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.