The Strict Higher Grothendieck Integral

Authors

Scott W. Dyer, University of Nebraska-LincolnFollow

Date of this Version

8-2015

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Mark Brittenham and Susan Hermiller. Lincoln, Nebraska: August, 2015

Copyrigt (c) 2015 Scott W. Dyer

Abstract

This thesis generalizes A. Grothendieck’s construction, denoted by an integral, of a fibered category from a contravariant pseudofunctor, to a construction for n- and even ∞-categories. Only strict higher categories are considered, the more difficult theory of weak higher categories being neglected. Using his axioms for a fibered category, Grothendieck produces a contravariant pseudofunctor from which the original fibered category can be reconstituted by integration. In applications, the integral is often most efficient, constructing the fibered category with its structure laid bare. The situation generalizes the external and internal definitions of the semidirect product in group theory: fibration is the internal notion, while the integral is a form of the external semidirect product. The strict higher integral functor is continuous, and under mild assumptions the integral n-categories produced are complete. The integral retains most formulae (like Fubini’s theorem) familiar from analytic geometry, providing a useful calculus for many applications in pure mathematics.

Adviser: Mark Brittenham and Susan Hermiller