Category theory over a minimalist or no foundation

Abstract

I call dynamic construction the vision of mathematics as a human enterprise. Truth is not universal and given, but local and constructed, which ultimately means a judicious management of certain and reliable information. The corresponding foundation is minimalist in assumptions, since it aims to maximize conceptual distinctions. However, in a dynamic view there is no single formal system or single language, however powerful, not even that of categories, to which the activity of mathematicians can be reduced. After a quick introduction to dynamic constructivism and to the minimalist foundation, I will illustrate some technical modifications and additions to category theory and topos theory that they suggest: mathematization of existential statements (related to the absence of the Law of Excluded Middle), abstract treatment of well-foundedness and of effectivity (related to the absence of Power Set Axiom and of the Axiom of (unique) Choice, respectively).