nLab tensor category

Idea

Some authors use “tensor category” essentially as a synonym for (symmetric) monoidal categories (e.g. Davydov 1998, Kashiwara & Schapira 2006, Def. 4.2.1).

These days, a tensor category is usually understood to be a monoidal category equipped with further “linear algebraic” properties and structure, hence with monoidal-structure given by a kind of tensor product in the original sense (i.e. actually being a universal bilinear map of sorts) whence the name.

Conventions differ, but at the very least one means

• a monoidal category,

which is at times required to be

• symmetric monoidal

  (e.g Deligne 1990 in (2.1.1); Davydov 1998 says “tensor category” for symmetric monoidal categories and “quasitensor category” for braided monoidal categories),

• (Ab, $\otimes$)-enriched or (Vect,$\otimes$)-enriched,

  to make an enriched monoidal category

and, in addition, often

• additive (symmetric) monoidal, so that the tensor product preserves finite direct sums,

• abelian (symmetric) monoidal, in which the tensor product preserves finite colimits in separate arguments,

• with dual objects, making a rigid monoidal category.

Properties

Tannaka theory, Deligne’s theorem, super-representation theory

Deligne's theorem on tensor categories (Deligne 02) establishes Tannaka duality between sufficiently well-behaved linear tensor categories in characteristic zero and supergroups, realizing these tensor categories as categories of representations of these supergroups.

Related concepts

• fusion category

• tensor triangulated category

• tensor (∞,1)-category

• 2-algebraic geometry

• tensor network

References

• Pierre Deligne, section 2 of: Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) 111-195 (pdf)

• Alexei Davydov: Monoidal categories and functors, Chapter 1 in: Monoidal Categories, J. Math. Sci.(New York) 88 (1998) 457-519 [doi:10.1007/BF02365309]

• Bojko Bakalov, Alexander Kirillov, Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001) [webpage, ams:ulect/21, pdf]

  (focus on Reshetikhin-Turaev construction of modular functors from modular tensor categories)

• Masaki Kashiwara, Pierre Schapira, Section 4 of: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

• Damien Calaque, Pavel Etingof, Lectures on tensor categories, IRMA Lectures in Mathematics and Theoretical Physics 12, 1-38 (2008) (arXiv:math/0401246)

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Topics in Lie theory and Tensor categories – 9 Tensor categories, Lecture notes (spring 2009) (pdf web)

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015) [ISBN:978-1-4704-3441-0, pdf]

• Alexei Davydov: Tensor categories, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2311.05789]

Deligne's theorem on tensor categories is due to

• Pierre Deligne, Catégorie Tensorielle, Moscow Math. Journal 2 (2002) no. 2, 227-248. (pdf)

Review in:

• Victor Ostrik, Tensor categories (after P. Deligne) (arXiv:math/0401347)

On quotients of tensor categories:

• Zhenbang Zuo, Gongxiang Liu. Quotient Category of a Multiring Category (2024). (arXiv:2403.06244).