Tripos theory
Tripos theory
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三重理论
DOI：
10.1017/s0305004100057534
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发表时间：
1980
影响因子：
0.8
通讯作者：
A. Pitts
中科院分区：
文献类型：
--
作者：
J.;M.;É.;Hyland;T. P.;Johnstone;A. Pitts
拓扑理论中最重要的构造之一是在一个区域（=完全Heyting代数）A上的束的范畴Shv (A)的构造。通常，这个范畴的对象被描述为“满足粘合条件的A上的束”；但是，正如Higgs(7)和Fourman和Scott(5)所观察到的，它们也可以被视为“用a值相等谓词构造的集合”（简称为“a值集合”）。从后者的观点来看，每个束都有一个正则表示作为“完全”a值集，这是这种情况的一个无关紧要的特征。在本文中，我们的目的是研究A必须具备的那些性质，以便我们能够构造A值集合的拓扑：我们将看到，有一个重要的方面，关于有限（命题）结构和无限（量词）结构之间的关系，其中通常的区域定义可以放宽，我们将给出一些例子（其中一些将在后面的论文中进行更充分的探讨），以表明这种放宽可能是有用的。
One of the most important constructions in topos theory ia that of the category Shv (A) of sheaves on a locale (= complete Heyting algebra) A. Normally, the objects of this category are described as ‘presheaves on A satisfying a gluing condition’; but, as Higgs(7) and Fourman and Scott(5) have observed, they may also be regarded as ‘sets structured with an A-valued equality predicate’ (briefly, ‘A-valued sets’). From the latter point of view, it is an inessential feature of the situation that every sheaf has a canonical representation as a ‘complete’ A-valued set. In this paper, our aim is to investigate those properties which A must have for us to be able to construct a topos of A-valued sets: we shall see that there is one important respect, concerning the relationship between the finitary (propositional) structure and the infinitary (quantifier) structure, in which the usual definition of a locale may be relaxed, and we shall give a number of examples (some of which will be explored more fully in a later paper (8)) to show that this relaxation is potentially useful.