If you take C to be the dual of the category of models of a theory in universal algebra, and F to be the free functor (in this case F is covariant), then you will find a lot of familiar facts. I like the mathematics in your posting for its simplicity, and the idea of using contravariant functors very good. Whether it rings bells in some people or not, it is certainly a nice and worthwhile stuff just as it is, write it for the arXiv, and may be you can publish it somewhere afterwards. best e.d. On 13/09/14 21:28, Dana Scott wrote:
If you have comments/suggestions, please reply to Mr. Kruckman. Thanks.
On Sep 13, 2014, at 10:02 AM, Alex Kruckman<kruckman@gmail.com> wrote:
Professor Scott,
In writing up some work I did with another graduate student, we?ve noticed that one argument is really a special case of a very general fact. It's easy to prove, and it's quite nice, but I've never seen it explicitly noted. Have you?
Here it is:
1. Suppose we have a contravariant functor F from Sets to some other category C which turns coproducts into products. This functor automatically has an adjoint, given by G(-) = Hom_C(-,F(1)), where 1 is the one element set. If you like, the existence of G is an instance of the special adjoint functor theorem, but it's also easy to check by hand. The key thing is that every set X can be expressed as the X-indexed coproduct of copies of the one element set, so we have (the = signs here are natural isomorphisms):
Hom_C(A,F(X)) = Hom_C(A,F(coprod_X 1)) = Hom_C(A,prod_X F(1)) = prod_X Hom_C(A,F(1)) = prod_X G(A) = Hom_Set(X,G(A))
2. Now let's say the category C is the category of algebras in some signature. Let's call algebras in the image of F "full", and let's say we're interested in the class K of subalgebras of full algebras. This class is closed under products and subalgebras, so if it's elementary, then it has an axiomatization by universal Horn sentences (i.e. it's a quasivariety), and moreover every algebra in the class is a subalgebra of a product of copies of F(1), so a universal Horn sentence is true of every algebra in the class if and only if it's true of F(1).
3. Okay, let's say we have an axiomatization T for K. Then we have a ?representation problem": given an algebra A satisfying T, embed it in some full algebra. Well, there's a canonical such embedding, given by the unit of the adjunction A -> F(G(A)). That is, A -> F(Hom_C(A,F(1))).
Examples of these observations include all the constructions of algebras from sets by powerset - the Stone representation theorem for Boolean algebras (minus the topology, of course), but also the representation theorems for lattices, semilattices, etc.
Thanks for taking the time to read this. Let me know if it rings a bell.
-Alex
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