Remark 2.5.6.14. When regarded as a functor from $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ to the category of simplicial sets, the functor $M_{\ast } \mapsto \mathrm{K}( M_{\ast } )$ fits into the paradigm of Variant 1.2.2.8: it is the functor $\operatorname{Sing}^{Q}_{\bullet }$ associated to the cosimplicial chain complex
\[ Q: \operatorname{{\bf \Delta }}\rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}}) \quad \quad [n] \mapsto \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}). \]