Limits, Colimits, and Spectra of Modelled Spaces
Limits, Colimits, and Spectra of Modelled Spaces
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建模空间的极限、余极限和谱
DOI：
10.1016/j.jpaa.2023.107414
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发表时间：
2023
影响因子：
0.8
通讯作者：
Aratake Hisashi
中科院分区：
文献类型：
--
作者：
古川賢;古川賢;古川賢;Aratake Hisashi
众所周知，(交换)环的Zariski谱的构造产生了环范畴和局部环空间范畴之间的对偶加合。在给出这类附加词的各种上下文中，有许多代数的谱的构造。Michel Coste用范畴逻辑的语言统一了它们，证明了对于适当的三元组(T0，T，Λ)(我们称之为空间Coste上下文)，每个T0-模型可以与一个T-模型空间相关联，并且这产生了T0-模型范畴与T-模型空间范畴和“容许”态射之间的对偶附加。然而，他的大部分校样仍然没有发表。本文介绍了一种类似于Zariski谱的T0-模型的谱结构，并给出了Coste伴随的一个新的证明。此外，我们还将T0-模型的谱推广到T0-模空间的相对谱，并证明了T0-模空间的相关范畴中极限和极限的存在性。例如，我们可以推论，茎是场的环空间范畴是完备的和上完备的。
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 However, most of his proofs remain unpublished.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, which is reminiscent of Zariski spectra, and give a new 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.