Kerodon

Example 2.5.2.6 (Differential Graded Algebras). A differential graded algebra is a (not necessarily commutative) graded ring $A_{\ast } = \{ A_ n \} _{n \in \operatorname{\mathbf{Z}}}$ equipped with a differential $\partial : A_{\ast } \rightarrow A_{\ast -1}$ satisfying $\partial ^2 = 0$ and the Leibniz rule $\partial (x \cdot y) = (\partial x) \cdot y + (-1)^{m} x \cdot (\partial y)$ for $x \in A_{m}$ and $y \in A_{n}$. If $\operatorname{\mathcal{C}}$ is a differential graded category containing an object $X$, then the composition law on $\operatorname{\mathcal{C}}$ endows the chain complex $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)_{\ast } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\ast }$ with the structure of a differential graded algebra. Conversely, for every differential graded algebra $(A_{\ast }, \partial )$, there is a unique differential graded category $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X \} $. In other words, the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{End}_{\operatorname{\mathcal{C}}}(X)_{\ast }$ induces a bijective correspondence