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Let $\mathbf C$ be a category with finite limits. Then a left exact functor $F\colon \mathbf C\to \mathbf{Set}$ is pro-representable and hence extends to the pro-completion $\mathbf C$. My question is whether it is true that the extension of $F$ preserves finite colimits whenever $F$ does, and if so what is a reference? I'm also curious about arbitrary colimits but for the application I care about, finite colimits (or even coequalizers) are enough.

flag	asked Jul 2 2011 at 1:01 Benjamin Steinberg 10.7k●1●14●49

	what do you mean here by the extension of $F$ to $pro(C)$ the pro-completion of $C$?? $\mathbf{Set}$ has cofiltered limits, so any functor $G \colon C \to \mathbf{Set}$ extends to the pro-completion (this is a free completion) regardless of $F$ being left exact or not. – Eduardo Pareja Tobes Jul 22 2011 at 16:35
	Under my assumptions F is representable on pro(C). I want to know whether this representable functor on pro(C) preserves finite colimits if F did on C. – Benjamin Steinberg Jul 22 2011 at 17:37
	Ok, I see what you mean; just a little clarification: by "this representable functor on $pro(C)$" you mean 1. $pro(C)(-,F)\colon pro(C)^{op} \to \mathbf{Set}$ or 2. $pro(C)(F,-)\colon pro(C) \to \mathbf{Set}$?? – Eduardo Pareja Tobes Jul 24 2011 at 19:13
	The covariant representable, i.e., option 2. – Benjamin Steinberg Jul 25 2011 at 18:27

Tim Porter · Answer 1 · 2011-12-17 21:01:06Z UTC

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I am not certain if he provides the answer to your question, but Dan Isaksen's paper: Calculating limits and colimits in pro-categories, Fundamenta Mathematicae 175 (2002) 175--194. is relevant I think. The point is that the reindexing lemmas in procategories provide a powerful tool for calculating limits and colimits.

answered Dec 17 2011 at 21:01

Tim Porter
3,735●6●14

	Tim, thanks. I will take a look. – Benjamin Steinberg Dec 18 2011 at 13:29
	No guarantees but it makes some good points. Reindexing lemmas are very important in pro-category theory in order to prove structural lifting results. – Tim Porter Dec 18 2011 at 15:07

Does the extension to the pro-completion of a left exact (finite) colimit preserving functor preserve (finite) colimits

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