论文标题
单体类别中的关键对象及其HOPF MONADS
Pivotal Objects in Monoidal Categories and Their Hopf Monads
论文作者
论文摘要
单型类别中的对象$ p $ $ \ mathcal {c} $，如果其左对二和右二对象是同构的，则称为关键。给定这样的对象和双$ q $的选择，我们以兼容的方式构造了与$ p $和$ p $和$ q $的对象的类别$ \ mathcal {c}（p，q）$。我们表明，当$ \ Mathcal {c} $关闭时，此类别将提起$ \ Mathcal {C} $的单体结构和$ \ Mathcal {C} $的封闭结构。如果$ \ MATHCAL {C} $具有合适的colimits，我们表明$ \ Mathcal {C}（p，q）$是monadic，从而在任意封闭的单体类别上构建了一个hopf monad家族$ \ Mathcal {c} $。我们还介绍了单体类别的关键覆盖物，并将我们的工作扩展到任意关键图。
An object $P$ in a monoidal category $\mathcal{C}$ is called pivotal if its left dual and right dual objects are isomorphic. Given such an object and a choice of dual $Q$, we construct the category $\mathcal{C}(P,Q)$, of objects which intertwine with $P$ and $Q$ in a compatible manner. We show that this category lifts the monoidal structure of $\mathcal{C}$ and the closed structure of $\mathcal{C}$, when $\mathcal{C}$ is closed. If $\mathcal{C}$ has suitable colimits we show that $\mathcal{C}(P,Q)$ is monadic and thereby construct a family of Hopf monads on arbitrary closed monoidal categories $\mathcal{C}$. We also introduce the pivotal cover of a monoidal category and extend our work to arbitrary pivotal diagrams.
