论文标题
建模空间的限制，colimit和光谱
Limits, Colimits, and Spectra of Modelled Spaces
论文作者
论文摘要
众所周知，（交换）环的Zariski光谱的构建产生了环类别与局部环空间类别之间的双重毗邻。在各种情况下，代数的光谱构造了许多构造。米歇尔·科斯特（Michel Coste）以分类逻辑的语言统一了它们，表明，对于适当的三重$（t_0，t，λ）$（我们称为空间cost cost的上下文），每种$ t_0 $ - 模型都可以与$ t $ t $模型的空间相关“可接受的”态度。但是，他的大多数证据仍未出版。在本文中，我们介绍了$ T_0 $模型的光谱的替代结构，并提供了Coste的另一个证明。此外，我们还将$ T_0 $模型的光谱扩展到$ T_0 $模型的空间的相对光谱，并证明存在涉及的模型空间类别中的极限和colimits。例如，我们可以推断出，茎是田地的环形空间的类别是完整的，并且可以完整。
It is well-known that the construction of Zariski spectra of (commutative) rings yields a dual adjunction between the category of rings and the category of locally ringed spaces. There are many constructions of spectra of algebras in various contexts giving such adjunctions. Michel Coste unified them in the language of categorical logic by showing that, for an appropriate triple $ (T_0,T,Λ) $ (which we call a spatial Coste context), each $ T_0 $-model can be associated with a $ T $-modelled space and that this yields a dual adjunction between the category of $ T_0 $-models and the category of $ T $-modelled spaces and "admissible" morphisms. However, most of his proofs remain unpublished. In this paper, we introduce an alternative construction of spectra of $ T_0 $-models and give another proof of Coste adjunction. Moreover, we also extend spectra of $ T_0 $-models to relative spectra of $ T_0 $-modelled spaces and prove the existence of limits and colimits in the involved categories of modelled spaces. We can deduce, for instance, that the category of ringed spaces whose stalks are fields is complete and cocomplete.
