Texas A&M Number Theory Seminar

Department of Mathematics
Milner 317
Thursdays, 1:00-2:00 PM

Title: Is it time for "object arrow" topology?

Abstract: This talk is part survey and part advocacy. A "Grothendieck topology" is a framework for a category, whose axioms and applications are closely related to constructions from point-set topology. This framework is the basis of a family of cohomology theories. Our thesis is that, with slight adjustment to the axioms, there are a variety of theorems of content which can be proved at the universal level. We argue that point-set topology has a general, "object-arrow" version.

Below are results which, in our opinion, can be exploited as diagrammatic facts:

1. Grothendieck created the fundamental group construct for schemes using diagrammatic properties. A slight expansion on standard theory provides for a universal formulation which applies to any category with a suitable notion of "connected object."

2. Profinite groups are inverse limits of finite groups. They have a structure theory which other classes of inverse limits lack. The universal construction dual to inverse limit is the colimit. Colimits of certain spaces of bundles also have useful structure, provided that underlying topologies are compact. We contend that the existence or absence of key propoerties can be explained with diagrams. The observation should enable mathematicians to predict in which context inversely directed systems (or their duals) can be used to construct classifying objects.

3. Many mathematical objects are characterized by pieces. A manifold is, essentially, something which consists of open subsets of R^n attached compatibly. Likewise, a scheme is, somehow, a composite of commutative rings. In algebraic geometry, the process by which "local" objects combine to make new "global" objects is referred to as descent. There is a universal theorem to the following effect: Start with a category C in which every object supports a Grothendieck topology. Then there is a canonical expansion to C^+ such that

• every object in C^+ is defined from a "descent" based on C-objects, and
• every diagram of C-objects of a certain kind (descent applies only to certain arrangements of data) "descends" to a C^+ object.
The category C^+ is canonical in the sense that it has functorial mapping properties.

Mathematicians experimented with many formulations for the fundamental object of algebraic geometry before consensus developed around the scheme. The universal approach not only models schemes, but explains why it is the "right" choice. (One can prove that the category of schemes is functorially equivalent to the universal C^++ for C the opposite category of rings.) The universal construct may simplify the development of topoi.