刻画（余）自由dagger范畴
Characterizing (Co)Free Dagger Categories
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中文总结 AI 辅助
本文通过内部dagger范畴结构，利用矩形带和之字形概念，分别给出了余自由与自由dagger范畴的充要刻画，并联系到伴随诱导的（余）单子（余）代数。
中文摘要 AI 辅助
对于任何范畴，在其上既存在一个自由dagger范畴，也存在一个余自由dagger范畴。一个自然的问题是：给定一个dagger范畴，在不指定外部基范畴的情况下，我们如何判断它是自由的还是余自由的？在本文中，我们通过内部dagger范畴结构给出了自由dagger范畴和余自由dagger范畴两者的刻画。为了刻画余自由dagger范畴，我们使用矩形带并证明：一个dagger范畴是余自由的当且仅当它关于矩形带是 enriched 的。对于自由dagger范畴，我们定义了之字形dagger范畴的概念，然后证明：一个dagger范畴是自由的当且仅当它是一个之字形dagger范畴。我们还证明了自由dagger范畴可以被刻画为自由dagger范畴伴随所诱导的余单子的余代数，类似地，余自由dagger范畴可以被刻画为余自由dagger范畴伴随所诱导的单子的代数。
英文摘要
For any category, there exists both a free dagger category and a cofree dagger category over it. A natural question to ask is: given a dagger category, how can we tell if it is free or cofree without specifying an external base category? In this paper, we provide characterizations of both free dagger categories and cofree dagger categories via internal dagger category structure. To characterize cofree dagger categories, we use rectangular bands and show that a dagger category is cofree if and only if it is enriched over rectangular bands. For free dagger categories, we define the notion of a zigzag dagger category, and then show that a dagger category is free if and only if it is a zigzag dagger category. We also show that free dagger categories can be characterized as the coalgebras of the induced comonad from the free dagger category adjunction, and similarly that cofree free dagger categories can be characterized as the algebras of the induced monad from the cofree dagger category adjunction.
