Hi, the following result appears to be folklore: Given a locally small category C, the full subcategory of [C^op,Set] on small presheaves (i.e. those presheaves that are colimits of a small diagram of representables) is a free cocomplete locally small category on C. I've seen and heard this result in many places, and know how to prove it, but is there a proof written out in the literature? And where did the statement first appear? Paul -- Paul Blain Levy School of Computer Science, University of Birmingham http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi Paul, This is Prop 1.45 in Adamek and Rosicky. Surely that's not the first appearance of the result, but I don't know the answer to this question. Cheers, Tom
Hi, the following result appears to be folklore:
Given a locally small category C, the full subcategory of [C^op,Set] on small presheaves (i.e. those presheaves that are colimits of a small diagram of representables) is a free cocomplete locally small category on C.
I've seen and heard this result in many places, and know how to prove it, but is there a proof written out in the literature? And where did the statement first appear?
Paul
-- Paul Blain Levy School of Computer Science, University of Birmingham http://www.cs.bham.ac.uk/~pbl
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Sam, Theorems on free (co)completion of categories and internal categories, with explicit constructions, are given in Charles Ehresmann's paper: "Sur l'existence de structures libres et de foncteurs adjoints" (Cahiers IX, 1967) reprinted in his "Oeuvres" http://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/C.E_Works.htm Part IV-1, where I have added some bibliographical notes (Comment 199, p. 368). These results have been generalized to free 'relative' (co)completions in our joint paper "Categories of sketched structures" (Cahiers, 1972), reprinted in the "Oeuvres" part IV-2 (407-517). Sincerely Andree [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Andree EHRESMANN -
Paul B Levy -
Tom Hirschowitz