5 M ar 2 00 8 Algebras of higher operads as enriched categories

We decribe the correspondence between normalised ω-operads in the sense of [1] and certain lax monoidal structures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category enriched in a lax monoidal category. Within the aforementioned correspondence, we provide also an equivalence between the algebras of a given normalised ωoperad, and categories enriched in globular sets for the induced lax monoidal structure. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)-category is something like a category enriched in weak n-categories, and in this paper we begin to reveal how such an intuition may be formulated in terms of the machinery of globular operads.