Ethic Duality: A Homological Framework for Primal-Dual Problems

TL;DR

This paper introduces a homological duality framework for primal-dual problems using derived functors, unifying various classical dualities and providing a systematic hierarchy of obstructions to exactness across multiple domains.

Contribution

It develops a novel homological duality framework based on Ext-groups that unifies primal-dual exactness criteria across diverse fields like optimization, graph theory, and dynamical systems.

Findings

Ext^1 vanishing characterizes primal-dual exactness in optimization.

Higher Ext-groups encode obstructions in various duality contexts.

Framework is stable under derived Morita equivalence and domain-independent.

Abstract

We develop a homological duality framework based on a contravariant functor $D=\operatorname{Hom}_E(-,R)$ with dualizing object $R$. A morphism is called ethic when it satisfies the canonical double-dual compatibility $D^2(f)\eta=\eta f$. In the derived setting, the functor $\mathrm{RHom}_E(-,R)$ produces a graded family of Ext-groups that measure all failures of this compatibility. The first layer $\operatorname{Ext}^1$ identifies primal-dual gaps, while higher $\operatorname{Ext}^k$ provide a systematic hierarchy of derived obstructions to exactness. This formulation specializes uniformly across several classical domains. In linear and conic optimization, Farkas- and Slater-type exactness criteria correspond to the vanishing of $\operatorname{Ext}^1$, and integer duality gaps coincide with torsion Ext-classes. In graph theory, Kirchhoff- and Baker-Norine-type dualities arise as instances of ethic exactness. In dynamical systems, the higher derived layers encode nonvanishing persistence phenomena. Additional examples include social-choice configurations, categorical factorization in scattering formalisms, coding-theoretic duality, and Bellman-type recurrences, all appearing as concrete instances of Ext-controlled exactness. All resulting invariants are stable under derived Morita equivalence and depend only on the dualizing pair $(E,R)$. The framework therefore provides a substrate-independent criterion for primal-dual exactness and a uniform homological description of its obstructions.