The Unapologetic Mathematician

Mathematics for the interested outsider

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

   

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