Boolean algebras, Morita invariance, and the algebraic K-theory of Lawvere theories
arXiv:2011.11755 · doi:10.1017/S0305004123000105
Abstract
The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we fully address the question of how Morita equivalence classes of Lawvere theories interact with algebraic K-theory. On the one hand, we show that the higher algebraic K-theory is invariant under passage to matrix theories. On the other hand, we show that the higher algebraic K-theory is not fully Morita invariant because of the behavior of idempotents in non-additive contexts: We compute the K-theory of all Lawvere theories Morita equivalent to the theory of Boolean algebras.
22 pages. This updated paper discusses the work on Morita equivalence of Lawvere theories that appeared in version one. In order to better highlight the two separate directions of the results in that first version, the material on assembly maps has been incorporated into a second paper, arXiv:2112.07003. Revised version; to appear in Math. Proc. Cambridge Philos. Soc