Example 7.6.2.21 (Homology as a Colimit). Let $\operatorname{\mathcal{C}}= \operatorname{Ch}(\operatorname{\mathbf{Z}})$ denote the category of chain complexes of abelian groups, which we regard as a differential graded category (see Example 2.5.2.5). Let $A$ be an abelian group, and let us abuse notation by identifying $A$ with its image in $\operatorname{\mathcal{C}}$ (by regarding it as a chain complex concentrated in degree zero). For every simplicial set $S$, let $\mathrm{N}_{\ast }(S; A)$ denote the normalized chain complex of $S$ with coefficients in $A$, given by the tensor product $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \boxtimes A$. Then the tautological map
satisfies condition $(2)$ of Proposition 7.6.2.20: in fact, for every object $M_{\ast } \in \operatorname{\mathcal{C}}$, precomposition with $f$ induces an isomorphism of chain complexes
It follows that the induced map $S \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( A, \mathrm{N}_{\ast }(S; A) )$ exhibits $\mathrm{N}_{\ast }(S;A)$ as a tensor product of $A$ by $S$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$. In particular, the chain complex $\mathrm{N}_{\ast }(S;A)$ can be viewed as a colimit of the constant diagram $S \rightarrow \{ A \} \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{Ch}(\operatorname{\mathbf{Z}}) )$.