On gauge symmetries, indiscernibilities, and groupoid-theoretical equalities

We shall analyze the intrinsic geometric structure of Yang-Mills theory from the standpoint provided by (what we shall call) the homotopic paradigm. This mathematical paradigm was mainly developed in the framework of (higher) category theory and homotopy type theory and relies on a groupoid-theoretical understanding of equality statements of the form a ​= ​b. From a philosophical perspective, we shall argue that the homotopic paradigm relies a) on a rejection of Leibniz's Principle of the Identity of Indiscernibles and b) on a constructivist understanding of propositions as types of proofs. We shall apply the homotopic reconceptualization of equalities to the equalities between the base points and the fibers of the fiber bundle associated to a Yang-Mills theory. We shall revisit in this framework the articulation (heuristically established in the framework of the so-called gauge argument) between gauge symmetries and the mathematical notion of connection. We shall argue that this homotopic-theoretic understanding of Yang-Mills theories paves the way toward an ontological interpretation of gauge symmetries, that is an interpretation according to which gauge symmetries-far from being nothing but a "surplus structure" resulting from a descriptive redundancy-rely on the intrinsic geometric structures of Yang-Mills theories.