Notation 2.5.1.3 (Cycles and Boundaries). Let $\operatorname{\mathcal{A}}$ be an abelian category (Definition ) and let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$. For each integer $n$, we let $\mathrm{Z}_{n}(C)$ denote the kernel of the boundary operator $\partial : C_ n \rightarrow C_{n-1}$, and $\mathrm{B}_ n(C)$ the image of the boundary operator $\partial : C_{n+1} \rightarrow C_ n$. We regard $\mathrm{Z}_ n(C)$ and $\mathrm{B}_{n}(C)$ as subobjects of $C_ n$. Note that we have $\mathrm{B}_{n}(C) \subseteq \mathrm{Z}_{n}(C)$ (this is a reformulation of the identity $\partial ^2 = 0$).

In the special case where $\operatorname{\mathcal{A}}= \operatorname{ Ab }$ is the category of abelian groups, we will refer to the elements of $C_{n}$ as *$n$-chains* of $C_{\ast }$, to the elements of $\mathrm{Z}_{n}(C)$ as *$n$-cycles* of $C_{\ast }$, and to the elements of $\mathrm{B}_{n}(C)$ as *$n$-boundaries* of $C_{\ast }$.