# nLab cartesian closed 2-category

Contents

### Context

#### 2-category theory

2-category theory

## Structures on 2-categories

#### Limits and colimits

limits and colimits

# Contents

## Idea

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, $\mathcal{B}$, with finite products and a cartesian closed structure: For any $A$, $B$ in $\mathcal{B}$, there is an exponential object $B^A$, an evaluation 1-morphism, $eval_{A, B}: B^A \times A \to B$, and for every $X$ an adjoint equivalence between $\mathcal{B}(X, B^A)$ and $\mathcal{B}(X \times A, B)$.

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

## Examples

{Examples}

The following 2-categories are cartesian closed:

## References

• Michael Makkai, Avoiding the axiom of choice in general category theory, Journal of Pure and Applied Algebra, 108(2):109 – 173, 1996.

On cartesian closed 2-categories of internal categories or internal groupoids:

Discussion of the example of generalised species:

Discussion of syntax for CC 2-categories in type theory:

Last revised on August 28, 2021 at 13:52:59. See the history of this page for a list of all contributions to it.