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Is there a standard reference for the theory (if it exists) of $\mathcal{V}$-enriched locally presentable categories? Here $\mathcal{V}$ is a cosmos. Does anything unexpected happens here in contrast to the case $\mathcal{V}=\text{Set}$ treated in "Locally Presentable And Accessible Categories" by Adamek & Rosicky? In particular I'm interested in the case $\mathcal{V} = \text{Cat}$.

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up vote 8 down vote accepted

The standard reference is this paper by Max Kelly.

Perhaps the most unexpected thing is how well the theory works!

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Thanks ! – Martin Brandenburg Jan 27 '11 at 10:09

The theory is also developed further in two papers by Borceux, Quinteiro and Rosický: Enriched accessible categories MR1419612 and A theory of enriched sketches MR1624638. In the second one they say that they are particularly interested in case $\mathcal{V} = \text{Cat}$ and that they "intend to study this in a further publication", but I don't know of such a publication.

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