Re: characterization of flp endofucntors on Set?

Re: characterization of flp endofucntors on Set?

[Thomas Streicher]

Dear Andr'e,

I think we can't do better than exhibiting Hom(L,-) as the directed
colimit of the Set(X,-) indexed by f : L -> X.

> I have a question regarding certain finite limit preserving functors Set-->Set.
>
> If L is a locale, then the functor Hom(L,-):Set-->Set preserves finite limits,
> where  Hom(L,X) denotes the set of  morphisms of locales L-->X for a discrete locale X.
> Is there is a simple characterization of these flp functors?

I think I should reveil the background of my question. Look at p.7 of my
https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/UniGround.pdf
for formulations of my conditions Tr1-Tr3. I think Hom(L,-) validates
(Tr1) and (Tr2) but presumably not (Tr3) (for EE = SS =Set).

I am looking for a functor F : Set->Set validating all 3 conditions
but not being equivalent to Id_Set.
Butmaybe there is none?

Best, Thomas


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Re: characterization of flp endofucntors on Set?

[Joyal, André]

To Thomas and Richard,

I have a question regarding certain finite limit preserving functors Set-->Set.

If L is a locale, then the functor Hom(L,-):Set-->Set preserves finite limits, 
where  Hom(L,X) denotes the set of  morphisms of locales L-->X for a discrete locale X. 
Is there is a simple characterization of these flp functors?

Best,
André



________________________________________
From: Thomas Streicher [streicher@mathematik.tu-darmstadt.de]
Sent: Tuesday, October 23, 2018 4:56 AM
To: Richard Garner
Cc: categories@mta.ca
Subject: categories: Re: characterization of flp endofucntors on Set?

Dear Richard,

thanks for your answer. Offline I have received a reply by Jonas Frey
which answers my question satisfactorily.
Let U be a Groth. universe then Lex(U,Set) consists of filtered/directed
colimits of representables. Accordingly, Lex(U,U) is equivalent to the
full subcat of Set^U on U-small directed colimits of representables.

But that sounds related to what Blass says, isn't it.

Best, Thomas



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Re: characterization of flp endofucntors on Set?

[Thomas Streicher]

Dear Richard,

thanks for your answer. Offline I have received a reply by Jonas Frey
which answers my question satisfactorily.
Let U be a Groth. universe then Lex(U,Set) consists of filtered/directed
colimits of representables. Accordingly, Lex(U,U) is equivalent to the
full subcat of Set^U on U-small directed colimits of representables.

But that sounds related to what Blass says, isn't it.

Best, Thomas


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Re: characterization of flp endofucntors on Set?

[Richard Garner]

To be clear: by "subdirect product" I mean "reduced product by a filter"
and in this case all the sets are the same so I suppose I should really
say "reduced power".
On Tue, Oct 23, 2018, at 10:58 AM, Richard Garner wrote:
> Dear Thomas,
>
> I don't know a complete answer to your question - I think there is
> probably not an entirely elementary characterisation. The partial
> answers I am aware of (which probably you are aware of too) have a lot
> to do with filters.>
> - Trnkova classifies in "On descriptive classification of set-functors
>   I" different kinds of limit-preserving endofunctor of Set. In
>   particular, she shows that an endofunctor of set preserves finite
>   limits if and only if it preserves finite products and is not the
>   reflector of Set into 0<=1.>
> - Blass in "Exact functors and measurable cardinals" shows that any
>   finite-limit preserving endofunctor of Set is a directed union of
>   subdirect products. Of course, this is not so far away from the "for-
>   free" characterisation of the flp endofunctors of Set as ind-objects
>   in Set^op.>
> - Rather trivially, an flp and finitary endofunctor of Set must be of
>   the form FA = { continuous maps X ---> disc(A) of finite support }
>   for some Stone space X, because flp finitary endofunctors of Set are
>   the same as flp functors Set_f-->Set, and these are all of the form
>   Stone(X,-):Set_f-->Set. I don't how to extend this even to
>   endofunctors of rank aleph_1.>
> My knowledge of the literature in this area is patchy, so I am also
> interested to see what other answers you might receive!>
> Richard
>
>

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Re: characterization of flp endofucntors on Set?

[Richard Garner]

Dear Thomas,

I don't know a complete answer to your question - I think there is
probably not an entirely elementary characterisation. The partial
answers I am aware of (which probably you are aware of too) have a lot
to do with filters.
- Trnkova classifies in "On descriptive classification of set-functors
   I" different kinds of limit-preserving endofunctor of Set. In
   particular, she shows that an endofunctor of set preserves finite
   limits if and only if it preserves finite products and is not the
   reflector of Set into 0<=1.
- Blass in "Exact functors and measurable cardinals" shows that any
   finite-limit preserving endofunctor of Set is a directed union of
   subdirect products. Of course, this is not so far away from the "for-
   free" characterisation of the flp endofunctors of Set as ind-objects
   in Set^op.
- Rather trivially, an flp and finitary endofunctor of Set must be of
   the form FA = { continuous maps X ---> disc(A) of finite support } for
   some Stone space X, because flp finitary endofunctors of Set are the
   same as flp functors Set_f-->Set, and these are all of the form Stone(X,-):Set_f--
   >Set. I don't how to extend this even to endofunctors of rank aleph_1.
My knowledge of the literature in this area is patchy, so I am also
interested to see what other answers you might receive!
Richard





On Mon, Oct 22, 2018, at 10:14 PM, Thomas Streicher wrote:
> One easily shows that up to isomorphism the functors from Set to Set
> which preserves small limits are up to iso of the form (-)^I for some> set I.
> Is there known a similarly elementary characterization of FINITE limit> preserving functors from Set to Set?
>
> Thomas
>

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characterization of flp endofucntors on Set?

[Thomas Streicher]

One easily shows that up to isomorphism the functors from Set to Set
which preserves small limits are up to iso of the form (-)^I for some
set I.
Is there known a similarly elementary characterization of FINITE limit
preserving functors from Set to Set?

Thomas


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