Higher Dimensional Enrichment

Lyubashenko has described enriched 2–categories as categories enriched over V–Cat, the 2–category of categories enriched over a symmetric monoidal V. Here I generalize this to a k–fold monoidal V. The latter is defined as by Balteanu, Fiedorowicz, Schwänzl and Vogt but with the addition of making visible the coherent associators α. The symmetric case can easily be recovered. The introduction of this paper proposes a recursive definition of V–n–categories and their morphisms. Then I consider the special case of V–2–categories and give the details of the proof that with their morphisms these form the structure of a 3–category.