Differential graded category

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 ⁡ ( A , B ) {\displaystyle \operatorname {Hom} (A,B)} , the morphisms from any object A to another object B of the category is a direct sum ⨁ n ∈ Z Hom n ⁡ ( A , B ) {\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 d : Hom n ⁡ ( A , B ) → Hom n + 1 ⁡ ( A , B ) {\displaystyle d\colon \operatorname {Hom} _{n}(A,B)\rightarrow \operatorname {Hom} _{n+1}(A,B)} , which has to satisfy d ∘ d = 0 {\displaystyle d\circ d=0} .

Source: Wikipedia — Differential graded category (CC BY-SA 4.0)

Differential graded category

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 ⁡ ( A , B ) {\displaystyle \operatorname {Hom} (A,B)} , the morphisms from any object A to another object B of the category is a direct sum ⨁ n ∈ Z Hom n ⁡ ( A , B ) {\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 d : Hom n ⁡ ( A , B ) → Hom n + 1 ⁡ ( A , B ) {\displaystyle d\colon \operatorname {Hom} _{n}(A,B)\rightarrow \operatorname {Hom} _{n+1}(A,B)} , which has to satisfy d ∘ d = 0 {\displaystyle d\circ d=0} .

Source: Wikipedia "Differential graded category" · CC BY-SA 4.0

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