Re: looking for a reference...

Re: looking for a reference...

[Dana Scott]

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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Re: Re: looking for a reference...

[Eduardo J. Dubuc]

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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