Internal parameterization of hyperconnected quotients

One of the most fundamental facts in topos theory is the internal parameterization of subtoposes: the bijective correspondence between subtoposes and Lawvere-Tierney topologies.In this paper, we introduce a new but elementary concept, "a local state classifier," and give an analogous internal parameterization of hyperconnected quotients (i.e., hyperconnected geometric morphisms from a topos).As a corollary, we obtain a solution to the Boolean case of the first problem of Lawvere's open problems.Contents 1 Introduction 263 2 Hyperconnected geometric morphism 268 3 Local state classifier 270 4 Internal parameterization of hyperconnected quotients 291 5 Examples and Applications 297 6 Conclusions and future works 303 A Internal semilattices and the universal filter 304 B Existence theorem for a local state classifier 309The author would like to thank Ryu Hasegawa and Hisashi Aratake for their suggestions and support,