2-category

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).

Definition

A 2-category C consists of:

  • A class of 0-cells (or objects) A, B, ....
  • For all objects A and B, a category \mathbf{C}(A,B). The objects f,g:A\to B of this category are called 1-cells and its morphisms \alpha:f\Rightarrow g are called 2-cells; the composition in this category is usually written \circ or \circ_1 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 \mathbf{C}(A,A), 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 \circ_0 : \mathbf{C}(A,B)\times\mathbf{C}(B,C)\to\mathbf{C}(A,C), called horizontal composition or composition along a 0-cell, which is associative and admits the identity 1 and 2-cells of idA as identities. The composition symbol \circ_0 is often omitted, the horizontal composite of 2-cells \alpha:f\Rightarrow g:A\to B and \beta:f'\Rightarrow g':B\to C being written simply as \beta\alpha:f'f\Rightarrow g'g:A\to C.
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