Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A closed bicategory is a bicategory $B$ admitting all right extensions and right lifts, equivalently a bicategory whose composition functor
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 Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971).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.
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)
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 10:28:37. See the history of this page for a list of all contributions to it.