What kind of linearly distributive category do polynomial functors form?

David I. Spivak, Priyaa Varshinee Srinivasan
{"title":"What kind of linearly distributive category do polynomial functors form?","authors":"David I. Spivak, Priyaa Varshinee Srinivasan","doi":"arxiv-2407.01849","DOIUrl":null,"url":null,"abstract":"This paper has two purposes. The first is to extend the theory of linearly\ndistributive categories by considering the structures that emerge in a special\ncase: the normal duoidal category $(\\mathsf{Poly} ,\\mathcal{y}, \\otimes,\n\\triangleleft )$ of polynomial functors under Dirichlet and substitution\nproduct. This is an isomix LDC which is neither $*$-autonomous nor fully\nsymmetric. The additional structures of interest here are a closure for\n$\\otimes$ and a co-closure for $\\triangleleft$, making $\\mathsf{Poly}$ a\nbi-closed LDC, which is a notion we introduce in this paper. The second purpose is to use $\\mathsf{Poly}$ as a source of examples and\nintuition about various structures that can occur in the setting of LDCs,\nincluding duals, cores, linear monoids, and others, as well as how these\ngeneralize to the non-symmetric setting. To that end, we characterize the\nlinearly dual objects in $\\mathsf{Poly}$: every linear polynomial has a right\ndual which is a representable. It turns out that the linear and representable\npolynomials also form the left and right cores of $\\mathsf{Poly}$. Finally, we\nprovide examples of linear monoids, linear comonoids, and linear bialgebras in\n$\\mathsf{Poly}$.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Category Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.01849","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

This paper has two purposes. The first is to extend the theory of linearly distributive categories by considering the structures that emerge in a special case: the normal duoidal category $(\mathsf{Poly} ,\mathcal{y}, \otimes, \triangleleft )$ of polynomial functors under Dirichlet and substitution product. This is an isomix LDC which is neither $*$-autonomous nor fully symmetric. The additional structures of interest here are a closure for $\otimes$ and a co-closure for $\triangleleft$, making $\mathsf{Poly}$ a bi-closed LDC, which is a notion we introduce in this paper. The second purpose is to use $\mathsf{Poly}$ as a source of examples and intuition about various structures that can occur in the setting of LDCs, including duals, cores, linear monoids, and others, as well as how these generalize to the non-symmetric setting. To that end, we characterize the linearly dual objects in $\mathsf{Poly}$: every linear polynomial has a right dual which is a representable. It turns out that the linear and representable polynomials also form the left and right cores of $\mathsf{Poly}$. Finally, we provide examples of linear monoids, linear comonoids, and linear bialgebras in $\mathsf{Poly}$.
多项式函数构成了哪种线性分布范畴?
本文有两个目的。第一个目的是通过考虑一个特例中出现的结构来扩展线性分布范畴的理论:在迪里希特和置换品下的多项式函数的正常二元范畴$(\mathsf{Poly} ,\mathcal{y}, \otimes,\triangleleft )$。这是一个既不是 $*$-autonomous 也不是完全对称的等效 LDC。这里我们感兴趣的附加结构是 $\otimes$ 的闭包和 $\triangleleft$ 的共闭包,这使得 $\mathsf{Poly}$ 成为一个非闭包 LDC,这是我们在本文中引入的一个概念。第二个目的是利用 $\mathsf{Poly}$ 作为例子和启示的来源,来说明在 LDCs 环境中可能出现的各种结构,包括对偶、核、线性单体等,以及这些结构如何泛化到非对称环境中。为此,我们描述了 $\mathsf{Poly}$ 中的线性对偶对象:每个线性多项式都有一个右对偶,它是可表示的。事实证明,线性多项式和可表示多项式也构成了 $\mathsf{Poly}$ 的左核和右核。最后,我们举例说明$\mathsf{Poly}$中的线性单项式、线性组合子和线性双元组。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信