Kerodon

Variant 2.5.5.15 (Relative Chain Complexes). Let $S_{\bullet }$ be a simplicial set and let $S'_{\bullet } \subseteq S_{\bullet }$ be a simplicial subset. Then we can identify the free simplicial abelian group $\operatorname{\mathbf{Z}}[ S'_{\bullet } ]$ with a simplicial subgroup of $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$. We let $\mathrm{C}_{\ast }( S,S'; \operatorname{\mathbf{Z}})$ and $\mathrm{N}_{\ast }(S,S'; \operatorname{\mathbf{Z}})$ denote the Moore complex and normalized Moore complex of the simplicial abelian group $\operatorname{\mathbf{Z}}[ S_{\bullet } ] / \operatorname{\mathbf{Z}}[ S'_{\bullet } ]$. By virtue of Proposition 2.5.5.11, these complexes have the same homology groups, which we denote by $\mathrm{H}_{\ast }(S,S'; \operatorname{\mathbf{Z}})$ and refer to as the relative homology groups of the pair $(S'_{\bullet } \subseteq S_{\bullet })$.