AI 中文总结
本文将余交换Hopf代数范畴的原模范性扩展到拟三角情形，通过修改对极概念等方法，研究双幺半范畴中Hopf幺半群相关性质，证明了一些分解和引理，得到多个范畴的原模范性及相关结果，并扩展了双交叉积。
AI 中文摘要
在这项工作中，我们将余交换Hopf代数范畴的原模范性扩展到拟三角情形。每个拟三角Hopf代数都有一个最小拟三角Hopf子代数，并且如我们所示，可被视为其双模的张量编织双幺半范畴中的余交换双幺半群。这使我们在双幺半范畴中Hopf幺半群的更广泛背景下研究原模范性。为此，我们对与一个反转相关的Böhm的对极概念进行了轻微修改，进一步细化了Böhm - Lack早期的概念。该框架使我们能够研究此情形下的Hopf幺半群、伽罗瓦和余伽罗瓦映射以及Hopf幺半群的分解。利用这些工具，我们证明了点的分解、分裂短五引理以及在任何具有反转的张量编织双幺半范畴中，余交换Hopf幺半群范畴中分裂满态射沿任意态射的拉回的存在性；因此该范畴是原模范的。作为应用，在温和假设下我们恢复了对称幺半范畴中余交换Hopf代数的原模范性，并且得到了在固定子对象下拟三角（分别地，三角）Hopf代数的余切片范畴的原模范性；在三角情形下，这可追溯到本文引入的广义内部群范畴。当固定子对象最小时，我们推断出其最小拟三角Hopf子代数与固定子对象同构的拟三角Hopf代数范畴的原模范性，我们将其解释为一个函子的本质纤维的原模范性。作为副产品，我们的结果将余交换Hopf代数的双交叉积扩展到了拟三角情形。
英文摘要
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 Böhm's notion of antipode associated with a reversion, further refining an earlier one due to Böhm-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.
