On Pseudofunctors Sending Groups to 2-Groups

Abstract

For a category \({\textsf{B}}\) with finite products, we first characterize pseudofunctors from \({\textsf{B}}\) to \(\mathbb {C}\textsf{at}\) whose associated opfibration is Cartesian monoidal. Among those, we then characterize the ones which extend to pseudofunctors from internal groups to 2-groups. If \({\textsf{B}}\) is additive, this is the case precisely when the associated opfibration has groupoidal fibres.