Universal Assembly and Cellular Loop Spaces on Regular CW Complexes

Abstract

We develop a regular CW analogue of the classical assembly formalism for chain complexes appearing in algebraic surgery theory. From the cell poset, we construct combinatorial path and loop objects using fences of comparable cells and prove that their classifying spaces recover the homotopy types of the ordinary based path and loop spaces. The resulting loop object carries a natural monoid structure, giving rise to a DG algebra defined directly from the cellular structure. For complexes of cellular cosheaves, we introduce a universal assembly functor to modules over the group ring of the fundamental group and study the localization determined by global equivalences. The associated homotopy category is identified with a Verdier quotient of the derived category of cellular cosheaves, and its fibrant objects are precisely the homotopy locally constant complexes. A single elementary cosheaf becomes a compact generator after localization, and its derived endomorphism DG algebra is identified with singular chains on the cellular loop space. Consequently, the localized theory admits a Morita description in terms of DG modules over the loop DG algebra. The formalism provides a regular CW counterpart of the classical delta-set approach to assembly in algebraic surgery theory due to Ranicki and Weiss.