A semifunctor is a homomorphism between semicategories, like a functor is a homomorphims between categories.


A semifunctor FF from a semicategory CC to a semicategory DD is a map sending each object xCx \in C to an object F(x)DF(x) \in D and each morphism f:xyf : x \to y in CC to morphism F(f):F(x)F(y)F(f) : F(x) \to F(y) in DD, such that

  • FF preserves composition: F(gf)=F(g)F(f)F(g\circ f) = F(g)\circ F(f) whenever the left-hand side is well-defined.

If CC is a category, then FF need not preserve its identity morphisms, but this axiom does require that it send them to idempotents in DD.


A mapping FF of a category into another category that sends id Xid_X to a nontrivial idempotent endomorphism of F(X)F(X) is a semifunctor but not a functor.

More generally, recall from semicategory that the forgetful functor U:CatSemicatU \colon Cat \to Semicat has a right adjoint GG, which sends a semicategory to its category of idempotents, or Karoubi envelope. Thus to give a semifunctor from a category CC to a (semi)category DD is the same as giving a functor from CC to the Karoubi envelope D¯\bar D of DD (but beware that this correspondence does not hold for natural transformations).

Revised on November 23, 2012 20:05:02 by Finn Lawler (