Smooth atlas with identity transitions on blowups of locally ordered precubical realizations

Establish the existence of a smooth atlas on the blowup \tilde{|P|_{lo}} of the locally ordered realization |P|_{lo} of any precubical set P—where the blowup is the euclidean local order formed by germs of n-dimensional traversals that contain their base point—with the property that every chart transition map on overlaps is the identity map.

Background

The paper constructs, via sheaf-theoretic methods, the blowup \tilde{X} of a locally ordered space X as a euclidean local order whose points are germs of n-dimensional traversals containing their base point, and shows a universal lifting property for local embeddings. For locally ordered realizations of precubical sets, they provide charts in examples where transition maps are identities and give a combinatorial description of the blowup.

In the conclusion, the authors aim to extend metric constructions from prior work on non-Hausdorff manifolds over graph products by equipping the blowup of locally ordered realizations of precubical sets with a smooth atlas. They explicitly conjecture that such an atlas can be chosen so that all chart transitions are identity maps, which would greatly simplify smooth structure and may facilitate defining a pseudometric reflecting execution times of program traces.