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By a unital ℓ-group we mean a lattice-ordered abelian group with a distinguished order unit. This paper is concerned with the category   f  p$\mathsf {U}_{\textup {fp}}$    of finitely presented unital ℓ-groups. Using the duality between   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   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.

Simplicial geometry of unital lattice-ordered abelian groups