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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