Differential Graded Category

Overview

A differential graded category stores morphisms as complexes rather than just sets or vector spaces. Passing to degree-zero cohomology produces an ordinary category, but discards information about homotopies and higher morphism degrees. Keeping that information is useful in derived algebra, deformation theory, and comparisons between algebraic models.

Prerequisites

Use cochain complexes, graded modules, chain maps, and categories. Fix a commutative base ring $k$. A cochain differential has degree $+1$ and square zero. Homogeneous morphisms have specified degrees, which control signs in compositions.

Definition and conventions

A small $k$-linear dg category $\mathcal A$ has a set of objects and, for each pair $X,Y$, a complex $\mathcal A(X,Y)$ of $k$-modules. Composition is a degree-zero map of complexes, is associative, and has closed degree-zero identity morphisms. For homogeneous composable $f,g$, this means

We use cohomological grading. Reversing the convention without changing signs gives a different presentation and may break the differential law.

The category $H^0(\mathcal A)$ has the same objects and morphisms $H^0(\mathcal A(X,Y))$. A dg category is an enhancement of a specified ordinary category only when an appropriate equivalence with that category is supplied.

Intuition

Closed degree-zero maps describe maps compatible with differentials. Boundaries identify maps that differ by a homotopy. Other degrees retain derived information that an ordinary category has no place to record. The enrichment organizes these data and their compositions together.

Examples and non-examples

A dg algebra is a dg category with one object. Ordinary $k$-algebras are special cases concentrated in degree zero.

For complexes $X,Y$, the degree-$n$ part of their morphism complex consists of graded maps $f:X^i\to Y^{i+n}$, with differential

A closed degree-zero element is a chain map. A degree-zero boundary is a null-homotopic map. Hence $H^0$ of the dg category of complexes is the homotopy category of complexes.

This homotopy category is generally not the derived category: quasi-isomorphisms must also be inverted. Nor is an arbitrary triangulated category automatically a dg category; it does not come equipped with morphism complexes.

Important properties

A dg functor is a quasi-equivalence if its maps on morphism complexes are quasi-isomorphisms and its induced functor on $H^0$ is essentially surjective. This compares both the retained cohomology and the ordinary categorical shadow.

The displayed differential on morphisms squares to zero because $d_X^2=d_Y^2=0$ and the cross terms cancel with their signs. That elementary calculation is the foundation for the enriched composition law.

Connections and further reading

Derived Morita theory compares dg categories through their module categories rather than only their objects. These distinctions matter when constructing enhancements or invariants. See Keller, Section 2.2 and Sections 3–4; this overview does not assert that all triangulated categories admit unique enhancements.