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1.1.6 The Geometric Realization of Simplicial Set

Let $X$ be a topological space. By definition, $n$-simplices of the simplicial set $\operatorname{Sing}_{\bullet }(X)$ are continuous maps $| \Delta ^{n} | \rightarrow X$. This observation determines a bijection

\[ \operatorname{Hom}_{ \operatorname{Top}}( | \Delta ^{n} |, X ) \simeq \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \Delta ^{n}, \operatorname{Sing}_{\bullet }(X) ). \]

We now consider a generalization of this construction, which can be applied to simplicial sets other than $\Delta ^ n$.

Definition 1.1.6.1. Let $S_{\bullet }$ be a simplicial set and let $Y$ be a topological space. We will say that a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{Sing}_{\bullet }(Y)$ exhibits $Y$ as a geometric realization of $S_{\bullet }$ if, for every topological space $X$, the composite map

\[ \operatorname{Hom}_{\operatorname{Top}}( Y, X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }( Y ), \operatorname{Sing}_{\bullet }(X) ) \xrightarrow { \circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{Sing}_{\bullet }(X) ) \]

is bijective.

Example 1.1.6.2. For each $n \geq 0$, the identity map $\operatorname{id}: | \Delta ^{n} | \simeq | \Delta ^{n} |$ determines an $n$-simplex of the simplicial set $\operatorname{Sing}_{\bullet }( | \Delta ^{n} |)$, which we can identify with a map of simplicial sets $\Delta ^{n} \rightarrow \operatorname{Sing}_{\bullet }( | \Delta ^ n | )$ which exhibits $| \Delta ^{n} |$ as a geometric realization of $\Delta ^ n$.

Notation 1.1.6.3. Let $S_{\bullet }$ be a simplicial set. It follows immediately from the definitions that if there exists a map $u: S_{\bullet } \rightarrow \operatorname{Sing}_{\bullet }(Y)$ which exhibits $Y$ as a geometric realization of $S_{\bullet }$, then the topological space $Y$ is determined up to homeomorphism and depends functorially on $S_{\bullet }$. We will emphasize this dependence by writing $| S_{\bullet } |$ to denote a geometric realization of $S_{\bullet }$ (by virtue of Example 1.1.6.2, this is compatible with our existing notation in the case where $S_{\bullet }$ is the standard $n$-simplex).

Every simplicial set admits a geometric realization:

Proposition 1.1.6.4. For every simplicial set $S_{\bullet }$, there exists a topological space $Y$ and a map $u: S_{\bullet } \rightarrow \operatorname{Sing}_{\bullet }(Y)$ which exhibits $Y$ as a geometric realization of $S_{\bullet }$.

Corollary 1.1.6.5. The singular simplicial set functor $\operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint, given by the geometric realization construction $S_{\bullet } \mapsto | S_{\bullet } |$.

Our starting point is the following formal observation:

Lemma 1.1.6.6. Let $\operatorname{\mathcal{J}}$ be a small category equipped with a functor $F: \operatorname{\mathcal{J}}\rightarrow \operatorname{Set_{\Delta }}$, which we will denote by $(J \in \operatorname{\mathcal{J}}) \mapsto (F(J)_{\bullet } \in \operatorname{Set_{\Delta }})$. Let $S_{\bullet } = \varinjlim _{J \in \operatorname{\mathcal{J}}} F(J)_{\bullet }$ be a colimit of $F$. If each of the simplicial sets $F(J)_{\bullet }$ admits a geometric realization $| F(J)_{\bullet } |$, then $S_{\bullet }$ also admits a geometric realization, given by the colimit $Y = \varinjlim _{J \in \operatorname{\mathcal{J}}} | F(J)_{\bullet } |$.

Proof. For each $J \in \operatorname{\mathcal{J}}$, choose a map $u_ J: F(J)_{\bullet } \rightarrow \operatorname{Sing}_{\bullet }( | F(J)_{\bullet } | )$ which exhibits $| F(J)_{\bullet } |$ as a geometric realization of $F(J)_{\bullet }$. We can then amalgamate the composite maps

\[ F(I)_{\bullet } \xrightarrow {u_ I} \operatorname{Sing}_{\bullet }( | F(I)_{\bullet } | ) \rightarrow \operatorname{Sing}_{\bullet }( Y ) \]

to a single map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{Sing}_{\bullet }( Y)$. We claim that $u$ exhibits $Y$ as a geometric realization of the simplicial set $S_{\bullet }$. Let $X$ be any topological space; we wish to show that the composite map

\[ \operatorname{Hom}_{ \operatorname{Top}}( Y, X ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet } (Y), \operatorname{Sing}_{\bullet }(X) ) \xrightarrow {\circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{Sing}_{\bullet }(X) ) \]

is bijective. This is clear, since this composite map can be written as an inverse limit of the bijections $\operatorname{Hom}_{ \operatorname{Top}}( | F(J)_{\bullet } |, X ) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( F(J)_{\bullet }, \operatorname{Sing}_{\bullet }(X) )$ determined by $u_ J$. $\square$

It is possible to deduce Proposition 1.1.6.4 and Corollary 1.1.6.5 in a completely formal way from Lemma 1.1.6.6, since every simplicial set can be presented as a colimit of simplices (see Proposition 1.1.6.18 below). However, we will instead give a less direct argument which yields some additional information about the structure of the topological spaces $| S_{\bullet } |$. We begin by studying simplicial subsets of the standard simplex $\Delta ^{n}$.

Notation 1.1.6.7. Let $n \geq 0$ be an integer and let $\operatorname{\mathcal{U}}$ be a collection of nonempty subsets of $[n] = \{ 0, 1, \ldots , n \} $. We will say that $\operatorname{\mathcal{U}}$ is downward closed if $\emptyset \neq I \subseteq J \in \operatorname{\mathcal{U}}$ implies that $I \in \operatorname{\mathcal{U}}$. If this condition is satisfied, we let $\Delta ^{n}_{\operatorname{\mathcal{U}}}$ denote the simplicial subset of $\Delta ^{n}$ whose $m$-simplices are nondecreasing maps $\alpha : [m] \rightarrow [n]$ for which the image of $\alpha $ is an element of $\operatorname{\mathcal{U}}$. Similarly, we set

\[ | \Delta ^{n} |_{\operatorname{\mathcal{U}}} = \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: \{ i \in [n]: t_{i} \neq 0 \} \in \operatorname{\mathcal{U}}\} . \]

Example 1.1.6.8. For each $n \geq 0$, the boundary $\operatorname{\partial \Delta }^{n}$ of Construction 1.1.2.6 is given by $\Delta ^{n}_{\operatorname{\mathcal{U}}}$, where $\operatorname{\mathcal{U}}$ is the collection of all nonempty proper subsets of $[n]$.

Example 1.1.6.9. For $0 \leq i \leq n$, the horn $\Lambda ^{n}_{i}$ of Construction 1.1.2.9 is given by $\Delta ^{n}_{\operatorname{\mathcal{U}}}$, where $\operatorname{\mathcal{U}}$ is the collection of all nonempty subsets of $[n]$ which are distinct from $[n]$ and $[n] \setminus \{ i \} $.

Exercise 1.1.6.10. Show that every simplicial subset of the standard $n$-simplex $\Delta ^{n}$ has the form $\Delta ^{n}_{\operatorname{\mathcal{U}}}$, where $\operatorname{\mathcal{U}}$ is some (uniquely determined) downward closed collection of nonempty subsets of $[n]$.

Proposition 1.1.6.11. Let $n$ be a nonnegative integer and let $\operatorname{\mathcal{U}}$ be a downward closed collection of nonempty subsets of $[n]$. Then the canonical map $\Delta ^{n} \rightarrow \operatorname{Sing}_{\bullet }( | \Delta ^{n} | )$ restricts to a map of simplicial sets $f_{\operatorname{\mathcal{U}}}: \Delta ^ n_{\operatorname{\mathcal{U}}} \rightarrow \operatorname{Sing}_{\bullet } ( | \Delta ^{n} |_{\operatorname{\mathcal{U}}} )$, which exhibits the topological space $| \Delta ^{n} |_{\operatorname{\mathcal{U}}}$ as a geometric realization of $\Delta ^{n}_{\operatorname{\mathcal{U}}}$.

Proof. We proceed by induction on the cardinality of $\operatorname{\mathcal{U}}$. If $\operatorname{\mathcal{U}}$ is empty, then the simplicial set $\Delta ^{n}_{\operatorname{\mathcal{U}}}$ and the topological space $| \Delta ^{n} |_{\operatorname{\mathcal{U}}}$ are both empty, in which case there is nothing to prove. We may therefore assume that $\operatorname{\mathcal{U}}$ is nonempty. Choose some $S \in \operatorname{\mathcal{U}}$ whose cardinality is as large as possible. Set

\[ \operatorname{\mathcal{U}}_0 = \operatorname{\mathcal{U}}\setminus \{ S \} \quad \quad \operatorname{\mathcal{U}}_1 = \{ T \subseteq S: T \neq \emptyset \} \quad \quad \operatorname{\mathcal{U}}_{01} = \operatorname{\mathcal{U}}_0 \cap \operatorname{\mathcal{U}}_1. \]

Our inductive hypothesis implies that the maps $f_{\operatorname{\mathcal{U}}_0}$ and $f_{ \operatorname{\mathcal{U}}_{01} }$ exhibit $| \Delta ^{n} |_{\operatorname{\mathcal{U}}_0}$ and $| \Delta ^{n} |_{\operatorname{\mathcal{U}}_{01}}$ as geometric realizations of $\Delta ^{n}_{\operatorname{\mathcal{U}}_0}$ and $\Delta ^{n}_{ \operatorname{\mathcal{U}}_{01} }$, respectively. Moreover, if $S = \{ i_0 < i_1 < \cdots < i_ m \} \subseteq [n]$, then we can identify $f_{\operatorname{\mathcal{U}}_1}$ with the tautological map $\Delta ^{m} \rightarrow \operatorname{Sing}_{\bullet } ( | \Delta ^{m} | )$, so that $f_{ \operatorname{\mathcal{U}}_1}$ exhibits $| \Delta ^{n} |_{\operatorname{\mathcal{U}}_1}$ as a geometric realization of $\Delta ^{n}_{\operatorname{\mathcal{U}}_1}$ by virtue of Example 1.1.6.2. It follows immediately from the definitions that the diagram of simplicial sets

\[ \xymatrix { \Delta ^{n}_{\operatorname{\mathcal{U}}_{01}} \ar [r] \ar [d] & \Delta ^{n}_{\operatorname{\mathcal{U}}_0} \ar [d] \\ \Delta ^{n}_{\operatorname{\mathcal{U}}_1} \ar [r] & \Delta ^{n}_{\operatorname{\mathcal{U}}} } \]

is a pushout square. By virtue of Lemma 1.1.6.6, we are reduced to proving that the diagram of topological spaces

\[ \xymatrix { |\Delta ^{n}|_{\operatorname{\mathcal{U}}_{01}} \ar [r] \ar [d] & |\Delta ^{n}|_{\operatorname{\mathcal{U}}_0} \ar [d] \\ |\Delta ^{n}|_{\operatorname{\mathcal{U}}_1} \ar [r] & |\Delta ^{n}|_{\operatorname{\mathcal{U}}} } \]

is also a pushout square. This is clear, since $| \Delta ^{n} |_{\operatorname{\mathcal{U}}_0}$ and $| \Delta ^{n} |_{\operatorname{\mathcal{U}}_1}$ are closed subsets of $| \Delta ^{n} |$ whose union is $| \Delta ^{n} |_{\operatorname{\mathcal{U}}}$ and whose intersection is $| \Delta ^{n} |_{\operatorname{\mathcal{U}}_{01} }$. $\square$

Example 1.1.6.12. Let $n$ be a nonnegative integer. Combining Example 1.1.6.8 with Proposition 1.1.6.11, we see that the inclusion map $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^{n}$ induces a homeomorphism from $| \operatorname{\partial \Delta }^{n} |$ to the boundary of the topological $n$-simplex $| \Delta ^{n} |$, given by

\[ \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: t_ j = 0 \text{ for some $j$} \} . \]

Example 1.1.6.13. Let $0 \leq i \leq n$. Combining Example 1.1.6.9 with Proposition 1.1.6.11, we see that the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^{n}$ induces a homeomorphism from $| \Lambda ^{n}_{i} |$ to the subset of $| \Delta ^{n} |$ given by

\[ \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: t_ j = 0 \text{ for some $j \neq i$} \} . \]

Proof of Proposition 1.1.6.4. Let $S_{\bullet }$ be a simplicial set. We first show that for each $n \geq -1$, the skeleton $\operatorname{sk}_{n}( S_{\bullet } )$ admits a geometric realization. The proof proceeds by induction on $n$, the case $n = -1$ being trivial (since $\operatorname{sk}_{-1}( S_{\bullet } )$ is empty). Let $S_{n}^{\mathrm{nd} }$ denote the collection of nondegenerate $n$-simplices of $S$. we note that Proposition 1.1.3.11 provides a pushout diagram

\[ \xymatrix { \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}( S_{\bullet } ) \ar [r] & \operatorname{sk}_{n}( S_{\bullet } ). } \]

Combining our inductive hypothesis, Example 1.1.6.2, Example 1.1.6.12, and Lemma 1.1.6.6, we deduce that $\operatorname{sk}_{n}( S_{\bullet } )$ admits a geometric realization $| \operatorname{sk}_{n} (S_{\bullet } ) |$ which fits into a pushout diagram of topological spaces

\[ \xymatrix { \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } | \operatorname{\partial \Delta }^{n} | \ar [r] \ar [d] & \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } |\Delta ^{n}| \ar [d] \\ |\operatorname{sk}_{n-1}( S_{\bullet } )| \ar [r] & | \operatorname{sk}_{n}( S_{\bullet } )|. } \]

Combining the equality $S_{\bullet } = \bigcup _{n} \operatorname{sk}_{n} (S_{\bullet } )$ of Remark 1.1.3.6 with Lemma 1.1.6.6, we deduce that the simplicial set $S_{\bullet }$ also admits a geometric realization, given by the direct limit $\varinjlim _{n} | \operatorname{sk}_{n}( S_{\bullet } ) |$. $\square$

Remark 1.1.6.14. The proof of Proposition 1.1.6.4 shows that the geometric realization $| S_{\bullet } |$ of a simplicial set $S_{\bullet }$ has a canonical realization as a CW complex, having one cell of dimension $n$ for each nondegenerate $n$-simplex $\sigma $ of $S_{\bullet }$; this cell is can be described explicitly as the image of the map

\[ | \Delta ^{n} | \setminus | \operatorname{\partial \Delta }^{n} | \hookrightarrow | \Delta ^{n} | \xrightarrow { \sigma } | S_{\bullet } |. \]

The proof of Proposition 1.1.6.4 also yields the following fact, which we will use repeatedly throughout this book:

Lemma 1.1.6.15. Let $\operatorname{\mathcal{U}}$ be a full subcategory of the category $\operatorname{Set_{\Delta }}$ of simplicial sets. If $\operatorname{\mathcal{U}}$ is closed under small colimits and contains the standard $n$-simplex $\Delta ^ n$ for each $n \geq 0$, then $\operatorname{\mathcal{U}}= \operatorname{Set_{\Delta }}$.

Remark 1.1.6.16. We can state Lemma 1.1.6.15 more informally as follows: the category $\operatorname{Set_{\Delta }}$ of simplicial sets is generated, under small colimits, by objects of the form $\Delta ^ n$. In fact, one can say more: it is freely generated (under small colimits) by the essential image of the Yoneda embedding

\[ \operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Set_{\Delta }}\quad \quad [n] \mapsto \Delta ^ n. \]

This is a general fact about presheaf categories, which we will return to in §.

We give two proofs of Lemma 1.1.6.15: one using the strategy of Proposition 1.1.6.4, and another using the formal properties of presheaf categories.

First Proof of Lemma 1.1.6.15. Set $S_{\bullet }$ be a simplicial set; we wish to show that $S_{\bullet }$ belongs to $\operatorname{\mathcal{U}}$. By virtue of Remark 1.1.3.6, we can identify $S_{\bullet }$ with the colimit $\varinjlim _{n} \operatorname{sk}_{n}( S_{\bullet } )$. Consequently, it will suffice to show that each skeleton $\operatorname{sk}_{n}(S_{\bullet })$ belongs to $\operatorname{\mathcal{U}}$. We may therefore assume without loss of generality that $S_{\bullet }$ has dimension $\leq n$, for some integer $n$. We proceed by induction on $n$, the case $n = -1$ being trivial (note that the empty simplicial set is the colimit of an empty diagram). To carry out the inductive step, we invoke Proposition 1.1.3.11 to choose a pushout diagram

\[ \xymatrix { \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}( S_{\bullet } ) \ar [r] & S_{\bullet } . } \]

It will therefore suffice to show that the simplicial sets $|\operatorname{sk}_{n-1}( S_{\bullet } )| $, $\underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } | \operatorname{\partial \Delta }^{n} |$, and $ \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } |\Delta ^{n}|$ belong to $\operatorname{\mathcal{U}}$. In the first two cases, this follows from our inductive hypothesis. In the third, it follows from our assumption that $\Delta ^ n$ belongs to $\operatorname{\mathcal{U}}$ and that $\operatorname{\mathcal{U}}$ is closed under coproducts. $\square$

Second Proof of Lemma 1.1.6.15. Let $S_{\bullet }$ be a simplicial set. We define a category $\operatorname{{\bf \Delta }}_{S}$ as follows:

  • The objects of $\operatorname{{\bf \Delta }}_{S}$ are pairs $([n], \sigma )$, where $[n]$ is an object of $\operatorname{{\bf \Delta }}$ and $\sigma $ is an $n$-simplex of $S_{\bullet }$.

  • A morphism from $([n], \sigma )$ to $([n'], \sigma ')$ in the category $\operatorname{{\bf \Delta }}_{S}$ is a nondecreasing function $f: [n] \rightarrow [n']$ with the property that the induced map $S_{n'} \rightarrow S_{n}$ carries $\sigma '$ to $\sigma $.

Via the Yoneda embedding $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Set_{\Delta }}$, we can identify $\operatorname{{\bf \Delta }}_{S}$ with the category whose objects are simplicial sets of the form $\Delta ^ n$ (for some $n \geq 0$), which are equipped with a map of simplicial sets $\Delta ^ n \rightarrow S_{\bullet }$. In particular, we have a canonical map of simplicial sets $\varinjlim _{ ([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S} } \Delta ^ n \rightarrow S_{\bullet }$. To prove Lemma 1.1.6.15, it suffices to observe that this map is an isomorphism. This is an elementary calculation which we leave to the reader (see § for more details). $\square$

Remark 1.1.6.17. Each of our proofs of Lemma 1.1.6.15 gives additional information that the other does not. Our first proof shows that every simplicial set $S_{\bullet }$ can be built as a colimit of standard simplices in a very specific way: namely, by forming pushouts along boundary inclusions $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ (for a more precise assertion, see the proof of Proposition 1.4.5.12). This extra information was used in the proof of Proposition 1.1.6.4 to show that the geometric realization $| S_{\bullet } |$ is a CW complex (and not merely a topological space which is colimit of disks). On the other hand, our second proof shows that every simplicial set $S_{\bullet }$ can be built in a single step as the colimit of a diagram of standard simplices (which can be chosen in a specific, canonical way).

In Chapter , we will encounter a number of variants of the geometric realization construction $S_{\bullet } \mapsto | S_{\bullet } |$, which arise as special cases of the following:

Proposition 1.1.6.18. Let $\operatorname{\mathcal{C}}$ be a category, let $Q^{\bullet }$ be a cosimplicial object of $\operatorname{\mathcal{C}}$, and let $\operatorname{Sing}^{Q}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the functor of Variant 1.1.5.3. If the category $\operatorname{\mathcal{C}}$ admits small colimits, then the functor $\operatorname{Sing}^{Q}_{\bullet }$ admits a left adjoint $\operatorname{Set_{\Delta }}\rightarrow \operatorname{\mathcal{C}}$, which we will denote by $S_{\bullet } \mapsto | S_{\bullet } |^{Q}$.

Proof. Let us say that a simplicial set $S_{\bullet }$ is good if the functor

\[ (C \in \operatorname{\mathcal{C}}) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{Sing}^{Q}_{\bullet }(C) ) \]

is corepresentable by an object of the category $\operatorname{\mathcal{C}}$ (in which case we denote the corepresenting object by $|S_{\bullet }|^{Q}$). It follows from Yoneda's lemma that the standard $n$-simplex $\Delta ^ n$ is good for each $n \geq 0$, with $| \Delta ^{n} |^{Q} \simeq Q^{n}$. If $\operatorname{\mathcal{C}}$ admits small colimits, then the proof of Lemma 1.1.6.6 shows that the collection of good simplicial sets is closed under small colimits. It now suffices to observe that every simplicial set $S_{\bullet }$ can be written as a small colimit of simplices (Lemma 1.1.6.15). $\square$