# The Unapologetic Mathematician

## Hom functors

Every locally small category $\mathcal{C}$ comes with a very interesting functor indeed. Given objects $A'$ and $A$ we can find the set $\hom_\mathcal{C}(A',A)$ of morphisms from $A'$ to $A$. I say that this is a functor $\hom:\mathcal{C}^{\rm op}\times\mathcal{C}\rightarrow\mathbf{Set}$.

We’ve given how $\hom$ behaves on pairs of objects. Now if we have morphisms $f':B'\rightarrow A'$ and $f:A\rightarrow B$ in $\mathcal{C}$ we need to construct a function $\hom_\mathcal{C}(f',f):\hom_\mathcal{C}(A',A)\rightarrow\hom_\mathcal{C}(B',B)$. Notice that the direction of the arrow in the first slot gets reversed, since we want the functor to be contravariant in that place. So, given a morphism $m\in\hom_\mathcal{C}(A',A)$ we define $\hom_\mathcal{C}(f',f)(m)=f\circ m\circ f'$.

Clearly if we pick a pair of identity morphisms, $\hom_\mathcal{C}(1_{A'},1_A)$ is the identity function on $\hom_\mathcal{C}(A',A)$. Also, if we take $f':B'\rightarrow A'$, $f:A\rightarrow B$, $g':C'\rightarrow B'$, and $g:B\rightarrow C$ in $\mathcal{C}$, then we can check $\left[\hom_\mathcal{C}(g',g)\circ\hom_\mathcal{C}(f',f)\right](m)=\hom_\mathcal{C}(g',g)(f\circ m\circ f')=$ $g\circ f\circ m\circ f'\circ g'=\hom_\mathcal{C}(f'\circ g',g\circ f)(m)$
Again, notice that the order of composition gets swapped in the first slot. Thus $\hom_\mathcal{C}$ is a functor, contravariant in the first slot and covariant in the second.

Not only do hom-functors exist for all locally small categories, but we’ll see that they have all sorts of special properties that will come in handy.

June 2, 2007 Posted by | Category theory | 2 Comments