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re:functors
[[User:PatrickD|PatrickD]] 15:31, 1 October 2009 (EDT)
[[User:PatrickD|PatrickD]] 15:31, 1 October 2009 (EDT)
:My article's not elegant at all! It's just a cleanup until the CP treatment of category theory gets redone (if ever; I don't think it's a high priority). Thanks for adding the CS definition -- I'd heard the term, but never knew what it was before! I think the ideal situation would be a single page about basic notions in category theory -- definition of category, examples, definition of functor, examples, definition of natural transformation, examples, .... It doesn't make sense to me for functor to have its own page. I'd put things like "pushout square" and "adjoint functor" on their own pages, if they were ever written, but I'm not sure they belong here.
:You're right on about contravariant functors. I probably should've mentioned it. Sometimes these days when people write "functor" they just mean covariant functor, and a controvariant functor from A is just regarded as a covariant functor from A^op (A with all arrows reversed). It's definitely more illuminating this way, and cohomology is indeed an example.
:The other example I'm thinking about adding is Hom(-,V) and Hom(V,-), both functors from Vect_k to Vect_k. Given a fixed vector space V, Hom(-,V) is the functor that sends an object W in Vect_k to the set of linear maps from W to V, which is itself a vector space. The first of these is contravariant and the second is covariant (or the other way, I always get it wrong), and it's easy to write down why. It's a nice pair of examples in an easy-to-understand category. --[[User:MarkGall|MarkGall]] 17:00, 1 October 2009 (EDT)