{"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}
引用次数: 0
Abstract
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.
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.