nLab homotopy hypothesis

Idea

The homotopy hypothesis is the hypothesis or assertion that

• ∞-groupoids are 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 complexes.

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 n-groupoid-construction with values in the corresponding (n,0)-categories which makes these equivalent to homotopy n-types.

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-groupoids used to model homotopy n-types is the appropriate n-category-theoretic notion of equivalence. It is in this way that, for instance, it is known that 1-groupoids 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 sets, 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 spaces by a 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

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

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, section on algebraic Kan complexes).

Realizations

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

For groupoids

Proposition

The 2-category Grpd of groupoids, functors, and natural transformations is equivalent to the 2-category of homotopy 1-types, continuous maps, and homotopy classes of homotopies.

See homotopy hypothesis for 1-types for more.

For 2-groupoids

(…) 2-groupoids model all homotopy 2-types (…)

strict 2-groupoids 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-groupoids (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‘s 2019 paper.

For Kan complexes

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.

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 presented (∞,1)-categories

• Top$\simeq$ ∞Grpd

(Quillen 67), see e.g. (Goerss-Jardine 96, section I.11, Joyal-Tierney 05, chapter I)

For algebraic Kan complexes

An algebraic Kan complex is an 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 associators 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.

Theorem

The transferred model structure on algebraic Kan complexes is 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

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

Remark

With the above homotopy hypothesis-theorem for Kan complexes 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 presented (∞,1)-categories (as discussed there)

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

But there is also a direct Quillen equivalence:

Definition

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

Fix any choice of retracts

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

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 $U \Pi_\infty = Sing$.

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.

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 $A$;

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

Proposition

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

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

This is (Nikolaus, prop. 3.4).

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.

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)

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 $|-|$.

Corollary

The adjunction

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

constitutes a Quillen equivalence

This is Nikolaus, corollary 3.6

Proof

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

For cubical sets

Also cubical sets 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.

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 monomorphisms.

This is Jardine, sections 3.

Theorem

The 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.

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.

Theorem

The homotopy category of cat-n-groups is equivalent to that of pointed connected homotopy n+1-types.

This is proven in (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

Proposition

There is realization/singular complex adjunction

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

for Segal groupoids,

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

Proof

This is lemma 6.3.21 and corollary 6.3.24 in (Pellissier)

For strict $\omega$-categories with weak inverses

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) it was argued that these are in fact sufficient, but a mistake in the argument was found in (Simpson, Cor. 5.2) (see also here).

The issue however is somewhat subtle, as highlighted by Voevodsky here. 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).

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, Gurski-Johnson-Osorno)

References and a bit of history.

The equivalence between the classical homotopy theory of topological spaces and the homotopy theory of Kan complexes is due to

• 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 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 here in New Spaces for Mathematics and Physics. These models included Kan complexes which now are interpreted as being one model for infinity groupoids, and weak Kan complexes, 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, 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:

• Simon Henry, Edoardo Lanari, On the homotopy hypothesis in dimension 3, Theory and Applications of Categories 39 26 (2023) 735-768 [arxiv/1905.05625, tac: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:

• Paul Goerss, Rick Jardine, section I.11 of Simplicial homotopy theory, 1996

• André Joyal, Myles Tierney An introduction to simplicial homotopy theory, 2005 (chapter I, more notes pdf)

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

• 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

• Carlos Simpson, Homotopy types of strict 3-groupoids (arXiv:math/9810059)

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

• Thomas Nikolaus, Algebraic models for higher categories (arXiv)

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)

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.

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)

• Simona Paoli, Internal categorical structures in homotopical algebra, to appear in Towards Higher Categories, eds. John Baez and Peter May (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 sets is discussed in

• Jardine, Model structure on cubical sets (pdf)

• 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

• David Ayala, John Francis, Nick Rozenblyum, A stratified homotopy hypothesis, arXiv

A version for spectra is discussed in

• Niles Johnson and Angelica Osorno, Modeling stable one-types, tac

• Nick Gurski, Niles Johnson, and Angelica Osorno, The 2-dimensional stable homotopy hypothesis, arxiv