tensor category (changes) in nLab

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Contents

Idea

A Some authors use “tensor category” essentially as a synonym for (tensor categorysymmetric ) is usually understood to be a monoidal category categories equipped (e.g. with further “ linear Davydov algebraic 1998 ” , properties Kashiwara & Schapira 2006, Def. 4.2.1 ). andstructure, 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.

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 typically at times required to be

• symmetric monoidal

  (and (e.g if it is only braided Deligne monoidal 1990 one in speaks (2.1.1); of aquasitensor categoryDavydov 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.

• fusion category

• tensor triangulated category

• tensor (∞,1)-category

• 2-algebraic geometry

• tensor network

References

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

• Bojko Alexei Bakalov Davydov , :Alexander KirillovMonoidal categories and functors , Chapter 1 in: Lectures Monoidal on Categories tensor categories and modular functors , University J. Lecture Math. Series Sci.(New York) 21 88 , Amer. (1998) Math. 457-519 Soc. [[doi:10.1007/BF02365309](https://doi.org/10.1007/BF02365309)] (2001) [[web](http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html),ams:ulect/21]

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

• Masaki Bojko Kashiwara Bakalov, Pierre Alexander Schapira Kirillov , Section 4 of:Categories and SheavesLectures on tensor categories and modular functors , Grundlehren University der Lecture Mathematischen Series Wissenschaften 332 21 , Springer Amer. (2006) Math. [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), Soc. (2001) [[webpage](http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html), pdf 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](https://link.springer.com/book/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](https://bookstore.ams.org/surv-205), pdf]

• Alexei Davydov , :Tensor categories, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [[arXiv:2311.05789](https://arxiv.org/abs/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).