cartesian closed 2-category




In the context of 2-category theory, a cartesian closed 2-category is a 2-category, \mathcal{B}, with finite products and a cartesian closed structure. For any AA, BB in \mathcal{B}, there is an exponential object B AB^A, an evaluation 1-arrow, eval A,B:B A×ABeval_{A, B}: B^A \times A \to B, and for every XX an adjoint equivalence between (X,B A)\mathcal{B}(X, B^A) and (X×A,B)\mathcal{B}(X \times A, B).

The concept was introduced in (Makkai 96). There is no connection to the concept of cartesian bicategory.


  • the 2-category of generalized species (FGHW).

  • the 2-category of cartesian distributors.

  • the 2-category of operads.


Discussion of the example of generalised species:

  • M. Fiore, N. Gambino, M. Hyland, G. Winskel, The cartesian closed bicategory of generalised species of structures, (pdf)

Last revised on July 2, 2020 at 06:37:45. See the history of this page for a list of all contributions to it.