  Loday's assembly maps approximate the K-theory of group rings by the K-theory
of the coefficient ring and the corresponding homology of the group. We present
a generalization that places both ingredients on the same footing. Building on
Elmendorf--Mandell's multiplicativity results and our earlier work, we show
that the K-theory of Lawvere theories is lax monoidal. This result makes it
possible to present our theory in a user-friendly way without using higher
categorical language. It also allows us to extend the idea to new contexts and
set up a non-abelian interpolation scheme, raising novel questions. Numerous
examples illustrate the scope of our extension.
