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Concept# Strict 2-category

Summary

In , a strict 2-category is a 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 over Cat (the , with the structure given by ).
The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou.
A 2-category C consists of:
A class of 0-cells (or ) 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 (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 1 and 2-cells of idA as identities. Here, associativity for means that horizontally composing twice to is independent of which of the two and are composed first. The composition symbol is often omitted, the horizontal composite of 2-cells and being written simply as .
The 0-cells, 1-cells, and 2-cells terminology is replaced by 0-morphisms, 1-morphisms, and 2-morphisms in some sources (see also ).
The notion of 2-category differs from the more general notion of a 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 idf.

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