The Canonical Complex of the Weak Order

We define and study the canonical complex of a finite semidistributive lattice L. It is simplicial on join or meet irreducible elements L which encodes each interval by recording representation its bottom element top element. This behaves properly with respect to quotients L, in sense that quotient subcomplex induced irreducibles uncontracted quotient. then describe combinatorially weak order permutations terms semi-crossing arc bidiagrams, formed superimposition two non-crossing diagrams N. Reading. provide explicit direct bijections between bidiagrams posets G. Châtel, V. Pilaud Pons. Finally, we an algorithm Kreweras maps any bidiagrams.