On split extensions of hoops

Internal actions have been introduced in [3] 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 [20] 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 [19]. 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.