Glutoses: a Generalization of Topos Theory

Abstract

A generalization of topos theory is proposed giving an abstract realization of such categories as, say, the categories of manifolds and of Grothendieck schemes on the one hand, and permitting one, on the other hand, a view on "non-commutative" or, more generally, "universal" algebraic geometry, which is alternative to already existing, and is closer, in some sense, to the classical Grothendieck's construction of commutative schemes. Another immediate application of the theory developed is construction of an extension of the category of Grothendieck schemes to the category of "etale schemes" containing together with any scheme every etale sheaf over it as well. The main result of this work is that for any presite satisfying some smallness conditions (existence of local sets of topological generators) there exists the universal "completion" of a presite to a glutos.