Simplicial geometry of unital lattice-ordered abelian groups

Abstract

Abstract By a unital ℓ-group we mean a lattice-ordered abelian group with a distinguished order unit. This paper is concerned with the category 𝖴 f p $\mathsf {U}_{\textup {fp}}$ of finitely presented unital ℓ-groups. Using the duality between 𝖴 f p $\mathsf {U}_{\textup {fp}}$ and a category of rational polyhedra, we will provide (i) a construction of finite limits and co-limits in 𝖴 f p $\mathsf {U}_{\textup {fp}}$ ; (ii) a Cantor–Bernstein–Schröder theorem for finitely presented unital ℓ-groups; (iii) a proof that the fibered product of finitely generated projective unital ℓ-groups is projective; (iv) a geometrical characterization of exact unital ℓ-groups.