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Came across this categorification of pro-object.
I’ve been sniffing around a categorified Stone/Isbell duality at the nCafe, hence the interest in 2-pro-objects. I see it hasn’t been taken up much. There’s a use in
to construct the étale fundamental pro-groupoid of a flat finitely presented algebraic stack.
I wonder what $2Pro(FP Cat)$ is for finitely-presented categories.
For what its worth in the original work on shape theory, predating Borsuk by a long time, D. E. Christie (1944) used a homotopy version of 2-pro-object to define his invariants. The idea of homotopy coherent pro-objects was used by Batanin in his work on strong shape theory and of course, comes into Lurie’s work.
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