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3.5 Comparison with Topological Spaces

Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets and let $\operatorname{Top}$ denote the category of topological spaces. In §1.1.7 and §1.1.8, we constructed a pair of adjoint functors

\[ \xymatrix@1{\operatorname{Set_{\Delta }} \ar@ <.4ex>[r]^-{| \bullet |} & \operatorname{Top}. \ar@ <.4ex>[l]^-{\operatorname{Sing}_{\bullet } }} \]

Our goal in this section is to prove that, after passing to homotopy categories, these functors are not far from being (mutually inverse) equivalences:

Theorem 3.5.0.1. The geometric realization functor $| \bullet |: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}$ induces an equivalence from the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ to the full subcategory of $\mathrm{h} \mathit{\operatorname{Top}}$ spanned by those topological spaces $X$ which have the homotopy type of a CW complex.

Theorem 3.5.0.1 is essentially due to Milnor (see [MR0084138]). We give a proof in §3.5.5, which has three main steps. The first of these is of a technical nature: we must show that geometric realization is well-defined at the level of homotopy categories (see Construction 3.5.5.1). Let $X$ and $Y$ be Kan complexes, and suppose that we are given a pair of morphisms $f_0, f_1: X \rightarrow Y$. If $f_0$ is homotopic to $f_1$ (in the category of Kan complexes), then there exists a morphism of simplicial sets $h: \Delta ^1 \times X \rightarrow Y$ satisfying $f_0 = h|_{ \{ 0\} \times X}$ and $f_1 = h|_{ \{ 1\} \times X}$. Passing to geometric realizations, we obtain a continuous function $|h|: | \Delta ^1 \times X | \rightarrow |Y|$. We would like to interpret $|h|$ as a homotopy from $|f_0|$ to $|f_1|$ (in the category of topological spaces). For this, we need to know that the comparison map

\[ | \Delta ^1 \times X | \rightarrow | \Delta ^1 | \times X \simeq [0,1] \times X \]

is a homeomorphism. In §3.5.2, we prove a more general assertion: for any pair of simplicial sets $A$ and $B$, the comparison map $| A \times B | \rightarrow |A| \times |B|$ is a bijection (Theorem 3.5.2.1), which is a homeomorphism if either $A$ or $B$ is finite (that is, if either $A$ or $B$ has only finitely many nondegenerate simplices; see Corollary 3.5.2.2).

The second step in the proof of Theorem 3.5.0.1 is to show that the geometric realization functor $| \bullet |: \mathrm{h} \mathit{\operatorname{Kan}} \rightarrow \mathrm{h} \mathit{\operatorname{Top}}$ is fully faithful (Proposition 3.5.5.2). This is equivalent to the assertion that for any Kan complex $X$, the unit map $u_{X}: X \rightarrow \operatorname{Sing}_{\bullet }( | X |)$ is a homotopy equivalence. More generally, we show in §3.5.4 that for any simplicial set $X$, the unit map $u_ X: X \rightarrow \operatorname{Sing}_{\bullet }(|X|)$ is a weak homotopy equivalence (Theorem 3.5.4.1). Our strategy is to reduce to the case where the simplicial set $X$ is finite, and to proceed by induction on the number of nondegenerate simplices of $X$. The inductive step will make use of excision (Theorem 3.4.6.1) to analyze the homotopy type of the Kan complex $\operatorname{Sing}_{\bullet }(|X|)$.

To complete the proof of Theorem 3.5.0.1, we must show that if $Y$ is a topological space, then the counit map $v_{Y}: | \operatorname{Sing}_{\bullet }(Y) | \rightarrow Y$ is a homotopy equivalence if and only if $Y$ has the homotopy type of a CW complex (Proposition 3.5.5.3). It follows formally from the preceding step that the map $v_{Y}$ is always a weak homotopy equivalence: that is, it induces a bijection on path components and an isomorphism on homotopy groups for any choice of base point (Corollary 3.5.4.2). We will complete the proof using a result of Whitehead which asserts that any weak homotopy equivalence between CW complexes is a homotopy equivalence (see Proposition 3.5.3.8 and Corollary 3.5.3.10), which we prove in §3.5.3.

Structure

  • Subsection 3.5.1: Digression: Finite Simplicial Sets
  • Subsection 3.5.2: Exactness of Geometric Realization
  • Subsection 3.5.3: Weak Homotopy Equivalences in Topology
  • Subsection 3.5.4: The Unit Map $u: X \rightarrow \operatorname{Sing}_{\bullet }(|X|)$
  • Subsection 3.5.5: Comparison of Homotopy Categories