{"title":"On the unicity of the theory of higher categories","authors":"C. Barwick, Christopher J. Schommer-Pries","doi":"10.1090/JAMS/972","DOIUrl":null,"url":null,"abstract":"<p>We axiomatise the theory of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis normal infinity comma n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(\\infty ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories. We prove that the space of theories of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis normal infinity comma n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(\\infty ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories is a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B left-parenthesis double-struck upper Z slash 2 right-parenthesis Superscript n\"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B(\\mathbb {Z}/2)^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that Rezk’s complete Segal <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Theta Subscript n\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Θ<!-- Θ --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\Theta _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces, Simpson and Tamsamani’s Segal <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories, the first author’s <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-fold complete Segal spaces, Kan and the first author’s <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-relative categories, and complete Segal space objects in any model of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis normal infinity comma n minus 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(\\infty , n-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis double-struck upper Z slash 2 right-parenthesis Superscript n\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(\\mathbb {Z}/2)^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":"1"},"PeriodicalIF":3.5000,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"26","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/JAMS/972","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 26
Abstract
We axiomatise the theory of (∞,n)(\infty ,n)-categories. We prove that the space of theories of (∞,n)(\infty ,n)-categories is a B(Z/2)nB(\mathbb {Z}/2)^n. We prove that Rezk’s complete Segal Θn\Theta _n spaces, Simpson and Tamsamani’s Segal nn-categories, the first author’s nn-fold complete Segal spaces, Kan and the first author’s nn-relative categories, and complete Segal space objects in any model of (∞,n−1)(\infty , n-1)-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of (Z/2)n(\mathbb {Z}/2)^n.
我们公理化了（∞，n）（infty，n）-范畴的理论。我们证明了（∞，n）（infty，n）-范畴的理论空间是一个B（Z/2）nB（\mathbb｛Z｝/2）^n。我们证明了Rezk的完全SegalΘn\Theta _n空间，Simpson和Tamsamani的Segal n n-范畴，第一作者的n n-折叠完全Segal空间，Kan和第一作者的nn-相对范畴，和（∞，n−1）（infty，n-1）-范畴的任何模型中的完全分段空间对象都满足我们的公理。因此，这些理论都是等价的，在（Z/2）n（\mathbb｛Z｝/2）^n的作用下是唯一的。
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