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Reid Barton (Dec 04 2020 at 23:48):Is there a standard reference for the following fact:
Let be a small category and a set of colimit cocones in ; then the category of presheaves on which take the cocones of to limit cones is the free cocompletion of subject to the condition that the colimit cones of are preserved.
Reid Barton (Dec 04 2020 at 23:48):(Preferably a reference where understanding the statement of the theorem is easier than proving it yourself. I'm looking at you Basic Concepts of Enriched Category Theory...)
John Baez (Dec 05 2020 at 01:29):You haven't seen nothing yet - wait 'til Advanced Concepts of Enriched Category Theory comes out. :upside_down:
Nathanael Arkor (Dec 05 2020 at 03:57):The standard reference is Theorem 6.23 of Basic concepts, but a more approachable reference may be Velebil–Adámek's A remark on conservative cocompletions of categories, which at least sketches the proof.
Nathanael Arkor (Dec 05 2020 at 04:00):(I moved John's comment, because it seemed clearly in reply to this topic.)
Reid Barton (Dec 05 2020 at 13:24):I think neither of these sources really contains the exact statement above, because they both construct the cocompletion by some kind of transfinite construction (e.g., closure under colimits), which is necessary if you want to adjoin only certain types of colimits. In this special case of a small category and adjoining all colimits, there's this direct description of the cocompletion, and it takes more work to extract this description from the general construction (which is not hard, but then it is not a hard theorem to begin with).
Reid Barton (Dec 05 2020 at 13:26):I came across https://mathoverflow.net/questions/349409/universal-property-of-the-cocomplete-category-of-models-of-a-limit-sketch,
and thereby Theorem 2.2.4 of https://arxiv.org/pdf/1105.3104.pdf, which might be the best reference.
Nathanael Arkor (Dec 05 2020 at 15:23):Ah, that's true. In that case, a reference is Proposition 11.4 and Theorem 11.5 of Fiore's Enrichment and Representation Theorems for Categories of Domains and Continuous Functions.
Reid Barton (Dec 06 2020 at 12:36):Perfect, thanks!