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arXiv 2202.005342022-02math.CT

关于 $\mathbf{Poly}$ 上范畴结构的参考

A reference for categorical structures on $\mathbf{Poly}$

David I. Spivak

备注：: 33 pages

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AI总结：

本文系统整理了多项式函子范畴 $\mathbf{Poly}$ 上的各类范畴结构，包括无穷多个幺半群结构、余闭结构及伴随，并探讨其相互作用。

摘要：

本文档收集了多项式函子范畴 $\mathbf{Poly}$ 上的一系列范畴结构。并不暗示此列表在任何意义上完整。其中包括：无穷多个幺半群结构，其中除一个外，其余都是对称、闭且对 $+$ 分配，其中若干以对偶方式相互作用；还包括一个右余闭结构和两个指标化左余闭结构；还包括 $\mathbf{Poly}$ 作为组成部分的各种伴随，包括自由单子与余自由余单子及其与各种幺半群结构的相互作用。

英文摘要：

In this document, we collect a list of categorical structures on the category $\mathbf{Poly}$ of polynomial functors. There is no implied claim that this list is in any way complete. It includes: infinitely many monoidal structures, all but one of which is symmetric, closed, and distributes over $+$, several of which interact duoidally; it also includes a right-coclosure and two indexed left coclosures; it also includes various adjunctions of which $\mathbf{Poly}$ is a part, including the free monad and cofree comonad and their interaction with various monoidal structures.

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