Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Categorical extension of Cayley's Theorem

  1. Aug 10, 2008 #1
    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.

    Many thanks in advance!
     
  2. jcsd
  3. Aug 10, 2008 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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)
     
  4. Aug 11, 2008 #3
    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.
     
  5. Aug 12, 2008 #4
    Hi all, I have figured out all the details of this problem. Thanks again.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Categorical extension of Cayley's Theorem
Loading...