Controlled K-theory I: Basic Theory

This paper provides a full controlled version of algebraic K-theory. There is a rich array of assembly maps; the controlled assembly isomorphism theorem identifying the controlled group with homology; and a statement of the stability theorem describing the behavior of the inverse limit as the control parameter goes to 0. There is also a careful treatment of spectral cosheaf homology and related tools, including an "iterated homology identity" giving a spectrum-level version of the Leray-Serre spectral sequence.