论文标题
布尔值有价值的型号，预示和Étalé空间
Boolean valued models, presheaves, and étalé spaces
论文作者
论文摘要
布尔值$ \ Mathcal {l} $的布尔值模型是$ \ Mathcal {l} $ - 结构的概括，其中我们允许$ \ Mathcal {l} $ - 关系符号可以由布尔真实值解释。例如，对于元素$ a，b \ in \ nathcal {m} $，带有$ \ mathcal {m} $ a $ \ mathsf {b} $ - 值$ \ mathcal {l} $ - 对于某些布尔algeean algeean algebra $ \ m athsf {b} $，$ n = b）$ note ant ann ant ant的结构$ \ mathsf {b} $。在本文中，我们介绍了由密集的Grothendieck拓扑引起的拓扑空间上的造成造成的造成拓扑表征。在产生我们的表征的方式上，我们还将拓扑空间之间的开放连续映射概念与完全布尔代数之间的完全同构形成相关联，以及布尔代数之间的同构同构的概念（例如，同源性的伴随伴侣，如果视为部分函数订单之间的函数订单/类别）。接下来，我们将这些拓扑/类别理论结果与布尔有价值模型的理论联系起来。我们通过Monro识别带有布尔值模型的石材空间上的拓扑预示性给我们提供了不同的证明，并带有带布尔值的带价型模型的滑轮（根据密集的Grothendieck拓扑结构）具有混合特性。我们还提供了一个确切的拓扑表征（所谓的丰满特性），其布尔值有价值的模型满足了Loś定理（即强迫定理的一般形式，Cohen -Scott，Sotovay，Solovay，Vopenka-为强迫方法中的特殊案例建立的cohen，Solovay，vopenka - 在集合理论中为特殊情况提供了）。然后，我们通过表明后者严格强大，将饱度属性与混合属性区分开。最后，我们给出了确切的分类表征，其前示对应于完整的布尔有价值模型，从其相关étalé空间的全球段的结构来看
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
