# Categorical extension of Cayley's Theorem

1. Aug 10, 2008

### Singularity

Hey PF gurus!

I read that Cayley's theorem can be extended to categories, i.e. that any category with a set of morphisms can be represented as a category with sets as objects and functions as morphisms. I was looking at the construction and for some reason I don't fully understand how they define the morphisms in the 'dual' category. If someone could please shed some light on this, I would appreciate it. But please don't post the whole proof of the representation result - I would like to try it out myself first.

2. Aug 10, 2008

### Hurkyl

Staff Emeritus
What do you mean by 'dual category' here?

Have you actually defined the idea of a category acting on a set? Or are you just constructing a subcategory of Set that is isomorphic to your category C? (Or equivalently, a faithful functor C-->Set that separates (is injective on) objects)

3. Aug 11, 2008

### Singularity

Hi Hurkyl. Thanks for the reply. I realise that the concept of dual category already exists in the literature, and it has a different meaning to the one I am asking here. Clearly I am looking for a functor from C to Set (as stated in the first post I restrict myself to categories with sets of morphisms). I am unsure if I should (could) check out the full and/or faithful properties. I went to a professor in my department and he showed me the basics of the construction. I filled in the gaps and showed that one can find such a functor. I will investigate further to see if this functor is faithful.

4. Aug 12, 2008

### Singularity

Hi all, I have figured out all the details of this problem. Thanks again.