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
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A sylleptic monoidal (weak) 2-category is a monoidal 2-category with a categorified sort of “commutativity” that lies in between a braiding and a symmetry.
That is, it is a 2-category equipped with a tensor product 2-functor which satisfies the first two in a hierarchy of conditions for being commutative up to equivalence. In the language of k-tuply monoidal n-categories, a sylleptic monoidal 2-category is a triply monoidal 2-category. As described there, this may be identified with a pointed 5-category with a single -morphism for . We can also say that it is a monoidal 2-category whose E1-algebra structure is refined to an E3-algebra structure.
sylleptic monoidal 2-category, sylleptic 3-group
Brian Day, Ross Street, Section 5 of: Monoidal Bicategories and Hopf Algebroids, Advances in Mathematics Volume 129, Issue 1, 15 July 1997, Pages 99-157 (doi:10.1006/aima.1997.1649)
Sjoerd E. Crans, Section 4 of: Generalized Centers of Braided and Sylleptic Monoidal 2-Categories, Advances in Mathematics, Volume 136, Issue 2, 25 June 1998, Pages 183-223 (doi:10.1006/aima.1998.1720)
Last revised on November 4, 2024 at 22:11:25. See the history of this page for a list of all contributions to it.