  The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable
orders defined on the set of Catalan objects of a given size. These lattices
are ordered by inclusion: the Stanley lattice is an extension of the Tamari
lattice which is an extension of the Kreweras lattice. The Stanley order can be
defined on the set of Dyck paths of size $n$ as the relation of \emph{being
above}. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck
paths. In a former article, the second author defined a bijection $\Phi$
between pairs of non-crossing Dyck paths and the realizers of triangulations
(or Schnyder woods). We give a simpler description of the bijection $\Phi$.
Then, we study the restriction of $\Phi$ to Tamari's and Kreweras' intervals.
We prove that $\Phi$ induces a bijection between Tamari intervals and minimal
realizers. This gives a bijection between Tamari intervals and triangulations.
We also prove that $\Phi$ induces a bijection between Kreweras intervals and
the (unique) realizers of stack triangulations. Thus, $\Phi$ induces a
bijection between Kreweras intervals and stack triangulations which are known
to be in bijection with ternary trees.
