1.2.3 The Geometric Realization of a Simplicial Set
Let $X$ be a topological space. By definition, $n$-simplices of the simplicial set $\operatorname{Sing}_{\bullet }(X)$ are continuous functions $| \Delta ^{n} | \rightarrow X$. Using Proposition 1.1.0.12, we obtain 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 observation, where we replace $\Delta ^{n}$ by an arbitrary simplicial set.
Definition 1.2.3.1. Let $S$ be a simplicial set and let $Y$ be a topological space. We will say that a map of simplicial sets $u: S\rightarrow \operatorname{Sing}_{\bullet }(Y)$ exhibits $Y$ as a geometric realization of $S$ if, for every topological space $X$, the composite map
\[ \operatorname{Hom}_{\operatorname{Top}}( Y, X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }( Y ), \operatorname{Sing}_{\bullet }(X) ) \xrightarrow { \circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{Sing}_{\bullet }(X) ) \]
is a bijection.
Example 1.2.3.2. 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 morphism of simplicial sets $u: \Delta ^{n} \rightarrow \operatorname{Sing}_{\bullet }( | \Delta ^ n | )$. It follows from Proposition 1.1.0.12 that $u$ exhibits the topological space $| \Delta ^{n} |$ as a geometric realization of the simplicial set $\Delta ^{n}$.
Notation 1.2.3.3. Let $S$ be a simplicial set. It follows immediately from the definitions that if there exists a map $u: S \rightarrow \operatorname{Sing}_{\bullet }(Y)$ which exhibits $Y$ as a geometric realization of $S$, then the topological space $Y$ is determined up to homeomorphism and depends functorially on $S$. We will emphasize this dependence by writing $| S |$ to denote a geometric realization of $S$. By virtue of Example 1.2.3.2, this is compatible with the convention of Notation 1.2.2.1 in the special case where $S = \Delta ^{n}$ is a standard simplex.
Every simplicial set admits a geometric realization:
Proposition 1.2.3.4. For every simplicial set $S$, there exists a topological space $Y$ and a map $u: S \rightarrow \operatorname{Sing}_{\bullet }(Y)$ which exhibits $Y$ as a geometric realization of $S$.
Corollary 1.2.3.5. The singular simplicial set functor $\operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint, given by the geometric realization construction $S \mapsto | S|$.
Our proof of Proposition 1.2.3.4 will make use of the following formal observation:
Lemma 1.2.3.6. Let $\operatorname{\mathcal{J}}$ be a small category equipped with a functor $S: \operatorname{\mathcal{J}}\rightarrow \operatorname{Set_{\Delta }}$. Suppose that, for each $J \in \operatorname{\mathcal{J}}$, the simplicial set $S(J)$ admits a geometric realization $| S(J) |$. Then the colimit $T = \varinjlim _{J \in \operatorname{\mathcal{J}}} S(J)$ also admits a geometric realization, given by the colimit $Y = \varinjlim _{J \in \operatorname{\mathcal{J}}} | S(J) |$ in the category of topological spaces.
Proof.
For each $J \in \operatorname{\mathcal{J}}$, choose a topological space $| S(J) |$ and a map $u_{J}: S(J) \rightarrow \operatorname{Sing}_{\bullet }( | S(J) | )$ which exhibits $| S(J) |$ as a geometric realization of $S(J)$. We can then amalgamate the composite maps
\[ S(J) \xrightarrow {u_ J} \operatorname{Sing}_{\bullet }( | S(J) | ) \rightarrow \operatorname{Sing}_{\bullet }( Y ) \]
to a single map of simplicial sets $u: T \rightarrow \operatorname{Sing}_{\bullet }( Y)$. We claim that $u$ exhibits $Y$ as a geometric realization of the simplicial set $T$. Let $X$ be any topological space; we wish to show that the composite map
\[ \operatorname{Hom}_{ \operatorname{Top}}( Y, X ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet } (Y), \operatorname{Sing}_{\bullet }(X) ) \xrightarrow {\circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( T, \operatorname{Sing}_{\bullet }(X) ) \]
is a bijection. This is clear, since this composite map can be written as an inverse limit of the bijections $\operatorname{Hom}_{ \operatorname{Top}}( | S(J) |, X ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S(J), \operatorname{Sing}_{\bullet }(X) )$ determined by the maps $u_ J$.
$\square$
It is possible to prove Proposition 1.2.3.4 in a completely formal way from Lemma 1.2.3.6, since every simplicial set can be presented as a colimit of simplices (see Proposition 1.2.3.15 below). However, we will instead give a less formal argument which yields some additional information about the structure of the geometric realization $|S|$. We begin by studying simplicial subsets of the standard simplex $\Delta ^{n}$.
Notation 1.2.3.7. 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 1.2.3.8. For each $n \geq 0$, the boundary $\operatorname{\partial \Delta }^{n}$ of Construction 1.1.4.10 is given by $\Delta ^{n}_{\operatorname{\mathcal{U}}}$, where $\operatorname{\mathcal{U}}$ is the collection of all nonempty proper subsets of $[n]$.
Exercise 1.2.3.9. 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 1.2.3.10. 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 1.2.3.2. It follows immediately from the definitions that the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{n}_{\operatorname{\mathcal{U}}_{01}} \ar [r] \ar [d] & \Delta ^{n}_{\operatorname{\mathcal{U}}_0} \ar [d] \\ \Delta ^{n}_{\operatorname{\mathcal{U}}_1} \ar [r] & \Delta ^{n}_{\operatorname{\mathcal{U}}} } \]
is a pushout square. By virtue of Lemma 1.2.3.6, we are reduced to proving that the diagram of topological spaces
\[ \xymatrix@R =50pt@C=50pt{ |\Delta ^{n}|_{\operatorname{\mathcal{U}}_{01}} \ar [r] \ar [d] & |\Delta ^{n}|_{\operatorname{\mathcal{U}}_0} \ar [d] \\ |\Delta ^{n}|_{\operatorname{\mathcal{U}}_1} \ar [r] & |\Delta ^{n}|_{\operatorname{\mathcal{U}}} } \]
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 1.2.3.11. Let $n$ be a nonnegative integer. Combining Example 1.2.3.8 with Proposition 1.2.3.10, 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
\[ \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: t_ j = 0 \text{ for some $j$} \} . \]
Proof of Proposition 1.2.3.4.
Let $S = S_{\bullet }$ be a simplicial set; we wish to show that $S$ admits a geometric realization $|S|$. We first show that for each $n \geq -1$, the $n$-skeleton $\operatorname{sk}_{n}( S )$ admits a geometric realization. The proof proceeds by induction on $n$, the case $n = -1$ being trivial (since $\operatorname{sk}_{-1}( S )$ is empty). Let $C$ denote the collection of nondegenerate $n$-simplices of $C$. we note that Proposition 1.1.4.12 provides a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in C }{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in C }{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}( S) \ar [r] & \operatorname{sk}_{n}( S). } \]
Combining our inductive hypothesis, Example 1.2.3.2, Example 1.2.3.11, and Lemma 1.2.3.6, we deduce that $\operatorname{sk}_{n}( S )$ admits a geometric realization $| \operatorname{sk}_{n} (S ) |$ which fits into a pushout diagram of topological spaces
\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in C }{\coprod } | \operatorname{\partial \Delta }^{n} | \ar [r] \ar [d] & \underset { \sigma \in C }{\coprod } |\Delta ^{n}| \ar [d] \\ |\operatorname{sk}_{n-1}( S )| \ar [r] & | \operatorname{sk}_{n}( S )|. } \]
Combining the equality $S = \bigcup _{n} \operatorname{sk}_{n} (S)$ of Remark 1.1.4.4 with Lemma 1.2.3.6, we deduce that the simplicial set $S$ also admits a geometric realization, given by the direct limit $\varinjlim _{n} | \operatorname{sk}_{n}( S ) |$.
$\square$
The proof of Proposition 1.2.3.4 also yields the following fact, which we will use often throughout this book:
Lemma 1.2.3.13. Let $\operatorname{\mathcal{U}}$ be a full subcategory of the category $\operatorname{Set_{\Delta }}$ of simplicial sets. Suppose that $\operatorname{\mathcal{U}}$ satisfies the following three conditions:
- $(1)$
Suppose we are given a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d] & Y \ar [d] \\ X' \ar [r] & Y', } \]
where $f$ is a monomorphism. If $X$, $Y$, and $X'$ belong to $\operatorname{\mathcal{U}}$, then $Y'$ belongs to $\operatorname{\mathcal{U}}$.
- $(2)$
Suppose we are given a sequence of monomorphisms of simplicial sets
\[ X(0) \hookrightarrow X(1) \hookrightarrow X(2) \hookrightarrow X(3) \hookrightarrow \cdots \]
If each $X(m)$ belongs to $\operatorname{\mathcal{U}}$, then the sequentual colimit $\varinjlim _{m} X(m)$ belongs to $\operatorname{\mathcal{U}}$.
- $(3)$
For each $n \geq 0$ and every set $I$, the coproduct $\coprod _{i \in I} \Delta ^{n}$ belongs to $\operatorname{\mathcal{U}}$.
Then every simplicial set belongs to $\operatorname{\mathcal{U}}$.
Proof.
Let $S$ be a simplicial set; we wish to show that $S$ belongs to $\operatorname{\mathcal{U}}$. By virtue of Remark 1.1.4.4, we can identify $S$ with the colimit $\varinjlim _{n} \operatorname{sk}_{n}( S )$. By virtue of $(2)$, it will suffice to show that each skeleton $\operatorname{sk}_{n}(S)$ belongs to $\operatorname{\mathcal{U}}$. We may therefore assume without loss of generality that $S$ has dimension $\leq n$, for some integer $n$. We proceed by induction on $n$. In the case $n=-1$, the simplicial set $S$ is empty and the desired result is a special case of $(3)$. To carry out the inductive step, we invoke Proposition 1.1.4.12 to choose a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in C }{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in C }{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}( S ) \ar [r] & S, } \]
where $C$ is the collection of nondegenerate $n$-simplices of $S$. By virtue of assumption $(1)$, it will suffice to show that the simplicial sets $\operatorname{sk}_{n-1}( S )$, $\underset { \sigma \in C}{\coprod } \operatorname{\partial \Delta }^{n}$, and $ \underset { \sigma \in C }{\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 assumption $(3)$.
$\square$
In Chapter 2, we will encounter a number of variants of the geometric realization construction $S \mapsto | S |$, which can be obtained from the following generalization of Corollary 1.2.3.5:
Proposition 1.2.3.15. 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 1.2.2.8. 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 $S \mapsto | S |^{Q}$.
Proof.
Let $S$ be a simplicial set; we wish to show that the functor
\[ \lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}\quad \quad C \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{Sing}^{Q}_{\bullet }(C) ) \]
is corepresentable by an object $|S|^{Q} \in \operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ admits small colimits, the collection of corepresentable functors from $\operatorname{\mathcal{C}}$ to $\operatorname{Set}$ is closed under the formation of small limits. Using Remark 1.1.3.13 (or Lemma 1.2.3.13), we can reduce to the case where $S = \Delta ^{n}$ is a standard simplex. In this case, the functor $\lambda $ is corepresented by the object $Q^{n} \in \operatorname{\mathcal{C}}$ (see Proposition 1.1.0.12).
$\square$
Proposition 1.2.3.18. Let $S$ be a simplicial set. The following conditions are equivalent:
- $(1)$
The geometric realization $| S |$ is a path-connected topological space.
- $(2)$
The geometric realization $| S |$ is a connected topological space.
- $(3)$
The simplicial set $S$ is connected, in the sense of Definition 1.2.1.6.
Proof.
The implication $(1) \Rightarrow (2)$ holds for any topological space. To prove that $(2) \Rightarrow (3)$, we observe that any decomposition $S \simeq S' \coprod S''$ into disjoint nonempty simplicial subsets determines a homeomorphism $| S | \simeq | S' | \coprod | S'' |$. We will complete the proof by showing that $(3) \Rightarrow (1)$. Let $\operatorname{{\bf \Delta }}_{S}$ denote the category of simplices of $S$ (Construction 1.1.3.9). We then have a commutative diagram of sets
\[ \xymatrix@R =50pt@C=50pt{ \varinjlim _{([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S} } | \Delta ^ n | \ar [r]^-{\sim } \ar [d] & | S | \ar [d] \\ \varinjlim _{ ([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S} } \pi _0( | \Delta ^ n | ) \ar [r] & \pi _0( | S | ), } \]
where the upper horizontal map is bijective (Remark 1.2.3.16) and the right vertical map is surjective. It follows that the lower horizontal map is also surjective. Since each of the topological spaces $| \Delta ^ n |$ is path connected, the colimit in the lower left can be identified with the set $\pi _0( S )$ (Remark 1.2.3.17). If $S$ is connected, the set $\pi _0(S)$ consists of a single element, so that $\pi _0( | S | )$ is also a singleton.
$\square$
Corollary 1.2.3.19. For every simplicial set $S$, we have a canonical bijection
\[ \pi _0(S ) \simeq \pi _0( | S | ). \]
Proof.
Writing $S$ as a disjoint union of connected components (Proposition 1.2.1.11, we can reduce to the case where $S$ is connected, in which case both sets have a single element (Proposition 1.2.3.18).
$\square$