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Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits? (in which T is a set of weights)

Thank you in advance

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    $\begingroup$ What are F-(weighted) limits? $\endgroup$
    – David Roberts
    Jul 14, 2015 at 13:48
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    $\begingroup$ What does "F" refer to? I don't recall F-categories appearing in Kelly's book. Do you simply mean that you're interested in limits / colimits weighted by a (co)presheaf? If so, are you interested in the enriched case? $\endgroup$
    – Tim Campion
    Jul 14, 2015 at 13:48
  • $\begingroup$ Sorry. I am aware of F-categories. But, instead, I was using a wrong notation for a set of weights. Thank you for your answer. I am going to take a look in both references. $\endgroup$
    – Fernando
    Jul 15, 2015 at 5:59

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An extensive study of such cocompletions (thus also their duals - completions), together with some characterisation results, is given in the paper:

"A Representation Results for Free Cocompletions" by J. Power, G.L. Cattani and G. Winskel

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In the case of enriched categories, a review of the free cocompletion theory can be found in Kelly and Schmitt's Notes on Enriched Categories with colimits of some class. I believe this is a more down-to-earth perspective than Power/Cattani/Winskel's 2-categorical approach.

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