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 ribbon category [Reshetikhin & Turaev (1990)] (also called a tortile category [Joyal & Street (1993)] or balanced rigid braided tensor category) is a monoidal category equipped with braiding , twist and duality that satisfy some compatibility conditions.
A braided monoidal category is a monoidal category equipped with a braiding , which is a natural isomorphism obeying the hexagon identities.
A braided monoidal category is rigid if, for every object , there exist objects and (called its right and left dual) and associated morphisms
obeying the zig-zag identities:
A twist on rigid braided monoidal category is a natural isomorphism from the identity functor to itself, with components for which
A ribbon category is a rigid braided monoidal category equipped with such a twist.
Nicolai Reshetikhin, Vladimir Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 1 (1990) [doi:10.1007/BF02096491]
AndrΓ© Joyal, Ross Street, Braided tensor categories, Advances in Mathematics 102 (1993) 20β78 [doi:10.1006/aima.1993.1055]
Mei Chee Shum, Tortile tensor categories, Journal of Pure and Applied Algebra 93 1 (1994) 57-110 [10.1016/0022-4049(92)00039-T]
Last revised on July 11, 2024 at 02:29:53. See the history of this page for a list of all contributions to it.