From Zero Object to ($\infty$,n)-Topos: A Categorical Foundation for Gauge Field Theory

This paper presents a rigorous construction of an $(\infty,1)$-topos from a zero object through a sequence of universal categorical constructions, and extends this to a general method for constructing $(\infty,n)$-topoi. Beginning with the terminal category containing only a zero object, we systematically build up categorical structure through free completions under finite (co)limits, additive closure, simplicial enrichment, localization, and presheaf formation. We prove that each step is functorial and preserves essential categorical properties, culminating in an $(\infty,1)$-topos that satisfies all the Giraud-Lurie axioms. The construction is then iterated to yield $(\infty,n)$-topoi for arbitrary $n$, using $n$-fold complete Segal spaces. We establish precise connections between our abstract construction and the cohesive $(\infty,1)$-topos used in mathematical physics, demonstrating how the logical structure required for gauge theory emerges from minimal categorical assumptions. This work provides both a pedagogical pathway through higher category theory and a foundational perspective on the categorical structures underlying quantum field theory, particularly Yang-Mills theory. By distinguishing between the abstract topos constructed here and concrete geometric realizations, we clarify which aspects of gauge theory depend only on general topos-theoretic axioms versus those requiring specific geometric content.