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

Algebraic $K$-theory Spectra and Factorisations of Analytic Assembly Maps