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.^[1]

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.

References

1. Tabuada, Gonçalo (2005), "Invariants additifs de DG-catégories", International Mathematics Research Notices (53): 3309–3339, doi:10.1155/IMRN.2005.3309, ISSN 1073-7928

• Keller, Bernhard (1994), "Deriving DG categories", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 27 (1): 63–102, ISSN 0012-9593, MR 1258406

External links

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