Differential graded category

In mathematics, especially homological algebra, a differential graded category or DG category for short, is a category whose morphism sets are endowed with the additional structure of a differential graded Z-module.

In detail, this means that Hom(A,B), the morphisms from any object A to another object B of the category is a direct sum $\oplus_{n \in \mathbf Z}Hom_n(A,B)$ and there is a differential d on this graded group, i.e. for all n a linear map $d: Hom_n(A,B) \rightarrow Hom_{n+1}(A,B)$, which has to satisfy $d \circ d = 0$. This is equivalent to saying that Hom(A,B) is a cochain complex. Furthermore, the composition of morphisms $Hom(A,B) \otimes Hom(B,C) \rightarrow Hom(A,C)$ is required to be a map of complexes, and for all objects A of the category, one requires d(id_A) = 0.

Examples

• Any additive category may be considered to be a DG-category by imposing the trivial grading (i.e. all Hom_n( − , − ) vanish for n ≠ 0) and trivial differential (d = 0).
• A little bit more sophisticated is the category of complexes $C(\mathcal A)$over an additive category $\mathcal A$. By definition,

$Hom_{C(\mathcal A), n} (A, B)$ is the group of maps $A \rightarrow B[n]$ which do not need to respect the differentials of the complexes A and B, i.e. $Hom_{C(\mathcal A), n} (A, B) = \Pi_{l \in \mathbf Z} Hom(A_l, B_{l+n})$. The differential of such a morphism $f = (f_l : A_l \rightarrow B_{l+n})$ of degree n is defined to be $f_{l+1} \circ d_A + (-1)^{n+1} d_B \circ f_l$, where d_A,d_B are the differentials of A and B, respectively.

• A DG-category with one object is the same as a DG-ring.

References

• Keller, Bernhard (1994), "Deriving DG categories", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série 27 (1): 63–102, ISSN 0012-9593, MR1258406, http://www.numdam.org/numdam-bin/fitem?id=ASENS_1994_4_27_1_63_0