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A tensor category is a category equipped with an operation similar to the tensor product in Ab.
The precise definition associated with the term “tensor category” varies somewhat in the literature.
It may mean any :
any monoidal category,
a symmetric monoidal category (and then a quasitensor category is a braided monoidal category),
an Ab-enriched or Vect-enriched (symmetric) monoidal category.
an additive (symmetric) monoidal category, so that the tensor preserves finite direct sums,
an abelian (symmetric) monoidal category, in which the tensor preserves finite colimits in separate arguments,
an abelian (symmetric) monoidal category with dual objects (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)
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Topics in Lie theory and Tensor categories – 9 Tensor categories, Lecture notes (spring 2009) (pdf web)
Deligne's theorem on tensor categories is due to
Review includes
Last revised on November 25, 2016 at 03:42:45. See the history of this page for a list of all contributions to it.