Categorical structures enriched in a quantaloid : categories and semicategories/
Categorical structures enriched in a quantaloid : categories and semicategories/
复制标题
DOI：
--
复制
发表时间：
2003
期刊：
影响因子：
--
通讯作者：
Isar Stubbe
中科院分区：
文献类型：
--
作者：
Isar Stubbe
本论文由两部分组成：一个在quantaloid中丰富的范畴理论的综合;和一个弱化的这个理论，它包括半范畴描述quantaloid上的有序层。一个综合的，并补充，结果在文献中有关理论的范畴丰富的quantaloid Q（作为特殊情况下的范畴丰富的bicategory）载于第一章。这个理论是建立在Q-范畴，函子和分配器，并包含这样的概念，例如，伴随函子，加权余极限，预层，Kan扩张，柯西完成和森田等价，等等。文献没有提供这些事项的概述，所以有必要在这里提供一个。然后，必要的理论发展到一个基本的描述“有序层的quantaloid Q”，此后被称为Q-订单。由于量子体上没有层拓扑，Q序不能被定义为这种拓扑中的有序对象。相反，Q-订单的范畴结构丰富的quantaloid Q的描述。在locale L（即~）上著名的有序层L上层拓扑中的有序对象）当然应该是一般理论的一个特例，把Q当作L的（一个对象的悬置）。于是，Q-范畴的理论必须被削弱，以包括“没有单位的范畴”，即Q-半范畴。但是，Q-半范畴承认一个方便的分配器演算，一个“正则性”的条件，必须施加。而对于那些正则Q-半范畴来说，要承认柯西完备化和森田等价的合理理论，就必须施加更强的“全正则性”条件。前一个概念之前已经针对丰富在对称么半群闭范畴中的半范畴进行了研究;后一个概念是新的，并且是通过直观清晰的“对象的稳定性”概念引入的。关键是，柯西完备全正则Q-半范畴正是Q-序;对于局部空间L，它们确实是L上层拓扑中的有序对象。这些Q-序的一个（bi）等价描述可以用在量子体Q的分裂幂等完备化中丰富的范畴来给出：一个在Q中丰富的全正则半范畴在精确意义上对应于一个在Q的分裂幂等完备化中丰富的范畴。这些结果再一次应用于区域L而不是量子群Q，从而深化了新卢万学派的工作，并使之与悉尼学派的工作相一致，即把区域上的（有序）层描述为丰富的范畴结构。扩展的介绍给出了一个紧凑而直观的介绍，在论文中所包含的发展。
This thesis consists of two parts: a synthesis of the theory of categories enriched in a quantaloid; and a weakening of this theory for it to include semicategories describing ordered sheaves on a quantaloid. A synthesis of, and supplements to, results in the literature concerning the theory of categories enriched in a quantaloid Q (as particular case of categories enriched in a bicategory) is contained in the first chapters. This theory is built with Q-categories, functors and distributors, and contains such notions as, for example, adjoint functors, weighted colimits, presheaves, Kan extensions, Cauchy completions and Morita equivalence, and so on. The literature does not provide an overview of these matters, so it was necessary to provide one here. Then the necessary theory is developed to arrive at an elementary description of ``ordered sheaves on a quantaloid Q', henceforth referred to as Q-orders. As there is no ``topos of sheaves on a quantaloid', Q-orders cannot be defined as ordered objects in such a topos. Instead a description of Q-orders as categorical structures enriched in the quantaloid Q is proposed. The well-known ordered sheaves on a locale L (i.e.~ordered objects in the topos of sheaves on L) should of course be a particular example of the general theory, taking Q to be the (one-object suspension of) L. Then it turns out that the theory of Q-categories has to be weakened to include ``categories without units', i.e. Q-semicategories. But for Q-semicategories to admit a convenient distributor calculus, a ``regularity' condition has to be imposed. And for those regular Q-semicategories to admit a reasonable theory of Cauchy completions and Morita equivalence, the even stronger condition of ``total regularity' has to be imposed. The former notion has been studied before for semicategories enriched in a symmetric monoidal closed category; the latter notion is new, and is introduced via the intuitively clear idea of ``stability of objects'. The point is then that precisely the Cauchy complete totally regular Q-semicategories are the Q-orders; for a locale L they are indeed the ordered objects in the topos of sheaves on L. A (bi)equivalent description of those Q-orders can be given in terms of categories enriched in the split-idempotent completion of the quantaloid Q: a totally regular semicategory enriched in Q corresponds in a precise sense to a category enriched in the split-idempotent completion of Q. Applying this once more to a locale L instead of a quantaloid Q, these results thus deepen the work of the Louvain-la-Neuve school, and reconcile it with that of the Sydney school, on the description of (ordered) sheaves on a locale as enriched categorical structures. The extended introduction gives a compact yet intuitive presentation of the developments contained in the thesis.