Kerodon

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3.5.2 Exactness of Geometric Realization

Our goal in this section is to study the exactness properties of the geometric realization functor $X \mapsto |X|$ of Definition 1.1.8.1. Our main result can be stated as follows:

Theorem 3.5.2.1. The geometric realization functor

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad X \mapsto |X| \]

preserves finite limits. In particular, for every diagram of simplicial sets $X \rightarrow Z \leftarrow Y$, the induced map $| X \times _{Z} Y | \rightarrow |X| \times _{|Z|} |Y|$ is a bijection.

Before giving the proof of Theorem 3.5.2.1, let us collect some consequences.

Corollary 3.5.2.2. Let $X$ and $Y$ be simplicial sets. Then the canonical map $\theta _{X,Y}: |X \times Y| \rightarrow |X| \times |Y|$ is a bijection. If either $X$ or $Y$ is finite, then $\theta $ is a homeomorphism.

Proof. The first assertion follows immediately from Theorem 3.5.2.1. If $X$ and $Y$ are both finite, then the product $X \times Y$ is also finite (Remark 3.5.1.6), so that the geometric realizations $|X|$, $|Y|$, and $|X \times Y|$ are compact Hausdorff spaces (Corollary 3.5.1.10). In this case, $\theta _{X,Y}$ is a continuous bijection between compact Hausdorff spaces, and therefore a homeomorphism.

Now suppose that $X$ is finite and $Y$ is arbitrary. Let $M = \operatorname{Hom}_{\operatorname{Top}}( |X|, |X \times Y| )$ denote the set of all continuous functions from $|X|$ to $|X \times Y|$, endowed with the compact-open topology. For every finite simplicial subset $Y' \subseteq Y$, the composite map

\[ |X| \times |Y'| \xrightarrow { \theta _{X,Y'}^{-1} } |X \times Y'| \hookrightarrow |X \times Y|, \]

determines a continuous function $\rho _{Y'}: |Y'| \rightarrow M$. Writing the geometric realization $|Y|$ as a colimit $\varinjlim _{Y' \subseteq Y} |Y'|$ (see Remark 3.5.1.8), we can amalgamate the functions $f_{Y'}$ to a single continuous function $\rho : |Y| \rightarrow M$. Our assumption that $X$ guarantees that the topological space $|X|$ is compact and Hausdorff, so the evaluation map

\[ \operatorname{ev}: |X| \times M \rightarrow |X \times Y| \quad \quad (x,f) \mapsto f(x) \]

is continuous (see Theorem ). We complete the proof by observing that the bijection $\theta _{X,Y}^{-1}$ is a composition of continuous functions

\[ |X| \times |Y| \xrightarrow { \operatorname{id}\times \rho } |X| \times M \xrightarrow {\operatorname{ev}} | X \times Y |, \]

and is therefore continuous. $\square$

Warning 3.5.2.3. Let $X$ and $Y$ be simplicial sets. In general, the comparison map $\theta _{X,Y}: |X \times Y| \rightarrow |X| \times |Y|$ need not be a homeomorphism if neither $X$ or $Y$ is assumed to be finite. For an explicit counterexample, we refer the reader to Section 5 of [MR48020].

Remark 3.5.2.4. Let $X$ and $Y$ be simplicial sets having at most countably many simplices of each dimension. Then the comparison map $\theta _{X,Y}: |X \times Y| \rightarrow |X| \times |Y|$ is a homeomorphism. For a proof, we refer the reader to [MR0084138].

Example 3.5.2.5. Let $X$ be a simplicial set and let $Y$ be a topological space, and let $\operatorname{Hom}_{\operatorname{Top}}( |X|, Y)_{\bullet }$ be the simplicial set defined in Example 2.4.1.5. For each $n \geq 0$, precomposition with the homeomorphism $|X \times \Delta ^ n| \rightarrow |X| \times | \Delta ^ n |$ induces a bijection

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Top}}( |X|, Y)_{n} & = & \operatorname{Hom}_{\operatorname{Top}}( |X| \times | \Delta ^ n |, Y) \\ & \simeq & \operatorname{Hom}_{\operatorname{Top}}( |X \times \Delta ^ n|, Y) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X \times \Delta ^ n, \operatorname{Sing}_{\bullet }(Y) ) \\ & = & \operatorname{Fun}(X, \operatorname{Sing}_{\bullet }(Y) )_{n}. \end{eqnarray*}

These bijections are compatible with face and degeneracy operators, and therefore determine an isomorphism of simplicial sets $\operatorname{Hom}_{\operatorname{Top}}( |X|, Y)_{\bullet } \rightarrow \operatorname{Fun}(X, \operatorname{Sing}_{\bullet }(Y) )$.

We now turn to the proof of Theorem 3.5.2.1. Our proof will make use of an explicit description of the underlying set of a geometric realization $|X|$ (see Remark 3.5.2.10) which given by Drinfeld in [MR2119142] (and also appears in unpublished work of Besser and Grayson).

Construction 3.5.2.6. Let $S$ be a finite subset of the unit interval $[0,1]$, and assume that $0,1 \in S$. For each $n \geq 0$, we let $| \Delta ^ n |_{S}$ denote the subset of the topological $n$-simplex

\[ | \Delta ^ n | = \{ (t_0, \ldots , t_ n) \in \operatorname{\mathbf{R}}_{\geq 0}^{n+1}: t_0 + t_1 + \cdots + t_ n = 1 \} \]

consisting of those tuples $(t_0, t_1, \ldots , t_ n)$ having the property that each of the partial sums $t_0 + t_1 + \cdots t_ i$ belongs to $S$. Note that these subsets are stable under the coface and codegeneracy operators of the cosimplicial topological space $| \Delta ^{\bullet } |$, so we can regard the construction $[n] \mapsto | \Delta ^ n |_{S}$ as a cosimplicial set.

By virtue of Proposition 1.1.8.22, the functor

\[ \operatorname{Set}\rightarrow \operatorname{Set_{\Delta }}\quad \quad (Y \mapsto ([n] \mapsto \operatorname{Hom}_{\operatorname{Set}}( | \Delta ^{n} |_{S}, Y))) \]

admits a left adjoint, which we will denote by $| \bullet |_{S}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ and refer to as the $S$-partial geometric realization. Concretely, this functor carries a simplicial set $X$ to the colimit $|X|_{S} = \varinjlim _{ \Delta ^{n} \rightarrow X} | \Delta ^{n} |_{S}$, where the colimit is indexed by the category of simplices $\int ^{\operatorname{{\bf \Delta }}} X$ of Construction 1.1.8.19.

Remark 3.5.2.7. For each integer $n \geq 0$, the topological $n$-simplex $| \Delta ^ n |$ can be identified with the filtered direct limit $\varinjlim _{S} | \Delta ^ n |_{S}$, where $S$ ranges over the collection of all finite subsets of $[0,1]$ which contain the endpoints $0$ and $1$ (which we regard as a partially ordered set with respect to inclusion). We therefore obtain a canonical isomorphism of cosimplicial sets $\varinjlim _{S} | \Delta ^{\bullet } |_{S} \xrightarrow {\sim } | \Delta ^{\bullet } |$. It follows that, for every simplicial set $X$, the canonical map $\varinjlim _{S} | X|_{S} \rightarrow |X|$ is a bijection.

Notation 3.5.2.8. Let $\operatorname{Lin}_{\neq \emptyset }$ denote the category whose objects are nonempty finite linearly ordered sets, and whose morphisms are nondecreasing functions. Note that, if $S$ is a finite subset of the unit interval $[0,1]$, then the complement $[0,1] \setminus S$ has finitely many connected components. Moreover, there is a unique linear ordering on the set $\pi _0( [0,1] \setminus S)$ for which the quotient map

\[ ( [0,1] \setminus S) \rightarrow \pi _0( [0,1] \setminus S) \]

is nondecreasing. We can therefore regard $\pi _0( [0,1] \setminus S)$ as an object of the category $\operatorname{Lin}_{\neq \emptyset }$.

Proposition 3.5.2.9. Let $S$ be a finite subset of the unit interval $[0,1]$ which contains $0$ and $1$. Then the cosimplicial set

\[ | \Delta ^{\bullet } |_{S}: \operatorname{{\bf \Delta }}\rightarrow \operatorname{Set}\quad \quad [n] \mapsto | \Delta ^ n |_{S} \]

is a corepresentable functor. More precisely, there exists a functorial bijection $| \Delta ^ n |_{S} \simeq \operatorname{Hom}_{ \operatorname{Lin}_{\neq \emptyset } }( \pi _0( [0,1] \setminus S), [n] )$.

Proof. Let $S = \{ 0 = s_0 < s_1 < \cdots < s_ k = 1 \} $ be a finite subset of the unit interval $[0,1]$ which contains $0$ and $1$. Let $n$ be a nonnegative integer and let $(t_0, \ldots , t_ n)$ be a point of $| \Delta ^ n |_{S}$. For every real number $u \in [0,1] \setminus S$, there exists a unique integer $0 \leq i \leq n$ satisfying

\[ t_0 + t_1 + \cdots + t_{i-1} < u < t_0 + t_1 + \cdots + t_{i}. \]

The construction $u \mapsto i$ defines a continuous nondecreasing function $([0,1] \setminus S) \rightarrow [n]$. This observation induces a bijection

\begin{eqnarray*} | \Delta ^ n |_{S} & \simeq & \{ \text{Continuous nondecreasing functions $f: [0,1] \setminus S \rightarrow [n]$} \} \\ & \simeq & \operatorname{Hom}_{ \operatorname{Lin}_{\neq \emptyset } }( \pi _0( [0,1] \setminus S), [n] ). \end{eqnarray*}

Explicitly, the inverse bijection carries a continuous nondecreasing function $f: [0,1] \setminus S \rightarrow [n]$ to the sequence

\[ ( \mu ( f^{-1} \{ 0\} ), \mu ( f^{-1} \{ 1\} ), \cdots , \mu ( f^{-1} \{ n\} ) ), \]

where

\[ \mu ( f^{-1} \{ i\} ) = \sum _{ (s_{j-1}, s_ j) \subseteq f^{-1} \{ i\} } (s_ j - s_{j-1} ) \]

denotes the measure of the inverse image $f^{-1} \{ i \} $. $\square$

Proof of Theorem 3.5.2.1. Let $U: \operatorname{Top}\rightarrow \operatorname{Set}$ denote the forgetful functor. We wish to show that the composite functor

\[ \operatorname{Set_{\Delta }}\xrightarrow { | \bullet | } \operatorname{Top}\xrightarrow {U} \operatorname{Set} \]

preserves finite limits. By virtue of Remark 3.5.2.7, we can write this composite functor as a filtered colimit of functors of the form $X \mapsto |X|_{S}$, where $S$ ranges over all finite subsets of the unit interval $[0,1]$ which contain $0$ and $1$. It will therefore suffice to show that each of the functors $X \mapsto |X|_{S}$ preserves finite limits. Using Proposition 3.5.2.9, see that $X \mapsto |X|_{S}$ can be identified with the evaluation functor $X \mapsto X_{m}$, where $m$ is chosen so that there is an isomorphism of linearly ordered sets $[m] \simeq \pi _0( [0,1] \setminus S)$. $\square$

Remark 3.5.2.10. Let $X$ be a simplicial set, which we view as a functor from $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the category of sets. Then $X$ admits a canonical extension to a functor $\operatorname{Lin}_{\neq \emptyset }^{\operatorname{op}} \rightarrow \operatorname{Set}$, given on objects by the construction $(I = \{ i_0 < i_1 < \cdots < i_ n \} ) \mapsto X_{n}$. Let us write $X(I)$ for the value of this extension on an object $I \in \operatorname{Lin}_{\neq \emptyset }$. Arguing as in the proof of Theorem 3.5.2.1, we obtain a canonical bijection

\[ \varinjlim _{S} X( [0,1] \setminus S) \simeq \varinjlim _{S} |X|_{S} \xrightarrow {\sim } X, \]

where the (filtered) colimit is taken over the collection of all finite subsets $S \subseteq [0,1]$ containing $0$ and $1$.