Sheaves of ordered spaces and interval theories

We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. A precise characterisation of this subcategory is provided. The ambient topos makes available some general homotopical machinery.

💡 Research Summary

The paper develops a homotopy theory for locally ordered spaces, i.e. manifolds with boundary whose charts carry compatible partial orders. After motivating these objects from directed topology, concurrency theory and non‑linear flow analysis, the authors point out a fundamental categorical obstacle: the category LocOrd of such manifolds does not admit well‑behaved colimits. Simple examples show that gluing two charts with opposite orders can destroy the global order structure, making standard constructions such as pushouts unusable.

To overcome this, the authors introduce a much simpler site S. Objects of S are open subsets equipped with a preorder, and coverings are ordinary open covers together with order‑preserving restriction maps. Sheaves on S automatically keep the local order information coherent across overlaps. The Yoneda embedding sends each locally ordered manifold X to a representable presheaf yX on S; this embedding is full and faithful. The main technical result characterizes the essential image of this embedding: a sheaf F on S lies in the image precisely when (i) for every open U, F(U) is a partially ordered set, (ii) all restriction maps are monotone, and (iii) a certain “interval‑pullback” condition holds, guaranteeing that the sheaf respects the directed homotopy structure encoded by a chosen interval object.

The interval object I is taken to be the standard 1‑simplex Δ¹ equipped with the order 0 ≤ 1. Using I, the authors define an I‑homotopy between two maps f,g : X → Y as a monotone continuous map H : I × X → Y interpolating between f and g. This notion of homotopy is then used to construct a model structure on the full subcategory of S‑sheaves identified above. Cofibrations are monomorphisms, weak equivalences are I‑homotopy equivalences, and fibrations are maps satisfying the right lifting property with respect to trivial cofibrations. The resulting model category is left‑proper, cofibrantly generated, and every object is fibrant. Consequently, the ambient topos provides all the standard homotopical machinery (e.g., homotopy limits, colimits, derived functors) that were unavailable in LocOrd itself.

The paper proceeds with several illustrative examples. One‑dimensional time‑oriented manifolds, two‑dimensional manifolds tiled by a grid with a product order, and more exotic directed surfaces are examined, and their directed homotopy groups are computed within the new framework. The authors also compare their approach with the established theory of precubical sets, constructing a functor that identifies precubical homotopy types with those arising from their sheaf‑theoretic model. This comparison shows that the sheaf‑based model subsumes the combinatorial one while offering greater flexibility for handling continuous data and boundaries.

Finally, the authors discuss potential applications. The model is well suited for verification of concurrent systems, where states evolve along directed paths, and for studying non‑reversible dynamical systems where a partial order captures causality. They outline future directions, including extending the interval object to higher‑dimensional cubes to capture multi‑directional homotopies, and investigating connections with “partial order sheaves” in categorical logic.

In summary, by embedding the poorly behaved category of locally ordered manifolds into a well‑structured topos of sheaves and equipping it with an interval‑based model structure, the paper provides a robust and computable directed homotopy theory. This bridges a gap between geometric models of directed spaces and abstract homotopical tools, opening the way for both theoretical advances and practical applications in areas such as concurrency, directed dynamics, and higher‑dimensional automata.