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 compact closed 2-category (also called an autonomous symmetric monoidal 2-category) is the (weak) 2-category-analog of the notion of compact closed category. That is, it is a symmetric monoidal 2-category in which all objects have duals.
For $V$ a cocomplete closed symmetric monoidal category, the bicategory $V Prof$ of small $V$-enriched categories and $V$-enriched profunctors is compact closed. (Day-Street)
For $C$ a 2-category with weak finite limits, the 2-category of spans in $C$ is compact closed. (Stay 13)
In line with the microcosm principle, internal to a compact closed 2-category one can define a notion of compact closed map pseudomonoid, which specializes in $V Prof$ (at least in the Cauchy complete case) to the usual notion of compact closed $V$-category. (Day-Street) To deal with non-Cauchy complete categories, one can instead talk about compact closed proarrow equipments.
Last revised on November 26, 2023 at 18:35:00. See the history of this page for a list of all contributions to it.