Categorical and Geometrical Methods in Physics

In this work we develop the higher categorical language aiming to apply it in the foundations of physics, following an approach based in works of Urs Schreiber, John Baez, Jacob Lurie, Daniel Freed and many others. The text has three parts. In Part I we introduce categorical language with special focus in algebraic topological aspects, and we discuss that it is not abstract enough to give a full description for the foundations of physics. In Part II we introduce the categorical process, which produce an abstract language from a concrete language. Examples are given, again focused on Algebraic Topology. In Part III we use the categorification process in order to construct arbitrarily abstract languages, the higher categorical ones, including the cohesive ∞-topos. An emphasis on the formalization of abstract stable homotopy theory is given. We discuss the reason why we should believe that cohesive ∞-topos are natural languages to use in order to attack Hilbert’s sixth problem.