1 Answers
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 Z {\displaystyle \mathbb {Z} } -module.
In detail, this means that Hom {\displaystyle \operatorname {Hom} } , the morphisms from any object A to another object B of the category is a direct sum
and there is a differential d on this graded group, i.e., for each n there is a linear map
which has to satisfy d ∘ d = 0 {\displaystyle d\circ d=0}. This is equivalent to saying that Hom {\displaystyle \operatorname {Hom} } is a cochain complex. Furthermore, the composition of morphisms Hom ⊗ Hom → Hom {\displaystyle \operatorname {Hom} \otimes \operatorname {Hom} \rightarrow \operatorname {Hom} } is required to be a map of complexes, and for all objects A of the category, one requires d = 0 {\displaystyle d=0}.