A differential graded category over a commutative ring 
is a category whose morphism
spaces are cochain complexes of 
-modules, with bilinear composition of morphisms compatible with
the differentials. For homogeneous morphisms 
and 
,
with 
of degree 
,
compatibility means
 |
The abbreviation is DG category. Its degree-zero homotopy category has the same objects and morphisms 
(Stacks Project, Definition 22.26.3).
For example, cochain complexes of modules form a DG category whose morphism complex contains maps of all degrees. Closed degree-zero morphisms are chain homomorphisms, and their boundaries identify chain-homotopic maps. DG enhancements retain information beyond the resulting ordinary derived category or homotopy category.
