On pseudofunctors sending groups to 2-groups

TL;DR

This paper characterizes pseudofunctors from a category with finite products to Cat that extend to 2-groups, focusing on conditions related to cartesian monoidal opfibrations and additive categories.

Contribution

It provides a characterization of pseudofunctors that extend to 2-groups, linking properties of opfibrations and the additive structure of the base category.

Findings

Pseudofunctors with cartesian monoidal opfibrations are characterized.

Extension to 2-groups occurs when the opfibration has groupoidal fibers in additive categories.

The paper offers criteria for when pseudofunctors from internal groups to 2-groups exist.

Abstract

For a category B with finite products, we first characterize pseudofunctors from B to Cat whose corresponding opfibration is cartesian monoidal. Among those, we then characterize the ones which extend to pseudofunctors from internal groups to 2-groups. If B is additive, this is the case precisely when the corresponding opfibration has groupoidal fibres.