Abstract zip data
Christopher Lang
TL;DR
The work addresses the problem of computing the orbit/topological structure of the quotient stack $[E\setminus G]$ arising from zip data and extends the refinement approach from $G$-zips to arbitrary groups with maps $\tau,\sigma$. The method defines a zip datum ${\mathcal Z}=(E,G,\tau,\sigma)$, introduces twists ${\mathcal Z}^x$ and refinements ${\mathcal Z}_1$, and uses a coarser equivalence $y\sim x$ with a central bijection ${\Psi}_{\mathcal Z}^x$ to relate refined and original classes; a rooted-forest model then describes the full orbit structure via successive double quotients, with $G/\sim_{\mathcal Z} \cong \lim R_n$ under stationarity. The contributions include invariant constructions $E_\infty$ and $G_\infty$, the formal refinement framework, and the forest-based description that generalizes Bruhat-type decompositions to assemble the topological space from simpler data. The results offer a broad, combinatorial toolkit for computing orbit decompositions in diverse zip-data settings, with potential applications to displays and moduli problems where similar quotient structures arise.
Abstract
The topological space of the stack of $G$-zips can be computed using a refinement process. We extend this refinement process to a more general framework and show that in many situations this process can be used to compute the equivalence classes of a certain equivalence relation, which in the case of $G$-zips is precisely the topological space.