Hom-functor (category theory)

Given any object of a locally-small category $𝐂$, one may construct several functors referred to as hom-functors.

(Covariant) hom-functor

For any fixed object $s$ of a locally-small category $𝐂$, the covariant hom-functor $𝐂[s\to -]:𝐂\to 𝐒𝐞𝐭$

• sends objects $r\in 𝐂$ to hom-sets $𝐂[s\to r]\in 𝐒𝐞𝐭$,

• and sends $𝐂$-morphisms $f:x\to y$ to functions $(g\mapsto gf):𝐂[s\to x]\to 𝐂[s\to y]$ post-composing their argument with $f$.

Considering the mapping $x,y\mapsto 𝐂[x\to y]$ naturally gives rise to the covariant hom-functor by fixing the first argument.

The covariant hom-functor makes a star appearance in the Haskell programming language, being the function type itself (or equivalently, the reader monad):

  instance Functor ((->) s) where
    fmap g f = g . f

Contravariant hom-functor

«TODO»

Considering the mapping $x,y\mapsto 𝐂[x\to y]$ naturally gives rise to the contravariant hom-functor by fixing the first argument.

Two-sided represented functor

If you fix neither side, you get the two-sided represented functor. See Two-sided represented functor.

References

• [cite:@nlab:hom-functor]

• [cite:@enwiki:1278466099]