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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?
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.
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.
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