# Strict 2-category

In category theory, a **strict 2-category** is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over **Cat** (the category of categories and functors, with the monoidal structure given by product of categories).

The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories, in 1965.[1] The more general concept of bicategory (or *weak* 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was discovered in 1968 by Jean Bénabou.[2]

## Definition

A 2-category **C** consists of:

- A class of
*0-cells*(or*objects*) A, B, .... - For all objects A and B, a category
. The objects
of this category are called
*1-cells*and its morphisms are called*2-cells*; the composition in this category is usually written or and called*vertical composition*or*composition along a 1-cell*. - For any object A there is a functor from the terminal category (with one object and one arrow) to
, that picks out the identity 1-cell id
_{A}on A and its identity 2-cell id_{idA}. In practice these two are often denoted simply by A. - For all objects A, B and C, there is a functor
, called
*horizontal composition*or*composition along a 0-cell*, which is associative and admits the identity 1 and 2-cells of id_{A}as identities. The composition symbol is often omitted, the horizontal composite of 2-cells and being written simply as .

The notion of 2-category differs from the more general notion of a bicategory in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in a bicategory it needs only be associative up to a 2-isomorphism. The axioms of a 2-category are consequences of their definition as **Cat**-enriched categories:

- Vertical composition is associative and unital, the units being the identity 2-cells id
_{f}. - Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells id
_{idA}on the identity 1-cells id_{A}. - The interchange law holds; i.e. it is true that for composable 2-cells

The interchange law follows from the fact that is a functor between hom categories. It can be drawn as a pasting diagram as follows:

= | = | |||||

Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both.

## Doctrines

In mathematics, a **doctrine** is simply a 2-category which is heuristically regarded as a system of theories. For example, algebraic theories, as invented by William Lawvere, is an example of a doctrine, as are multi-sorted theories, operads, categories, and toposes.

The objects of the 2-category are called *theories*, the 1-morphisms
are called *models* of the A in B, and the 2-morphisms are called *morphisms between models.*

The distinction between a 2-category and a doctrine is really only heuristic: one does not typically consider a 2-category to be populated by theories as objects and models as morphisms. It is this vocabulary that makes the theory of doctrines worth while.

For example, the 2-category **Cat** of categories, functors, and natural transformations is a doctrine. One sees immediately that all presheaf categories are categories of models.

As another example, one may take the subcategory of **Cat** consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms. This is the doctrine of multi-sorted algebraic theories. If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those categories that are generated under products by a single object.

Doctrines were discovered by Jonathan Mock Beck.

## See also

## References

- Charles Ehresmann, Catégories et structures, Dunod, Paris 1965.
- Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77.

## Footnotes

*Generalised algebraic models*, by Claudia Centazzo.