Noncommutative Covering Theory via Fiber Functors: Extension Classes, Monodromy, and the UMIN Framework

We establish two complementary results in noncommutative covering theorywithin the framework of stable infinity-categories. **Theorem A** (unconditional): The Čech cocycle construction and theGrothendieck construction are equivalent geometric operations. From anylocally constant infinity-functor F : U → ∞-Grpd, global non-triviality[F] ≠ 0 in H¹(U, Aut(V)) is equivalent to the Grothendieck construction∫F → U carrying non-trivial monodromy. This result requires no additionalassumptions and no homotopy type theoretic machinery. This statement canbe viewed as a reformulation of the standard correspondence betweenlocally constant ∞-sheaves, Aut(V)-torsors, and monodromy representations. **Theorem B** (conditional): Under a natural correspondence assumption(Assumption A) — the existence of a natural mapΦ_X : Ext¹(X,X) → H¹(U, Aut(V)) compatible with classifying maps ofextensions — the full three-way equivalence holds: non-triviality of [F],non-splitting of the extension class ε ∈ Ext¹(X,X), and non-trivialmonodromy of ∫F are equivalent under Assumption A.The complete naturality of Φ_X in the stable infinity-categorical settingis an open problem; it is verified in the HoTT/infinity-topos setting ofthe companion paper (DOI: 10.5281/zenodo.19385944). **Central principle (UMIN)**:The covering does not pre-exist and produce monodromy; rather, the failureof splitting gives rise to covering structures. In this sense, obstructionclasses provide the generative mechanism for global geometry, reversingthe classical logical order. The novelty lies in three specific contributions:(i) the explicit construction of F as a locally constant infinity-functor,making the Grothendieck construction the central geometric object;(ii) the three-way equivalence as a unified obstruction principle,clarifying the logical order from non-splitting to covering to monodromy;(iii) the stable infinity-categorical setting extending the HoTT frameworkof the companion paper to a broader context. As a purely illustrative example (not used in any proof), the quaternionHopf bundle η : S⁷ → S⁴ is discussed as a UMIN covering, with non-trivialhigher monodromy indexed by the Hopf invariant one theorem (Adams, 1960). **MSC 2020**: 18N60, 55P20, 55R05, 18G80 **Related work**: This paper extends the companion preprint(DOI: 10.5281/zenodo.19385944) from the HoTT framework to the stableinfinity-categorical setting. **Agda formalization**: Partial Cubical Agda formalization available athttps://github.com/Psypher33/UMIN(using the --cubical and --guardedness flags)