  We investigate the notion of involutive weak globular $\omega$-categories via
Jacque Penon's approach. In particular, we give the constructions of a free
self-dual globular $\omega$-magma, of a free strict involutive globular
$\omega$-category, over an $\omega$-globular set, and a contraction between
them. The monadic definition of involutive weak globular $\omega$-categories is
given as usual via algebras for the monad induced by a certain adjunction. In
our case, the adjunction is obtained from the "free functor" that associates to
every $\omega$-globular set the above contraction. Some examples of involutive
weak globular $\omega$-categories are also provided.
