Variant 7.1.2.18 (Cohomology as a Limit). Let $A$ be an abelian group, let $S$ be a simplicial set, and let
denote the normalized cochain complex of $S$ with coefficients in $A$. Applying Proposition 7.1.2.16 to the differential graded category $\operatorname{Ch}(\operatorname{\mathbf{Z}})^{\operatorname{op}}$ (and using Remark 7.1.2.4), we see that the tautological chain map $\mathrm{N}_{\ast }( S; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}})}( \mathrm{N}^{\ast }(S;A), A )_{\ast }$ induces a morphism of simplicial sets
which exhibits $\mathrm{N}^{\ast }(S;A)$ as a power of $A$ by $S$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{Ch}(\operatorname{\mathbf{Z}}) )$. In particular, $\mathrm{N}^{\ast }(S;A)$ can be viewed as a limit of the constant diagram $S \rightarrow \{ A\} \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{Ch}(\operatorname{\mathbf{Z}}) )$.