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 $d(id_A) = 0$.

See also

• Differential algebra
• Graded (mathematics)
• Graded category

Examples

• Any additive category may be considered to be a DG-category by imposing the trivial grading (i.e. all $\mathrm{Hom}_n(-,-)$ 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, $\mathrm{Hom}_{C(\mathcal A), n} (A, B)$ 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. $\mathrm{Hom}_{C(\mathcal A), n} (A, B) = \Pi_{l \in \mathbf Z} \mathrm{Hom}(A_l, B_{l+n})$. 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. This applies to the category of complexes of quasi-coherent sheaves on a scheme over a ring.
• A DG-category with one object is the same as a DG-ring. A DG-ring over a field is called DG-algebra, or differential graded algebra.

Further properties

The category of small dg-categories can be endowed with a model category structure such that weak equivalences are those functors that induce an equivalence of derived categories.

Given a dg-category C over some ring R, there is a notion of smoothness and properness of C that reduces to the usual notions of smooth and proper morphisms in case C is the category of quasi-coherent sheaves on some scheme X over R.

External links

• http://ncatlab.org/nlab/show/dg-category