Kerodon

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Example 2.5.6.5. Let $M_{\ast }$ be a chain complex. Then we have canonical isomorphisms

\[ \mathrm{K}_{0}( M_{\ast } ) = \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}})}( \mathrm{N}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}}), M_{\ast } ) = \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}})}( \operatorname{\mathbf{Z}}[0], M_{\ast }) = \mathrm{Z}_0( M ). \]

In other words, we can identify vertices of the simplicial set $\mathrm{K}(M_{\ast })$ with $0$-cycles of the chain complex $M_{\ast }$.