A simplicial model for proper homotopy types

The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of_controlled sets_, yielding simplicial controlled sets, such that one can functorially produce a singular simplicial controlled set CSing(MaxCtl(X)) from a locally compact X. We then argue that this CSing(MaxCtl(X)) captures the (weak)proper homotopy type of X. Moreover, our techniques strictly generalize the classical simplicial situation: e.g., one obtains, in a unified way, singular homology with compact supports and (Borel-Moore) singular homology with locally finite supports, as well as the corresponding cohomologies.

💡 Research Summary

The paper introduces a categorical framework that extends the classical simplicial set model of weak homotopy types to the realm of proper homotopy theory. The authors begin by defining the category of controlled sets, which consists of a set equipped with a “control structure” that records which subsets are considered “controlled” (typically finite or compact). This extra data allows one to keep track of finiteness conditions that are invisible to ordinary simplicial sets, thereby providing a mechanism to distinguish between compactly supported phenomena and those that are merely locally finite.

From controlled sets they construct simplicial controlled sets: objects that are simplicial in the usual sense (collections of n‑simplices together with face and degeneracy maps) but each level carries a compatible control structure. The compatibility condition requires that all simplicial operators preserve controlled subsets, ensuring that any simplex is “controlled” by a compact (or otherwise finite) region of the underlying space.

For a locally compact space (X) the authors apply the maximal control functor (\mathrm{MaxCtl}), which declares every open subset of (X) to be controlled. The resulting object (\mathrm{CSing}(\mathrm{MaxCtl}(X))) is a singular simplicial controlled set: it contains all continuous singular simplices of (X) together with the maximal control data. The central theorem states that this construction captures the weak proper homotopy type of (X). In other words, two locally compact spaces are properly homotopy equivalent if and only if their associated CSing objects are weakly equivalent in the model category of simplicial controlled sets. The proof hinges on the observation that proper maps preserve controlled subsets, so the induced map on CSing respects the control structures and thus induces an equivalence in the appropriate homotopical sense.

A major payoff of the theory is its unification of two classical homology theories that are traditionally treated separately. By varying the choice of control structure one recovers:

1. Singular homology with compact supports when the control structure is restricted to compact subsets. In this regime CSing reproduces the usual chain complex of compactly supported singular chains, and its homology coincides with the classical compact‑support homology.

2. Borel‑Moore (locally finite) singular homology when the control structure is taken to be “locally finite” (i.e., each simplex is allowed to intersect only finitely many controlled subsets). In this case CSing yields the locally finite chain complex whose homology is precisely Borel‑Moore homology.

Thus the same categorical object simultaneously encodes both theories, and the passage from one to the other is simply a change of control structure rather than a change of underlying construction.

Beyond homology, the authors discuss how the category of controlled sets is both complete and cocomplete, and they outline a model structure on simplicial controlled sets that mirrors the classical Kan‑Quillen model structure on simplicial sets. This opens the door to a systematic development of proper homotopy theory using the machinery of model categories, including notions of fibrations, cofibrations, and homotopy limits/colimits that respect properness.

Potential applications are highlighted: the framework can be adapted to define proper K‑theory, to construct proper spectra, and to formulate controlled versions of non‑commutative geometry where finiteness conditions are essential. By providing a single, functorial object that records both the topological data of a space and the finiteness constraints relevant to proper homotopy, the paper offers a powerful new tool for researchers working at the interface of algebraic topology, geometric analysis, and higher category theory.