Categorical Ambidexterity

TL;DR

This paper proves a unifying ambidexterity result for certain $infty$-categories, extending known phenomena and employing Stefanich's universal property to encode ambidexterity coherently.

Contribution

It introduces a general ambidexterity theorem for $infty$-categories with colimits, unifying previous results and employing a novel proof technique.

Findings

Unifies limits and colimits in presentable $infty$-categories.

Extends $infty$-semiadditivity to broader contexts.

Provides a new proof approach using Stefanich's universal property.

Abstract

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 $\pi$-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.