nLab tensor category

Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Category theory

Contents

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 algebraicproperties 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

which is at times required to be

and, in addition, often

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.

References

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:

On quotients of tensor categories:

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

Last revised on August 24, 2024 at 17:08:20. See the history of this page for a list of all contributions to it.