There are a number of constructions involving categories that behave similarly to semidirect products of groups. These include Artin glueings of frames (viewed as thin categories) or toposes, and the decomposition of the category of cocommutative Hopf algebras (over a field of characteristic zero) into the categories of Lie algebras and groups (by the Cartier-Kostant-Milnor-Moore theorem). The basic idea is that for every morphism in the 1-dimensional split extension diagram we have a left adjoint functor between categories. Moreover, the splitting is left adjoint to the cokernel map. In the 1-categorical setting, the good case is when the kernel and the splitting are jointly extremally epic; in the 2-dimensional setting, we instead ask that right adjoints of the kernel and the splitting are jointly conservative. If we restrict our consideration to regular pointed protomodular categories, we find that every adjoint extension is conservative in this sense, and hence the 2-category of regular pointed protomodular categories exhibits a 2-dimensional analogue of protomodularity. A similar (but dual) result holds for toposes.

This is joint work with Nelson Martins-Ferreira and Ülo Reimaa.

(The talk  will be 30 minutes long.)