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Categorical Ambidexterity

Shay Ben-Moshe

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 $π$-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.

Categorical Ambidexterity

Abstract

We prove an ambidexterity result for -categories of -categories admitting a collection of colimits. This unifies and extends two known phenomena: the identification of limits and colimits of presentable -categories indexed by a space, and the -semiadditivity of the -category of -categories with -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.

Paper Structure

This paper contains 2 sections, 1 theorem, 1 equation.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem A

Let $\mathcal{C}_\bullet\colon X \to \mathrm{Cat}_\mathcal{K}$ be a diagram indexed by a space $X \in \mathcal{K}_0$. There is a canonical equivalence

Theorems & Definitions (1)

  • Theorem A: \ref{['lim-colim']}