  The aim of this paper is to give an alternative construction of Street's
cosimplicial object of orientals, based on an idea of Burroni that orientals
are free algebras for some algebraic structure on strict $\omega$-categories.
More precisely, following Burroni, we define the notion of an expansion on an
$\omega$-category and we show that the forgetful functor from strict
$\omega$-categories endowed with an expansion to strict $\omega$-categories is
monadic. By iterating this monad starting from the empty $\omega$-category, we
get a cosimplicial object in strict $\omega$-categories. Our main contribution
is to show that this cosimplicial object is the cosimplicial objects of
orientals. To do so, we prove, using Steiner's theory of augmented directed
chain complexes, a general result for comparing polygraphs having same
generators and same linearized sources and targets.
