nLab closed bicategory

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 Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971).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

Examples

• Prof
• The bicategory $Span(E)$ is closed if and only if the category $E$ is locally cartesian closed; see (Day 1974, Proposition 4.1).

Related entries

• symmetric bicategory

• An extension system is to a closed bicategory what a closed category is to a monoidal category.

• locally internal category

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$]$