Generalisations of Loday’s Assembly Maps for Lawvere’s Algebraic Theories

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 generalisation 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 nonabelian interpolation scheme, raising novel questions. Numerous examples illustrate the scope of our extension.

Boolean algebras, Morita invariance and the algebraic K-theory of Lawvere theories

“…Nonetheless, the specific setting of Lawvere theories balances rigidity and flexibility in a way that suggests it to be particularly amenable to homological stability questions as well. Additional motivation for the algebraic K-theory of Lawvere theories, in the form of multiplicative matters and applications to assembly maps, is discussed in [ 5 ].…”