The Canonical Complex of the Weak Order

We study the canonical complex of a finite semidistributive lattice L, a simplicial complex which encodes each interval [x, y] of L by recording simultaneously the canonical join representation of x and the canonical meet representation of y, and behaves properly with respect to lattice quotients of L. We then describe combinatorially the canonical complex of the weak order on permutations in terms of semi-crossing arc bidiagrams, formed by the superimposition of two non-crossing arc diagrams of N. Reading. Finally, we provide an algorithm to describe the Kreweras maps in any lattice quotient of the weak order in terms of semi-crossing arc bidiagrams.