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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
F
:
C
→
D
{\displaystyle F\colon C\to D}
consists of
a function
F
:
Ob
C
→
Ob
D
{\displaystyle F\colon {\text{Ob}}C\to {\text{Ob}}D}
, and
for each pair of objects
c
,
c
′
∈
Ob
C
{\displaystyle c,c'\in {\text{Ob}}C}
, a functor
F
c
,
c
′
:
Hom
C
(
c
,
c
′
)
→
Hom
D
(
F
c
,
F
c
′
)
{\displaystyle F_{c,c'}\colon {\text{Hom}}_{C}(c,c')\to {\text{Hom}}_{D}(Fc,Fc')}
such that each
F
c
,
c
′
{\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
^ 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 n Lab