超格上的Tannaka理论
Tannaka Theory over Sup-Lattices
摘要：
本文的主要结果是在超格范畴上构造了一个与任意Grothendieck topos相关联的tannakian context，并由此获得了tannakian表示论的新结果（见我们的介绍），这些是不同的，不太一般，因为他们假设存在伽罗瓦闭包和工作在伽罗瓦拓扑，而不是任意拓扑。相反，我们在谈论伽罗瓦论时，指的是Joyal和Tierney的任意拓扑的扩展，这对得到本论文的结果至关重要。与Grothendieck topos相关的tannakian上下文是通过取其局部覆盖关系的过程获得的。然后，通过对伽罗瓦理论和Tannaka理论的构造的调查和详尽的比较，我们证明了它们的基本识别定理的等价性（见第8节）。由于Grothendieck topos的（bi）关系范畴已由Carboni和Walters刻画，得到了一个新的非线性型tannakian定理（定理8.12），它与迄今已知的定理有本质的不同.
英文摘要：
The main result of this thesis is the construction of a tannakian context over the category of sup-lattices, associated with an arbitrary Grothendieck topos, and the attainment of new results in tannakian representation theory from it. Although many results were obtained and published historically linking Galois and Tannaka theory (see our introduction), these are different and less general since they assume the existence of Galois closures and work on Galois topos rather than on arbitrary topos. Instead we, when talking about Galois theory, mean the extension to arbitrary topos of Joyal and Tierney, critical to get the results of this thesis. The tannakian context associated with a Grothendieck topos is obtained through the process of taking relations of its localic cover. Then, through an investigation and exhaustive comparison of the constructions of the Galois and Tannaka theories, we prove the equivalence of their fundamental recognition theorems (see section 8). Since the (bi)categories of relations of a Grothendieck topos were characterized by Carboni and Walters, a new recognition-type tannakian theorem (theorem 8.12) is obtained, essentially different from those known so far.
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