Semifunctors

Idea

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

Definition

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

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

If $C$ is a category, then $F$ need not preserve its identity morphisms, but this axiom does require that it send them to idempotents in $D$.

Examples

A mapping $F$ of a category into another category that sends ${\mathrm{id}}_X$ to a nontrivial idempotent endomorphism of $F(X)$ is a semifunctor but not a functor.

More generally, recall from semicategory that the forgetful functor $U:\mathrm{Cat}\to \mathrm{Semicat}$ has a right adjoint $G$, which sends a semicategory to its category of idempotents, or Karoubi envelope. Thus to give a semifunctor from a category $C$ to a (semi)category $D$ is the same as giving a functor from $C$ to the Karoubi envelope $\overline{D}$ of $D$ (but beware that this correspondence does not hold for natural transformations).