Algebras of higher operads as enriched categories

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads (Batanin, Adv Math 136:39-103, 1998) to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n∈+∈1)-operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. 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. In this paper we reveal how such an intuition may be formulated in terms of globular operads.