Involutive Weak Globular $\omega$-categories

We investigate the notion of involutive weak globular $\omega$-categories via T.Leinster’s approach: as algebras for the initial contracted globular operad in the bicategory of globular collections induced by the Cartesian monad of the free involutive strict $\omega$-category functor on globular $\omega$-sets. An apparently more restrictive notion of involutive weak globular $\omega$-categories as algebras for the initial operadic-contraction in the bicategory of globular contracted-collections induced by the previous Cartesian monad (where here the operadic multiplications and units satisfy further compatibility axioms with the contractions) is also considered.