Definition 2.5.2.1 (Differential Graded Categories). A differential graded category $\operatorname{\mathcal{C}}$ consists of the following data:
- $(1)$
A collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$, whose elements we refer to as objects of $\operatorname{\mathcal{C}}$. We will often abuse notation by writing $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an element of $\operatorname{Ob}(\operatorname{\mathcal{C}})$.
- $(2)$
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a chain complex $(\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }, \partial )$. For each integer $n$, we refer to the elements of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$ as morphisms of degree $n$ from $X$ to $Y$.
- $(3)$
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and every pair of integers $m,n \in \operatorname{\mathbf{Z}}$, a function
\[ c_{Z,Y,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)_{m+n}, \]which we will refer to as the composition law. Given a pair of morphisms $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$ and $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n}$, we will often denote the image $c_{Z,Y,X}(g,f) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{m+n}$ by $g \circ f$ or $gf$.
- $(4)$
For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{0}$, which we will refer to as the identity morphism.
These data are required to satisfy the following conditions:
The composition law on $\operatorname{\mathcal{C}}$ is associative in the following sense: for every triple of elements $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{\ell }$, $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$, and $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n}$, we have an equality $h \circ (g \circ f) = (h \circ g) \circ f$ (in the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z)_{\ell +m+n}$).
The composition law on $\operatorname{\mathcal{C}}$ is unital on both sides: for every element $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$, we have $\operatorname{id}_{Y} \circ f = f = f \circ \operatorname{id}_{X}$.
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition maps $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{n} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)_{m+n}$ are bilinear and satisfy the Leibniz rule of Exercise 2.5.1.15. In other words, we have
\[ g \circ (f + f') = (g \circ f) + (g \circ f') \quad \quad (g + g') \circ f = (g \circ f) + (g' \circ f) \]\[ \partial (g \circ f) = (\partial g) \circ f + (-1)^{n} g \circ (\partial f). \]