Notation 2.5.0.2. 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

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 $C_{\ast }$).