A category-theoretic approach to Boolean-valued models of set theory
Let \({\mathbb{B}}\) be a complete Boolean algebra. Scott and Solovay constructed a \({\mathbb{B}}\)-valued model of set theory \(V^{({\mathbb{B}})}\); in this paper a category-theoretic translation of \(V^{({\mathbb{B}})}\) is given, in the form of a \({\mathbb{B}}\)-valued model of category theory. The usual category-theoretic translation of \(V^{({\mathbb{B}})}\), namely the category of sheaves Shv(\({\mathbb{B}})\), appears as an image of the \({\mathbb{B}}\)- valued model. The \({\mathbb{B}}\)-valued model lives in a category MOD(\({\mathbb{B}})\), which is intended to be the category of all \({\mathbb{B}}\)- valued models. The last part of the paper investigates Easton's construction, which is the construction of \(V^{({\mathbb{B}})}\) for a `large' \({\mathbb{B}}\). The construction (in MOD(\({\mathbb{B}}))\) of the \({\mathbb{B}}\)- valued model of category theory can still be carried out in this case, though the construction of Shv(\({\mathbb{B}})\) fails.
scientific article; zbMATH DE number 4016897