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John Baez (Sep 04 2025 at 14:13):I seem to recall a reference to something like this on the nLab but I can't find it:
Suppose X is some class of colimits closed under composition - e.g. a finite colimit of finite colimits is a finite colimit. Then the 'free category with X-colimits' on a category C is the full subcategory of the presheaf category on C consisting of X-colimits of representables.
Does anyone know where this has been precisely stated and proved?
John Baez (Sep 04 2025 at 14:15):I happen to need it in just two cases: where X is 'finite colimits' and where X is 'finite coproducts'.
Nathanael Arkor (Sep 04 2025 at 15:48):Theorem 5.35 of Kelly's Basic Concepts.
John Baez (Sep 04 2025 at 16:21):Great, thanks! That terrifying book... okay, good, that theorem statement looks locally comprehensible.
John Baez (Sep 04 2025 at 16:29):Excellent, that's just what I need!
By the way, all I needed was this: a proof that
is the same as
and that both are equivalent to the category of finite sets over the set . This should do the job.
I just needed this fact for something....