L\dashv R)$-[[adjunct]] is a commuting diagram as on the right here:

$$
  \array{
    A &\longrightarrow& R(X)
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}}
    \\
    B &\longrightarrow& R(Y)
  }
  \;\;\;\;\;\;
  \,,
  \;\;\;\;\;\;
  \array{
    L(A) &\longrightarrow& X
    \\
    {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}}
    \\
    L(B) &\longrightarrow& Y
  }
  \,.
$$

If $L$ preserves acyclic cofibrations, then the diagram on the right has a [[lift]], and so the $(L\dashv R)$-[[adjunct]] of that lift is a lift of the left diagram. This shows that $R(g)$ has the [[right lifting property]] against all acylic cofibrations and hence is a fibration.
Conversely, if $R$ preserves fibrations, the same argument run from right to left gives that $L$ preserves acyclic fibrations.

Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.

=--

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

For $\mathcal{C}, \mathcal{D}$ two [[model categories]],  a [[Quillen adjunction]] (def.\ref{QuillenAdjunction})

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

is called a **[[Quillen equivalence]]** if the following equivalent conditions hold.

1. The [[right derived functor]] of $R$ (via prop. \ref{ConditionsOnQuillenAdjunctionAreIndeedEquivalent}, corollary \ref{LeftAndRightDerivedFunctors}) is an [[equivalence of categories]]

   $$
     \mathbb{R}R \colon Ho(\mathcal{C}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{D})
     \,.
   $$

1. The [[left derived functor]] of $L$ (via prop. \ref{ConditionsOnQuillenAdjunctionAreIndeedEquivalent}, corollary \ref{LeftAndRightDerivedFunctors}) is an [[equivalence of categories]]

   $$
     \mathbb{L}L \colon Ho(\mathcal{D}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{C})
     \,.
   $$

1. For every cofibrant object $d \in \mathcal{D}$ and every fibrant object $c \in \mathcal{C}$, a morphism $d \longrightarrow R(c)$ is a weak equivalence precisely if its [[adjunct]] morphism $L(c) \to d$ is

   $$
      \frac{
        d \overset{\in W_{\mathcal{D}}}{\longrightarrow} R(c)
      }{
         L(d) \overset{\in W_{\mathcal{C}}}{\longrightarrow} c
      }
      \,.
   $$

1. For every cofibrant object $d\in \mathcal{C}$, the [[derived adjunction unit]], hence the composite

   $$
     d \longrightarrow R(L(d)) \longrightarrow R(L(d)^{fib})
   $$

   (of the [[adjunction unit]] with any [[fibrant replacement]]) is a [[weak equivalence]].

  1. For every fibrant object $c \in \mathcal{C}$ the [[derived adjunction counit]], hence the composite

     $$
       L(R(c)^{cof}) \longrightarrow L(R(c)) \longrightarrow c
     $$

     (of the [[adjunction counit]] with any [[cofibrant replacement]]) is a [[weak equivalence]].

=--

+-- {: .num_prop #QuillenAdjunctionInducesAdjunctionOnHomotopyCategories}
###### Proposition
**([[derived adjunction]])**

For $\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D}$ a [[Quillen adjunction]] between [[model categories]], also the corresponding left and right [[derived functors]] form a pair of [[adjoint functors]]

$$
  Ho(\mathcal{C})
    \underoverset
      {\underset{\mathbb{R}R}{\longrightarrow}}
      {\overset{\mathbb{L}L}{\longleftarrow}}
      {\bot}
  Ho(\mathcal{D})
$$

between the corresponding [[homotopy category of a model category|homotopy categories]].

Moreover, the [[adjunction unit]] and [[adjunction counit]] of this derived adjunction are the images of the [[derived adjunction unit]] and [[derived adjunction counit]] of the original [[Quillen adjunction]].

=--

([Quillen 67, I.4 theorem 3](model+category#Quillen67))

## Further properties

This construction extends to a [[double pseudofunctor]]

$$
  Ho \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)
$$

on the [[double category of model categories]] ([this Prop.](double+category+of+model+categories#HomotopyDoublePseudofunctor)).

## Related concepts

* [[monoidal homotopy category of a monoidal model category]]

## References

The original account:

* {#Quillen67} [[Daniel Quillen]], Chapter I, _Axiomatic homotopy theory_ in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

Review:

* {#DwyerSpalinski95} [[William Dwyer]], J. Spalinski, _[[Homotopy theories and model categories]]_ ([pdf](http://folk.uio.no/paularne/SUPh05/DS.pdf)) in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_ 1995

* {#GoerssJardine99} [[Paul Goerss]], [[Rick Jardine]], section II.1 of _[[Simplicial homotopy theory]]_ Birkhäuser 1999, 2009

* *[[Introduction to Homotopy Theory]]*, [this Section](Introduction+to+Homotopy+Theory#TheHomotopyCategory)

  > (where the above material is mostly taken from)

[[!redirects homotopy categories of model categories]]


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

#Contents#
* table of contents
{:toc}

## Idea

The [[homotopy category]] of an [[(∞,1)-category]] $\mathcal{C}$ is its [[decategorification]] to an ordinary [[category]] obtained by identifying 1-[[morphisms]] that are connected by a [[2-morphism]].

If the [[(∞,1)-category]] $\mathcal{C}$ is presented by a [[category with weak equivalences]] $C$ (for instance as the [[simplicial localization]] $\mathcal{C} = L C$) then the notion of [[homotopy category]] of $C$ (where the weak equivalences are universally turned into [[isomorphism]]s) coinicides with that of $\mathcal{C}$:

$$
  Ho(\mathcal{C}) \simeq Ho(C)
  \,.
$$

## Definition

The component-wise definition depend on the chosen model for $(\infty,1)$-categories, as either

* [[simplicially enriched category]]/ [[topologically enriched categories]]

* [[complete Segal space]]

* [[Segal category]]

* [[quasi-category]]

### In terms of simplicially enriched categories

The **homotopy category** $h C$ of a [[sSet]]-[[enriched category]] $C$ (equivalently of a [[Top]]-[[enriched category]]) is hom-wise the image under the functor

$$
  \pi_0 : sSet \to Set
 \,,
$$

which sends each [[simplicial set]] to its 0th [[homotopy group|homotopy set]]
of [[connected]] components, i.e. to the set of [[path component]]s:

$$
  Hom_{h C}(A,B) := \pi_0(Hom_C(A,B))
  \,.
$$

Sometimes it is useful to regard this after all as an enriched category, but now enriched over the [[homotopy category]] of the standard [[model structure on simplicial sets]] $sSet_{Quillen}$, which is the _homotopy category of an $(\infty,1)$-category_ of [[∞Grpd]].

Let $\mathbf{h}: sSet \to Ho(sSet)$ be the localization functor to the [[homotopy category]] (of a [[category with weak equivalences]]).  This functor is a [[monoidal functor]] by the fact that the cartesian monoidal product is a left-[[Quillen bifunctor]] for $sSet$, which means that since every object in $sSet$ is cofibrant, it preserves weak equivalences in both arguments and hence descends to the homotopy category

$$
  \array{
    sSet \times sSet &\stackrel{\times}{\to}& sSet
    \\
    \downarrow^{\mathrlap{\mathbf{h} \times \mathbf{h}}}
    &&
    \downarrow^{\mathrlap{\mathbf{h}}}
    \\
    Ho(sSet) \times Ho(sSet) &\stackrel{\mathbf{h}(\times)}{\to}&
    Ho(sSet)
  }
  \,.
$$

This inuces a canonical functor $h:sSet Cat\to Ho(sSet) Cat$ which is given by the identity on objects and: $Map_{h C}(A,B):=\mathbf{h} Map_{C}(A,B)$.   Then since $Hom_C(A,B)=Hom_{sSet}(\Delta^0,Map_{C}(A,B))$, it is easy to see that $Hom_{hC}(A,B)=Hom_{Ho(sSet)}(\mathbf{h} \Delta^0, \mathbf{h} Map_C(A,B))=\pi_0 Map_C(A,B)$.

### In terms of complete Segal spaces and Segal categories

Similar, but more complicated, definitions work for [[complete Segal space]]s and [[Segal category|Segal categories]].

### In terms of quasi-categories

For [[quasi-category|quasi-categories]], one can write down a definition similar to those of $sSet$-enriched categories.

Viewing $C$ as a [[simplicial set]], the homotopy category $hC$ can also be described as its [[fundamental category]] $\tau_1(C)$, i.e. the image of $C$ by the [[left adjoint]] $\tau_1 : SSet \to Cat$ of the [[nerve]] functor $N$.

## Properties

### Brown representability
 {#BrownRepresentability}

The [[Brown representability theorem]] characterizes [[representable functors]] on homotopy categories of $(\infty,1)$-categories:

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

Let $\mathcal{C}$ be a [[locally presentable (∞,1)-category]], [[compactly generated (∞,1)-category|generated]] by a set

$$
  \{S_i \in \mathcal{C}\}_{i \in I}
$$

of [[compact object in an (infinity,1)-category|compact objects]] (i.e. every object of $\mathcal{C}$ is an [[(∞,1)-colimit]] of the objects $S_i$.)

If each $S_i$ admits the structure of a [[cogroup]] object in the [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(\mathcal{C})$, then a [[functor]]

$$
  F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set
$$

(from the [[opposite category|opposite]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathcal{C}$ to [[Set]])

is [[representable functor|representable]] precisely if it satisfies these two conditions:

1. $F$ sends small [[coproducts]] to [[products]];

1. $F$ sends [[(∞,1)-pushouts]] $X \underset{Z}{\sqcup}Y$ to [[epimorphisms]], i.e. the canonical morphisms into the [[fiber product]]

   $$
     F\left(X \underset{Z}{\sqcup}Y\right)
     \stackrel{epi}{\longrightarrow}
     F(X) \underset{F(Z)}{\times} F(Y)
   $$

   are [[surjections]].

=--

([Lurie "Higher Algebra", theorem 1.4.1.2](#LurieHigherAlgebra))

## References

* [[Jacob Lurie]], [Section 1.2.3, p. 33](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=33)  of [[Higher Topos Theory]]

* {#LurieHigherAlgebra} [[Jacob Lurie]], section 1.4.1 of _[[Higher Algebra]]_

[[!redirects homotopy categories of (infinity,1)-categories]]

[[!redirects homotopy category of an (∞,1)-category]]
[[!redirects homotopy categories of (∞,1)-categories]]