That’s ’absolutely’ neat, complete, and reflects my limits. Thank you!

For my own reference:

“absolute” means that the weight has an adjoint

That this gives the expected result is a theorem due to Street.

]]>Yes. The short answer is that an object is Cauchy complete if every left adjoint proarrow into it is representable, and the Cauchy completion is the universal map into a Cauchy complete object. This way of phrasing Cauchy completeness can equivalently be stated as the existence of all absolute colimits, where “colimit” has its usual meaning inside a proarrow equipment (/ framed bicategory) and “absolute” means that the weight has an adjoint. Of course, in an arbitrary equipment, not every object may have a Cauchy completion.

]]>Has anyone worked out the universal property of the Cauchy completion of an enriched category?

What I have in mind is a definition that makes sense in any (suitable) framed bicategory, such as $\mathcal{V}$-Prof. Is there a notion of when an object in a framed bicategory has all absolute limits, formulated in terms of something like absolute Kan extensions? If so, is there a definition of Cauchy completion in these terms?

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