Analytic functors between presheaf categories over groupoids

The paper studies analytic functors between presheaf categories. Generalising results of A. Joyal [11] and R. Hasegawa [9] 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.