2-category

Encyclopedia : 2 : 2C : 2CA : 2-category

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In category theory, a 2-category is a category with "morphisms between morphisms". It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure induced by the composition).

More explicitly, a 2-category C consists of:

• A class of 0-cells (or objects) A, B, ....
• For all objects A and B, a category [\mathbf(A,B)]. The objects [f: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 written [\bullet] and called vertical composition.
• For all objects A, B and C, there is a functor [\circ : \mathbf(B,C)\times\mathbf(A,B)\to\mathbf(A,C)], called horizontal composition, which is associative and admits the identity 2-cells of id_A as identities.

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.

See also

• n-category

 

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