Lattice homomorphisms between weak orders

We classify surjective lattice homomorphisms $W\to W^{\mathrm{\prime }}$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\mathrm{\Theta }$ on $W$ such that the quotient $W\mathrm{/}\mathrm{\Theta }$ is isomorphic to $W^{\mathrm{\prime }}$. Surprisingly, surjective homomorphisms exist quite generally: They exist if and only if the diagram of $W^{\mathrm{\prime }}$ is obtained from the diagram of $W$ by deleting vertices, deleting edges, and/or decreasing edge labels. A surjective homomorphism $W\to W^{\mathrm{\prime }}$ is determined by its restrictions to rank-two standard parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter groups, this determination in rank two is nontrivial. Indeed, from the combinatorial lattice theory point of view, all of these classification results should appear unlikely a priori. As an application of the classification of surjective homomorphisms between weak orders, we also obtain a classification of surjective homomorphisms between Cambrian lattices and a general construction of refinement relations between Cambrian fans.