mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a 2-functor is a morphism between

2-categories 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 catego ...

. They may be defined formally using

enrichment Enrichment may refer to: * Behavioral enrichment, the practice of providing animals under managed care with stimuli such as natural and artificial objects * Data enrichment, appending or enhancing data with relevant context from other sources, se ...

by saying that a 2-category is exactly a ''Cat''-enriched category and a 2-functor is a ''Cat''-functor. Explicitly, if ''C'' and ''D'' are 2-categories then a 2-functor

F\colon C\to D

consists of * a function

F\colon \text C\to \text D

, and * for each pair of objects

c,c'\in C

a functor

F_\colon \text_(c,c')\to\text_D(Fc,Fc')

such that each

F_

strictly preserves identity objects and they commute with horizontal composition in ''C'' and ''D''. See for more details and for lax versions.

References

Functors Higher category theory {{categorytheory-stub