nLab dg-category

Idea

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.

Definition

A dg-category over a commutative ring $k$ is an (infinity,1)-category enriched in the (infinity,1)-category of chain complexes of $k$-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.

Properties

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 $H k$. The latter is equivalent, at least morally, to the (infinity,1)-category of $k$-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

• homotopy category of a dg-category.
• equivalence of dg-categories
• dg-modules, perfect dg-modules
• derived dg-category
• dg-Yoneda embedding
• pretriangulated dg-category
• dg-localization, dg-quotient?
• dg-nerve
• Waldhausen K-theory of a dg-category
• semi-topological K-theory of a dg-category
• derived moduli stack of objects in a dg-category

Related concepts

• enhanced triangulated category
• A-infinity category
• spectral category
• stable (infinity,1)-category

References

Historically, dg-categories were introduced in

• G. M. Kelly, Chain maps inducing zero homology maps, Proc. Cambridge Philos. Soc. 61 (1965), 847–854, doi:10.1017/S0305004100039207

whilst their modern development can be traced to

• A. I. Bondal?, Mikhail Kapranov, Enhanced triangulated categories, Матем. Сборник, Том 181 (1990), No.5, 669–683 (Russian); transl. in USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93–107, (MR91g:18010) (Bondal-Kapranov Enhanced triangulated categories pdf)

A bicategory refinement

• Yuki Imamura. A formal categorical approach to the homotopy theory of dg categories (2024). (arXiv:2405.07873).

Overviews

For concise reviews of the theory, see section 1 of

• A. Beilinson, V. Vologodsky, DG guide to Voevodsky’s motives.

as well as the introduction and appendices to

• V. Drinfeld, DG quotients of DG categories, pdf.

For longer surveys, see

• Bernhard Keller, On differential graded categories, International Congress of Mathematicians. II (2006) 151-190, Eur. Math. Soc., Zürich [arXiv:math/0601185]

and

• Bertrand Toën, Lectures on dg-categories (pdf).

Homotopy theory of dg-categories

The homotopy theory of dg-categories is studied in

• Gonçalo Tabuada, Théorie homotopique des DG-caté́gories, 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

• Lee Cohn, Differential Graded Categories are k-linear Stable Infinity Categories (arXiv:1308.2587)

(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.

• Rune Haugseng, Rectification of enriched infinity-categories, arXiv:1312.3881.

Derived 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)

• D. Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, Adv. Math. 302 (2014) doi

Other aspects

• Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102 (numdam)

• Dmitry Tamarkin, What do dg-categories form?,

  Compos. Math. 143 (2007), no. 5, 1335–1358.

• M. Batanin, What do dg-categories form (after Tamarkin), talks at Paris 7 and Australian category seminar (abstract), math.CT/0606553

• Oren Ben-Bassat, Jonathan Block, Cohesive DG categories I: Milnor descent, arxiv/1201.6118