布尔值模型、预层和概形空间
Boolean valued models, presheaves, and étalé spaces
AI总结：
本文研究布尔值模型、预层和概形空间之间的关系，通过拓扑和范畴论方法，提出布尔值模型满足Loś定理的特征化条件，并区分完整性性质与混合性质。
摘要：
布尔值模型对于一个签名L是L-结构的推广，在其中我们允许L-关系符号被布尔真值所解释。例如，对于元素a,b∈M，其中M是B-值的L-结构，对于某些布尔代数B，(a=b)可能既不真也不假，而是得到B中的中间真值。在本文中，我们引入了预层在拓扑空间上由密集格罗滕迪克拓扑诱导的sheafification过程的拓扑特征化。在产生我们的特征化的过程中，我们还把拓扑空间之间的开放连续映射的概念与完全布尔代数之间的完全同态的概念，以及布尔代数之间的伴随同态的概念（例如，一个同态如果作为偏序/范畴之间的函子，具有左伴随）联系起来。接下来，我们将这些拓扑/范畴论的结果与布尔值模型的理论联系起来。我们给出了Monro识别Stone空间上的拓扑预层与布尔值模型，以及根据密集格罗滕迪克拓扑的sheaves与具有混合性质的布尔值模型的另一种证明。我们还给出了一个精确的拓扑特征化（所谓的完整性性质），说明哪些布尔值模型满足Loś定理（即forcing定理的一般形式，由Cohen--Scott、Solovay、Vopenka等人为集合论中的特殊情形所建立）。然后，我们通过显示后者严格更强，将完整性性质与混合性质分开。最后，我们以其相关概形空间的全局截面结构的特征化，给出了哪些预层对应于完整的布尔值模型。
英文摘要：
Boolean valued models for a signature $\mathcal{L}$ are generalizations of $\mathcal{L}$-structures in which we allow the $\mathcal{L}$-relation symbols to be interpreted by boolean truth values. For example, for elements $a,b\in\mathcal{M}$ with $\mathcal{M}$ a $\mathsf{B}$-valued $\mathcal{L}$-structure for some boolean algebra $\mathsf{B}$, $(a=b)$ may be neither true nor false, but get an intermediate truth value in $\mathsf{B}$. In this paper we introduce a topological characterization of the sheafification process for presheaves on topological spaces induced by the dense Grothendieck topology. On the way to produce our characterization, we also relate the notion of open continuous mapping between topological spaces to that of complete homomorphism between complete boolean algebras, and to that of adjoint homomorphism between boolean algebras (e.g. an homomorphism which has a left adjoint, if seen as a functor between partial orders/categories). Next we link these topological/category theoretic results to the theory of boolean valued models. We give a different proof of a result by Monro identifying topological presheaves on Stone spaces with boolean valued models, and sheaves (according to the dense Grothendieck topology) with boolean valued models having the mixing property. We also give an exact topological characterization (the so called fullness property) of which boolean valued models satisfy Loś Theorem (i.e. the general form of the Forcing Theorem which Cohen -- Scott, Solovay, Vopenka -- established for the special case given by the forcing method in set theory). Then we separate the fullness property from the mixing property, by showing that the latter is strictly stronger. Finally we give an exact categorical characterization of which presheaves correspond to full boolean valued models in terms of the structure of global sections of their associated étalé space
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