Re: characterization of flp endofucntors on Set?

[Richard Garner <richard.garner@mq.edu.au>]

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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