It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations. © 2005 Elsevier B.V. All rights reserved.
Führmann, C., & Pym, D. (2006). Order-enriched categorical models of the classical sequent calculus. Journal of Pure and Applied Algebra, 204(1), 21–78. https://doi.org/10.1016/j.jpaa.2005.03.016