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
Examples/classes:
Types
Related concepts:
A ribbon category [Reshetikhin & Turaev (1990)] (also called a tortile category [Joyal & Street (1993), Shum 1994, Selinger 2011 Β§4.7] or balanced rigid braided tensor category) is a monoidal category equipped with braiding , twist and duality that satisfy some compatibility conditions.
Recall that:
A braided monoidal category is a monoidal category equipped with a braiding , which is a natural isomorphism obeying the hexagon identities.
A rigid monoidal category is a braided monoidal category (Def. ) where for every object , there exist objects and (called its right and left dual) and associated morphisms
obeying the zig-zag identities:
Now:
A twist on rigid braided monoidal category (Def. ) is a natural isomorphism from the identity functor to itself, with components for which
A ribbon category (tortile category) is a rigid braided monoidal category equipped with such a twist.
A functor between ribbon categories is a ribbon functor (tortile functor) if it preserves all this structure up to isomorphism.
(Shum's theorem)
The category of framed oriented tangles is equivalently the free ribbon category generated by a single object.
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]
Vladimir Turaev, Β§I.1 in: Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, de Gruyter & Co. (1994) [doi:10.1515/9783110435221, pdf]
Mei Chee Shum, Tortile tensor categories, Journal of Pure and Applied Algebra 93 1 (1994) 57-110 [10.1016/0022-4049(92)00039-T]
David N. Yetter: Functorial Knot Theory β Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants, Series on Knots and Everything 26, World Scientific (2001) [doi:10.1142/4542]
Peter Selinger, Β§4.7 in: A survey of graphical languages for monoidal categories, Springer Lecture Notes in Physics 813 (2011) 289-355 [arXiv:0908.3347, doi:10.1007/978-3-642-12821-9_4]
Lecture notes:
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