Example 10.2.1.7 (Simplicial Abelian Groups). Let $\operatorname{ Ab }$ denote the category of abelian groups. By virtue of the Dold-Kan correspondence (Theorem 2.5.6.1), there is an equivalence of categories $\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{ Ab }) \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})_{\geq 0}$, which carries each simplicial abelian group $A_{\bullet }$ to its normalized Moore complex
Under this equivalence, the coequalizer of the pair of face operators $d^{1}_0, d^{1}_1: A_1 \rightrightarrows A_0$ can be identified with the $0$th homology group $\mathrm{H}_{0}( \mathrm{N}_{\ast }(A) ) = \operatorname{coker}( \partial : \mathrm{N}_{1}(A) \rightarrow \mathrm{N}_{0}(A) )$, or alternatively with the homotopy group $\pi _0( A_{\bullet } )$ (see Exercise 3.2.2.22). Using Exercise 10.2.1.6 we see that the $\pi _0(A)$ can be regarded as a geometric realization of $A_{\bullet }$ in the category of abelian groups. In particular, the forgetful functor $\operatorname{ Ab }\rightarrow \operatorname{Set}$ commutes with the formation of geometric realizations (this is a special case of a more general phenomenon, which we will return to in ยง
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