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I would like to show that the category of sets and spans between them, seen as a $(2,1)$-category, is Cauchy complete, i.e. has splitting of (homotopicaly coherent) idempotent.

Ideally I would also like to prove that the retract of a set $X$ in this bi-category are all subsets of $X$, and that all this also works if we replace set by a category with finite limits.

I think I have a poof of this (If I'm not mistaken), but it is very long and technical and I wanted to know if there is literature about this, or if there is a simpler or more conceptual way to obtain these results.

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  • $\begingroup$ I think I worked something out here but ... erm ... I definitely skipped some bits that seemed technical :) $\endgroup$ – Tim Campion Mar 30 '20 at 18:29

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