AI 中文总结
本文在辫子幺半范畴中证明：双幺半群为单侧Hopf幺半群且对映体为反同态，当且仅当自由Hopf模函子是Frobenius的，并给出自由单侧Hopf幺半群的构造及例子。
AI 中文摘要
证据表明，在类双代数结构上存在类似对映体的映射，与相关的自由Hopf模函子的Frobenius性质之间存在紧密联系。本文证明，在满足少数温和假设的任意辫子幺半范畴中，一个双幺半群是单侧Hopf幺半群且对映体是双幺半群反同态，当且仅当自由Hopf模函子是Frobenius的。我们对此类结构的兴趣源于在辫子非平凡的情境中发现了单侧Hopf幺半群的真实例子。事实上，我们在具有一些非限制性附加假设的任意对称幺半范畴中，提供了余幺半群上的自由单侧Hopf幺半群的构造。我们给出了该构造的若干例子，并描述了由此产生的单侧（有时是双侧）Hopf幺半群。
英文摘要
Evidence suggests a tight connection between the existence of antipode-like maps on bialgebra-like structures, and the Frobenius property for the associated free Hopf module functor. In this paper, we prove that in any braided monoidal category satisfying few mild assumptions, a bimonoid is a one-sided Hopf monoid and the antipode is a bimonoid anti-homomorphism, if and only if the free Hopf module functor is Frobenius. Our interest in these structures stems from having found genuine examples of one-sided Hopf monoids in contexts where the braiding is non-trivial. In fact, we provide a construction of the free one-sided Hopf monoid over a comonoid in any symmetric monoidal category with some non-restrictive additional assumptions. We present several examples of this construction, and describe the resulting one-sided (sometimes two-sided) Hopf monoids.
Comments39 pages. The replacement is due to an issue with the zref-clever package
