  We define and study the canonical complex of a finite semidistributive
lattice $L$. It is the simplicial complex on the join or meet irreducible
elements of $L$ which encodes each interval of $L$ by recording the canonical
join representation of its bottom element and the canonical meet representation
of its top element. This complex behaves properly with respect to lattice
quotients of $L$, in the sense that the canonical complex of a quotient of $L$
is the subcomplex of the canonical complex of $L$ induced by the join or meet
irreducibles of $L$ uncontracted in the quotient. 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. We provide explicit direct bijections
between the semi-crossing arc bidiagrams and the weak order interval posets of
G. Ch\^atel, V. Pilaud and V. Pons. 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.
