The geometry of sheaves on sites

Abstract

In this work, we study doing geometry on sheaves on sites. Categories of our sites consist of objects that are building blocks for a given geometry. Generalized spaces then will be sheaves on these sort of sites. Next we introduce the notion of varieties, and show the relationship between certain class of varieties known as diffeologies with the category of smooth manifolds. Along the way, the notion of schemes will be generalized as a variety on symmetric monoidal categories. And we show how a differential geometric construction on a site can be translated to a construction on generalized spaces.