nLab compact closed 2-category

Idea

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.

Examples

1. 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)

2. For $C$ a 2-category with weak finite limits, the 2-category of spans in $C$ is compact closed. (Stay 13)

Internalization

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.

Related concepts

• monoidal 2-category
• compact closed category

References

• Mike Stay, Compact Closed Bicategories (arXiv:1301.1053)

• Brian Day and Ross Street, Monoidal bicategories and Hopf algebroids