Monad interleaving: A construction of the operad for Leinster’s weak ωω-categories

We show how to “interleave” the monad for operads and the monad for contractions on the category CollColl of collections, to construct the monad for the operads-with-contraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit mutual stability conditions that enable us to alternate the dimension-by-dimension free functors. Hence we give an explicit construction of a left adjoint for the forgetful functor OWC⟶CollOWC⟶Coll, from the category of operads-with-contraction to the category of collections. By applying this to the initial (empty) collection, we obtain explicitly an initial operad-with-contraction, whose algebras are by definition the weak ωω-categories of Leinster.