homotopy hypothesis

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

## Idea

The _homotopy hypothesis_ is the [[hypothesis]] or [[assertion]] that

* [[∞-groupoids]] are [[equivalence of (infinity,1)-categories|equivalent]] to the [[simplicial localization]] of [[topological spaces]] at their [[weak homotopy equivalences]],

or rather the stronger statement that

* [[n-groupoids]] are equivalent to this localization of those topological spaces which are [[homotopy n-types]] for all [[extended natural numbers]] $n \in \bar{\mathbb{N}}$

and moreover

* this equivalence is induced by the [[fundamental ∞-groupoid]] construction.

What this actually means in detail depends on which definition of [[∞Grpd]] is being used and to which precise incarnation of [[Top]] it is being compared to.

There are some definitions of [[∞-groupoids]] for which the homotopy hypothesis is a proven _theorem_, notably for the usual definition via [[Kan complexes]] (see at *[[classical model structure on simplicial sets]]*).

Depending on where in the spectrum between [[geometric definitions of higher categories]] and [[algebraic definitions of higher categories]] a given definition of $\infty$-groupoids is located, the statement may be more or less obvious.

For instance there is some justification for _defining_ an $\infty$-groupoid to _be_ equivalently a topological space considered modulo weak homotopy equivalence (see at *[[classical model structure on topological spaces]]*). For this definition the homotopy hypothesis is of course a tautology.

A definition of $\infty$-groupoid that is still very geometrical but much more combinatorial is that given by [[Kan complexes]]. For these the homotopy hypothesis has a (non-trivial but fairly tractable) proof. The equivalence between Kan complexes and [[CW-complexes]] obtained this way is at the heart of all traditional [[homotopy theory]].

A genuine algebraic definition of $\infty$-groupoids for which the homotopy theory has a (non-trivial but tractable) proof is given by [[algebraic Kan complex]]es.

However for other algebraic definitions of $\infty$-groupoids not much indication for how to prove the homotopy hypothesis is known. The definition of [[Trimble ∞-category]] stands out as an algebraic definition that has the notion of [[fundamental ∞-groupoid]] built right into it, but also here it seems unclear at the moment how to make progress with proving the homotopy hypothesis.

In fact, generally the homotopy hypothesis is regarded as a _consistency condition_ for definitions in [[higher category theory]]:

* Any definition of [[(n,r)-category]] should be such that there is a [[fundamental infinity-groupoid|fundamental n-groupoid]]-construction with values in the corresponding [[(n,0)-categories]] which makes these equivalent to [[homotopy n-type]]s.

One way to justify this condition is by recourse to the proven cases of the homotopy hypothesis: experience shows that the collections of all three models --  topological spaces, Kan complexes, algebraic Kan complexes -- provide a model for [[∞Grpd]] that supports the general abstract [[higher category theory]] -- specifically [[(∞,1)-category theory]] -- that one expects in analogy to how [[Set]] supports ordinary [[category]] theory.  Any other definition of $\infty$-groupoids is hoped/required to reproduce this, and hence is hoped/required to satisfy the homotopy hypothesis.

Apart from having different models of $\infty$-groupoids that lend themselves more or less to a comparison with topological spaces, there is also the issue as to how to conceive of the notion of _equivalence_ between $\infty$-groupoids.

The usual, unstated, implication is that the notion of _equivalence_ of [[n-groupoid]]s used to model [[homotopy n-type]]s is the appropriate *[[n-category]]-theoretic* notion of [[equivalence of categories|equivalence]].  It is in this way that, for instance, it is known that 1-[[groupoid]]s model homotopy 1-types (see below).

The reason this is important to specify is that there are other notions of equivalence on categorical structures which model homotopy types in other ways.  For example, if we declare a functor between categories to be a weak equivalence iff its [[nerve]] is a weak equivalence of [[simplicial set]]s, then _all_ homotopy types can be modeled by 1-categories in this way; see the [[Thomason model structure]] for 1-categories.

Finally, in analogy to the homotopy hypothesis, there are also attempts to relate general [[(∞,n)-categories]] (not necessarily groupoidal) to [[directed topological space]]s by a [[fundamental (∞,1)-category|fundamental (∞,n)-category]]-construction. There have been claims that a _directed homotopy hypothesis_ can be proven, but at the moment there does not seem to be a published statement.

## Abstract statement {#AbstractStatement}

The following general abstract statement of the homotopy hypothesis is often useful to make explicit.

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

There is an [[equivalence of (∞,1)-categories]]

$(\Pi \dashv |-|) : $ [[Top]] $\simeq$ [[∞Grpd]].

=--

This statement can be formulated, holds true and is proven  below at least for the standard definitions of these two [[(∞,1)-categories]] (see [section on Kan complexes](#ForKanComplexes), [section on algebraic Kan complexes](#ForAlgebraicKanComplexes)).

## Realizations

We discuss various different definitions of [[n-groupoid]]s and [[∞-groupoid]]s and the formulation and proof of the homotopy hypothesis for them, to the extent that it is available.

### For groupoids

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

The [[2-category]] [[Grpd]] of [[groupoid]]s, [[functor]]s, and [[natural transformation]]s is equivalent to the 2-category of [[homotopy n-type|homotopy 1-types]], [[continuous map]]s, and [[homotopy]] classes of homotopies.

=--

See [[homotopy hypothesis for 1-types]] for more.

### For 2-groupoids
 {#For2Groupoids}

(...) [[2-groupoid]]s model all [[homotopy n-type|homotopy 2-type]]s (...)

[[strict 2-groupoid]]s suffice (...) (but note that strict 2-functors are not sufficient to model all maps between 2-types)

### For Gray-groupoids

It is known that not all homotopy 3-types can be modeled by strict 3-groupoids, but that [[Gray-groupoid]]s ([[semi-strict infinity-category|semi-strict 3-groupoids]]) suffice; the obstruction is the [[Whitehead product]] which arises from a nontrivial [[interchanger]].

###  For Grothendieck ∞-groupoids

For a review of progress on proving the homotopy hypothesis for Grothendieck's original conception of ∞-groupoid in _Pursuing Stacks_, as later developed by [[Georges Maltsiniotis]], see [Henry and Lanari](#HenryLanari)'s 2019 paper.

### For Kan complexes {#ForKanComplexes}

We write $sSet_{Quillen}$ for [[sSet]] equipped with the [[classical model structure on simplicial sets]]. The cofibrant-fibrant objects in $sSet_{Quillen}$ are precisely the [[Kan complexes]].

Also we write $Top_{Quillen}$ for [[Top]] equipped with the [[classical model structure on topological spaces]].

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

There is a [[Quillen equivalence]]

$$
  (|-| \dashv Sing) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen}
  \,,
$$

where the [[left adjoint]] $|-|$ is [[geometric realization]] and the [[right adjoint]] $Sing(-)$ is forming the [[singular simplicial complex]] $Sing(-)$.

This induces an [[equivalence of (∞,1)-categories]] of the corresponding [[presentable (infinity,1)-category|presented]] [[(∞,1)-categories]]

* [[Top]] $\simeq$ [[∞Grpd]]

=--

([Quillen 67](#Quillen67)), see e.g. ([Goerss-Jardine 96, section I.11](#GoerssJardine96), [Joyal-Tierney 05, chapter I](#JoyalTierney05))

### For algebraic Kan complexes {#ForAlgebraicKanComplexes}

An [[algebraic Kan complex]] is an [[algebraic definition of higher categories|algebraic definition of higher groupoids]] obtained by taking the ordinary definition of [[Kan complex]] and equipping these with _choices_ of [[horn]]-fillers. These choices encode specified composition operations, specified [[associator]]s for these, specified _pentagonators_ and so on.

Algebraic Kan complexes constitute a genuine algebraic model in that they are precisely the [[algebras over a monad]] on [[sSet]].

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

The [[transferred model structure|transferred]] [[model structure on algebraic fibrant objects|model structure on algebraic Kan complexes]] is [[Quillen equivalence|Quillen equivalent]] to the standard [[model structure on simplicial sets]]

$$
  (F \dashv U) : Alg Kan \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
   sSet_{Quillen}
  \,.
$$

=--

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

The proof is spelled out at [[model structure on algebraic fibrant objects]].

=--

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

With the above [homotopy hypothesis-theorem for Kan complexes](ForKanComplexes) this gives a zig-zag of Quillen equivalences between $Alg Kan$ and $Top$

$$
  Alg Kan
    \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
   sSet_{Quillen}
     \stackrel{\overset{|-|}{\to}}{\underset{Sing}{\leftarrow}}
    Top
  \,.
$$

This already yields the homotopy hypothesis for algebraic Kan complexes at the level of the corresponding [[presentable (∞,1)-category|presented (∞,1)-categories]] (as discussed there)

$$
  Alg C^\circ \simeq Top^\circ
  \,.
$$

=--

But there is also a direct Quillen equivalence:

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

Write $\Delta^n$ and $\Lambda^n_k$ for the topological $n$-[[simplex]] and its $k$-[[horn]].

Fix any choice of [[retract]]s

$$
  R(n,k) : \Delta^n \to \Lambda^n_k
$$

for all topological [[horn]] inclusions $\Lambda^n_i \hookrightarrow \Delta^n$.

For $X$ any [[topological space]] equip the [[singular simplicial complex]] $Sing X$ with the stucture of an [[algebraic Kan complex]] by taking the filler of $\Lambda_k[n] \to Sing X$ to be given by the $(|-| \dashv Sing)$-[[adjunct]] of $\Delta^n \stackrel{R(n,k)}{\to} \Lambda^n_k \to X$. Write $\Pi_\infty(X) \in Alg Kan$ for the resulting algebraic Kan complex.

This construction constitutes a functor

$$
  \Pi_\infty(-) : Top \to Alg C
  \,,
$$

with $U \Pi_\infty = Sing$.

=--

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

The choices of fillers in $\Pi_\infty(X)$ may be thought of as explicit choice of reparameterizations of paths in $X$. These choices are arbitrary, but by the general statement at [[model structure on algebraic fibrant objects]], any two chocies yield equivalent objects.

=--

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

Given choices $R(-,-)$ of horn retracts as above, define a functor

$$
  |-|_r : Alg Kan \to Top
$$

called **reduced geometric realization** by taking it on an object $A \in Alg Kan$ to be given by the [[coequalizer]]

$$
  |A|_r := \lim_{\to}(\coprod_{\Lambda[n]_k \to A} \Delta^n \stackrel{\to}{\to} |U A|)
  \,,
$$

where $|U A|$ is the ordinary [[geometric realization]] of the underlying simplicial set of $A$ and where the two maps are

1. the image under $|-|$ of the distinguished fillers $\Delta[n] \to U A$ of $A$;

1. the composite $\Delta^n \stackrel{R(n,k)}{\to} \Lambda^n_k \to |U X|$ .

=--

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

The functor $|-|_r$ is [[left adjoint]] to $\Pi_\infty$.

$$
  (|-|_r \dashv \Pi_\infty)
  \,.
$$

=--

This is ([Nikolaus, prop. 3.4](#Nikolaus)).

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

We check the hom-isomorphism. A morphism $f : |A|_r \to X$ is
by definition of the coequalizer the same as a map $\tilde f : |A| \to X$ such that for each horn $h : \Lambda[n]_k \to A$ with distinguished filler $\hat h : \Delta[n] \to A$ the composites

$$
  \Delta^n \stackrel{R(n,k)}{\to} \Lambda^n_k
  \stackrel{|h|}{\to} |U A|
  \stackrel{\tilde f}{\to}
  X
$$

and

$$
  \Delta^n \stackrel{|\hat h|}{\to} |U A| \stackrel{\tilde f}{\to} X
$$

are equal. This means equivalently that the $(|-| \dashv Sing)$-[[adjunct]] $\tilde \tilde f : U A \to Sing X$ sends distinguished fillers in $A$ to distinguished fillers in $\Pi_\infty(X)$ and is hence a morphism in $Alg Kan$.

This construction shows that the map $(f : |A|_r \to X) \mapsto (\tilde \tilde f : A \to \Pi_\infty(X))$ thus obtained is a bijection.

=--

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

We have an identity:

$$
  \array{
    && Alg Kan
    \\
    & {}^{\mathllap{U}}\swarrow
    & \Downarrow^{=}&
    \nwarrow^{\mathrlap{\Pi_\infty}}
    \\
    sSet &&\underset{Sing}{\leftarrow}&& Top
  }
$$

and a [[natural isomorphism]]

$$
  \array{
    && Alg Kan
    \\
    & {}^{\mathllap{F}}\nearrow
    & \Downarrow^{\simeq}&
    \searrow^{\mathrlap{|-|_r}}
    \\
    sSet &&\underset{|-|}{\to}&& Top
  }
  \,.
$$

=--

This is ([Nikolaus, corollary 3.5](#Nikolaus))

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

The identity is evident by definition of $\Pi_\infty$.

Using this, we have that

$$
  (|-|_r \circ F \dashv U \circ \Pi_\infty = Sing)
  \,.
$$

So $|-|_r \circ F$ is another [[left adjoint]] to $Sing$ and hence naturally isomorphism to $|-|$.

=--

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

The adjunction

$$
  (|-|_r \dashv \Pi_\infty)
  :
  Alg C \to Top
$$

constitutes a [[Quillen equivalence]]

=--

This is [Nikolaus, corollary 3.6](#Nilkolaus)

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

By the above theorem and the 2-out-of-3-property of Quillen equivalences.

=--

### For cubical sets {#Cubical}

Also [[cubical set]]s may serve as a model for homotopy theory.

There is an evident [[simplicial set]]-valued [[functor]]

$$
  \Box \to sSet
$$

from the [[cube category]] to [[sSet]], which sends the cubical $n$-cube to the simplicial $n$-cube

$$
  \mathbf{1}^n \mapsto (\Delta[1])^{\times n}
  \,.
$$

Similarly there is a canonical [[Top]]-valued functor

$$
  \Box \to Top
$$

$$
  \mathbf{1}^n \mapsto (\Delta^1_{Top})^n
  \,.
$$

The corresponding [[nerve and realization]] [[adjunction]]

$$
  (|-| \dashv Sing_\Box) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing_\Box}{\to}} Set^{\Box^{op}}
$$

is the cubical analogue of the simplicial nerve and realization discussed [above](#ForKanComplexes).

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

There is a [[model structure on cubical sets]] $Set^{\Box^{op}}$ whose

* weak equivalences are the morphisms that become weak equivalences under geometric realization $|-|$;

* cofibrations are the [[monomorphism]]s.

=--

This is [Jardine, sections 3.](#Jardine)

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

The [[unit of an adjunction|unit of the adjunction]]

$$
  A \to Sing_\Box(|A|)
$$

is a weak equivalence in $Set^{\Box}$ for every cubical set $A$.

The counit of the adjunction

$$
  |Sing_\Box X| \to X
$$

is a weak equivalence in $Top$ for every topological space $X$.

It follows that we have an [[equivalence of categories]] induced on the [[homotopy categories]]

$$
  Ho(Top) \simeq Ho(Set^{\Box^{op}})
  \,.
$$

=--

This is [Jardine, theorem 29, corollary 30](#Jardine).

### For $cat^n$ groups and $n+1$-fold groupoids

Loday's notion of a [[cat-n-group]] corresponds to the connected version of an $n+1$-fold groupoid.  We will restrict our discussion to that connected case.

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

The [[homotopy category]] of [[cat-n-group]]s is equivalent to that of [[pointed object|pointed]] [[connected]] [[homotopy n-type|homotopy n+1-type]]s.

=--

This is proven in ([Loday](#Loday)). (There are some glitches in his proof and these were fixed by various authors (Steiner, Gilbert, ..) and then detailed proofs were given by Bullejos, Cegarra, Duskin and separately, using the equivalent formulation of crossed n-cubes, by Porter. Detailed references and some more commentary is at [[cat-n-group]].)

### For Segal-groupoids

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

There is realization/singular complex [[adjunction]]

$$
  (|-| \dashv sSing) : Top \to SegalGrpd
$$

for [[Segal groupoid]]s,

Its unit is an equivalence of [[Segal categories]] and its counit a [[weak homotopy equivalence]] of topological spaces.

=--

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

This is lemma 6.3.21 and corollary 6.3.24 in ([Pellissier](#Pellissier))

=--

### For strict $\omega$-categories with weak inverses
 {#ForStrictOmegaCategoriesWithWeakInverses}

While [[strict omega-groupoids]] in the sense of
[[strict omega-categories]] with _strict_ inverses are far from modelling all homotopy types, strict $\omega$-categories with all _weak_ inverses come closer. In ([Kapranov-Voevodsky](#KapranovVoevodsky)) it was argued that these are in fact sufficient, but a mistake in the argument was found in ([Simpson, Cor. 5.2](#Simpson)) (see also [here](Simpson%27s+conjecture#History)).

The issue however is somewhat subtle, as highlighted by Voevodsky [here](http://ncatlab.org/nlab/show/homotopy+type+theory#VoevodskyIASTalk2014).
For more on this see at _[[Simpson's conjecture]]_.

## Generalizations

### For stratified spaces

For a [[categorified]] version which finds an equivalence for $(\infty, 1)$-categories, see [[stratified homotopy hypothesis]] ([Ayala-Francis-Rozenblyum](#AFR15)).

### For spectra

For a stabilized version which finds an equivalence between stable homotopy types (i.e. [[spectra]]) and symmetric monoidal $\infty$-groupoids, see [[stable homotopy hypothesis]] ([Johnson-Osorno](#JO12), [Gurski-Johnson-Osorno](#GJO17))

## References and a bit of history.
 {#References}

The equivalence between the [[classical model structure on topological spaces|classical homotopy theory of topological spaces]] and the [[classical model structure on simplicial sets|homotopy theory of Kan complexes]] is due to

* {#Quillen67} [[Daniel Quillen]], _Homotopical Algebra_, LNM 43, Springer, (1967)

Later in

* [[Alexander Grothendieck]], _[[Pursuing Stacks]]_, 1983  (including and especially the 'letter to [[Daniel Quillen]]' given  on the first few pages),

it was argued that there were [[algebraic definitions of higher categories|algebraic definitions of higher groupoids]] that could be put forward,  so that the resulting objects ought to have a homotopy theory equivalent to the classical homotopy theory of topological spaces, and, moreover, would provide useful tools in the interpretation of [[non-abelian cohomology]] classes. This extended ideas that Grothendieck had explored in letters to [[Larry Breen]] in the mid 1970s in which he had given a sketch of a theory of $n$-stacks and their relation with the homotopy $n$-type of a space or more generally a topos.

At this stage, (in the  1970s and early 1980s) more _geometric_ or _combinatorial_ definitions of infinity categories were not yet available, or, perhaps more accurately, had been discovered, but were not _recognised_ as having such an infinity categoric interpretation; see [[Tim Porter]]'s Letter to Grothendieck (16 June 1983) and the discussion [[Spaces as infinity-groupoids.pdf|here:file]]  in [[New Spaces for Mathematics and Physics]]. These models included Kan complexes which now are interpreted as being one model for infinity groupoids, and [[quasicategory|weak Kan complex]]es, as put forward by Boardman and Vogt, which give one of the main models for (weak) $(\infty,1)$-categories. From this perspective, Quillen's result can be seen as being a precursor of one form of the Homotopy Hypothesis.

The name "Homotopy Hypothesis" for this statement is due to

* [[John Baez]], _The Homotopy Hypothesis_, 2007 ([web](http://math.ucr.edu/home/baez/homotopy/), [pdf](http://math.ucr.edu/home/baez/homotopy/homotopy.pdf))

More recently [[Georges Maltsiniotis]] has revived the approach to $\infty$-groupoids that Grothendieck initiated in _Pursuing Stacks_ - see [[Grothendieck ∞-groupoids]] - and work has begun on proving the homotopy hypothesis for Grothendieck ∞-groupoids.  For a review of progress so far, see:

* {#HenryLanari} [[Simon Henry]], [[Edoardo Lanari]], *On the homotopy hypothesis in dimension 3*, Theory and Applications of Categories **39** 26 (2023) 735-768 [[arxiv/1905.05625](https://arxiv.org/abs/1905.05625), [tac:39-26](http://www.tac.mta.ca/tac/volumes/39/26/39-26abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/39/26/39-26.pdf)].

These authors have reduced the hypothesis to a conjecture which has so far been proved only up to dimension 3, hence the title of their paper.

Technical reviews of Quillen's proof of a version of the homotopy hypothesis include:

* {#GoerssJardine96} [[Paul Goerss]], [[Rick Jardine]], section I.11 of _[[Simplicial homotopy theory]]_, 1996

* {#JoyalTierney05} [[André Joyal]], [[Myles Tierney]] _An introduction to simplicial homotopy theory_, 2005  ([chapter I](http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01), more notes [pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern47.pdf))

The homotopy hypothesis for strict $\omega$-categories with weak inverses is discussed in

* {#KapranovVoevodsky} [[Mikhail Kapranov]], [[Vladimir Voevodsky]], _$\infty$-groupoids and homotopy types_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 32 no. 1 (1991), p. 29-46

but a mistake in the argument was pointed out in Corollary 5.2 of

* {#Simpson} [[Carlos Simpson]], _Homotopy types of strict 3-groupoids_ ([arXiv:math/9810059](http://arxiv.org/abs/math/9810059))

The homotopy hypothesis for [[algebraic Kan complexes]] is established and discussed in

* [[Thomas Nikolaus]], _Algebraic models for higher categories_ ([arXiv](http://arxiv.org/abs/1003.1342))
{#Nikolaus}

For progress on

The homotopy hypothesis for [[Segal groupoids]] is formulated in section 6.3.4 of

* Regis Pellissier, _Weak enriched categories - Categories enrichies faibles_ PhD Thesis (2002) ([arXiv:math/0308246](http://arxiv.org/abs/math/0308246))

Models of homotopy $n$-types by $Cat^n$-groups are discussed in

* [[Jean-Louis Loday]], _Spaces with finitely many non-trivial homotopy groups_, J. Pure Appl. Algebra 24 (1982), 179-202.
{#Loday}

More literature on models of homotopy types by strict higher groupoids is at

* [[Ronnie Brown]], _Computing Homotopy Types Using Crossed $N$-Cubes of Groups_ ([arXiv](http://arxiv.org/abs/math/0109091))

* [[Simona Paoli]], _Internal categorical structures in homotopical algebra_, to appear in _[[johnbaez:Towards Higher Categories]]_, eds. [[John Baez]] and [[Peter May]]  ([pdf](http://www.maths.mq.edu.au/~simonap/paoli_IMA.pdf))

The first paper,  as its title suggests, has an emphasis on using higher groupoids for computation of homotopical invariants, in fact by applying higher homotopy van Kampen Theorems.  These theorems lead to algebraic colimit arguments in algebraic topology, implying  results, often nonabelian,  not obtainable by other methods. It is also remarkable that the precision of these results requires the use of strict structures, whereas the current emphasis in higher category theory is on non strict structures.

The homotopy theory of [[cubical set]]s is discussed in

* Jardine, _Model structure on cubical sets_ ([pdf](http://hopf.math.purdue.edu/Jardine/cubical2.pdf))
{#Jardine}

* Maltsiniotis, G. La catégorie cubique avec connexions est une catégorie
  test stricte. Homology, Homotopy Appl. 11, (2) (2009) 309--326.

Cubical methods are also essential in

*  R. Brown, P.J. Higgins, R. Sivera,  _Nonabelian algebraic
topology: filtered spaces, crossed complexes, cubical homotopy
groupoids_, EMS Tracts in Mathematics Vol. 15, 703 pages. (August
2011).

A version for [[stratified spaces]] is discussed in

* {#AFR15} [[David Ayala]], [[John Francis]], [[Nick Rozenblyum]], *A stratified homotopy hypothesis*, [arXiv](http://arxiv.org/abs/1502.01713)

A version for [[spectra]] is discussed in

* {#JO12} [[Niles Johnson]] and [[Angelica Osorno]], *Modeling stable one-types*, [tac](http://tac.mta.ca/tac/volumes/26/20/26-20abs.html)

* {#GJO17} [[Nick Gurski]], [[Niles Johnson]], and [[Angelica Osorno]], *The 2-dimensional stable homotopy hypothesis*, [arxiv](https://arxiv.org/abs/1712.07218)


homotopy hypothesis for 1-types

# Contents
* table of contents
{: toc}

## Introduction

We give a proof that [[groupoid | groupoids]] model [[homotopy 1-type | homotopy 1-types]]. More precisely, we construct a Quillen equivalence between a model structure on the category of groupoids, and a certain model structure on the category of cubical sets.

## Preliminaries

+-- {: .num_note}
###### Notation

We make use throughout of the notation of [[category of cubes]], [[cubical set]], [[cubical truncation, skeleton, and co-skeleton]], and [[fundamental groupoid of a cubical set and the cubical nerve of a groupoid]].

=--

## One direction of the equivalence

+-- {: .num_prop #PropositionPi1NToIdIsNatIso}
###### Proposition

The adjunction natural transformation $\epsilon : \Pi^{\leq 2}_{1} \circ N^{\leq 2} \rightarrow id$ is a natural isomorphism.

=--

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

Let $\mathcal{A}$ be a groupoid. Let $r(\mathcal{A}) : \mathcal{A} \rightarrow \Pi^{\leq 2}_{1} \circ N^{\leq 2}$ be the functor defined as follows.

1) On objects it is the identity.

2) To an arrow $f : a_{0} \rightarrow a_{1}$ of $\mathcal{A}$, we associate the arrow

$$
   \array{
      a_{0}  &  \overset{f}{\rightarrow} & a_{1} \overset{id}{\leftarrow} & a_{1}
   }
$$

of $\Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A})$.

Because of 1 b) ii) of [Notation 3](fundamental+groupoid+of+a+cubical+set+and+the+cubical+nerve+of+a+groupoid#NotationFundamentalGroupoid) of [[fundamental groupoid of a cubical set and the cubical nerve of a groupoid]], the following diagram of commutative squares illustrates that $r(g \circ f) = r(g) \circ r(f)$, where $f : a_{0} \rightarrow a_{1}$ and $g : a_{1} \rightarrow a_{2}$ are arrows of $\mathcal{A}$.

$$
   \array{
      a_{0}         & \overset{f}{\rightarrow}          & a_{1}        & \overset{id}{\rightarrow} & a_{1}        & \overset{g}{\rightarrow}   & a_{2}         & \overset{id}{\leftarrow} & a_{2} \\
      id \downarrow &                                   & \downarrow g &                           & \downarrow g &                            & \downarrow id &                          & \downarrow id \\
      a_{0}         & \underset{g \circ f}{\rightarrow} & a_{2}        & \underset{id}{\leftarrow} & a_{2}        & \underset{id}{\rightarrow} & a_{2}         & \underset{id}{\leftarrow} & a_{2}
   }
$$

That $r(id) = id$ follows immediately from 1 b) i) of [Notation 3](fundamental+groupoid+of+a+cubical+set+and+the+cubical+nerve+of+a+groupoid#NotationFundamentalGroupoid) of [[fundamental groupoid of a cubical set and the cubical nerve of a groupoid]].

It is clear that $\epsilon(\mathcal{A}) \circ r(\mathcal{A})$ is $id(\mathcal{A})$.

Moreover, because of 1 b) ii) of [Notation 3](fundamental+groupoid+of+a+cubical+set+and+the+cubical+nerve+of+a+groupoid#NotationFundamentalGroupoid) of [[fundamental groupoid of a cubical set and the cubical nerve of a groupoid]], the following diagram of commutative squares illustrates that $r(\mathcal{A}) \circ \epsilon(\mathcal{A})$ is $id(\mathcal{A})$.

$$
   \array{
      a_{0}         & \overset{f_{1}}{\rightarrow} & a_{1}        & \overset{g_{1}}{\leftarrow} & a_{2}        & \cdots & a_{2n-2}     & \overset{f_{n}}{\rightarrow} & a_{2n-1}              & \overset{g_{n}}{\leftarrow} & a_{2n} \\
      id \downarrow &                              & \downarrow A &                              & \downarrow B &        & \downarrow C &                              & \downarrow g_{n}^{-1} &                             & \downarrow id \\
      a_{0}         & \underset{D}{\rightarrow}    & a_{2n}       & \underset{id}{\leftarrow} & a_{2n}          & \cdots & a_{2n}       & \underset{id}{\rightarrow} & a_{2n}                  & \underset{id}{\leftarrow}   & a_{2n}
   }
$$

Here

$$
   \array{
      a_{0}  &  \overset{f_{1}}{\rightarrow} & a_{1} \overset{g_{1}}{\leftarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} \overset{g_{n}}{\leftarrow} & a_{2n}
   }
$$

is any zig-zag of arrows of $\mathcal{A}$, and the arrows A, B, C, and D are given as follows.

A) $g_{n}^{-1} \circ f_{n} \circ \cdots g_{2}^{-1} \circ f_{2} \circ g_{1}^{-1}$

B) $g_{n}^{-1} \circ f_{n} \circ \cdots g_{2}^{-1} \circ f_{2}$

C) $g_{n}^{-1} \circ f_{n}$

D) $g_{n}^{-1} \circ f_{n} \circ \cdots g_{2}^{-1} \circ f_{2} \circ g_{1}^{-1} \circ f_{1}$

We also make use of 1 b) i) of [Notation 3](fundamental+groupoid+of+a+cubical+set+and+the+cubical+nerve+of+a+groupoid#NotationFundamentalGroupoid) of [[fundamental groupoid of a cubical set and the cubical nerve of a groupoid]] in identifying

$$
    \array{a_{0} & \underset{D}{\rightarrow} & a_{2n} & \underset{id}{\leftarrow} & a_{2n} & \cdots & a_{2n} & \underset{id}{\rightarrow} & x_{2n} & \underset{id}{\leftarrow} & a_{2n}}
$$

with

$$
    \array{a_{0} & \underset{D}{\rightarrow} & a_{2n} & \underset{id}{\leftarrow} a_{2n}}
$$

as required.

=--

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

The adjunction natural transformation $\Pi_{1} \circ N \rightarrow id$ is a natural isomorphism.

=--

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

Since $tr_{2}$ is fully faithful, the adjunction natural transformation $id \rightarrow tr_{2} cosk_{2}$ is a natural isomorphism. The corollary follows from this and Proposition \ref{PropositionPi1NToIdIsNatIso}.

=--

## Other direction of the equivalence

To be written.


homotopy image

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _homotopy image_ generalizes the notion of [[image]] of a morphism in a [[category]] to that of a morphism in a [[presentable (infinity,1)-category]] or [[model category]].

## Definition

### In a model category

One of the definitions of the [[image]] of a morphism $f : c \to d$ is in terms of universal [[subobject]]s -- i.e. universal [[monomorphism]]s -- through which  $f$ factors.

This definition can be generalized to the context of [[(infinity,1)-category|(infinity,1)-categories]] [[presentable (infinity,1)-category|presented]] by a [[model category]].

+-- {: .num_defn}
###### Definition (homotopy image)

Let $C$ be an $S$ [[enriched model category]] satisfying some assumptions... .

* A morphism $f : c \to d$ in $C$ is called a **homotopy monomorphism**  if the universal morphism $Id \times Id : c  \to c \times^h_d c$ into its [[homotopy pullback]]  along itself is an [[isomorphism]] in the [[homotopy category]].

* The **homotopy image** of $f$ is a factorization of $f$ into a cofibration $c \to f(c)$ followed by a homotopy monomorphism $f(c) \to d$

  * such that for any other such factorization $c \to e \to d$ there exists a unique morphism $f(c) \to e$ in the [[homotopy category]] making the obvious triangles commute.

=--

### In an $(\infty,1)$-topos

The above definition of _homotopy monomorphism_ [[presentable (infinity,1)-category|presents]] precisely the notion of _[[monomorphism in an (∞,1)-category]]_ : a [[(-1)-truncated]] morphism.
Because ([[Higher Topos Theory|HTT, lemma 5.5.6.15]]) a morphism is [[(-1)-truncated]] precisely if its [[diagonal]] is [[(-2)-truncated]], hence is an [[equivalence in an (∞,1)-category|equivalence]].

Therefore in an [[(∞,1)-topos]] the homtopy image of a morphism is a presentation for the [[n-connected/n-truncated factorization system|(-1)-connected/(-1)-truncated factorization]] of the morphism.

See _[[∞-image]]_.

## Related concepts

* [[image]], [[coimage]]

## References

A definition for model categories is def. 2.36 in

* [[Clark Barwick]], _On (enriched) left Bousfield localization of model categories_ ([arXiv](http://arxiv.org/abs/0708.2067))

For the definition in $(\infty,1)$-topos theory see the references at [[n-connected/n-truncated factorization system]].


homotopy in 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

The extra structure of a [[model category]] over a [[category with weak equivalences]] induces concrete constructions for expressing [[homotopy]] between [[morphisms]]. These lead in particular to an explicit construction of the [[homotopy category of a model category]].

## Definition

+-- {: .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 cyclinder object $Cyl(X)$ is analogous to the [[tensoring]]  with a cylinder object $I\odot X$.  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]]:

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

=--

## Properties

### Basic lemmas

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

If
$
  X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X
$
is a good [[cylinder object]] for a cofibrant object $X$ def. \ref{PathAndCylinderObjectsInAModelCategory}, then both components $i_0, i_1 \colon X \to Cyl(X)$ are acyclic cofibrations.

Dually, if
$
   X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X
$

is a good path object for a fibrant object $X$, then both component $p_0,p_1 \colon Path(X)\to X$ are acyclic fibrations.

=--

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

We discuss the first case, the second is [[formal dual|formally dual]]. First observe that the two inclusions $X \to X \sqcup X$ are cofibrations, since they are the [[pushout]] of the cofibration $\emptyset \to X$. This implies that $i_0$ and $i_1$ are composites of two cofibrations

$$
  i_0, i_1
   \;\colon\;
  X \overset{\in Cof}{\longrightarrow} X\sqcup X
  \overset{\in Cof}{\longrightarrow}
$$

and hence are themselves cofibrations. That they are in addition weak equivalences follows from [[two-out-of-three]] applied to the identity

$$
  id_X \;\colon\;
  X \overset{\in W}{\longrightarrow}
  Cyl(X)
  \overset{i_0}{\longrightarrow}
  X
  \,.
$$

implied by the fact that the cylinder by definition factors the [[codiagonal]].

=--

The following says that the choice of cylinder/path objects in def. \ref{LeftAndRightHomotopyInAModelCategory} is irrelevant as long it is "good".

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

For $\eta \colon f \Rightarrow_L g \colon X \to Y$ a [[left homotopy]] in some [[model category]], def. \ref{LeftAndRightHomotopyInAModelCategory}, such that $Y$ is a fibrant object, then for $Cyl(X)$ any choice of _good_ [[cylinder object]] for $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, there is a [[commuting diagram]] of the form

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

Dually, for $\eta \colon f \Rightarrow_R g \colon X \to Y$ a [[right homotopy]], def. \ref{LeftAndRightHomotopyInAModelCategory}, such that $X$ is cofibrant, then for $Path(X)$ any choice of _good_ [[path object]] for $X$, def. \ref{PathAndCylinderObjectsInAModelCategory}, there is a [[commuting diagram]] of the form

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

=--

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

We discuss the first statement, the second is [[formal dual|formally dual]].
Let $\eta \colon \hat X \longrightarrow Y$ be the given left homotopy with respect to a given cylinder object $\hat X$ of $X$. Factor it as

$$
  \eta
    \;\colon\;
  \hat X \overset{\in Cof}{\longrightarrow} Z \overset{\in W \cap Fib}{\longrightarrow}
  Y
  \,.
$$

Then find liftings $\ell$ and $k$ in the following two [[commuting diagrams]]

$$
  \array{
    X \sqcup X &\overset{}{\longrightarrow}& \hat X &\longrightarrow& Z
    \\
    \downarrow && & {}^{\mathllap{\ell}}\nearrow & \downarrow
    \\
    Cyl(X) &\longrightarrow& &\longrightarrow& Y
  }
  \;\;\;\;\;
  \,,
  \;\;\;\;\;
  \array{
    \hat X &\overset{\eta}{\longrightarrow}& Y
    \\
    \downarrow &{}^{\mathllap{k}}\nearrow& \downarrow
    \\
    Z &\longrightarrow& \ast
  }
  \,.
$$

Now the composite $\eta \coloneqq k \circ \ell$ is of the required kind,

$$
  \array{
    X \sqcup X &\overset{}{\longrightarrow}& \hat X &\longrightarrow& Z &\overset{k}{\longrightarrow}& Y
    \\
    \downarrow &&&  {}^{\mathllap{\ell}}\nearrow &
    \\
    Cyl(X) &\longrightarrow&
  }
  \,.
$$

=--

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

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

* Let $X$ be cofibrant. If there is a [[left homotopy]] $f \Rightarrow_L g$ then there is also a [[right homotopy]] $f \Rightarrow_R g$ (def. \ref{LeftAndRightHomotopyInAModelCategory}) with respect to any chosen good path object.

* Let $Y$ be fibrant. If there is a [[right homotopy]] $f \Rightarrow_R g$ then there is also a [[left homotopy]] $f \Rightarrow_L g$ with respect to any chosen good cylinder object.

=--

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

We discuss the first case, the second is [[formal dual|formally dual]].
Let $\eta \colon Cyl(X) \longrightarrow Y$ be the given left homotopy. By lemma \ref{GoodCylinderObjectsSupportEveryLeftHomotopyAndDually} we may assume without restriction that $Cyl(X)$ is _good_ in the sense of def. \ref{PathAndCylinderObjectsInAModelCategory}, for otherwise replace it by one that is. With this, lemma \ref{ComponentsOfGoodCylinderOfCofibrantAreAcyclicCofibrationsAndDually} implies that we have a lift $h$ in the following [[commuting diagram]]

$$
  \array{
    X &\overset{i \circ f}{\longrightarrow}& Path(Y)
    \\
    {}^{\mathllap{i_0}}_{\mathllap{\in W \cap Cof}}\downarrow
    &{}^{\mathllap{h}}\nearrow&
    \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}}
    \\
    Cyl(X)
    &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y
  }
  \,,
$$

where on the right we have the chosen path space object. Now the composite $\tilde \eta \coloneqq h \circ i_1$ is a right homotopy as required.

$$
  \array{
    && && Path(Y)
    \\
    && &{}^{\mathllap{h}}\nearrow&
    \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}}
    \\
    X &\overset{i_1}{\longrightarrow}& Cyl(X)
    &\underset{(f \circ p,\eta)}{\longrightarrow}&
    Y \times Y
  }
  \,.
$$

=--

### Equivalence relation

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

For $X$ a cofibrant object in a [[model category]] and $Y$ a [[fibrant object]], then the [[relations]] of [[left homotopy]] $f \Rightarrow_L g$ and of [[right homotopy]] $f \Rightarrow_R g$ (def. \ref{LeftAndRightHomotopyInAModelCategory}) on the [[hom set]] $Hom(X,Y)$ coincide and are both [[equivalence relations]].

=--

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

That both relations coincide under the (co-)fibrancy assumption follows directly from lemma \ref{LeftHomotopyWithCofibrantDomainImpliesRightHomotopyAndDually}.

To see that left homotopy with domain $X$ is a [[transitive relation]] first use lemma \ref{GoodCylinderObjectsSupportEveryLeftHomotopyAndDually} to obtain that every left homotopy is exhibited by a _good_ cylinder object $Cyl(X)$ and then lemma \ref{ComponentsOfGoodCylinderOfCofibrantAreAcyclicCofibrationsAndDually} to see that the cofiber coproduct $Cyl(X)\underset{X}{\sqcup} Cyl(X)$ in

$$
  \array{
     && && X
     \\
     && && \downarrow^{\mathrlap{i_0}}
     \\
     && X &\underoverset{\in W}{i_1}{\longrightarrow}& Cyl(X)
     \\
     && {}^{\mathllap{i_0}}\downarrow &(po)& \downarrow
     \\
     X &\underset{i_1}{\longrightarrow}&
    Cyl(X) &\underset{\in W}{\longrightarrow}& Cyl(X) \underset{X}{\sqcup} Cyl(X)
     \\
     && &{}_{\mathllap{}}\searrow& & \searrow
     \\
     && && \underset{\in W}{\longrightarrow} &\longrightarrow& X
  }
$$

is again a [[cylinder object]], def. \ref{PathAndCylinderObjectsInAModelCategory}. The [[symmetric relation|symmetry]] and [[reflexive relation|reflexivity]] of the relation is obvious.

=--

## References

See the references at _[[model category]]_.


homotopy invariance

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

#Contents#
* table of contents
{:toc}

## Idea

A [[functor]] on [[spaces]] (e.g. some [[cohomology]] functor) is called "homotopy invariant" if  it does not distinguish between a space $X$ and the space $X \times I$, where $I$ is an [[interval]]; equivalently if it takes the same value on [[morphisms]] which are related by a ([[left homotopy|left]]) [[homotopy]].

The term _homotopy invariant_ may also refer to the refinement of [[invariants]] to [[homotopy theory]], hence to [[homotopy fixed points]].

## Examples

A [[generalized (Eilenberg-Steenrod) cohomology]]-functor is by definition homotopy invariant, but for instance its refinement to [[differential cohomology]] in general no longer is.

## Related concepts

* [[homeomorphism invariant]]

* [[characteristic class]]

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

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

[[!redirects homotopy invariance]]

[[!redirects homotopy invariants]]

[[!redirects homotopy invariant]]


homotopy inverse

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

#Contents#
* table of contents
{:toc}

## Definition

Let $\sim$ be the [[relation]] of being [[homotopy|homotopic]] (for example between [[morphisms]] in the [[category]] [[Top]]). Let $f:X\to Y$ and $g:Y\to X$ be two morphisms. We say that $g$ is a **left homotopy inverse** to $f$ or that $f$ is a **right homotopy inverse** to $g$ if $g\circ f\sim id_X$. A **homotopy inverse** of $f$ is a map which is simultaneously a left and a right homotopy inverse to $f$.

$f$ is said to be a [[homotopy equivalence]] if it has a left and a right homotopy inverse. In that case we can choose the left and right homotopy inverses of $f$ to be equal. To show this denote by $g_L$ the left and by $g_R$ the right homotopy inverse of $f$. Then

$$ g_L \sim g_L\circ (f\circ g_R) = (g_L\circ f)\circ g_R \sim g_R. $$

Hence

$$ f\circ g_L\sim f\circ g_R\sim id_X, $$

therefore $g_L$ is not only a left, but also a right, homotopy inverse to $f$.

This makes sense in any category equipped with an [[equivalence relation]] $\sim$, which is compatible with the composition (and with the equality of morphisms).

## Examples

* A [[functor]] which is an [[equivalence of categories]] is/has a homotopy inverse with respect to the notion of homotopy given by [[natural isomorphisms]].

* A [[weak homotopy equivalence]] of [[topological spaces]] between [[CW-complexes]] is a [[homotopy equivalence]] and hence has a homotopy inverse.

* A [[weak homotopy equivalence]] of [[simplicial sets]] between [[Kan complexes]] is a [[homotopy equivalence]] and hence has a homotopy inverse.

[[!redirects homotopy inverses]]


homotopy Kan extension

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

#Contents#
* table of contents
{:toc}

## Idea

Homotopy Kan extensions are models/presentations for [[(∞,1)-Kan extension]]s -- i.e. [[Kan extension]]s in an [[(∞,1)-category]] theory -- in terms of [[homotopical category]] theory and [[enriched category theory]].

As a special case they reduce to [[homotopy limit]]s and [[homotopy colimit]]s, which in turn are models for [[limit in a quasi-category|(∞,1)-categorical limits and colimits]].

## In combinatorial simplicial model categories

The following describes homotopy Kan extensions in the context of [[combinatorial simplicial model categories]], i.e. $sSet_{Quillen}$-[[enriched model categories]] whose underlying ordinary model category is [[combinatorial model category|combinatorial]]. The discussion goes through verbatim also with $sSet_{Quillen}$ replaced by any [[excellent model category]].

### Recollection of ordinary global Kan extensions

Recall the global definition of ordinary [[Kan extension]]s: for $A$ a [[category]] and $p : C \to C'$ a [[functor]] between  [[small categories]], we have the [[functor categories]] $[C,A]$ and $[C',A]$ and precomposition with $p$ induces a functor

$$
  p^* : [C',A] \to [C,A]
  \,.
$$

If $A$ has all limits and colimits, then this functor has a [[left adjoint]] $Lan_p$ and a [[right adjoint]] $Ran_p$

$$
  (Lan_p \dashv p^* \dashv Ran_p)
  :=
  (p_! \dashv p^* \dashv p_*)
  :
  [C,A]
  \stackrel{\overset{Lan_p}{\to}}{\stackrel{\overset{p^*}{\leftarrow}}{\underset{Ran_p}{\to}}}
  [C',A]
  \,.
$$

These are the left and right [[Kan extension]] functors.

The following definition is the straightforward evident generalization of this from plain categories to [[simplicial model categories]].

### Definition

Let $A$ be a [[combinatorial simplicial model category]].
Let $C, C'$ be [[small category|small]] [[simplicially enriched categories]]. Write $[C,A]$ and $[C',A]$ for the corresponding [[enriched functor categories]]. Notice that these carry the injective and the projective [[model structure on functors]] $[C,A]_{inj}$ and $[C,A]_{proj}$, which themselves are combinatorial simplicial model categories.

Let

$$
  f : C \to C'
$$

be an [[sSet]]-[[enriched functor]]. Let

$$
  f^* : [C'.A] \to [C,A]
$$

be the [[sSet]]-[[enriched functor]] induced by precomposition with $f$.

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

The functor $f^*$ has both an [[sSet]]-[[left adjoint]] $f_!$ as well as a [[right adjoint]] $f_*$

$$
  (f_! \dashv f^* \dashv f_*) \; : \;
  [C,A]
  \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
  [C',A]
$$

and

* $(f_! \dashv f^*)$ is a [[Quillen adjunction]] for
  the projective [[model structure on functors]]

  $$
    (f_! \dashv f^*) : [C,A]_{proj} \stackrel{\to}{\leftarrow} [C',A]_{proj}
    \,,
  $$

* $(f^* \dashv f_*)$ is a [[Quillen adjunction]]
  for the injective [[model structure on functors]]

  $$
    (f^*\dashv f_*) : [C,A]_{inj} \stackrel{\to}{\leftarrow} [C',A]_{inj}
    \,,
  $$

If $f : C \to C'$ is a weak equivalence in the [[model structure on sSet-categories]] then these are [[Quillen equivalence]]s.

Now

* the right [[derived functor]]

  $$
    R f_* : ([C,A]_{inj})^\circ \to ([C',A]_{inj})^\circ
  $$

  is the **homotopy right Kan extension** functor;

* the left [[derived functor]]

  $$
    L f_{!} : ([C,A]_{proj})^\circ \to ([C',A]_{proj})^\circ
  $$

  is the **homotopy left Kan extension** functor.

For the special case that $C' = *$ we have

* the right [[derived functor]]

  $$
    R f_* : ([C,A]_{inj})^\circ \to ([*,A]_{inj})^\circ =\simeq A^\circ
  $$

  is the **[[homotopy limit]]** functor;

* the left [[derived functor]]

  $$
    L f_{!} : ([C,A]_{proj})^\circ \to ([*,A]_{proj})^\circ = A^\circ
  $$

  is the **[[homotopy colimit]]** functor.

=--

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

The statement of the Quillen adjunctions appears as [[Higher Topos Theory|HTT, prop A.3.3.7]]. The statement about the Quillen equivalences as [[Higher Topos Theory|HTT, prop A.3.3.8]].

=--

In fact homotopy Kan extension forms as above forms a [[Quillen adjoint triple]]. See [there](Quillen+adjoint+triple#QuillenAdjointTripleHomotopyKanExtension).

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

Since, intrinsically, Kan extensions, as every [[universal construction]], are supposed to be only defined up to weak equivalence, it is sometimes useful to make the extra freedom of choosing any weakly equivalent object explicit by the following definition.

Given $F \in [C,A]$ and $G \in [C',A]$ and a morphism $\eta : G \to f_* F$, we say that $\eta$ **exhibits $G$ as a homotopy right Kan extension** of $F$ if for some injectively fibrant replacement $F \to \hat F$ the composite morphism $G \to f_* F \to f_* \hat F$ is a weak equivalence.

So $f_* \hat F$ here is a homotopy Kan extension as produced by the derived functor, while $G$ may be a more general object, weakly equivalent to it.

=--

### Properties

#### Derived-hom into a ho-Kan extension is a ho-Kan extention {#HomIntoKanExt}

Recall that for $F : C \to A$ an ordinary [[functor]] between ordinary [[categories]], its ordinary [[limit]] $\lim_\leftarrow F$ is characterized by the fact that for every object $a \in A$ the set $Hom(a, \lim_\leftarrow F)$ is the limit in [[Set]] of the functor $C \to A \stackrel{Hom_A(a,-)}{\to} Set$. So all ordinary limits are determined by limits in [[Set]].

The analogous statement here is that all homotopy limits are determined by homotopy limits in $sSet_{Quillen}$.

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

Let $F \in [C,A]$ and $G \in [C',A]$ be fibrant in the projective [[model structure on functors]]. Then a morphism $\eta : G \to f_* F$ exhibits $G$ as a homotopy right Kan extension of $F$ precisely if for each cofibrant $a \in A$ -- equivalently for each fibrant-cofibrant $a \in A$ -- the morphism

$$
  \eta_a : A(a,G(-)) \to  A(a, f_* F(-))
$$

exhibits $A(a,G(-))  \in [C',sSet]$ as a homotopy right Kan extension of $A(a,F(-)) \in [C,sSet]$.

=--

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

This appears as [[Higher Topos Theory|HTT, prop. A.3.3.12]].

First notice that a  replacement $F \stackrel{\simeq}{\to} \hat F$ in $[C,A]_{inj}$ by a fibrant $\hat F$ induces a weak equivalence $A(a,F(-)) \to A(a,\hat F(-))$ for all cofibrant $a \in A$, since $F$ is assumed projectively fibrant and using the properties of [[derived hom-space]]s in an [[enriched model category]].

Therefore we may assume without loss of generality that $F$ is already injectively fibrant. Then it also follows that for all cofibrant $a \in A $ we have that $A(a,F(-)) \in [C,sSet]_{inj}$ is fibrant: because $A(a,(-))$ having [[right lifting property]] against all acyclic cofibrations $H \to H'$ in $[C,sSet]_{inj}$

$$
  \array{
    H &\to & A(a,F(-))
    \\
    \downarrow & \nearrow
    \\
    H'
  }
$$

is equivalent, by the [[sSet]]-[[copower|tensor]]-adjunction in $A$, to $F$ itself having the right lifting property against the map from $A \cdot H : c \mapsto H(c)\cdot A$ to $H' \cdot A$

$$
  \array{
    H\cdot a &\to & F
    \\
    \downarrow & \nearrow
    \\
    H'\cdot a
  }
  \,.
$$

But since tensoring in $A$ with [[sSet]] is a left [[Quillen bifunctor]] by definition of [[enriched model category]], we have that tensoring the cofibrant $a$ with an acyclic cofibration of simplicial sets produces an acaclic cofibration in $A$, so that $H \cdot a \to H'\cdot a$ is an acyclic cofibration in $[C,sSet]_{inj}$. But by the previous remark $F$ is (can assumed to be) injectively fibrant, hence the lift exists. Hence $A(a,F(-))$ is indeed injectively fibrant.

With this in hand, we have now the following equivalent restatement of the claim:

$\eta$ is a weak equivalence precisely if $\eta_a$ is for all cofibrant (or cofibrant and fibrant) $a$.

The implication $(\eta we) \Rightarrow (\eta_a we)$ follows because in the [[enriched model category]] $A$, the functor $A(a,F(-))$ out of the cofibrant objectwise $a$ into the fibrant $F(-)$  preserves weak equivalences.

Conversely, if $\eta_a: A(a,G(-)) \to A(a,f_* F(-))$ is a weak equivalence for all fibrant and cofibrant $a$, then for all $c \in C$ $\eta(c) : G(c) \to f_* F(c)$ is a weak equivalence for all $c \in C$ by the [[Yoneda lemma]], for instance in the $Ho(sSet)$-enriched [[homotopy category]] $Ho(A)$ of $A$: a morphism in $Ho(A)$ is an iso if homming all other objects into it produces an isomorphism.

=--

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

Notice that the statement makes sense in the full $sSet$-subcategory $A^\circ$ on fibrant-cofibrant objects of $A$, without needing any further mentioning on the model category structure on $A$, only that on $sSet_{Quillen}$ is involved. This allows to define homotopy Kan extensions in arbitrary Kan-complex enriched categories, which may or may not arise as $A^\circ$ for A a simplicial model category. This is discussed below.

=--

## In Kan-complex enriched categories {#InKanCplCat}

We obtain a notion of homotopy Kan extension that does not depend on any model category structure or even on weak equivalences anymore, but takes place entirely just in Kan-complex enriched categories.

### Definition

The above characterization of homotopy Kan extensions in simplicial combinatorial model categories $A$ in terms of homotopy Kan extensions in $sSet$ only involves [[hom-object]]s of the form $A(a,c)$, where $a$ is cofibrant and $c$ is fibrant. So it involves only the [[derived hom-space]]s of $A$, which are [[Kan complex]]es.

Accordingly, this characterization makes sense for $A$ any locally fibrant $sSet_{Quillen}$-enriched category, i.e. for every Kan-complex-enriched category:

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

For $A$ a [[Kan complex]]-[[enriched category]] and $f : C \to C'$ an [[enriched functor]] of small [[sSet]]-[[enriched categories]],
given $F \in [C,A]$ and $G \in [C',A]$ we say a morphism $\eta : G \to f_* F$, **exhibits $G$ as a homotopy right Kan extension** if for all $a \in A$ the morphism

$$
  \eta_a : A(a,G(-)) \to  A(a,f_* F(-))
$$

exhibits $A(a,G(-)) : C' \to sSet_{Quillen}$ as a homotopy right Kan extension of $A(a,F(-)) : C \to sSet_{Quillen}$.

=--

### Homotopy limits and colimits {#Holims}

If the diagram category $C'$ is the terminal $sSet$-category, the left and right homotopy Kan extension along $f : C \to {*}$ is the [[homotopy limit]] and [[homotopy colimit]], respectively.

#### Characterization in terms of hom-adjuncts

In thae case that we are homotopy Kan extending to the point, if $\eta : G \to f_* F$ exhibits a right homotopy Kan extension, $G \in A$ is  a single object of $A$ and by adjunction this corresponds to a [[natural transformation]] $f^* G = const G \to F$, whose components are a collection of morphisms

$$
  \{ \eta_c : G \to F(c) \}_{c \in C}
$$

in $A$. Then

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

The fact that $\eta$ exhibits a right homotopy Kan extension is equivalent to the statement that for all $a \in A$ the morphism

$$
  A(a, G) \to \lim_{\leftarrow} A(a,F(-))
$$

induced by composing with the $\{\eta_c\}$ _exhibits $A(a,G)$ as a homotopy limit_  of $A(a,F(-))$ in $sSet_{Quillen}$, in the above sense.

Since $\lim_{\leftarrow} A(a,F(-)) = [C,sSet](const a,F)$ is an isomorphissm, this in turn is equivalent to the statemeent that

$$
  A(a, G) \to [C,sSet](const a, F)
$$

exhibits that homotopy Kan extension.

=--

Analogously for homotopy colimits.

#### Relation to quasi-categorical limits and colimits

The above considerations can be used to show that under the [[homotopy coherent nerve]], homotopy colimits in a Kan-complex enriched categories as defined above are [[limit in a quasi-category|quasi-categorical colimits]]:

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

For $C$ and $A$ Kan-complex-enriched categories and $F \in [C,A]$, a morphism $\eta : F \to const_q$ exhibits $q \in A$ as a homotopy colimit in $A$ in the above sense precisely  if for $N(f) : N(C) \to N(A)$ the corresponding morphism of [[quasi-categories]] under the [[homotopy coherent nerve]] and $N(f)^\triangleright : N(C)^\triangleright \to N(A)$ the extension to cones given by $\eta$, $N(f)^{\triangleright}$ is a [[limit in a quasi-category|quasi-categorical colimit diagram]].

=--

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

This is [[Higher Topos Theory|HTT, theorem 4.2.4.1]]. Some details on the proof are discussed at [[limit in a quasi-category]].

=--

## In terms of derivators

The notion of [[derivator]] is largely a tool for handling homotopy Kan extensions. See there for details.

## Properties

### Pointwise homotopy Kan extensions
 {#Pointwise}

Under suitable conditions (but typically) homotopy Kan extensions may be computed [pointwise](Kan+extension#Pointwise) by [[homotopy colimits]].

Discussion of pointwise homotopy Kan extensions in [[cofibration categories]] is in [Radulescu & Banu 2006, theorem 9.6.5](#Radulescu-Banu06). This is reviewed in the context of [[model categories]] in [Cisinski 2009, prop. 1.14](#Cisinski09). In the more general context of [[relative categories]] discussion is in [Gonzales 2011, section 4](#Gonzales11).

See also at *[(∞,1)-Kan extension -- Properties -- Pointwise](/%28infinity%2C1%29-Kan+extension#Pointwise)*.

## References

General theory of homotopy Kan extensions is discussed in

* [[Jacob Lurie]], section A.3.3 of _[[Higher Topos Theory]]_

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]],
  [[Jeff Smith]], _[[Homotopy Limit Functors on Model Categories
and Homotopical Categories]]_ , volume 113 of Mathematical Surveys and Monographs

* [[Bruno Kahn]], [[Georges Maltsiniotis]], _[[Structures de Dérivabilité]]_

* [[Jean-Marc Cordier]] and [[Tim Porter]], _Homotopy Coherent Category Theory_, Trans. Amer. Math. Soc. 349 (1997) 1-54.

A list of basic properties is in

* [[Samuel Isaacson]], section 4 of _Exercises on homotopy colimits_ ([pdf](http://math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf))

Pointwise [[homotopy Kan extensions]] are discussed in

* {#Radulescu-Banu06} Andrei Radulescu-Banu, _Cofibrations in Homotopy Theory_ ([arXiv:0610009](http://arxiv.org/abs/math/0610009))

* {#Cisinski09} [[Denis-Charles Cisinski]], *Images directes cohomologiques dans les catégories de modèles*, Ann. Math. Blaise Pascal **10** (2003) 195-244 [[numdam:AMBP_2003__10_2_195_0](http://www.numdam.org/item/AMBP_2003__10_2_195_0)]

 * [[Denis-Charles Cisinski]], _Locally constant functors_, Math. Proc. Camb. Phil. Soc. (2009), 147, 593 ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/lcmodcat3.pdf))

* {#Gonzales11} Beatriz Rodriguez Gonzalez, section 4 of _Realizable homotopy colimits_ ([arXiv:1104.0646](http://arxiv.org/abs/1104.0646))

[[!redirects homotopy Kan extensions]]
[[!redirects left homotopy Kan extension]]
[[!redirects left homotopy Kan extensions]]
[[!redirects homotopy left Kan extension]]
[[!redirects homotopy left Kan extensions]]
[[!redirects right homotopy Kan extension]]
[[!redirects right homotopy Kan extensions]]
[[!redirects homotopy right Kan extension]]
[[!redirects homotopy right Kan extensions]]


homotopy Kan fibration

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of *homotopy Kan fibration* is the evident [[homotopy theory|homotopy-theoretic]] generalization ([[(infinity,1)-category|$(\infty,1)$-]][[categorification]]) of the notion of [[Kan fibration]]: where a [[Kan fibration]] is a morphism of [[simplicial sets]] (hence: [[simplicial objects]] [[internalization|internal]] to [[Sets]]) which satisfies the [[right lifting property]] against [[horn]] inclusions, so a homotopy Kan fibration is a morphism of [[simplicial object in an (infinity,1)-category|simplicial]] [[infinity-groupoids|$\infty$-groupoids]] ("[[simplicial spaces]]", [[bisimplicial sets]]) which satisfies a *homotopy*-lifting property along all [[horn]] inclusions.

Just like [[Kan fibrations]] serve as the [[fibrations]] in the [[classical model structure on simplicial sets]], so homotopy Kan fibrations are the fibrations in the archetypical [[model (infinity,1)-category|model $\infty$-category]]-structure on [[simplicial object in an (infinity,1)-category|simplicial]] [[infinity-groupoids|$\infty$-groupoids]].

## Definition

\begin{definition}\label{HomotopyKanFibration}
A morphism $f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet$ is a *homotopy Kan fibration* if for all [[positive numbers]] $n \in \mathbb{N}_+$ and all $0 \leq k \leq n$ the induced map

\[
  \label{HomotopyTheoreticKanFibration}
  X(\Delta^n)
  \xrightarrow{\;\;}
  X(\Lambda^n_k)
    \underset
      { Y(\Lambda^n_k) }
      {\times^h}
  Y(\Delta^n)
\]

(into the [[homotopy fiber product]] of the space of space of [[horns|$(n,k)$-horns]] in $X$ with that of [[simplex|$n$-simplices]] in $Y$)

is [[surjective]] on [[connected components]],

hence in that for all solid homotopy-commutative squares as follows, there exists a dashed lift up to homotopy (with the left morphism being the [[horn|$(n,k)$-horn]]-inclusion in [[simplicial sets]] regarded as degree-wise [[discrete topological spaces]]):

\begin{tikzcd}
        \Lambda^n_k
        \ar[
          rr,
          "{\ }"{below, name=s1, pos=.7}
        ]
        \ar[d]
        &&
        \mathrm{X}_\bullet
        \ar[
          d,
          "{f_\bullet}"{right}
        ]
        \\
        \Delta^n
        \ar[
          rr,
          "{\ }"{above, name=t2, pos=.3}
        ]
        \ar[
          urr,
          dashed,
          "{\ }"{above, name=t1},
          "{\ }"{below, name=s2}
        ]
        &&
        \mathrm{Y}_{\bullet}
        \mathrlap{\,,}
        %
        \ar[
          from=s1,
          to=t1,
          Rightarrow,
          dashed
        ]
        \ar[
          from=s2,
          to=t2,
          Rightarrow,
          dashed
        ]
\end{tikzcd}

\end{definition}

([Lurie 2011, Def. 3](#Lurie11); [Mazel-Gee 2014, p. 2](#MazelGee14))

## Examples

For $G \,\in\, Grp(Grpd_\infty)$ an [[infinity-group|$\infty$-group]], consider a homomorphism of $G$-[[infinity-action|$\infty$-actions]]

\begin{tikzcd}
  X
    \ar[r]
  &
  X /\!/ G
  \ar[d]
  \\
  &
  B G
\end{tikzcd}

(...)

## Properties

### General

* Homotopy Kan fibrations participate in a [[pullback-power axiom]] with (acyclic) homotopy cofibrations as expected in a [[monoidal model category]] ([Mazel-Gee 2015, Def. 4.3 with Exp. 5.3](#MazelGee15))

### Relation to homotopy colimits (geometric realization)

See at *[[geometric realization of simplicial topological spaces]]* the section *[Preservation of homotopy limits](geometric+realization+of+simplicial+topological+spaces#PreservationOfHomotopyLimits)*.

## Related concepts

* [[Kan fibration]]

* [[model (infinity,1)-category|model $\infty$-category]]

## References

* {#Lurie11} [[Jacob Lurie]], *Simplicial spaces*, Def. 3 in: *[Algebraic L-theory and Surgery](https://www.math.ias.edu/~lurie/287x.html)*, 2011 ([pdf](https://www.math.ias.edu/~lurie/287xnotes/Lecture7.pdf))

* {#MazelGee14} [[Aaron Mazel-Gee]], _Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces_ [[arXiv:1412.8411](https://arxiv.org/abs/1412.8411)]

* {#MazelGee15} [[Aaron Mazel-Gee]], *Model ∞-categories II: Quillen adjunctions*, New York Journal of Mathematics **27** (2021) 508-550.  [[arXiv:1510.04392](https://arxiv.org/abs/1510.04392), [nyjm:27-21](http://nyjm.albany.edu/j/2021/27-21.html)]

[[!redirects homotopy Kan fibrations]]


homotopy level

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

#Contents#
* table of contents
{:toc}

## Idea

The term **homotopy level** (or **h-level**) -- originating in [[homotopy type theory]] -- is another name for the notion of [[truncated object|truncation]] (particularly in [[(∞,1)-categories]] and their
[[internal logic|internal language]] of [[homotopy type theory]]) in which the numbering is offset by 2:

a [[homotopy n-type]] is a [[type]] of homotopy level $n+2$.

This offset in counting enables it to "start" at 0 rather than (-2), which is convenient when defining it by induction over the natural numbers in type theory.

## Definition ##

A [[type]] $A$ has a __homotopy level__ or __h-level__ of $n$ if the type $hasHLevel(n, A)$ is inhabited, for [[natural numbers|natural number]] $n:\mathbb{N}$.  $hasHLevel(n, A)$ is inductively defined as

$$hasHLevel(0, A) \coloneqq isContr(A)$$
$$hasHLevel(s(n), A) \coloneqq \prod_{a:A} \prod_{b:A} hasHLevel(n, a =_A b)$$

### Rules for hasHLevel

Suppose the dependent type theory has [[identity types]], [[contraction types]], and a [[natural numbers type]], but no [[dependent product types]] or [[dependent sum types]]. Then one could still define the type family $\mathrm{hasHLevel}(n, A)$ for all types $A$ in the context of the natural numbers variable $n:\mathrm{N}$, with the following rules:

Formation rules for hasHLevel types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash \mathrm{Contr}_A(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{hasHLevel}(0, A) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, n:\mathbb{N}, x:A, y:A \vdash \mathrm{hasHLevel}(n, x =_A y) \; \mathrm{type}}{\Gamma, n:\mathbb{N} \vdash \mathrm{hasHLevel}(s(n), A) \; \mathrm{type}}$$

Introduction rules for hasHLevel types:
$$\frac{\Gamma, x:A \vdash b(x):\mathrm{Contr}_A(x) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:\mathrm{Contr}_A[a/x]}{\Gamma, n:\mathbb{N} \vdash (a, b):\mathrm{hasHLevel}(0, A)}$$

$$\frac{\Gamma, n:\mathbb{N}, x:A, y:A \vdash p(n, x, y):\mathrm{hasHLevel}(n, x =_A y)}{\Gamma, n:\mathbb{N} \vdash \lambda(x).\lambda(y).p(n, x, y):\mathrm{hasHLevel}(s(n), A)}$$

Elimination rules for hasHLevel types:
$$\frac{\Gamma \vdash z:\mathrm{hasHLevel}(0, A)}{\Gamma \vdash \pi_1(z):A} \qquad \frac{\Gamma \vdash z:\mathrm{hasHLevel}(0, A)}{\Gamma \vdash \pi_2(z):\mathrm{Contr}_A(\pi_1(z))}$$

$$\frac{\Gamma, n:\mathbb{N} \vdash p:\mathrm{hasHLevel}(s(n), A) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma, n:\mathbb{N} \vdash p(n, a, b):\mathrm{hasHLevel}(n, a =_A b)}$$

Computation rules for hasHLevel types:
$$\frac{\Gamma, x:A \vdash b(x):\mathrm{Contr}_A(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_{\mathrm{hasHLevel} 1}^0:\pi_1(a, b) =_A a} \qquad \frac{\Gamma, x:A \vdash b(x):\mathrm{Contr}_A(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_{\mathrm{hasHLevel} 2}^0:\pi_2(a, b) =_{\mathrm{Contr}_A(a)} b}$$

$$\frac{\Gamma, n:\mathrm{N}, x:A, y:A \vdash p(n, x, y):\mathrm{hasHLevel}(n, x =_A y) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma, n:\mathrm{N} \vdash \beta_{\mathrm{hasHLevel}}^s:(\lambda(x).\lambda(y).p(n, x, y))(a, b) =_{\mathrm{hasHLevel}(n, a =_A b)} p(n, a, b)}$$

Uniqueness rules for hasHLevel types:
$$\frac{\Gamma \vdash z:\mathrm{hasHLevel}(0, A)}{\Gamma \vdash \eta_{\mathrm{hasHLevel}}^0:z =_{\mathrm{hasHLevel}(0, A)} (\pi_1(z), \pi_2(z))}$$

$$\frac{\Gamma, n:\mathbb{N} \vdash p:\mathrm{hasHLevel}(s(n), A)}{\Gamma, n:\mathbb{N} \vdash \eta_{\mathrm{hasHLevel}}^s:p(n) =_{\mathrm{hasHLevel}(s(n), A)} \lambda(x).\lambda(y).p(n, x, y)}$$

## Examples ##

* Every [[proposition]] has a homotopy level of $1$.

* Every [[set]] has a homotopy level of $2$.

The correspondence between the various terminologies is indicated in the following table:

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

## Referencees ##

The notion of "homotopy levels" (as a notion in [[type theory]]) originates with:

* {#Voevodsky10} [[Vladimir Voevodsky]], *Univalent Foundations Project* (2010) [[pdf](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/univalent_foundations_project.pdf), [[Voevodsky-UFP2010.pdf:file]]]

Further discussion:

* [[Guillaume Brunerie]], *Truncations and truncated higher inductive types* (2012) [[blog post](https://homotopytypetheory.org/2012/09/16/truncations-and-truncated-higher-inductive-types/)]

Discussion with [[Agda]]:

* [[1lab]], *[h-Levels](https://1lab.dev/1Lab.HLevel.html)*

Synonymous discussion, but under the name *[[homotopy n-type|$n$-types]]*, is in

* [[Univalent Foundations Project]], §7 of: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* [[Nicolai Kraus]], *Truncation levels in Homotopy Type Theory*, Nottingham (2015) [[pdf](https://eprints.nottingham.ac.uk/28986/1/thesis.pdf), [eprints:28986](https://eprints.nottingham.ac.uk/28986)]

See also:

* [[Thierry Coquand]], Ayberk Tosun, *Formal Topology in Univalent Foundations*, Ch. 6 in:  *Proof and Computation II -- From Proof Theory and Univalent Mathematics to Program Extraction and Verification*, Wold Scientific (2021) [[doi:10.1142/12263](https://doi.org/10.1142/12263)]

[[!redirects homotopy level]]
[[!redirects homotopy levels]]
[[!redirects h-level]]
[[!redirects h-levels]]

[[!redirects hlevel]]
[[!redirects hlevels]]


homotopy lifting property

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

#Contents#
* table of contents
{:toc}

## Idea

In classical [[homotopy theory]], a _homotopy lifting property_ is a condition satisfied by [[continuous map]]s between [[topological space]]s.
More generally, such a condition may appear in more general context of a [[category]] with [[product]]s and with an [[interval object]] $I$:

The [[Eckmann-Hilton duality|Eckmann–Hilton dual]] of the homotopy lifting property is the [[homotopy extension property]].

## Definition

Let $C$ be a [[category]] with [[product]]s and with [[interval object]] $I$.

A [[morphism]] $E \to B$ has the homotopy lifting property if it has the [[right lifting property]] with respect to all morphisms of the form $(Id,0) :  Y \to Y \times I$.

This means that for all commuting squares

$$
  \array{
    Y &\stackrel{f}\to& E
    \\
    \downarrow &&  \downarrow^p
    \\
    Y\times I &\stackrel{F}{\to}& B
  }
$$

there exists a morphism $\sigma : Y \times I \to E$ such that both triangles in

$$
  \array{
    Y &\stackrel{f}\to& E
    \\
    \downarrow &{}^\sigma\nearrow&  \downarrow^p
    \\
    Y\times I &\stackrel{F}{\to}& B
  }
$$
commute.

For $Y = *$ a [[generator]] this can be rephrased as saying that the
universal morphism $E^I \to B^I \times_B E$ induced by the commuting square

$$
  \array{
    E^I &\to& E
    \\
    \downarrow && \downarrow
    \\
    B^I &\to& B
  }
$$

is an [[epimorphism]]. If it is even an [[isomorphism]] then the lift $\sigma$ exists _uniquely_ .

The homotopy lifting property is an instance of a [[right lifting property]].

## Examples

### For topological spaces

Here the ambient category is $C = $ [[Top]] and the [[interval object]] is the [[topological interval]] $I = [0,1]$.

A [[continuous map]] $p:E\to B$ of [[topological spaces]] satisfies the **homotopy lifting property** (or *covering homotopy property*) with respect to a space $Y$ if for every [[commuting square]] in $Top$

$$
  \array{
    Y &\stackrel{f}\to& E
    \\
    \downarrow^{\sigma_0} &{}^{\tilde{F}}\nearrow&  \downarrow^p
    \\
    Y\times I &\stackrel{F}{\to}& B
  }
  \,.
$$

there is a diagonal such that the entire diagram commutes. The map $\sigma_0:Y\to Y\times I$ is given by $y\mapsto (y,0)$ for $y\in Y$.

A map $p$ is a **[[Hurewicz fibration]]** if it satisfies the homotopy lifting property with respect to all spaces $X$. A map $p$ is a **[[Serre fibration]]** if it satisfies the homotopy lifting property with respect to all disks (equivalently, all topological [[cube]]s).

There are weaker notions than the usual homotopy lifting property. For example, in the notion of [[Dold fibration]] one requires in the above diagram that the lower triangle is commutative while the upper one is commutative only up to a homotopy. Alternatively, one can characterize the Dold fibrations by the delayed homotopy lifting property, where instead the notion of [[delayed homotopy]] is used, but the lift makes the division of the square strictly commutative.

### For metric spaces

For [[metric spaces]], there is also a weaker notion, the [[approximate homotopy lifting property]].

### For quasi-categories

Morphism between [[quasi-categories]] that are left [[fibrations of quasi-categories]] satisfy the homotopy lifting property with respect to $\Delta[0] \hookrightarrow \Delta[1]$

[[!redirects covering homotopy property]]

[[!redirects right homotopy lifting property]]
[[!redirects left homotopy lifting property]]