$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Theorem admits many generalizations. For example, if $\operatorname{\mathcal{A}}$ is an abelian category (Definition ), then a variant of Construction supplies an equivalence of categories

\[ \mathrm{N}_{\ast }: \{ \text{Simplicial objects of $\operatorname{\mathcal{A}}$} \} \rightarrow \operatorname{Ch}(\operatorname{\mathcal{A}})_{\geq 0}, \]

where $\operatorname{Ch}(\operatorname{\mathcal{A}})_{\geq 0}$ denotes the category of (nonnegatively graded) chain complexes with values in $\operatorname{\mathcal{A}}$ (see Theorem ). For more general categories $\operatorname{\mathcal{A}}$, one can think of the category of simplicial objects $\operatorname{\mathcal{A}}_{\Delta } = \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{A}})$ as a replacement for the category of chain complexes $\operatorname{Ch}(\operatorname{\mathcal{A}})_{\geq 0}$, which is better behaved in “non-additive” situations.