谱位点再探
The spectral site revisited
AI总结：
本文从位点理论角度重新审视Coste的谱构造，详析其几何性质，给出谱伴随的新证明，并通过命题代数实例将一阶理论的分类topos实现为谱。
摘要：
我们给出了Coste首次引入的谱构造的位点理论解释。我们详细考察了谱的几何性质，特别是当将其应用于几何中不同类别的对象和映射时，它会产生哪类topos。我们还给出了集合值模型的谱伴随和结构层的分类性质的新证明。我们还讨论了“大谱”与“小谱”之间的对立，以及与一些典范分类位点的关系。然后我们描述了模型化topos的一般情形。我们还证明了局部模型化topos的谱在其基上是局部的，并由此推导出一般情形下谱伴随的新证明。最后，我们给出了命题代数世界中的几个例子，特别是将一阶理论在逻辑各片段中的分类topos恢复为某些适当模型化topos的谱。
英文摘要：
We give the site-theoretic account of the spectral construction as first introduced by Coste. We provide a detailed examination of the geometric properties of the spectrum, in particular what classes of topoi it produces when applied to the different classes of objects and maps in a geometry. We also give a new proof of the spectral adjunction for set-valued models and the classifying property of the structure sheaf. We also discuss the opposition between ``gros and petit" spectra and relation with some canonical classifying sites. We then describe the general case of a modelled topos. We also prove that the spectrum of a locally modelled topos is local over its base and deduce a new proof of the spectral adjunction for the general case. We finally give several examples from the world of propositional algebras, and in particular recover classifying topoi of first order theories in various fragment of logic as spectra of some suitably modelled topoi.
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