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

TL;DR

This paper investigates the relationship between Morita equivalence of Lawvere theories and their algebraic K-theory, revealing invariance under matrix constructions but not full Morita invariance due to idempotent behavior.

Contribution

It establishes the invariance of higher algebraic K-theory under matrix theories and computes K-theory for theories Morita equivalent to Boolean algebras, clarifying Morita invariance limits.

Findings

Higher algebraic K-theory is invariant under passage to matrix theories.

Higher algebraic K-theory is not fully Morita invariant in non-additive contexts.

Computed K-theory for Lawvere theories Morita equivalent to Boolean algebras.

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.