A perspective on non-commutative frame theory
A perspective on non-commutative frame theory
复制标题
非交换框架理论的视角
DOI：
10.1016/j.aim.2017.02.028
复制
发表时间：
2017
影响因子：
1.7
通讯作者：
Kudryavtseva G
中科院分区：
文献类型：
--
作者：
Kudryavtseva G
本文将框架理论的基本结果推广到一个非交换的环境中，在这个环境中，locales的角色被etale localic范畴所取代。这涉及到Quantale理论和半群理论的思想，特别是Ehresmann半群，限制半群和逆半群。我们证明了几个主要结果。首先，我们建立了完全限制幺半群范畴与étale局部范畴范畴之间的对偶。幺半群和范畴之间的关系是由一类称为限制量子框架的量子介导的。这一结果的基础上的工作佩德罗雷森德之间的联系伪群和étale局部广群，但在这个过程中，我们都推广和简化：例如，我们不需要对合，此外，我们使他的结果函。一个更广泛的类的量子，称为乘法Ehresmann量子框架，被放入对应的那些局部范畴的乘法结构映射是半开的，所有其他的结构映射是开放的。我们还向下投影到拓扑空间，并由此将locales和拓扑空间之间的经典邻接扩展到etale localic范畴和etale拓扑范畴之间的邻接。事实上，通过改变态射，我们得到了几个映射。就像在交换的情况下，我们限制这些空间清醒和相干谱等价的解释。将凝聚框架与分配格之间的经典等价性推广到凝聚完备限制幺半群与分配限制半群之间的等价性。由此，我们导出了分配限制半群与谱代数拓扑范畴之间的几个对偶。我们还专门这些设置的拓扑范畴是cancellative或群胚的对偶。因此，我们的方法联系、统一和扩展了劳森和伦茨以及雷森德在工作中所采取的方法。
This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by étale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We prove several main results. To start with, we establish a duality between the category of complete restriction monoids and the category of étale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups and étale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. A wider class of quantales, called multiplicative Ehresmann quantal frames, is put into a correspondence with those localic categories where the multiplication structure map is semiopen, and all the other structure maps are open. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction between étale localic categories and étale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectral étale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work by Lawson and Lenz and by Resende.
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DOI：
10.1016/j.aim.2009.09.001
发表时间：
2009-03
期刊：
arXiv: Rings and Algebras
影响因子：
--
作者：
B. Steinberg
通讯作者：
B. Steinberg
DOI：
10.1007/978-1-84800-281-4
发表时间：
2008-12
期刊：
--
影响因子：
--
作者：
O. Ganyushkin;V. Mazorchuk
通讯作者：
O. Ganyushkin;V. Mazorchuk
影响因子：
1.7
作者：
M. Lawson;D. Lenz
通讯作者：
D. Lenz
DOI：
--
发表时间：
2007
期刊：
影响因子：
--
作者：
R. Exel
通讯作者：
R. Exel
作为滤波器群的反半群的 ÉTALE 群
影响因子：
0.7
作者：
M. Lawson;S. Margolis;B. Steinberg
通讯作者：
B. Steinberg