2008/12/05 by Viêt-Trung Luu, Luu, Viêt-Trung
Mathematics · #18G30 #55N10 (Secondary) #55P57 (Primary) #55U10 #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.0812.1138
openalex publication_date 2008/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category ofcontrolled 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.