Re: characterization of flp endofucntors on Set?

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

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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