Assembly and Morita invariance in the algebraic K-theory of Lawvere theories.

The algebraic K-theory of Lawvere theories provides a context for the systematic study of the stable homology of the automorphism groups of algebraic structures, such as the symmetric groups, the general linear groups, the automorphism groups of free groups, and many, many more. We develop this theory and present a wealth of old and new examples to compare our non-linear setting to the theories of modules over rings via assembly maps. For instance, a new computation included here is that of the algebraic K-theory of the Lawvere theory of Boolean algebras and all theories Morita equivalent to it, in terms of the stable homotopy groups of spheres. We give a comprehensive discussion of Morita invariance: The higher algebraic K-theory of Lawvere theories is invariant under passage to matrix theories, but, in general, not under idempotent modifications. We also prove that algebraic K-theory is a monoidal functor on the category of Lawvere theories with the Kronecker product as its monoidal product. This result enables us to embed the classical assembly maps in algebraic K-theory into our framework and discuss many other examples and extensions.