Generalizations of Loday's assembly maps for Lawvere's algebraic theories

TL;DR

This paper generalizes Loday's assembly maps to Lawvere theories, showing their K-theory is lax monoidal, which simplifies the theory and enables new applications and interpolation schemes.

Contribution

It introduces a framework that places the components of Loday's assembly maps on equal footing and extends the concept to Lawvere theories, broadening the scope of algebraic K-theory.

Findings

K-theory of Lawvere theories is lax monoidal

Extended assembly maps to new algebraic contexts

Provided numerous illustrative examples

Abstract

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.