Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
The notion of cartesian closed 2-category is the analog in 2-category theory (hence the categorification) of the notion of cartesian closed category in category theory:
A cartesian closed 2-category is a 2-category, , with finite products and a cartesian closed structure: For any , in , there is an exponential object , an evaluation 1-morphism, , and for every an adjoint equivalence between and .
The concept was introduced in (Makkai 96). There is no connection to the concept of cartesian bicategory.
{Examples}
The following 2-categories are cartesian closed:
the 2-categories Cat() of internal categories or Grpd of internal groupoids internal to a cartesian closed category (e.g. Niefield & Pronk 2019);
the 2-category of cartesian distributors;
the 2-category of coloured operads and bimodules.
On cartesian closed 2-categories of internal categories or internal groupoids:
Susan B. Niefield, Dorette A. Pronk, Internal groupoids and exponentiability, Cahiers de topologie et géométrie différentielle catégoriques, LX 4 (2019) (pdf)
Enrico Ghiorzi, Section 1.2 of: Complete internal categories (arXiv:2004.08741)
Discussion of the example of generalised species:
On the example of coloured operads:
Discussion of syntax for cartesian closed 2-categories in type theory:
Marcelo Fiore, Philip Saville, A type theory for cartesian closed bicategories, (arXiv:1904.06538)
Philip Saville, Cartesian closed bicategories: type theory and coherence, (PhD thesis, arXiv:2007.00624)
Last revised on August 20, 2024 at 08:14:31. See the history of this page for a list of all contributions to it.