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, Schw¨anzl and Vogt but with the addition of making visible the coherent associators _i. 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.
Forcey, Stefan, "Higher Dimensional Enrichment" (2006). Mathematical Sciences Faculty Research. 6.