Generalizations of Loday's assembly maps for Lawvere's algebraic theories
arXiv:2112.07003 · doi:10.1017/S1474748022000603
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.
24 pages. This paper overlaps with the first version of arXiv:2011.11755, which we have split into two separate papers in order to highlight the two separate directions of results. The current work contains the results on assembly maps and arXiv:2011.11755 focuses on Morita equivalence results. Revised version; to appear in J. Inst. Math. Jussieu