In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded ${\displaystyle \mathbb {Z} }$-module.

In detail, this means that ${\displaystyle \operatorname {Hom} (A,B)}$, the morphisms from any object A to another object B of the category is a direct sum

${\displaystyle \bigoplus _{n\in \mathbb {Z} }\operatorname {Hom} _{n}(A,B)}$

and there is a differential d on this graded group, i.e., for each n there is a linear map

${\displaystyle d\colon \operatorname {Hom} _{n}(A,B)\rightarrow \operatorname {Hom} _{n+1}(A,B)}$,

which has to satisfy ${\displaystyle d\circ d=0}$. This is equivalent to saying that ${\displaystyle \operatorname {Hom} (A,B)}$ is a cochain complex. Furthermore, the composition of morphisms ${\displaystyle \operatorname {Hom} (A,B)\otimes \operatorname {Hom} (B,C)\rightarrow \operatorname {Hom} (A,C)}$ is required to be a map of complexes, and for all objects A of the category, one requires ${\displaystyle d(\operatorname {id} _{A})=0}$.