Unifying duality theorems for width parameters in graphs and matroids I. Weak and strong duality

We prove a general duality theorem for width parameters in combinatorial structures such as graphs and matroids. It implies the classical such theorems for path-width, tree-width, branch-width and rank-width, and gives rise to new width parameters with associated duality theorems. The dense substructures witnessing large width are presented in a unified way akin to tangles, as orientations of separation systems satisfying certain consistency axioms.