Homological algebra

homological algebra


nonabelian homological algebra


Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories


Enriched category theory



Differential graded categories or dg-categories are linear analogues of spectral categories. In other words they are linear stable (infinity,1)-categories. It is common and useful to view them as enhanced triangulated categories.


A dg-category over a commutative ring kk is an (infinity,1)-category enriched in the (infinity,1)-category of chain complexes of kk-modules. Equivalently, it is an ordinary category strictly enriched in chain complexes (see Haugseng 13).

Hence a dg-category is a category with mapping complexes of morphisms between any two objects. By taking the homologies of these chain complexes in degree zero, one gets an ordinary category, called the homotopy category of a dg-category. Notice that a dg-category with a single object is the same thing as a dg-algebra.


The (infinity,1)-category of dg-categories

The Dwyer-Kan model structure on dg-categories presents the (infinity,1)-category of dg-categories.

Relation to stable \infty-categories

By the stable Dold-Kan correspondence, the (infinity,1)-category of dg-categories is equivalent to the (infinity,1)-category of (infinity,1)-categories enriched in the symmetric monoidal (infinity,1)-category of modules over the Eilenberg-Mac Lane spectrum HkH k. The latter is equivalent, at least morally, to the (infinity,1)-category of kk-linear stable (infinity,1)-categories.

More precisely, it is shown in Cohn 13 that the Morita model structure on dg-categories presents the (infinity,1)-category of idempotent complete linear stable (infinity,1)-categories.

Aspects of dg-categories


Historically, dg-categories were introduced in

  • G. M. Kelly, Chain maps inducing zero homology maps, Proc. Cambridge Philos. Soc. 61 (1965), 847–854,

whilst their modern development can be traced to


For concise reviews of the theory, see section 1 of

as well as the introduction and appendices to

For longer surveys, see

  • Bernhard Keller, On differential graded categories International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Zürich, 2006. (arXiv)


Homotopy theory of dg-categories

The homotopy theory of dg-categories is studied in

  • Gonçalo Tabuada, Homotopy theory of DG categories, Thesis, Paris, 2007, pdf.

  • Gonçalo Tabuada, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 15–19.

The equivalence with the homotopy theory of stable (infinity,1)-categories is discussed in

(Note that the proof works over any ring, even though it is stated there for characteristic zero.)

In the following it is shown that the homotopy theory of (infinity,1)-categories enriched in the (infinity,1)-category of chain complexes is equivalent to the homotopy theory of ordinary categories strictly enriched in chain complexes.

Noncommutative geometry

The following references discuss the use of dg-categories in derived noncommutative algebraic geometry and noncommutative motives.

  • Gonçalo Tabuada, Invariants additifs de DG-catégories, Int. Math. Res. Not. 2005, no. 53, 3309–3339; Addendum in Int. Math. Res. Not. 2006, Art. ID 75853, 3 pp. ; Erratum in Int. Math. Res. Not. IMRN 2007, no. 24, Art. ID rnm149, 17 pp.

  • Marco Robalo, Théorie homotopique motivique des espaces noncommutatifs, pdf.

  • S. Mahanta, Noncommutative geometry in the framework of differential graded categories, (arXiv:0805.1628)

Other aspects

Revised on February 9, 2015 17:09:24 by Tim Porter (