Contents

# 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 Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971).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

Last revised on July 11, 2021 at 00:33:17. See the history of this page for a list of all contributions to it.