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 -module. In detail, this means that, 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. This is equivalent to saying that is a cochain complex. Furthermore, the composition of morphisms is required to be a map of complexes, and for all objects A of the category, one requires.
Examples
Any additive category may be considered to be a DG-category by imposing the trivial grading and trivial differential.
A little bit more sophisticated is the category of complexes over an additive category. By definition, is the group of maps which do notneed to respect the differentials of the complexes A and B, i.e.,
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.
A DG category C is called pre-triangulated if it has a suspension functor and a class of distinguished triangles compatible with the suspension, such that its homotopy category Ho is a triangulated category. A triangulated category T is said to have a dg enhancementC if C is a pretriangulated dg category whose homotopy category is equivalent to T. dg enhancements of an exact functor between triangulated categories are defined similarly. In general, there need not exist dg enhancements of triangulated categories or functors between them, for example stable homotopy category can be shown not to arise from a dg category in this way. However, various positive results do exist, for example the derived categoryD of a Grothendieck abelian categoryA admits a unique dg enhancement.
