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
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 typically required to be
(and if it is only braided monoidal one speaks of a quasitensor category),
(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) pp. 111-195 (pdf)
Bojko Bakalov, Alexander Kirillov, Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001) [web, ams:ulect/21]
(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]
Deligne's theorem on tensor categories is due to
Review in:
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