{"title":"非平凡宇宙和宇宙序列","authors":"Roland Coghetto","doi":"10.2478/forma-2022-0005","DOIUrl":null,"url":null,"abstract":"Summary Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck’s definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice “Univers” (par N. Bourbaki) de l’Exposé I. “PREFAISCE-AUX”) [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML’s definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of “trivial” and “non-trivial” Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay2. The following result links the universes U0 (FinSETS) and U1 (SETS): Grothendieck Universe ω=Grothendieck Universe U0=U1 {\\rm{Grothendieck}}\\,{\\rm{Universe}}\\,\\omega = {\\rm{Grothendieck}}\\,{\\rm{Universe}}\\,{{\\bf{U}}_0} = {{\\bf{U}}_1} Before turning to the last section, we establish some trivial propositions allowing the construction of sets outside the considered universe. The last section is devoted to the construction, in Tarski-Grothendieck, of a tower of universes indexed by the ordinal numbers (See 8. Examples, Grothendieck universe, ncatlab.org [24]). Grothendieck’s universe is referenced in current works: “Assuming the existence of a sufficient supply of (Grothendieck) univers”, Jacob Lurie in “Higher Topos Theory” [17], “Annexe B – Some results on Grothendieck universes”, Olivia Caramello and Riccardo Zanfa in “Relative topos theory via stacks” [13], “Remark 1.1.5 (quoting Michael Shulman [30])”, Emily Riehl in “Category theory in Context” [29], and more specifically “Strict Universes for Grothendieck Topoi” [14].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"61 1","pages":"53 - 66"},"PeriodicalIF":1.0000,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Non-Trivial Universes and Sequences of Universes\",\"authors\":\"Roland Coghetto\",\"doi\":\"10.2478/forma-2022-0005\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Summary Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck’s definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice “Univers” (par N. Bourbaki) de l’Exposé I. “PREFAISCE-AUX”) [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML’s definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of “trivial” and “non-trivial” Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay2. 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引用次数: 1
摘要
宇宙是一个概念,从Mizar数学库(MML)创建之初就以几种形式存在(Universe, Universe_closure, Universe)[25],然后是后来的the_universe_of,[33],以及最近的定义GrothendieckUniverse[26],[11],[11]。这些定义在许多文章[28,33,8,35],[19,32,31,15,6],以及[34,12,20,22,21],[27,2,3,23,16,7,4,5]中都很有用。本文利用Mizar系统[9][10],简单地证明了[26]中定义的Grothendieck对宇宙的定义与Artin、Grothendieck和Verdier(第0章Univers et Appendice“Univers”(par N. Bourbaki) de l ' exposeise I.“prefaisse - aux”)对宇宙的原始定义是一致的[1],以及关于宇宙的MML的不同定义是如何相互关联的。我们还证明了Mac Lane([18])引入的宇宙定义与MML的定义是兼容的。虽然宇宙可能是空的,但我们考虑了非空宇宙的性质,完成了[25]中所证明的性质。根据Robert M. Solovay2的概念,我们引入了“平凡”和“非平凡”宇宙的概念,这取决于它们是否包含集合ω (NAT)。以下结果将宇宙U0 (FinSETS)和U1 (SETS)联系起来:Grothendieck Universe ω=Grothendieck Universe U0=U1 {\rm{Grothendieck}}\,{\rm{Universe}}\,\omega = {\rm{Grothendieck}}\,{\rm{Universe}}\,{{\bf{U}}_0} = {{\bf{U}}_1}在进入最后一节之前,我们建立了一些微不足道的命题,允许在考虑的宇宙之外构造集合。最后一节致力于塔斯基-格罗滕迪克的宇宙塔的构建,以序数为索引(见8)。示例,Grothendieck宇宙,ncatlab.org[24])。Grothendieck的宇宙在当前的著作中被引用:“假设存在足够的(Grothendieck)宇宙”,Jacob Lurie在“高等拓扑理论”[17],“附件B -关于Grothendieck宇宙的一些结果”,Olivia Caramello和Riccardo Zanfa在“通过堆的相对拓扑理论”[13],“注释1.1.5(引用Michael Shulman[30])”,Emily Riehl在“语境中的范畴论”[29],更具体地说是“Grothendieck拓扑的严格宇宙”[14]。
Summary Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck’s definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice “Univers” (par N. Bourbaki) de l’Exposé I. “PREFAISCE-AUX”) [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML’s definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of “trivial” and “non-trivial” Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay2. The following result links the universes U0 (FinSETS) and U1 (SETS): Grothendieck Universe ω=Grothendieck Universe U0=U1 {\rm{Grothendieck}}\,{\rm{Universe}}\,\omega = {\rm{Grothendieck}}\,{\rm{Universe}}\,{{\bf{U}}_0} = {{\bf{U}}_1} Before turning to the last section, we establish some trivial propositions allowing the construction of sets outside the considered universe. The last section is devoted to the construction, in Tarski-Grothendieck, of a tower of universes indexed by the ordinal numbers (See 8. Examples, Grothendieck universe, ncatlab.org [24]). Grothendieck’s universe is referenced in current works: “Assuming the existence of a sufficient supply of (Grothendieck) univers”, Jacob Lurie in “Higher Topos Theory” [17], “Annexe B – Some results on Grothendieck universes”, Olivia Caramello and Riccardo Zanfa in “Relative topos theory via stacks” [13], “Remark 1.1.5 (quoting Michael Shulman [30])”, Emily Riehl in “Category theory in Context” [29], and more specifically “Strict Universes for Grothendieck Topoi” [14].
期刊介绍:
Formalized Mathematics is to be issued quarterly and publishes papers which are abstracts of Mizar articles contributed to the Mizar Mathematical Library (MML) - the basis of a knowledge management system for mathematics.