  In this article we use existing machinery to define connective $K$-theory
spectra associated to topological ringoids. Algebraic $K$-theory of discrete
ringoids, and the analytic $K$-theory of Banach categories are obtained as
special cases.
  As an application, we show how the analytic assembly maps featuring in the
Novikov and Baum-Connes conjectures can be factorised into composites of
assembly maps resembling those appearing in algebraic $K$-theory and maps
coming from completions of certain topological ringoids into Banach categories.
These factorisations are proved by using existing characterisations of assembly
maps along with our unified picture of algebraic and analytic $K$-theory.
