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I am wondering whether any enriched versions of ind- and pro- completions have been studied? I can not find any literature on them, even though I believe people (most likely the Australian school) must have cared about them in the previous years.

I would be grateful for any reference.

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    $\begingroup$ Have you looked at [Kelly, Structures defined by finite limits in the enriched context] or [Borceux and Quinteriro, Enriched accessible categories]? $\endgroup$
    – Zhen Lin
    Commented May 16, 2016 at 11:03

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The recent series of papers by Lack and Tendas are good references for enriched accessible categories:

You can also find relevant older references among the citations in these papers.

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