free cocompletion

free cocompletion

[Paul B Levy]

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


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Re: free cocompletion

[Tom Hirschowitz]

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

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Re: Free cocompletion

[Andree EHRESMANN]

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

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Free cocompletion

[Sam Dean]

I would also be interested in whether someone has written down the proof in
the additive setting: for a preadditive category A the category of additive
functors  (A^op,Ab) is the free cocompletion of A with respect to additive
functors.

Maybe something general has been written somewhere.

Many thanks
Sam


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