Protomodularity of cocommutative Hopf monoids in duoidal categories and quasitriangular Hopf algebras

Abstract

In this work, we extend the protomodularity of the category of cocommutative Hopf algebras to the quasitriangular setting. Every quasitriangular Hopf algebra admits a minimal quasitriangular Hopf subalgebra and, as we show, can be regarded as a cocommutative bimonoid in the tensor-braided duoidal category of bimodules over it. This leads us to investigate protomodularity in the broader context of Hopf monoids in duoidal categories. To this end, we adopt a slight modification of B\"ohm's notion of antipode associated with a reversion, further refining an earlier one due to B\"ohm-Lack. This framework allows us to study Hopf monoids in this setting, Galois and co-Galois maps, and the factorization of Hopf monoids. Using these tools, we prove a factorization of points, the Split Short Five Lemma, and the existence of pullbacks of split epimorphisms along arbitrary morphisms in the category of cocommutative Hopf monoids with monic unit in any tensor-braided duoidal category with a reversion; hence this category is protomodular. As applications, we recover the protomodularity of cocommutative Hopf algebras in symmetric monoidal categories under mild assumptions, and we obtain that of the coslice category of quasitriangular (resp. triangular) Hopf algebras under a fixed subobject; in the triangular case, this can be traced back to a category of generalized internal groups, introduced in the present work. When the fixed subobject is minimal, we infer the protomodularity of the category of quasitriangular Hopf algebras whose minimal quasitriangular Hopf subalgebra is isomorphic to the fixed subobject, which we interpret as the protomodularity of an essential fibre of a functor. As a byproduct, our results extend the double cross product of cocommutative Hopf algebras to the quasitriangular setting.