math.CT — Category Theory

Higher Galois Recognition and Parameter Descent

Contributed by Zhenyu Sun

Let $n\geq0$ and let $\mathcal X$ be a bounded coherent $\infty$-topos. Suppose that finitely many noninitial coherent $n$-truncated objects cover its terminal object, and that every coherent $n$-truncated object in each of their slices has a finite covering family of sections. We identify $\mathcal X^{\mathrm{coh}}_{\leq n}$ with $\operatorname{Fun}(M,\mathcal S_{\pi,\leq n})$ for a $\pi$-finite $(n+1)$-type $M$, recovered by all finite models preserving finite limits, finite coproducts, and effective epimorphisms. Descent uses degrees through $n+2$ for objects and $n+1$ for maps; both bounds are sharp for finite homotopy actions. Every exact functor $F$ between recognized categories has a universal parameter $P_F=F_!(1)$ and a canonical factorization through its slice with fully faithful second functor. This parameter characterizes full faithfulness, conservativity, and all exact slice factorizations. For successive abelian Postnikov layers, we identify the space of marked exact lifts with a path space of Eilenberg--Mac Lane sections with local coefficients. The obstruction, classes of lifts, and relative automorphisms occur in degrees $r+1$, $r$, and $r-1$. We compute a twisted parity obstruction and, for odd primes, classify cyclic self-equivalence components by a $p$-adic valuation. A nonzero Postnikov class can persist in the identity component. For continuous finite families over a profinite $(n+1)$-type, we prove that a uniform finite-section atlas exists precisely when the complete model space is $\pi$-finite. Exact functors from finite model semantics, together with their lift spaces, satisfy finite-stage descent, while explicit profinite examples separate objectwise trivialization from uniform parameter elimination.

Author comments: 49 pages. Second paper in the series Model groupoids and finite homotopy semantics. Includes two verification scripts and their outputs.