Question in Category Theory

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Diogo Dias
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Question in Category Theory

Postby Diogo Dias » Mon Dec 01, 2014 4:31 am UTC

Hi. I have two problems and I need some help or ideias on how to solve them.

Suppose I have the following Category:

- the objects are structures (X, Cn)

i) X is any set
ii) Cn is a map from the power set of X to the power set of X

- The arrows are defined as follows:

Given (X, Cn) and (X', Cn'), an arrow t goes from X to X', and is injective, such that for all A subset of X, t(Cn(A))=Cn'(t(A)).

Now, consider the following definition:
Given (X, Cn), and A subset of X, Cnp(A)= U{Cn(A')/A' subset of X, and Cn(A')≠ X}

We define the functor F as
i) F(X, Cn) = (X, Cnp)
ii) F(t) = t.

My problems are:

1) Is this functor idempotent, that is, F(F(Cn))=F(Cn) and F(F(t))=F(t)?
2) How can I define product in these categories?

Thanks!

mike-l
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Re: Question in Category Theory

Postby mike-l » Fri Dec 05, 2014 4:08 am UTC

If I'm reading this right, Cnp is independent of it's input so is a constant function?

In which case it would seem F is not idempotent. Suppose Cn is the identity. Then Cnp is identically X, thereby applying F again gives Cnpp identically the empty set.

Products are also troublesome. The subcategory of objects where Cn is the identity is just the category of sets and injective maps, which does not have products (though it does have coproducts)

I may be misreading (or just wrong) on one or both of these, I'm pretty out of practise
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z4lis
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Re: Question in Category Theory

Postby z4lis » Fri Dec 05, 2014 2:26 pm UTC

Something seems wrong with your definition of Cnp as written. It must be defined on all subsets of X, and the way you have it written looks like a union indexed by A' which are subsets of X with Cn(A')=X. This union may be empty, and in any case the expression is independent of A.
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Diogo Dias
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Joined: Mon Dec 01, 2014 3:51 am UTC

Re: Question in Category Theory

Postby Diogo Dias » Sat Dec 06, 2014 10:49 pm UTC

Hi. Thanks for the help.
You are right. There's a mistake with the definition of Cnp.
The right definition is:
Cnp(A)= U{Cn(A')/A' subset of A, and Cn(A')≠ X}

I'm a logician student and the ideia behind this is that each structure represents a logical system, and the functor creates another category with just the 'consistent' sets of the original logic.
I'm new with category theory, so I'm having trouble translate these ideas in this framework.

How can I define coproducts?

Thanks again for the help!

mike-l
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Joined: Tue Sep 04, 2007 2:16 am UTC

Re: Question in Category Theory

Postby mike-l » Sat Dec 06, 2014 11:12 pm UTC

Coproducts should probably just be disjoint union on everything. Products still won't work for the same reason I mentioned above.

Idempotence might be true with the new definition. Some obvious attmepts to find counterexamples fail. I'll give some thought to it

Edit: Let X ={1,2}, Cn{1}={1}, Cn{} = {2}

Then Cnp {1} = X and Cnp{} = {2}
Cnpp {1} = {2}

So still not idempotent. Are there maybe any restrictions on Cn?
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Diogo Dias
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Joined: Mon Dec 01, 2014 3:51 am UTC

Re: Question in Category Theory

Postby Diogo Dias » Sun Dec 07, 2014 2:56 am UTC

Thanks again!
I'm trying not to impose any restriction on Cn, to mantain things really abstract. But maybe it's necessary, to get some interesting results, to impose some properties like monotonicity, inclusion and idempotency, that is, to suppose that the Cn is Tarskian. Do you think this will help?

mike-l
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Joined: Tue Sep 04, 2007 2:16 am UTC

Re: Question in Category Theory

Postby mike-l » Mon Dec 08, 2014 5:51 am UTC

I'm not sure exactly what that means. The problem with idempotence right now is essentially only the example I posted, if Cn applied to all subsets of a subset yields the entire set, but is only able to do so using elements from the entire subset.

So one condition that would make F idempotent is that: If A is no empty and if x is in Cn(A) then x is in Cn({y}) for some y in A.

I'm not sure if this is a reasonable condition or not. There are weaker conditions as well, but they are more complicated.
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Diogo Dias
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Joined: Mon Dec 01, 2014 3:51 am UTC

Re: Question in Category Theory

Postby Diogo Dias » Thu Dec 11, 2014 12:55 pm UTC

In your example, why Cnpp(1)={2}?

mike-l
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Joined: Tue Sep 04, 2007 2:16 am UTC

Re: Question in Category Theory

Postby mike-l » Thu Dec 11, 2014 7:57 pm UTC

Cnpp{1} includes Cnp{}, but Cnp{1} is excluded as it is all of X
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