关于霍夫曼-斯特赖歇尔宇宙

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Steve Awodey
{"title":"关于霍夫曼-斯特赖歇尔宇宙","authors":"Steve Awodey","doi":"10.1017/s0960129524000203","DOIUrl":null,"url":null,"abstract":"We take another look at the construction by Hofmann and Streicher of a universe <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline1.png\"/> <jats:tex-math> $(U,{\\mathcal{E}l})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for the interpretation of Martin-Löf type theory in a presheaf category <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline2.png\"/> <jats:tex-math> $[{{{\\mathbb{C}}}^{\\textrm{op}}},\\textsf{Set}]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. It turns out that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline3.png\"/> <jats:tex-math> $(U,{\\mathcal{E}l})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> can be described as the <jats:italic>nerve</jats:italic> of the classifier <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline4.png\"/> <jats:tex-math> $\\dot{{\\textsf{Set}}}^{\\textsf{op}} \\rightarrow{{\\textsf{Set}}}^{\\textsf{op}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for discrete fibrations in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline5.png\"/> <jats:tex-math> $\\textsf{Cat}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where the nerve functor is right adjoint to the so-called “Grothendieck construction” taking a presheaf <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline6.png\"/> <jats:tex-math> $P :{{{\\mathbb{C}}}^{\\textrm{op}}}\\rightarrow{\\textsf{Set}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to its category of elements <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline7.png\"/> <jats:tex-math> $\\int _{\\mathbb{C}} P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also consider change of base for such universes, as well as universes of structured families, such as fibrations.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Hofmann–Streicher universes\",\"authors\":\"Steve Awodey\",\"doi\":\"10.1017/s0960129524000203\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We take another look at the construction by Hofmann and Streicher of a universe <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0960129524000203_inline1.png\\\"/> <jats:tex-math> $(U,{\\\\mathcal{E}l})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for the interpretation of Martin-Löf type theory in a presheaf category <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0960129524000203_inline2.png\\\"/> <jats:tex-math> $[{{{\\\\mathbb{C}}}^{\\\\textrm{op}}},\\\\textsf{Set}]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. It turns out that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0960129524000203_inline3.png\\\"/> <jats:tex-math> $(U,{\\\\mathcal{E}l})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> can be described as the <jats:italic>nerve</jats:italic> of the classifier <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0960129524000203_inline4.png\\\"/> <jats:tex-math> $\\\\dot{{\\\\textsf{Set}}}^{\\\\textsf{op}} \\\\rightarrow{{\\\\textsf{Set}}}^{\\\\textsf{op}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for discrete fibrations in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0960129524000203_inline5.png\\\"/> <jats:tex-math> $\\\\textsf{Cat}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where the nerve functor is right adjoint to the so-called “Grothendieck construction” taking a presheaf <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0960129524000203_inline6.png\\\"/> <jats:tex-math> $P :{{{\\\\mathbb{C}}}^{\\\\textrm{op}}}\\\\rightarrow{\\\\textsf{Set}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to its category of elements <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0960129524000203_inline7.png\\\"/> <jats:tex-math> $\\\\int _{\\\\mathbb{C}} P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also consider change of base for such universes, as well as universes of structured families, such as fibrations.\",\"PeriodicalId\":49855,\"journal\":{\"name\":\"Mathematical Structures in Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-09-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Structures in Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1017/s0960129524000203\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/s0960129524000203","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0

摘要

我们再来看一下霍夫曼和施特莱歇尔构建的宇宙 $(U,{\mathcal{E}l})$ 对于马丁-洛夫类型理论在预设范畴 $[{{{\mathbb{C}}^{\textrm{op}}},\textsf{Set}]$ 中的解释。事实证明,$(U,{{mathcal{E}l}})$ 可以被描述为分类器 $\dot{{\textsf{Set}}}^{textsf{op}} 的神经。\其中神经函子与所谓的 "格罗thendieck 构造 "是右邻接的,所谓的 "格罗thendieck 构造 "是将预叶 $P :{{{textrm{op}}}^{textsf{Set}}$ 取为其元素类别 $int _{{mathbb{C}} 。P$ .我们还考虑了这类宇宙的基底变化,以及结构族的宇宙,如纤维。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On Hofmann–Streicher universes
We take another look at the construction by Hofmann and Streicher of a universe $(U,{\mathcal{E}l})$ for the interpretation of Martin-Löf type theory in a presheaf category $[{{{\mathbb{C}}}^{\textrm{op}}},\textsf{Set}]$ . It turns out that $(U,{\mathcal{E}l})$ can be described as the nerve of the classifier $\dot{{\textsf{Set}}}^{\textsf{op}} \rightarrow{{\textsf{Set}}}^{\textsf{op}}$ for discrete fibrations in $\textsf{Cat}$ , where the nerve functor is right adjoint to the so-called “Grothendieck construction” taking a presheaf $P :{{{\mathbb{C}}}^{\textrm{op}}}\rightarrow{\textsf{Set}}$ to its category of elements $\int _{\mathbb{C}} P$ . We also consider change of base for such universes, as well as universes of structured families, such as fibrations.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
×
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