Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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 monoidal double category is a double category equipped with a tensor product (cf. monoidal 2-category), compatible with both tight and loose morphisms.
Formally, a monoidal double category is a pseudomonoid in the (cartesian monoidal) 2-category of double categories, (lax) double functors and loose natural transformations?.
Recall a double category is an internal (pseudo)category in Cat, hence is given by two categories (category of loose arrows) and (category of objects), plus source and target functors , a loose identity functor , and a loose composition functor . Clearly this data has to satisfy the properties of the composition of a category, thus provides identities for the , which in turn is associative. In Shulman 2010 moreover, these properties are only satisfied up to coherent isomorphism, making loose arrows form a bicategory.
A monoidal double category is a double category equipped with double functors
and with double natural transformations? , , satisfying the usual coherence laws of a pseudomonoid (pentagon and triangle, see monoidal category).
As often happens for structure on internal categories, these arise as structures on the categories of loose arrows and that of objects when the source, identity and target maps respect it.
Specifically, a monoidal double category arises when both and are monoidal categories and
and of course these isomorphisms satisfy coherence axioms one can find in Shulmanβ10.
A braiding on a monoidal double category is a natural transformation which satisfies the coherence laws of the braiding of a pseudomonoid (see braided monoidal category)
A symmetric monoidal double category is a braided monoidal double category whose braiding satisfies .
Mike Shulman, Def. 2.9 in: Constructing Symmetric Monoidal Double Categories (2010) [arxiv:1004.0993]
Linde Wester Hansen and Mike Shulman. Constructing symmetric monoidal bicategories functorially (2019) [arxiv:1910.09240]
Last revised on September 3, 2023 at 17:08:52. See the history of this page for a list of all contributions to it.