Hom functors
Every locally small category 
comes with a very interesting functor indeed. Given objects 
and 
we can find the set 
of morphisms from to . I say that this is a functor 
.
We’ve given how 
behaves on pairs of objects. Now if we have morphisms 
and 
in we need to construct a function 
. 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 
we define 
.
Clearly if we pick a pair of identity morphisms, 
is the identity function on . Also, if we take , , 
, and 
in , then we can check
=\hom_\mathcal{C}(g',g)(f\circ m\circ f')=](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Chom_%5Cmathcal%7BC%7D%28g%27%2Cg%29%5Ccirc%5Chom_%5Cmathcal%7BC%7D%28f%27%2Cf%29%5Cright%5D%28m%29%3D%5Chom_%5Cmathcal%7BC%7D%28g%27%2Cg%29%28f%5Ccirc+m%5Ccirc+f%27%29%3D&bg=e6e6e6&fg=333333&s=0&c=20201002)
Again, notice that the order of composition gets swapped in the first slot. Thus 
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.
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June 2, 2007 -
Posted by John Armstrong |
Category theory
The Unapologetic Mathematician 
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