A simplicial model for proper homotopy types

TL;DR

This paper introduces a new simplicial controlled set framework that captures the proper homotopy type of locally compact spaces, generalizing classical homology theories with a unified approach.

Contribution

It develops a category of controlled sets and constructs a functorial simplicial controlled set that encodes proper homotopy types, extending classical simplicial methods.

Findings

Captures proper homotopy types via simplicial controlled sets

Unifies singular homology with compact and locally finite supports

Generalizes classical simplicial homotopy and homology theories

Abstract

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.