closed bicategory

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 Closed bicategories were introduced by Lawvere in unpublished lecture notes *Closed categories and biclosed bicategories* (1971).

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**.

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

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

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

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