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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 ${\oplus }_{n\in \mathbf{Z}}Ho{m}_n(A,B)$ and there is a differential d on this graded group, i.e. for all n a linear map $d:Ho{m}_n(A,B)\to Ho{m}_{n+1}(A,B)$, 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 $Hom(A,B)\otimes Hom(B,C)\to Hom(A,C)$ is required to be a map of complexes, and for all objects A of the category, one requires $d(i{d}_A)=0$.

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 ${\mathrm{H}\mathrm{o}\mathrm{m}}_n(-,-)$ 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, ${\mathrm{H}\mathrm{o}\mathrm{m}}_{C(\mathcal{𝒜}),n}(A,B)$ is the group of maps $A\to B[n]$ which do not need to respect the differentials of the complexes A and B, i.e. ${\mathrm{H}\mathrm{o}\mathrm{m}}_{C(\mathcal{𝒜}),n}(A,B)={\mathrm{\Pi }}_{l\in \mathbf{Z}}\mathrm{H}\mathrm{o}\mathrm{m}(A_l,B_{l+n})$. The differential of such a morphism $f=(f_l:A_l\to B_{l+n})$ 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 . 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.

References

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