A Note on Laminar Presentations and Finite Chain Models for Ample Groupoid Homology
A finite free chain model for the homology of a Hausdorff ample groupoid \(G\) is cut out of the Matui complex by a finite clopen partition of each nerve space. We ask what such a model must supply and isolate three requirements: a single model that is a quasi-isomorphism, a chain of models that is a complete system with ranks and transition maps read off the presenting data, and a bound on the number of atoms in terms of the presenting cover. An \(n\)-Helly axiom on the presenting covers supplies none of them, being satisfied by every finite atomization for every \(n\), and a \(1\)-Helly…
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