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The strict higher Grothendieck integral
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.^
Dyer, S. W, "The strict higher Grothendieck integral" (2015). ETD collection for University of Nebraska - Lincoln. AAI3715728.