University Seminar: Logic Across Disciplines-An intuitive description of toposes, toposes as a description of intuitionism

Title:   An intuitive description of toposes, toposes as a description of intuitionism

Speaker:    Tslil Clingman, Johns Hopkins University

Date and Time: Thursday, November 30,  2017, 2:30pm-3:30pm

Place: Rome Hall (801 22^nd Street), Room 771

Abstract:   In this talk we will begin by examining, by way of example, the notion of a topos, a category exhibiting certain structural properties satisfied in particular by the familiar category of sets and functions but many strange and wonderful examples besides. We will then observe how the presence of certain structural properties naturally equips a topos with `internal' and `external' models intuitionistic propositional logic. From here we will explore the `Mitchell-Bènabou' language of a topos -- a framework which will allow us to redevelop much of the usual language of set theory internal to any topos. That is, we will see how we may make sense of forming objects of a topos via comprehensions, quantifiers and many other appropriate tools as in the case of "surjections(X,Y) = {f ∈ Y^X | ∀y∈Y ∃x∈X [f(x)=y]}" for the (object of) surjections from X to Y. That objects defined in this manner are in factthe ones we desired depends further on the semantics of internal language, the so termed Kripke-Joyal semantics of a topos. Time allowing we will develop this more fully and explore the relation this bears to the forcing arguments of Cohen.