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-trivialityF ≠ 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 mapΦX: 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 ΦX 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)