nLab
closed bicategory

Contents

Contents

Definition

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

x,y,z:B(y,z)×B(x,y)B(x,z){\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 Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971).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.

Examples

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.