Kerodon

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

Warning 2.5.6.23. Let $A_{\bullet }$ be a semisimplicial abelian group. Then we can still apply Construction 2.5.6.16 to define a subcomplex $\widetilde{ \mathrm{N} }_{\ast }(A)$ of the Moore complex $\mathrm{C}_{\ast }(A)$ (note that the definition of $\widetilde{ \mathrm{N} }_{\ast }(A)$ refers only to the face maps of $A_{\bullet }$). However, it is generally not true that the inclusion map $\widetilde{ \mathrm{N} }_{\ast }(A) \hookrightarrow \mathrm{C}_{\ast }(A)$ induces an isomorphism on homology unless $A_{\bullet }$ can be promoted to a simplicial abelian group.