缩回图类别及其在构造λ演算模型中的应用

Q4 Mathematics
Hirofumi Yokouchi
{"title":"缩回图类别及其在构造λ演算模型中的应用","authors":"Hirofumi Yokouchi","doi":"10.1016/S0019-9958(86)80017-1","DOIUrl":null,"url":null,"abstract":"<div><p>This paper deals with categorical models of the λ-calculus. We generalize the inverse limit method Scott used for his construction of <em>D</em><sub>∞</sub>, and introduce order-enriched ccc's, retraction map categories and ɛ-categories. An order-enriched ccc is a cartesian closed category <strong>C</strong> equipped with a partial order relation ⩽ on the set of the arrows. A retraction map category of <strong>C</strong> is <strong>R</strong>=(<strong>R</strong>, ⩽, <em>i</em>, <em>j</em>), where ⩽ is a partial order relation on the set |<strong>C</strong>| of all the objects of <strong>C</strong>, <strong>R</strong> is the category of the poset (|<strong>C</strong>|, ⩽), and <em>i</em> and <em>j</em> are functors from <strong>R</strong> to <strong>C</strong> and from <strong>R</strong><sup>op</sup> to <strong>C</strong> that satisfy the conditions: (1) <em>j a</em>, <em>b</em> ∘ <em>i a</em>, <em>b</em> ⩾ id<em><sub>a</sub></em> and (2) <em>i a</em>, <em>b</em> ∘ <em>j a</em>, <em>b</em> ⩽ id<em><sub>b</sub></em> for every arrow <em>a</em>, <em>b</em>: <em>a</em> → <em>b</em> in <strong>R</strong> (i.e., <em>a</em>⩽<em>b</em>). The ɛ-category <strong>E</strong>=<strong>E</strong>(<strong>C</strong>, <strong>R</strong>) of <strong>C</strong> w.r.t. <strong>R</strong> is the category whose objects are ideals of (|<strong>C</strong>|, ⩽) and whose arrows are ideals of (<strong>C</strong>, ⊑), where ⩽ is the partial order relation in <strong>R</strong> and ⊑ is the partial order relation defined by <em>f</em> ⊑ <em>g</em> iff dom(<em>f</em>)⩽dom(<em>g</em>), cod(<em>f</em>)⩽cod(<em>g</em>) in <strong>R</strong> and <em>f</em>⩽<em>j a</em>, <em>b</em> ∘ <em>g</em> ∘ <em>i</em>(<em>a</em>, <em>b</em> in <strong>C</strong>. We show that every ɛ-category <strong>E</strong>=<strong>E</strong>(<strong>C</strong>, <strong>R</strong>) is also an order-enriched ccc. Moreover when <strong>E</strong> and <strong>R</strong> satisfy a particular condition, <strong>E</strong>(<strong>C</strong>, <strong>R</strong>) has a reflexive object. For example, if there is an ideal <em>U</em> of (|<strong>C</strong>|, ⩽) satisfying the following conditions, then <em>U</em> is isomorphic to <em>U<sup>U</sup></em> in <strong>E</strong> and a λ-algebra is constructed from <strong>E</strong> and <em>U</em>: (1) for every pair of <em>a</em>, <em>b</em> ∈ <em>U</em>, <em>U</em> contains <em>b<sup>a</sup></em>, and (2) for every <em>c</em> ∈ <em>U</em>, there are <em>a</em>, <em>b</em> ∈ <em>U</em> such that <em>c</em> ∈ <em>b<sup>a</sup></em>. We reconstruct <em>P</em><sub>ω</sub> and <em>D</em><sub>∞</sub> using ɛ-categories.</p></div>","PeriodicalId":38164,"journal":{"name":"信息与控制","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1986-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0019-9958(86)80017-1","citationCount":"1","resultStr":"{\"title\":\"Retraction map categories and their applications to the construction of lambda calculus models\",\"authors\":\"Hirofumi Yokouchi\",\"doi\":\"10.1016/S0019-9958(86)80017-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This paper deals with categorical models of the λ-calculus. We generalize the inverse limit method Scott used for his construction of <em>D</em><sub>∞</sub>, and introduce order-enriched ccc's, retraction map categories and ɛ-categories. An order-enriched ccc is a cartesian closed category <strong>C</strong> equipped with a partial order relation ⩽ on the set of the arrows. A retraction map category of <strong>C</strong> is <strong>R</strong>=(<strong>R</strong>, ⩽, <em>i</em>, <em>j</em>), where ⩽ is a partial order relation on the set |<strong>C</strong>| of all the objects of <strong>C</strong>, <strong>R</strong> is the category of the poset (|<strong>C</strong>|, ⩽), and <em>i</em> and <em>j</em> are functors from <strong>R</strong> to <strong>C</strong> and from <strong>R</strong><sup>op</sup> to <strong>C</strong> that satisfy the conditions: (1) <em>j a</em>, <em>b</em> ∘ <em>i a</em>, <em>b</em> ⩾ id<em><sub>a</sub></em> and (2) <em>i a</em>, <em>b</em> ∘ <em>j a</em>, <em>b</em> ⩽ id<em><sub>b</sub></em> for every arrow <em>a</em>, <em>b</em>: <em>a</em> → <em>b</em> in <strong>R</strong> (i.e., <em>a</em>⩽<em>b</em>). The ɛ-category <strong>E</strong>=<strong>E</strong>(<strong>C</strong>, <strong>R</strong>) of <strong>C</strong> w.r.t. <strong>R</strong> is the category whose objects are ideals of (|<strong>C</strong>|, ⩽) and whose arrows are ideals of (<strong>C</strong>, ⊑), where ⩽ is the partial order relation in <strong>R</strong> and ⊑ is the partial order relation defined by <em>f</em> ⊑ <em>g</em> iff dom(<em>f</em>)⩽dom(<em>g</em>), cod(<em>f</em>)⩽cod(<em>g</em>) in <strong>R</strong> and <em>f</em>⩽<em>j a</em>, <em>b</em> ∘ <em>g</em> ∘ <em>i</em>(<em>a</em>, <em>b</em> in <strong>C</strong>. We show that every ɛ-category <strong>E</strong>=<strong>E</strong>(<strong>C</strong>, <strong>R</strong>) is also an order-enriched ccc. Moreover when <strong>E</strong> and <strong>R</strong> satisfy a particular condition, <strong>E</strong>(<strong>C</strong>, <strong>R</strong>) has a reflexive object. For example, if there is an ideal <em>U</em> of (|<strong>C</strong>|, ⩽) satisfying the following conditions, then <em>U</em> is isomorphic to <em>U<sup>U</sup></em> in <strong>E</strong> and a λ-algebra is constructed from <strong>E</strong> and <em>U</em>: (1) for every pair of <em>a</em>, <em>b</em> ∈ <em>U</em>, <em>U</em> contains <em>b<sup>a</sup></em>, and (2) for every <em>c</em> ∈ <em>U</em>, there are <em>a</em>, <em>b</em> ∈ <em>U</em> such that <em>c</em> ∈ <em>b<sup>a</sup></em>. We reconstruct <em>P</em><sub>ω</sub> and <em>D</em><sub>∞</sub> using ɛ-categories.</p></div>\",\"PeriodicalId\":38164,\"journal\":{\"name\":\"信息与控制\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1986-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0019-9958(86)80017-1\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"信息与控制\",\"FirstCategoryId\":\"1093\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019995886800171\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"信息与控制","FirstCategoryId":"1093","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019995886800171","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 1

摘要

本文讨论了λ微积分的范畴模型。我们推广了Scott用于构造D∞的逆极限方法,并引入了富序ccc、缩回映射范畴和i -范畴。富序ccc是在箭头集合上具有偏序关系的笛卡尔闭范畴C。收缩映射一类C R = (R,⩽,i, j),其中⩽是一个偏序关系在C | |所有对象的C, R是偏序集的类别(C | |⩽)和i和j函数子从罗普R C和C,满足条件:(1)j,我∘A, b⩾ida和(2)我,b∘j A, b⩽idb对于每一个箭头,b: A→b R(也就是说,⩽b)。的ɛ类别E = E (C, R) C R关于类别的对象是理想的(C | |⩽)和箭的理想(C,⊑),其中⩽是偏序关系在R和⊑是偏序关系定义为f⊑g iff dom (f)⩽dom (g),鳕鱼(f)⩽鳕鱼(g)在R和f⩽j a, b∘g∘我(a, b, C,我们表明,每个ɛ类别E = E (C, R)也是一个order-enriched ccc。当E和R满足特定条件时,E(C, R)有一个自反对象。例如,如果有一个理想U (|C|,≤)满足下列条件,则U与E中的UU同构,由E和U构造出λ代数:(1)对于每一对a, b∈U, U包含ba;(2)对于每一对C∈U,存在a, b∈U使得C∈ba。我们用i -范畴重构了Pω和D∞。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Retraction map categories and their applications to the construction of lambda calculus models

This paper deals with categorical models of the λ-calculus. We generalize the inverse limit method Scott used for his construction of D, and introduce order-enriched ccc's, retraction map categories and ɛ-categories. An order-enriched ccc is a cartesian closed category C equipped with a partial order relation ⩽ on the set of the arrows. A retraction map category of C is R=(R, ⩽, i, j), where ⩽ is a partial order relation on the set |C| of all the objects of C, R is the category of the poset (|C|, ⩽), and i and j are functors from R to C and from Rop to C that satisfy the conditions: (1) j a, bi a, b ⩾ ida and (2) i a, bj a, b ⩽ idb for every arrow a, b: ab in R (i.e., ab). The ɛ-category E=E(C, R) of C w.r.t. R is the category whose objects are ideals of (|C|, ⩽) and whose arrows are ideals of (C, ⊑), where ⩽ is the partial order relation in R and ⊑ is the partial order relation defined by fg iff dom(f)⩽dom(g), cod(f)⩽cod(g) in R and fj a, bgi(a, b in C. We show that every ɛ-category E=E(C, R) is also an order-enriched ccc. Moreover when E and R satisfy a particular condition, E(C, R) has a reflexive object. For example, if there is an ideal U of (|C|, ⩽) satisfying the following conditions, then U is isomorphic to UU in E and a λ-algebra is constructed from E and U: (1) for every pair of a, bU, U contains ba, and (2) for every cU, there are a, bU such that cba. We reconstruct Pω and D using ɛ-categories.

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
信息与控制
信息与控制 Mathematics-Control and Optimization
CiteScore
1.50
自引率
0.00%
发文量
4623
期刊介绍:
×
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