Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 3.1.4.8. Let $X$ be a simplicial set, let $M_{\ast }$ be a chain complex of abelian groups, and let $\mathrm{K}(M_{\ast })$ denote the associated Eilenberg-MacLane space (Construction 2.5.6.3). Suppose we are given a pair of morphisms $f,g: X \rightarrow \mathrm{K}(M_{\ast })$ in the category of simplicial sets, which we can identify with morphisms $f',g': \mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$ in the category of chain complexes (Corollary 2.5.6.13); here $\mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}})$ denotes the normalized Moore complex of $X$ (Construction 2.5.5.9). The following conditions are equivalent:

$(1)$

The morphisms $f$ and $g$ are homotopic, in the sense of Definition 3.1.4.1.

$(2)$

The chain maps $f'$ and $g'$ are chain homotopic, in the sense of Definition 2.5.0.5.

To prove this, we note that $(1)$ is equivalent to the assertion that there is a homotopy from $f$ to $g$ (since $\mathrm{K}(M_{\ast })$ is a Kan complex; see Remark 2.5.6.4): that is, a map of simplicial sets $h: \Delta ^{1} \times X \rightarrow \mathrm{K}(M_{\ast })$ satisfying $h|_{ \{ 0\} \times X} = f$ and $h|_{ \{ 1\} \times X} = g$. By virtue of Corollary 2.5.6.13, this is equivalent to the existence of a chain map $h': \mathrm{N}_{\ast }( \Delta ^1 \times X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$ which is compatible with $f'$ and $g'$. For any such chain map $h'$, the composition

\[ \mathrm{N}_{\ast }( \Delta ^1 ) \boxtimes \mathrm{N}_{\ast }( X; \operatorname{\mathbf{Z}}) \xrightarrow { \mathrm{EZ} } \mathrm{N}_{\ast }( \Delta ^1 \times X) \xrightarrow {h'} M_{\ast } \]

determines a chain homotopy from $f'$ to $g'$ (where $\mathrm{EZ}$ denotes the Eilenberg-Zilber homomorphism of Example 2.5.7.12). More explicitly, this chain homotopy is given by the map of graded abelian groups

\[ \mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast +1} \quad \quad \sigma \mapsto h'( \tau \triangledown \sigma ), \]

where $\tau $ is the generator of $\mathrm{N}_{1}( \Delta ^1 ) \simeq \operatorname{\mathbf{Z}}$ and $\triangledown $ is the shuffle product of Construction 2.5.7.9. This proves that $(1)$ implies $(2)$. Conversely, if $(2)$ is satisfied, then there exists a chain map $u: \mathrm{N}_{\ast }( \Delta ^1 ) \boxtimes \mathrm{N}_{\ast }( X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$ compatible with $f'$ and $g'$, and we can verify $(1)$ by taking $h'$ to be the composite map

\[ \mathrm{N}_{\ast }( \Delta ^1 \times X; \operatorname{\mathbf{Z}}) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }( \Delta ^1 ) \boxtimes \mathrm{N}_{\ast }( X; \operatorname{\mathbf{Z}}) \xrightarrow {u} M_{\ast } \]

where $\mathrm{AW}$ is the Alexander-Whitney homomorphism of Construction 2.5.8.6.