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

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.

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

“…It follows from Proposition 5.6 that Z ⊗ Boole is the theory of modules over the trivial ring, and the algebraic K-theory spectrum K(Z ⊗ Boole) is contractible. On the other hand, we can use the computation of the algebraic K-theory of the theory of Boolean algebras in [5,Corollary 5.2]. The result implies that the source K(Z) ∧ K(Boole) of the assembly map is not contractible because…”

“…Lawvere theories are a fundamental tool for encoding algebraic structures, first introduced in Lawvere’s thesis [30] and now widely used throughout algebra, logic and related disciplines. We review the basic notions and our notation for Lawvere theories, using the same language as in the prequel [5]. In addition, we also discuss Kronecker products.…”

“…In this article, we offer a new perspective on assembly maps in algebraic K-theory, one that leads to several generalisations. This perspective comes from constructing the three constituent spectra K(Z), Σ ∞ + (BG) and K(ZG) as part of the same framework: they are all algebraic K-theory spectra K(T ) of suitable Lawvere theories T, as defined in our earlier work [5] and recalled below. As a first approximation, we can think of Lawvere theories as generalisations of rings.…”

“…In the following Section 2, we recall Lawvere theories and their Kronecker products, which generalise tensor products of rings (Section 2.1). To make this paper self-contained, we also recall the definition of their algebraic K-theory from our earlier work [5] in Section 2.2. Section 3 is the heart of the paper and contains the main results on multiplicativity.…”

“…Here, the theory of abelian groups is given by the modules over the ring of integers and is the theory of abelian group objects in T , which is always given by modules over a ring (see [51, Theorem 13.2] and the discussion in Example 2.7 below). These assembly maps can be equivalences, as happens for Cantor algebras, where the rings in question are the Leavitt algebras (see our Theorem 5.4, based on [48]), or they can fail to be even rationally injective, as happens for Boolean algebras (see our Theorem 5.7, based on [5]), where they are null-homotopic. Specialising further to the theory T of actions by a fixed discrete group G , we recover Loday’s assembly map (1.1) for the group G .…”