With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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
which is at times required to be
(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, )-enriched or (Vect,)-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.
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.
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
Review in:
On quotients of tensor categories:
Last revised on August 24, 2024 at 17:08:20. See the history of this page for a list of all contributions to it.