2-functor

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 ${\displaystyle F\colon C\to D}$ consists of

• a function ${\displaystyle F\colon {\text{Ob}}C\to {\text{Ob}}D}$, and
• for each pair of objects ${\displaystyle c,c'\in {\text{Ob}}C}$, a functor ${\displaystyle F_{c,c'}\colon {\text{Hom}}_{C}(c,c')\to {\text{Hom}}_{D}(Fc,Fc')}$

such that each ${\displaystyle F_{c,c'}}$ strictly preserves identity objects and they commute with horizontal composition in C and D.

See ^[3] for more details and for lax versions.

References

1. Kelly, G. M.; Street, Ross (1974). "Review of the elements of 2-categories". In Kelly, Gregory M. (ed.). Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973. Lecture Notes in Mathematics. Vol. 420. Springer. pp. 75–103. doi:10.1007/BFb0063101. ISBN   978-3-540-06966-9. MR   0357542.
2. G. M. Kelly. Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, (10), 2005.
3. 2-functor at the nLab