2-category
From Wikipedia, the free encyclopedia
In category theory, a 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).
Contents |
[edit] 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 idA on A and its identity 2-cell ididA. 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 2-cells of idA 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-)morphisms is required to be strictly associative, whereas in a bicategory it need 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 idf.
- Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells A = ididA on the identity 1-cells idA.
- The interchange law holds; i.e. it is true that for composable 2-cells α,β,γ,δ
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.
[edit] 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 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 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 invented by J. M. Beck.
[edit] See also
[edit] References
- Generalised algebraic models, by Claudia Centazzo.
