Categorical Ambidexterity

We prove an ambidexterity result for $\infty$-categories of $\infty$-categories admitting a collection of colimits. This unifies and extends two known phenomena: the identification of limits and colimits of presentable $\infty$-categories indexed by a space, and the $\infty$-semiadditivity of the $\infty$-category of $\infty$-categories with $π$-finite colimits proven by Harpaz. Our proof employs Stefanich’s universal property for the higher category of iterated spans, which encodes ambidexterity phenomena in a coherent fashion.

💡 Research Summary

The paper “Categorical Ambidexterity” establishes a unified ambidexterity theorem for the ∞‑category Cat_K of ∞‑categories that admit a prescribed collection K of colimits and functors preserving them. The author shows that for any diagram C· : X → Cat_K indexed by a space X belonging to a subcategory K₀ of spaces (closed under pullbacks and containing the point), the colimit and limit of the diagram are canonically equivalent: \