Adjoint split extensions and generalized split extensions

n the last talk, we explored a notion of extension for categories, which was at the same time exhibited as an extension on the level of categories, but also as a natural family of extensions on the level of objects. In this talk we will consider adjoint split extensions of categories, and observe that on the level of objects, we get an associated natural family of split extensions in a suitably generalized sense, reducing to the usual notion of split extension under sufficient assumptions. The usefulness of adjoint split extensions is due to there being an associated notion of semidirect product of categories. These notions have been considered before in especially well-behaved cases, such as for frames [FM2020] or toposes [FMS2023], but not in general, as far as we can tell.

[FM2020] P. Faul, G. Manuell, Artin glueings of frames as semidirect products, J. Pure Appl. Algebra 224 (2020), no.8, Paper No. 106334.
[FMS2023] P. Faul, G. Manuell, J. Siqueira, Artin glueings of toposes as adjoint split extensions, J. Pure Appl. Algebra 227 (2023), no.5, Paper No. 107273.