Analytic functors between presheaf categories over groupoids

TL;DR

This paper characterizes analytic functors between presheaf categories over groupoids, extending previous results from sets to groupoids, and establishes a 2-category structure for these functors.

Contribution

It generalizes the concept of analytic functors from sets to presheaf categories over groupoids, providing new characterizations and a 2-category framework.

Findings

Analytic functors preserve filtered colimits, quasi-pullbacks, and cofiltered limits.

Two equivalent characterizations of analytic functors over groupoids.

Small groupoids and their analytic functors form a 2-category.

Abstract

The paper studies analytic functors between presheaf categories. Generalising results of A. Joyal and of R. Hasegawa for analytic endofunctors on the category of sets, we give two characterisations of analytic functors between presheaf categories over groupoids: (i) as functors preserving filtered colimits, quasi-pullbacks, and cofiltered limits; and (ii) as functors preserving filtered colimits and wide quasi-pullbacks. The development establishes that small groupoids, analytic functors between their presheaf categories, and quasi-cartesian natural transformations between them form a 2-category.