Representable Functors

Take your favorite category C. For each object A there is a functor,



from op-C to Set. (This is a contravariant functor and a lower index. NOT the same convention used for contravariant vectors in physics). We could do the other one by switching A and -. That tells you what happens to objects. What happens to morphisms, the only things that you can do, pre and post composition respectively.

In fact the assignment from A to the functor is also functorial. It is a functor from C to the functor category Fun(C-op,Set) where morphisms are natural transformations. I hear you like morphisms so we put some morphisms on morphisms. That is just pretentious abstract nonsense talk for associativity of reading functions.

This functor is the Yoneda embedding, It embeds the category C into the category Fun[C-op,Set]. Why bother translating these trivialities into such foul language?

The reason is that say we have a functor we want to understand like the functor from Rings to Set that takes a ring to the zero set of some polynomial where the variables are taken from that ring.



This functor is isomorphic to one of those above. Namely take A to be



So instead of understanding the equation, understand this functor. Things that seem artificial to do to the equation are more readily explained in the category language. So essentially this language shows you are not pulling all your constructions out of your ass.

We will use this to explain why manifolds are shiny by describing the category of smooth or analytic manifolds.