Kerodon

Remark 2.5.1.6. Let $C_{\ast }$ be a chain complex with values in an abelian category $\operatorname{\mathcal{A}}$. In practice, the homology objects $\mathrm{H}_{\ast }(C)$ are often primary objects of interest, while the chain complex $C_{\ast }$ itself plays an ancillary role. The terminology of Definition 2.5.1.5 emphasizes this perspective: a chain map $f: C_{\ast } \rightarrow D_{\ast }$ which induces an isomorphism on homology should allow us to view the chain complexes $C_{\ast }$ and $D_{\ast }$ as “the same” for many purposes (this idea is the starting point for Verdier's theory of derived categories, which we will discuss in §).