2.5.5 Digression: The Homology of Simplicial Sets
Among the most useful invariants studied in algebraic topology are the singular homology groups $\mathrm{H}_{\ast }(X; \operatorname{\mathbf{Z}})$ of a topological space $X$. These are defined as the homology groups of the singular chain complex
\[ \cdots \xrightarrow {\partial } \mathrm{C}_{3}(X; \operatorname{\mathbf{Z}}) \xrightarrow {\partial } \mathrm{C}_{2}(X; \operatorname{\mathbf{Z}}) \xrightarrow {\partial } \mathrm{C}_1(X; \operatorname{\mathbf{Z}}) \xrightarrow {\partial } \mathrm{C}_0(X; \operatorname{\mathbf{Z}}), \]
where $\mathrm{C}_ n(X; \operatorname{\mathbf{Z}})$ denotes the free abelian group generated by the set $\operatorname{Hom}_{\operatorname{Top}}( | \Delta ^ n |, X )$ of singular $n$-simplices of $X$, and the boundary operator $\partial $ is given by the formula
\[ \partial : \mathrm{C}_ n(X; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{C}_{n-1}(X; \operatorname{\mathbf{Z}}) \quad \quad \partial (\sigma ) = \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i(\sigma ). \]
We can therefore view the passage from the topological space $X$ to its homology $\mathrm{H}_{\ast }(X; \operatorname{\mathbf{Z}})$ as proceeding in four stages:
We first extract from the topological space $X$ its singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ (Construction 1.2.2.2).
We then replace $\operatorname{Sing}_{\bullet }(X)$ by the simplicial abelian group $\operatorname{\mathbf{Z}}[ \operatorname{Sing}_{\bullet }(X) ]$, carrying each object $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the free abelian group $\operatorname{\mathbf{Z}}[ \operatorname{Sing}_{n}(X) ]$ generated by the set $\operatorname{Sing}_{n}(X)$.
We next regard the abelian groups $\{ \operatorname{\mathbf{Z}}[ \operatorname{Sing}_{n}(X) ] \} _{n \geq 0}$ as the terms of a chain complex $(C_{\ast }(X; \operatorname{\mathbf{Z}}), \partial )$, where the differential $\partial $ is given by the alternating sum of the face operators of the simplicial abelian group $\operatorname{\mathbf{Z}}[ \operatorname{Sing}_{\bullet }(X) ]$.
For each integer $n$, we define $\mathrm{H}_{n}(X; \operatorname{\mathbf{Z}})$ to be the $n$th homology group of the chain complex $(C_{\ast }(X; \operatorname{\mathbf{Z}}), \partial )$ (Definition 2.5.1.4).
In other words, the functor $X \mapsto \mathrm{H}_{n}(X; \operatorname{\mathbf{Z}})$ factors as a composition
\[ \operatorname{Top}\xrightarrow {\operatorname{Sing}_{\bullet }} \operatorname{Set_{\Delta }}\xrightarrow { \operatorname{\mathbf{Z}}[-]} \operatorname{Ab_{\Delta }}\xrightarrow {\mathrm{C}_{\ast }} \operatorname{Ch}(\operatorname{\mathbf{Z}}) \xrightarrow { \mathrm{H}_{n}} \operatorname{ Ab }, \]
where $\operatorname{Ab_{\Delta }}$ denotes the category of simplicial abelian groups and $\mathrm{C}_{\ast }: \operatorname{Ab_{\Delta }}\rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$ is given by the following:
Construction 2.5.5.1 (The Moore Complex). Let $A_{\bullet }$ be a semisimplicial abelian group (Definition 1.1.1.2). For each $n \geq 1$, we define a group homomorphism $\partial : A_{n} \rightarrow A_{n-1}$ by the formula
\[ \partial (\sigma ) = \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i(\sigma ), \]
where $d^{n}_{i}: A_{n} \rightarrow A_{n-1}$ is the $i$th face operator (Construction 1.1.1.4). For $n \geq 2$ and $\sigma \in A_{n}$, we compute
\begin{eqnarray*} \partial ^2( \sigma ) & = & \partial ( \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i(\sigma ) ) \\ & = & \sum _{i = 0}^{n} \sum _{j = 0}^{n-1} (-1)^{i+j} (d^{n-1}_{j} d^{n}_ i)(\sigma ) \\ & = & 0 \end{eqnarray*}
where the final equality follows from the identity $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ for $0 \leq i < j \leq n$ (see Remark 1.1.1.7). We let $\mathrm{C}_{\ast }(A)$ denote the chain complex of abelian groups given by
\[ \mathrm{C}_{n}(A) = \begin{cases} A_{n} & \text{ if } n \geq 0 \\ 0 & \text{otherwise,} \end{cases} \]
where the differential is given by $\partial $. We will refer to $\mathrm{C}_{\ast }(A)$ as the Moore complex of the semisimplicial abelian group $A_{\bullet }$.
If $A_{\bullet }$ is a simplicial abelian group, we let $\mathrm{C}_{\ast }(A)$ denote the Moore complex of the semisimplicial abelian group underlying $A_{\bullet }$ (Remark 1.1.1.3).
Definition 2.5.5.2 (Homology of Simplicial Sets). Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$ denote the simplicial abelian group freely generated by $S_{\bullet }$. We let $\mathrm{C}_{\ast }( S; \operatorname{\mathbf{Z}})$ denote the Moore complex of $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$. We will refer to $\mathrm{C}_{\ast }(S; \operatorname{\mathbf{Z}})$ as the chain complex of $S_{\bullet }$. For each integer $n$, we denote the $n$th homology group of $\mathrm{C}_{\ast }(S; \operatorname{\mathbf{Z}})$ by $\mathrm{H}_{n}(S; \operatorname{\mathbf{Z}})$ and refer to it as the $n$th homology group of $S$ (with coefficients in $\operatorname{\mathbf{Z}}$).
Example 2.5.5.3. Let $X$ be a topological space. Then the singular chain complex $\mathrm{C}_{\ast }(X; \operatorname{\mathbf{Z}})$ is the chain complex of the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$. In particular, the homology groups of the simplicial set $\operatorname{Sing}_{\bullet }(X)$ are the usual singular homology groups of the topological space $X$.
Example 2.5.5.4. Let $S_{\bullet } = \Delta ^{0}$ be the standard $0$-simplex. Then $S_{\bullet }$ is a simplicial set having a single simplex of each dimension. Consequently, the chain complex $\mathrm{C}_{\ast }( S; \operatorname{\mathbf{Z}})$ is given by $\operatorname{\mathbf{Z}}$ in each nonnegative degree. For $n > 0$, the differential $\operatorname{\mathbf{Z}}\simeq \mathrm{C}_{n}(S; \operatorname{\mathbf{Z}}) \xrightarrow {\partial } \mathrm{C}_{n-1}(S;\operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}$ is given by multiplication by the integer
\[ \sum _{i = 0}^{n} (-1)^{i} = \begin{cases} 0 & \text{ if $n$ is odd } \\ 1 & \text{ if $n$ is even, } \end{cases} \]
as indicated in the diagram
\[ \cdots \rightarrow \operatorname{\mathbf{Z}}\xrightarrow {0} \operatorname{\mathbf{Z}}\xrightarrow {1} \operatorname{\mathbf{Z}}\xrightarrow {0} \operatorname{\mathbf{Z}}\xrightarrow {1} \operatorname{\mathbf{Z}}\xrightarrow {0} \operatorname{\mathbf{Z}}\xrightarrow {1} \operatorname{\mathbf{Z}}\xrightarrow {0} \operatorname{\mathbf{Z}}. \]
It follows that the homology groups of $S_{\bullet }$ are given by
\[ \mathrm{H}_{n}( S; \operatorname{\mathbf{Z}}) = \begin{cases} \operatorname{\mathbf{Z}}& \text{ if $n = 0$ } \\ 0 & \text{ otherwise.} \end{cases} \]
Note that although the homology of the simplicial set $S_{\bullet } = \Delta ^0$ is concentrated in degree zero, the chain complex $\mathrm{C}_{\ast }(S; \operatorname{\mathbf{Z}})$ is not. Essentially, this is because $S_{\bullet }$ has degenerate simplices in each dimension $n > 0$ which do not contribute to its homology. This is a special case of a more general phenomenon.
Notation 2.5.5.5. Let $A_{\bullet }$ be a simplicial abelian group. For each $n \geq 0$, let $\mathrm{D}_{n}(A)$ denote the subgroup of $\mathrm{C}_{n}(A) = A_ n$ generated by the images of the degeneracy operators $\{ s^{n-1}_{i}: A_{n-1} \rightarrow A_ n \} _{0 \leq i \leq n-1}$. By convention, we set $\mathrm{D}_{n}(A) = 0$ for $n < 0$.
Proposition 2.5.5.6. Let $A_{\bullet }$ be a simplicial abelian group. For every positive integer $n$, the boundary operator $\partial : \mathrm{C}_{n}(A) \rightarrow \mathrm{C}_{n-1}(A)$ carries the subgroup $\mathrm{D}_{n}(A)$ into $\mathrm{D}_{n-1}(A)$. Consequently, we can regard $\mathrm{D}_{\ast }(A)$ as a subcomplex of the Moore complex $\mathrm{C}_{\ast }(A)$.
Proof.
Choose an element $\sigma \in \mathrm{D}_{n}(A)$; we wish to show that $\partial (\sigma )$ belongs to $\mathrm{D}_{n-1}(A)$. Without loss of generality, we may assume that $\sigma = s^{n-1}_{i}(\tau )$ for some $0 \leq i \leq n-1$ and some $\tau \in A_{n-1}$. We now compute
\begin{eqnarray*} \partial (\sigma ) & = & \sum _{ j =0 }^{n} (-1)^{j} d^{n}_ j(\sigma ) \\ & = & (\sum _{j=0}^{i-1} (-1)^{j} d^{n}_ j s^{n-1}_ i \tau ) + (-1)^{i} d^{n}_ i s^{n-1}_ i \tau + (-1)^{i+1} d^{n}_{i+1} s^{n-1}_ i \tau + (\sum _{j = i+2}^{n} (-1)^{j} d^{n}_ j s^{n-1}_ i \tau ) \\ & = & (\sum _{j < i} (-1)^{j} s^{n-2}_{i-1} d^{n-1}_ j \tau ) + (-1)^{i} \tau + (-1)^{i+1} \tau + (\sum _{j = i+2}^{n} (-1)^{j} s^{n-2}_{i} d^{n-1}_{j-1}\tau ) \\ & \in & \operatorname{im}( s^{n-2}_{i-1} ) + \operatorname{im}( s^{n-2}_ i ) \\ & \subseteq & \mathrm{D}_{n-1}(A). \end{eqnarray*}
$\square$
Construction 2.5.5.7 (The Normalized Moore Complex: First Construction). Let $A_{\bullet }$ be a simplicial abelian group. We let $\mathrm{N}_{\ast }(A)$ denote the chain complex given by the quotient $\mathrm{C}_{\ast }(A) / \mathrm{D}_{\ast }(A)$, where $\mathrm{C}_{\ast }(A)$ is the Moore complex of Construction 2.5.5.1 and $\mathrm{D}_{\ast }(A) \subseteq \mathrm{C}_{\ast }(A)$ is the subcomplex of Proposition 2.5.5.6. We will refer to $\mathrm{N}_{\ast }(A)$ as the normalized Moore complex of the simplicial abelian group $A_{\bullet }$.
Put more informally, the normalized Moore complex $\mathrm{N}_{\ast }(A)$ of a simplicial abelian group $A_{\bullet }$ is obtained the Moore complex $\mathrm{C}_{\ast }(A)$ by forming the quotient by degenerate simplices of $A_{\bullet }$.
Construction 2.5.5.9 (The Normalized Chain Complex of a Simplicial Set). Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$ be the simplicial abelian group freely generated by $S_{\bullet }$. We let $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ denote the normalized Moore complex of $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$. This chain complex can be described more concretely as follows:
For each integer $n \geq 0$, we can identify $\mathrm{N}_{n}(S)$ with the free abelian group generated by the set $S_{n}^{\mathrm{nd}}$ of nondegenerate $n$-simplices of $S_{\bullet }$.
The boundary map $\partial : \mathrm{N}_{n}(S) \rightarrow \mathrm{N}_{n-1}(S)$ is given by the formula
\[ \partial (\sigma ) = \sum _{i=0}^{n} (-1)^{i} \begin{cases} d^{n}_ i(\sigma ) & \text{ if $d^{n}_ i(\sigma )$ is nondegenerate } \\ 0 & \text{otherwise.} \end{cases} \]
We will refer to $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ as the normalized chain complex of the simplicial set $S_{\bullet }$.
Example 2.5.5.10. Let $S_{\bullet } = \Delta ^0$ be the standard $0$-simplex. Then the normalized chain complex $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ can be identified with abelian group $\operatorname{\mathbf{Z}}$, regarded as a chain complex concentrated in degree zero. Note that the calculation of Example 2.5.5.4 shows that the quotient map $\mathrm{C}_{\ast }(S; \operatorname{\mathbf{Z}}) \twoheadrightarrow \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ induces an isomorphism on homology.
Example 2.5.5.10 is a special case of the following:
Proposition 2.5.5.11. For every simplicial abelian group $A_{\bullet }$, the quotient map $\mathrm{C}_{\ast }(A) \twoheadrightarrow \mathrm{N}_{\ast }(A)$ is a quasi-isomorphism of chain complexes: that is, it induces an isomorphism on homology groups.
We will give the proof of Proposition 2.5.5.11 in ยง2.5.6 (see Proposition 2.5.6.21).
Example 2.5.5.13. Let $S_{\bullet }$ be a simplicial set. It follows from Proposition 2.5.5.11 that the quotient map $\mathrm{C}_{\ast }( S; \operatorname{\mathbf{Z}}) \twoheadrightarrow \mathrm{N}_{\ast }( S; \operatorname{\mathbf{Z}})$ induces an isomorphism on homology. In particular, the homology groups $\mathrm{H}_{\ast }(S; \operatorname{\mathbf{Z}})$ of the simplicial set $S_{\bullet }$ (in the sense of Definition 2.5.5.2) can be computed by means of the normalized chain complex $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$. This has various practical advantages. For example, if $S_{\bullet }$ is a simplicial set of dimension $\leq d$, then the chain complex $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ is concentrated in degrees $\leq d$. It follows that the homology groups $\mathrm{H}_{\ast }(S; \operatorname{\mathbf{Z}})$ are also concentrated in degrees $\leq d$, which is not immediately obvious from the definition (note that the chain complex $\mathrm{C}_{\ast }(S; \operatorname{\mathbf{Z}})$ is never concentrated in degrees $\leq d$, except in the trivial case where $S_{\bullet }$ is empty).
Example 2.5.5.14. Let $S_{\bullet } = \operatorname{N}_{\bullet }(Q)$ be the nerve of a partially ordered set $Q$. Suppose that $Q$ has a least element $e$, which determines a map of simplicial sets $i: \Delta ^{0} \rightarrow S_{\bullet }$ which is right inverse to the projection map $q: S_{\bullet } \rightarrow \Delta ^{0}$. Passing to normalized chain complexes, we obtain chain maps
\[ \widehat{i}: \operatorname{\mathbf{Z}}[0] \simeq \mathrm{N}_{\ast }(\Delta ^0; \operatorname{\mathbf{Z}}) \hookrightarrow \mathrm{N}_{\ast }( S_{\bullet }; \operatorname{\mathbf{Z}}) \quad \quad \widehat{q}: \mathrm{N}_{\ast }(S_{\bullet }; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }( \Delta ^{0}; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}[0]. \]
We claim that $\widehat{i}$ and $\widehat{q}$ are chain homotopy inverse to one another. In one direction, this is clear: the composition $\widehat{q} \circ \widehat{i}$ is equal to the identity. We complete the proof by constructing a chain homotopy from the composite map $\widehat{i} \circ \widehat{q}$ to the identity $\operatorname{id}$ on $\mathrm{N}_{\ast }(S_{\bullet }; \operatorname{\mathbf{Z}})$. This chain homotopy is given by a collection of maps $h_{m}: \mathrm{N}_{m}( S; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{m+1}( S; \operatorname{\mathbf{Z}})$, given on nondegenerate simplices by the construction
\[ ( q_0 < q_1 < \cdots < q_ m ) \mapsto \begin{cases} 0 & \text{ if } q_0 = e \\ ( e < q_0 < q_1 < \cdots < q_ m ) & \text{ otherwise. } \end{cases} \]
In particular, if $Q$ is a partially ordered set with a least element, then the homology groups of the nerve $S_{\bullet } = \operatorname{N}_{\bullet }(Q)$ are given by
\[ \mathrm{H}_{\ast }(S; \operatorname{\mathbf{Z}}) = \begin{cases} \operatorname{\mathbf{Z}}& \text{ if $\ast = 0$} \\ 0 & \text{ otherwise. }\end{cases} \]
Variant 2.5.5.15 (Relative Chain Complexes). Let $S_{\bullet }$ be a simplicial set and let $S'_{\bullet } \subseteq S_{\bullet }$ be a simplicial subset. Then we can identify the free simplicial abelian group $\operatorname{\mathbf{Z}}[ S'_{\bullet } ]$ with a simplicial subgroup of $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$. We let $\mathrm{C}_{\ast }( S,S'; \operatorname{\mathbf{Z}})$ and $\mathrm{N}_{\ast }(S,S'; \operatorname{\mathbf{Z}})$ denote the Moore complex and normalized Moore complex of the simplicial abelian group $\operatorname{\mathbf{Z}}[ S_{\bullet } ] / \operatorname{\mathbf{Z}}[ S'_{\bullet } ]$. By virtue of Proposition 2.5.5.11, these complexes have the same homology groups, which we denote by $\mathrm{H}_{\ast }(S,S'; \operatorname{\mathbf{Z}})$ and refer to as the relative homology groups of the pair $(S'_{\bullet } \subseteq S_{\bullet })$.