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In mathematics, specifically, in category theory, a 2-functor is a morphism between 2-categories.[1] They may be defined formally using enrichment by saying that a 2-category is exactly a Cat-enriched category and a 2-functor is a Cat-functor.[2]
Explicitly, if C and D are 2-categories then a 2-functor
consists of
- a function
, and
- for each pair of objects
, a functor ![{\displaystyle F_{c,c'}\colon {\text{Hom}}_{C}(c,c')\to {\text{Hom}}_{D}(Fc,Fc')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b562dd4fa5fe9a015b29c721c7926bde134002b)
such that each
strictly preserves identity objects and they commute with horizontal composition in C and D.
See [3] for more details and for lax versions.
- ^ Kelly, G.M.; Street, R. (1974). "Review of the elements of 2-categories". Category Seminar. 420: 75–103.
- ^ G. M. Kelly. Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, (10), 2005.
- ^ 2-functor at the nLab