On split extensions of hoops

Internal actions have been introduced by F. Borceux, G. Janelidze, and G. M. Kelly as a means to generalize the connection between actions and split extensions from groups and Lie algebras to arbitrary semi-abelian categories. However, in certain settings such as Orzech categories of interest internal actions are often expressed in terms of external actions, i.e., via a set of maps which satisfy a certain set of identities. In this talk, we are gonna study external actions and split extensions in the category Hoops of hoops, with a focus on those split extensions which strongly splits. Split extensions with strong section in the category Hoops can be described in terms of strong external actions. We prove that there is bijection between the set EAct_ss(B, X) of strong external actions of B on X and the set SplExt_ss(B, X) of isomorphism classes of split extensions of B by X that strongly splits.