3.6.5 Comparison of Homotopy Categories
Our goal in this section is to carry out the proof of Theorem 3.6.0.1. We begin with an elementary application of the results of ยง3.6.2.
Construction 3.6.5.1 (Geometric Realization as a Simplicial Functor). Let $X$ and $Y$ be simplicial sets and let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Fun}(X,Y)$, which we identify with a morphism $\Delta ^ n \times X \rightarrow Y$. By virtue of Corollary 3.6.2.2, the geometric realization of $\sigma $ can be identified with a continuous function
\[ |\sigma |: | \Delta ^{n} | \times |X| \rightarrow |Y|, \]
which we can view as an $n$-simplex of the simplicial set $\operatorname{Hom}_{\operatorname{Top}}(|X|,|Y|)_{\bullet }$ parametrizing continuous functions from $X$ to $Y$ (see Example 2.4.1.5). This construction is compatible with face and degeneracy operators, and therefore determines a morphism of simplicial sets $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{Top}}( |X|, |Y| )_{\bullet }$. Allowing $X$ and $Y$ to vary, we obtain a simplicial structure on the geometric realization functor $| \bullet |: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}$.
Proposition 3.6.5.2. Let $X$ and $Y$ be simplicial sets. If $Y$ is a Kan complex, then the comparison map
\[ \theta : \operatorname{Fun}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{Top}}( |X|, |Y| )_{\bullet } \]
of Construction 3.6.5.1 is a homotopy equivalence of Kan complexes.
Proof.
Using Example 3.6.2.5, we can identify $\theta $ with the morphism
\[ \operatorname{Fun}(X,Y) \rightarrow \operatorname{Fun}( X, \operatorname{Sing}_{\bullet }(|Y| ) ) \]
given by postcomposition with the unit map $u_{Y}: Y \rightarrow \operatorname{Sing}_{\bullet }(|Y|)$. By virtue of Theorem 3.6.4.1, the map $u_{Y}$ is a weak homotopy equivalence. Since $Y$ and $\operatorname{Sing}_{\bullet }(|Y|)$ are Kan complexes, we conclude that $u_{Y}$ is a homotopy equivalence (Proposition 3.1.6.13). It follows that $\theta $ is also a homotopy equivalence (it admits a homotopy inverse, given by postcomposition with any homotopy inverse to $u_{Y}$).
$\square$
Proposition 3.6.5.3. Let $X$ be a topological space. The following conditions are equivalent:
- $(1)$
The counit map $| \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ is a homotopy equivalence of topological spaces.
- $(2)$
There exists a Kan complex $Y$ and a homotopy equivalence of topological spaces $| Y | \rightarrow X$.
- $(3)$
There exists a simplicial set $Y$ and a homotopy equivalence of topological spaces $|Y| \rightarrow X$.
- $(4)$
There exists a homotopy equivalence of topological spaces $X' \rightarrow X$, where $X'$ is a CW complex.
Proof.
The implication $(1) \Rightarrow (2)$ follows from the observation that $\operatorname{Sing}_{\bullet }(X)$ is a Kan complex (Proposition 1.2.5.8), the implication $(2) \Rightarrow (3)$ is trivial, and the implication $(3) \Rightarrow (4)$ follows from Remark 1.2.3.12. To complete the proof, it will suffice to show that if $X$ has the homotopy type of a CW complex, then the counit map $v: | \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ is a homotopy equivalence. By virtue of Proposition 3.6.3.8, it will suffice to show that $v$ is a weak homotopy equivalence, which follows from Corollary 3.6.4.2.
$\square$
Corollary 3.6.5.4. Let $f: Y \rightarrow Z$ be a continuous function between topological spaces. The following conditions are equivalent:
- $(1)$
The function $f$ is a weak homotopy equivalence (Definition 3.6.3.1).
- $(2)$
For every simplicial set $S$, the induced map $\operatorname{Fun}( S, \operatorname{Sing}_{\bullet }(Y) ) \rightarrow \operatorname{Fun}(S, \operatorname{Sing}_{\bullet }(Z) )$ is a homotopy equivalence of Kan complexes.
- $(3)$
For every simplicial set $S$, the induced map $\operatorname{Hom}_{\operatorname{Top}}( |S|, Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Top}}( |S|, Z)_{\bullet }$ is a homotopy equivalence of Kan complexes.
- $(4)$
For every topological space $X$ which has the homotopy type of a CW complex, the induced map $\operatorname{Hom}_{\operatorname{Top}}( X, Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Top}}( X, Z)_{\bullet }$ is a homotopy equivalence of Kan complexes.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 3.1.6.17, the equivalence $(2) \Leftrightarrow (3)$ from Example 3.6.2.5, and the equivalence $(3) \Leftrightarrow (4)$ from Proposition 3.6.5.3.
$\square$
Proof of Theorem 3.6.0.1.
Using Construction 3.6.5.1, we see that the geometric realization functor $| \bullet |: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}$ induces a functor of homotopy categories $| \bullet |: \mathrm{h} \mathit{\operatorname{Kan}} \rightarrow \mathrm{h} \mathit{\operatorname{Top}}$. It follows from Proposition 3.6.5.2 that this functor is fully faithful, and from Proposition 3.6.5.3 that its essential image consists of those topological spaces $X$ which have the homotopy type of a CW complex.
$\square$
Using Theorem 3.6.4.1, we can also give a purely topological characterization of the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (which does not make reference to the theory of simplicial sets).
Corollary 3.6.5.6. Let $\operatorname{\mathcal{C}}$ be a category, and let $\operatorname{\mathcal{E}}' \subseteq \operatorname{Fun}( \operatorname{Top}, \operatorname{\mathcal{C}})$ be the full subcategory spanned by those functors $F: \operatorname{Top}\rightarrow \operatorname{\mathcal{C}}$. which carry weak homotopy equivalences of topological spaces to isomorphisms in the category $\operatorname{\mathcal{C}}$. Then:
- $(a)$
For every functor $F \in \operatorname{\mathcal{E}}'$, the composite functor
\[ \operatorname{Kan}\xrightarrow { | \bullet | } \operatorname{Top}\xrightarrow {F} \operatorname{\mathcal{C}} \]
factors uniquely as a composition $\operatorname{Kan}\twoheadrightarrow \mathrm{h} \mathit{\operatorname{Kan}} \xrightarrow { \overline{F} } \operatorname{\mathcal{C}}$.
- $(b)$
The construction $F \mapsto \overline{F}$ induces an equivalence of categories $\operatorname{\mathcal{E}}' \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}})$.
We can state Corollary 3.6.5.6 more informally as follows: the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes can be obtained from the category of topological spaces $\operatorname{Top}$ by formally adjoining inverses to all weak homotopy equivalences.
Proof of Corollary 3.6.5.6.
Let $\operatorname{\mathcal{E}}\subseteq \operatorname{Fun}( \operatorname{Kan}, \operatorname{\mathcal{C}})$ be the full subcategory spanned by those functors $F: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$ which carry homotopy equivalences of Kan complexes to isomorphisms in $\operatorname{\mathcal{C}}$. By virtue of Corollary 3.1.7.7, it will suffice to show that precomposition with the geometric realization functor $| \bullet |: \operatorname{Kan}\rightarrow \operatorname{Top}$ induces an equivalence of categories $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$. We claim that this functor has a homotopy inverse $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$, given by precomposition with the functor $\operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Kan}$. This follows from the following pair of observations:
For every functor $F: \operatorname{Top}\rightarrow \operatorname{\mathcal{C}}$, the counit map $\overline{F} \circ \operatorname{Sing}_{\bullet } \rightarrow F$ is an isomorphism when $F$ belongs to $\operatorname{\mathcal{E}}'$ (since, for every topological space $X$, the counit map $| \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ is a weak homotopy equivalence; see Corollary 3.6.4.2).
For every functor $F_0: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$, the unit map $F_0 \rightarrow \overline{ F_0 \circ \operatorname{Sing}_{\bullet } }$ is an isomorphism (since, for every simplicial set $Y$, the unit map $Y \rightarrow \operatorname{Sing}_{\bullet }( |Y| )$ is a weak homotopy equivalence of simplicial sets, and therefore induces a homotopy equivalence of topological spaces $|Y| \rightarrow | \operatorname{Sing}_{\bullet }( |Y| ) |$).
$\square$