A Čech--Stokes Pushout Groupoid: a Log/Kummer Betti Presenter for Stokes Torsors

Abstract: Let $X$ be a complex manifold and let $D\subset X$ be a simple normal crossings divisor. Fix a Kummer level $n\ge 1$ and an irregular type $Φ$ along $D$, and consider the logarithmic Betti boundary $X^{\log}_n$ (the level-$n$ Kato--Nakayama space). We construct a strictly $1$-categorical, cover-based Betti presenter for Stokes torsors by building a small strict Čech--Stokes collar groupoid on a punctured logarithmic collar of $D$, together with a canonical projection $π$ to the underlying sector Čech groupoid. The central observation is that the appropriate collar Stokes moduli is the groupoid of strict functorial sections $\operatorname{Sec}(π)$ (equivalently, $\mathrm{St}Φ$-torsors on the sector cover), yielding a fully explicit presenter that remains small and strictly functorial. We then glue the collar presenter to the ordinary Čech presenter of $U:=X\setminus D$ through a strict diagram of small groupoids and an explicit $2$-pushout presenter $G{Φ,n}$. We prove that global Stokes objects, defined intrinsically as a $2$-fiber product, are computed by a strict torsorial gluing problem, i.e.\ by Stokes-corrected torsorial representations of $G_{Φ,n}$. In arbitrary SNC depth we solve the local corner presentation problem by an explicit chamber-complex model on the angular torus: Stokes torsors are nonabelian cocycles on the $1$-skeleton with relations read off from $2$-cells. The overall construction is canonical up to Morita equivalence with a contractible space of identifications (invariant under stratified subdivision), and it is compatible with Kummer descent along all normal crossings strata via $(μ_n)^k$-equivariant local models and descent to homotopy fixed points.