论文标题
追溯了Monads和Hopf Monads
Traced Monads and Hopf Monads
论文作者
论文摘要
追踪的单子是追踪的对称单类类别的单子，将追踪的对称单体结构提升为其eilenberg-moore类别。一个长期存在的问题是提供追溯的单子的特征，而不明确提及艾伦贝格 - 摩尔类别。另一方面，对称的Hopf Monad是对称的双子座，其融合操作员可逆。对于紧凑的封闭类别，对称的Hopf Monad恰恰是将紧凑型封闭结构提升为Eilenberg-Moore类别的单调。由于紧凑的封闭类别和跟踪的对称单类类别密切相关，因此自然而然地询问Hopf Monads和Traded Monads之间的关系是什么。在本文中，我们在追踪的单体类别上介绍了痕量固定的Hopf Monads，可以在不提及Eilenberg-Moore类别的情况下进行表征。本文的主要定理是，当且仅当它是痕量的霍普夫·蒙达德（Monad）时，对称的hopf monad是一个追溯的单元。我们提供了许多痕量型Hopf Monad的示例，例如由Cocommutative Hopf代数或紧凑型封闭类别中的任何对称的Hopf Monad引起的示例。我们还解释了如何使用Conway操作员来表达痕量的笛卡尔单类别类别，而痕量的hopf monad则可以表达，而对于痕迹的cocartesian单类别类别，任何痕量相连的hopf monad都是一个依恋的单元。我们还提供了不是Hopf Monad的追溯奇迹的示例，以及不是痕量固定的对称的Hopf Monad。
A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore category. The main theorem of this paper is that a symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad. We provide many examples of trace-coherent Hopf monads, such as those induced by cocommutative Hopf algebras or any symmetric Hopf monad on a compact closed category. We also explain how for traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator, while for traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We also provide separating examples of traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent.
