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

Notation Let $\operatorname{\mathcal{A}}$ be an additive category. Then a chain complex $(C_{\ast }, \partial )$ with values in $\operatorname{\mathcal{A}}$ can be graphically represented by a diagram

\[ \cdots \rightarrow C_2 \xrightarrow { \partial _2} C_1 \xrightarrow { \partial _1} C_0 \xrightarrow { \partial _0 } C_{-1} \xrightarrow { \partial _{-1} } C_{-2} \rightarrow \cdots \]

in which each successive composition is equal to zero. We will generally abuse terminology by identifying $( C_{\ast }, \partial )$ with the underlying collection $C_{\ast } = \{ C_ n \} _{n \in \operatorname{\mathbf{Z}}}$, which we will refer to as a graded object of $\operatorname{\mathcal{A}}$. We view $\partial = \{ \partial _{n} \} _{n \in \operatorname{\mathbf{Z}}}$ as an endomorphism of $C_{\ast }$ which is homogeneous of degree $-1$, which we refer to as the differential or the boundary operator of the chain complex $C_{\ast }$. We will generally abuse notation by omitting the subscript from the expression $\partial _ n$; that is, we denote each of the boundary operators $C_{n} \rightarrow C_{n-1}$ by the same symbol $\partial $ (or $\partial _{C}$, when we need to emphasize its association with the particular chain complex $C_{\ast }$).