A category-theoretic approach to boolean-valued models of set theory

AbstractLet B be a complete Boolean algebra. Scott and Solovay constructed a B-valued model of set theory V(B); in this paper a category-theoretic translation of V(B) is given, in the form of a B-valued model of category theory. The usual category-theoetic translation of V(B), namely the category of sheaves Shv(B), appears as an image of the B-valued model. The B-valued model lives in a category MOD(B), which is intended to be the category of all B-valued models.The last part of the paper investigates Easton's construction, which is the construction of V(B) for a ‘large’ B. The construction (in MOD(B)) of the B-valued model of category theory can still be carried out in this case, though the construction of Shv(B) fails