Quasitopoi, logic and Heyting-valued models
This paper deals with various facets of quasitopoi, which are relevant to the interpretation and modelling of first-order logic in quasitopoi. The first section deals with topologies. A topology \(\sigma\) on a quasitopos Q is strong iff the \(\sigma\)-closure of every mono is a strong mono. If \(\sigma\) is strong, the \(\sigma\)-sheaves, \(sh_{\sigma}(Q)\), form a topos. The strong topology which assigns to every mono the strong monic part of its epic/strong monic factorization has as its sheaves what the author calls the coarse objects of the quasitopos Q. A quasitopos is called solid iff it has disjoint coproducts. Solid quasitopoi behave well with respect to several topos theoretic constructions. In particular, if Q is solid and \(X\in Q\), then Q/X is again solid and if \({\mathcal C}\) is an internal category in Q, then \(Q^{{\mathcal C}}\) is a solid quasitopos. Next, the author describes the interpretation of first order logic in quasitopoi. It is analogous to the case of topoi, except that predicates must be interpreted as strong monos rather than arbitrary monos. Finally, if H is a complete Heyting algebra, a solid quasitopos of models, Mod(H), is described. The topos of coarse objects of Mod(H) turns out to be precisely sh(H).
scientific article; zbMATH DE number 3961916