# Retracts in the bicategory of spans

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.

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