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 twist, or balance, in a braided monoidal category is a natural isomorphism from the identity functor on to itself satisfying the following compatibility condition with the braiding :
A balanced monoidal category is a braided monoidal category equipped with such a balance.
Beware that there is an un-related notion of balanced categories.
In the language of string diagrams, the balancing is represented by a 360-degree twist:
Every symmetric monoidal category is balanced in a canonical way. In fact, the identity natural transformation on the identity functor of is a balance on . In this way, the twist can be seen as a way of “controlling” the non-symmetric behavior of the braiding.
A braided rigid monoidal category is balanced if and only if it is a pivotal category, but a balanced monoidal category need not be rigid (cf. Selinger 2011, Lem. 4.20).
The original definition:
The above definition follows:
See also:
Springer Lecture Notes in Physics 813 (2011) 289-355 [arXiv:0908.3347, doi:10.1007/978-3-642-12821-9_4]
Last revised on February 21, 2024 at 05:25:13. See the history of this page for a list of all contributions to it.