Variable Quantities and Variable Structures in Topoi

Abstract

This chapter discusses variable quantities and variable structures in topoi. The chapter presents the conceptual basis for topoi in mathematical experience with variable sets and discusses a formal theory of variable abstract sets as a relativized foundation for geometry and analysis. It also focuses on sheaves of continuous maps. The chapter presents two aspects of sheaf theory not yet sufficiently incorporated into general topoi theory. There has long been in geometry and differential equations the idea that the category of families of spaces smoothly parametrized by a given space X is similar in many respects to the category of spaces itself. Traditionally, the set theory has emphasized the constancy of sets, and both nonstandard analysis and forcing method involve passing from a system of supposedly constant sets to a new system.