# nLab closed bicategory

Contents

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## Definition

A closed bicategory is a bicategory $B$ admitting all right extensions and right lifts, equivalently a bicategory whose composition functor

${\circ}_{x, y, z} \colon B(y,z) \times B(x,y) \to B(x,z)$

participates in a two-variable adjunction. Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971).

## Remarks

A closed bicategory is a horizontal categorification of a closed monoidal category. It is not to be confused with a closed monoidal bicategory, which is a vertical categorification of the same concept.

Dually, a bicategory admitting all left extensions and lifts is called a coclosed bicategory, and is analogously the horizontal categorification of a coclosed monoidal category?. A bicategory admitting all (right and left) extensions and lifts is a biclosed bicategory.

## References

• Brian Day, Limit spaces and closed span categories, Lecture Notes in Mathematics, 420, 1974 (doi:10.1007/BFb0063100)

• Renato Betti, Robert F. C. Walters, Closed bicategories and variable category theory, Universita degli Studi di Milano (1985), reprinted in: Reprints in Theory and Applications of Categories, 26 (2020) 1-27 $[$tac:tr26$]$

Last revised on May 26, 2022 at 06:28:37. See the history of this page for a list of all contributions to it.