nLab
compact closed 2-category

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

2-Category theory

Contents

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 VV a cocomplete closed symmetric monoidal category, the bicategory VProfV Prof of small VV-enriched categories and VV-enriched profunctors is compact closed. (Day-Street)

  2. For CC a 2-category with weak finite limits, the 2-category of spans in CC 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 VProfV Prof (at least in the Cauchy complete case) to the usual notion of compact closed VV-category. (Day-Street) To deal with non-Cauchy complete categories, one can instead talk about compact closed proarrow equipments.

References

Last revised on October 10, 2017 at 17:10:37. See the history of this page for a list of all contributions to it.