nLab weak homotopy equivalence

Idea

A weak homotopy equivalence is a map between topological spaces or simplicial sets or similar which induces isomorphisms on all homotopy groups. (The analogous concept in homological algebra is called a quasi-isomorphism.)

The localization or simplicial localization of the categories Top and sSet at the weak homotopy equivalences used as weak equivalences yields the standard homotopy category Ho(Top) and Ho(sSet) or the (∞,1)-category of ∞-groupoids/homotopy types, respectively.

Weak homotopy equivalences are named after homotopy equivalences. They can be identified with homotopy equivalences after replacing the domains by a resolution. The corresponding notions in homological algebra are quasi-isomorphisms and chain homotopy-equivalences.

From another perspective, the notion of weak homotopy equivalence is ‘observational’, in that a map is a weak homotopy equivalence if when we look at it through the observations that we can make of it using homotopy groups or even the fundamental infinity-groupoid, it looks like an equivalence. In contrast, homotopy equivalence is more ‘constructive’; in that $f$ is a homotopy equivalence if there exists an inverse for it (up to homotopy, of course). Note that both of these notions are weaker than mere isomorphism of topological spaces (homeomorphism) and so can be considered examples of weak equivalences.

There are actually two related concepts here: whether two spaces are weakly homotopy equivalent and whether a map between spaces is a weak homotopy equivalence. The former is usually defined in terms of the latter.

Definition (for topological spaces and simplicial sets)

Definition

For $X, Y \in$ Top or $\in$ sSet two topological spaces or simplicial sets, a continuous function or simplicial map $f : X \to Y$ between them is called a weak homotopy equivalence if

1. $f$ induces an isomorphism of connected components (path components in the case of topological spaces)

   $$\Pi_0(f) \colon \Pi_0(X) \stackrel{\simeq}{\to} \Pi_0(Y)$$

   in Set;

2. for all points $x \in X$ and for all $(1 \leq n) \in \mathbb{N}$ $f$ induces an isomorphism on homotopy groups

   $$\pi_n(f,x) \colon \pi_n(X,x) \stackrel{\simeq}{\to} \pi_n(Y,f(x))$$

   in Grp.

Remark

If $X$ and $Y$ are path-connected, then (1) is trivial, and it suffices to require (2) for a single (arbitrary) $x$, but in general one must require it for at least one $x$ in each path connected component.

Remark

It is not enough to require that each pair of homotopy groups are isomorphic. For example, let $Y = S^1 \vee S^2$, and let $X$ be its double cover, which glues two spheres at the north and south poles of a circle. Then there are isomorphisms on the fundamental group $\pi_1(X) \simeq \mathbb{Z} \simeq \pi_1(Y)$, although the map $\pi_1(X) \to \pi_1(Y)$ induced by $f$ is not an isomorphism. And by the properties of covering spaces, all higher homotopy groups are isomorphic. But these two are not weakly homotopy equivalent.

Remark

There are many alternative definitions of weak homotopy equivalences. A simplicial map $f$ is a weak equivalence of simplicial sets if and only if $Ex^\infty f$ is a simplicial homotopy equivalence if and only if $Hom(f,A)$ is a simplicial homotopy equivalence for any Kan complex $A$ if and only if $f$ has a right relative-homotopy-lifting property with respect to the maps $\partial\Delta^n\to\Delta^n$ if and only if $f$ is a composition of an acyclic cofibration (i.e., a map with a left lifting property with respect to all maps with a right lifting property with respect to horn inclusions) and an acyclic fibration (i.e., a map with a right lifting property with respect to inclusions $\partial\Delta^n\to\Delta^n)$. A continuous map $f$ is a weak equivalence of topological spaces if and only if $|Sing(f)|$ is a homotopy equivalence of topological spaces if and only if $Hom(A,f)$ is a homotopy equivalence for any CW-complex $A$ if and only if $f$ has a right relative-homotopy-lifting property with respect to the maps $S^{n-1}\to D^n$ if and only if $f$ is a composition of an acyclic Serre cofibration (a retract of a relative CW-complex) and an acyclic Serre fibration. Both functors $|-|$ and $Sing$ preserve and reflect weak equivalences, so any of the two classes defines the other.

Definition

The homotopy category of Top with respect to weak homotopy equivalences is Ho(Top)${}_{whe}$.

Remark

Accordingly, weak homotopy equivalences are the weak equivalences in the standard Quillen model structure on topological spaces and the Quillen model structure on simplicial sets, and also in the mixed model structure.

Properties

Equivalent characterizations

Proposition

A continuous map $f : X \to Y$ is a weak homotopy equivalence precisely if for all $n \in \mathbb{N}$ and for all commuting diagrams of continuous maps of the form

$$\array{ S^{n-1} &\to& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\to& Y } \,,$$

where the left morphism is the inclusion of the $(n-1)$-sphere as the boundary of the $n$-ball, there exists a continuous map $\sigma : D^n \to X$ that makes the resulting upper triangle commute and such that the lower triangle commutes up to a homotopy

$$\array{ S^{n-1} &&\to&& X \\ \\ \downarrow && \nearrow & \swArrow& \downarrow^{\mathrlap{f}} \\\\ D^n &&\to&& Y }$$

which is constant along $S^{n-1} \hookrightarrow D^n$.

In this form the statement and its proof appears in (Jardine) (where it is also generalized to weak equivalences in a model structure on simplicial presheaves). See also around (Lurie, prop. 6.5.2.1). The relevant arguments are spelled out in (May, section 9.6). A variant is called the HELP lemma in (Vogt).

Relation to homotopy equivalences

Proposition

Every homotopy equivalence is a weak homotopy equivalence.

Proof

It requires a little bit of thought to prove this, because $f$ and its homotopy inverse $g$ need not preserve any chosen basepoint. But for any $x\in X$ and any $n\ge 1$, we have a commuting diagram

$$\array{ \pi_n(X,x) & & \to & & \pi_n(X,g(f(x))) \\ & \searrow && \nearrow && \searrow \\ && \pi_n(Y,f(x)) && \longrightarrow && \pi_n\big( Y,f(g(f(x))) \big) }$$

in which the two horizontal morphisms are isomorphisms, because $g f$ and $f g$ are homotopic to identities. Hence, by the two-out-of-six property for isomorphisms, the diagonal morphhisms are also all isomorphisms.

Proposition

Conversely, any weak homotopy equivalence between m-cofibrant spaces (spaces that are homotopy equivalent to CW complexes) is a homotopy equivalence.

Relation to homotopy types

We discuss the equivalence relation generated by weak homotopy equivalence, called (weak) homotopy type. For the “abelianized” analog of this situation see at quasi-isomorphism the section Relation to homology type.

Proposition

The existence of a weak homotopy equivalence from $X$ to $Y$ is a reflexive and transitive relation on Top, but it is not a symmetric relation.

Proof

Reflexivity and transitivity are trivially checked. A counterexample to symmetry is example below.

But we can consider the genuine equivalence relation generated by weak homotopy equivalence:

Definition

We say two spaces $X$ and $Y$ have the same (weak) homotopy type if they are equivalent under the equivalence relation generated by weak homotopy equivalence.

Remark

Equivalently this means that $X$ and $Y$ have the same (weak) homotopy type if there exists a zigzag of weak homotopy equivalences

$$X \leftarrow \to\leftarrow \dots \to Y \,.$$

This in turn is equivalent to saying that $X$ and $Y$ become isomorphic in the homotopy category Ho(Top)/Ho(sSet) with the weak homotopy equivalences inverted.

Remark

Two spaces $X$ and $Y$ may have isomorphic homotopy groups without being weak homotopy equivalence: for this all the isomorphisms must be induced by an actual map $f : X \to Y$, as in the above definition.

However, if, roughly, one remembers, how all the homotopy groups act on each other, then this is enough information to exhibit the full homotopy type. This collection of data is called the Postnikov tower decomposition of a homotopy type.

Relation to free homotopy sets

For $K, X \,\in\, TopSp$, write $Map(K, X)$ for their mapping space, i.e. not considering or respecting any basepoint. For example, $Map(S^1, Y)$ is the free loop space of $Y$, in contrast to the based loop space

$$\Omega_x X \xhookrightarrow{\; fib_x(ev_\ast) \;} Map(S^1, X) \overset{\; ev_x \;}{\twoheadrightarrow} X$$

for any base point $x \,\in\, X$.

Moreover, when $K$ is a CW complex, write

(1)$${[K,X]} \,\coloneqq\, \tau_0 \,Map(K,X)\,$$

for the free homotopy set of maps from $K$ to $X$, hence the set of homotopy classes of map $K \to X$, hence the set of connected components of the mapping space.

Proposition

(weak homotopy equivalences detected on free homotopy sets)
For $f \,\colon\, X \to Y$, a continuous function between connected topological spaces, the following are equivalent:

1. $f$ is a weak homotopy equivalence

   $$\underset{n \in \mathbb{N}}{\forall} \; \;\; p_n(X) \underoverset {\simeq} {f_\ast} {\longrightarrow} \pi_n(Y) \,;$$

2. $f$ induces an isomorphism on all free homotopy sets (1) out of CW-complexes:

   $$\underset{ K \in CWCplx}{\forall} \;\; [K, X] \underoverset {\simeq} {f_\ast} {\longrightarrow} [K, Y] \,;$$

• $f$ induces

  1. an isomorphism on all free homotopy sets (1) out of $K =$any n-sphere of positive dimension,

     $$\underset{ n \in \mathbb{N}_+ }{\forall} \;\; [S^n, X] \underoverset {\simeq} {f_\ast} {\longrightarrow} [S^n, Y] \,.$$

  2. a surjection on the free homotopy set out of the wedge sum of circles indexed by (the set underlying) the fundamental group of $Y$:

     $$\big[ \underset{\pi_1(Y)}{\vee} S^1 ,\, X \big] \overset{\; f_\ast \;}{\twoheadrightarrow} \big[ \underset{\pi_1(Y)}{\vee} S^1 ,\, Y \big]$$

(Matumoto, Minami and Sugawara 1984, Thm. 2)

Proof

The implication $(1) \Rightarrow (2)$ is a standard/classical conclusion in homotopy theory, for example it is a small special case of the fact that $Map(K,-)$ out of any cell complex $K$ preserves weak homotopy equivalences (this Prop.).

The implication $(2) \Rightarrow (3)$ is trivial, as the conditions in (3) are just a special case of the condition in (2).

So the point of the statement is that (3) is already sufficient to recover (1). This is the content of Matumoto, Minami and Sugawara 1984, Thm. 1 & Lem. 1.3.

For other kinds of spaces

A map of simplicial sets is called a weak homotopy equivalence equivalently if its geometric realization is a weak homotopy equivalence of topological spaces, as above. (Since the geometric realization of any simpicial set is a CW complex, in this case its geometric realization is actually a homotopy equivalence.)

Likewise, a functor between small categories is sometimes said to be a weak homotopy equivalence if its nerve is a weak homotopy equivalence of simplicial sets, hence of topological spaces after geometric realization of categories. These are the weak equivalences in the Thomason model structure on categories (not the canonical model structure). The statement of Quillen's theorem A and Quillen's theorem B in in this contex.

Similarly, one can define weak homotopy equivalences between any sort of object that has a geometric realization, such as a cubical set, a globular set, an n-category, an n-fold category, and so on.

Note that in some of these cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of “homotopy equivalence” from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times \Delta^1 \to X$ and $Y\times \Delta^1 \to Y$ relating $f g$ and $g f$ to identities.

A different direction of generalization is the notion of a homotopy equivalence of toposes.

Examples

Of non-reversible weak homotopy equivalences

We discuss examples of weak homotopy equivalences that have no weak homotopy equivalence going the other way, according to prop. above.

Example

Let $S^1 \in$ Top denote the ordinary circle and $\mathbb{S}$ the pseudocircle.

There is a continuous function $S^1 \to \mathbb{S}$ which is a weak homotopy equivalence, hence in particular $\pi_1(\mathbb{S}) \simeq \mathbb{Z}$. But every continuous map the other way round has to induce the trivial map on $\pi_1$.

This is the simplest in a class of counter-examples discussed in (McCord).

Related concepts

• equality

• isomorphism

• equivalence

• weak equivalence

• homotopy equivalence, weak homotopy equivalence

  • rational homotopy equivalence

  • n-equivalence

  • equivariant weak homotopy equivalence

  • stable weak homotopy equivalence

• Bousfield equivalence

• homotopy equivalence of toposes

• equivalence in an (∞,1)-category

• equivalence of (∞,1)-categories

References

A general account is for instance in section 9.6 of

• Peter May, A Concise Course in Algebraic Topology

The characterization of weak homotopy equivalences by lifts up to homotopy seems is in

• J. F. Jardine, Simplicial Presheaves, Journal of Pure and Applied Algebra 47, 1987, no.1, 35-87.

• Rainer Vogt, The HELP-Lemma and its converse in Quillen model categories (arXiv:1004.5249)

For related and general discussion see also section 6.5 of

• Jacob Lurie, Higher Topos Theory

Examples for the non-symmetry of the weak homotopy equivalence relation are in

• Michael McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. Volume 33, Number 3 (1966), 465-474. (euclid.dmj/1077376525)

See also:

• Topospaces-Wiki, Weak homotopy equivalence of topological spaces

Detection in terms of free homotopy sets:

• Takao Matumoto, Norihiko Minami, Masahiro Sugawara, On the set of free homotopy classes and Brown’s construction, Hiroshima Math. J. 14(2): 359-369 (1984) (doi:10.32917/hmj/1206133043)