Differential graded category

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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(idA) = 0.

[edit] Examples

  • Any additive category may be considered to be a DG-category by imposing the trivial grading (i.e. all Homn( − , − ) 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[n] \rightarrow B 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 dA,dB are the differentials of A and B, respectively.

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