Lyubashenko has described enriched 2-categories as categories enriched over V-Cat, the 2-category of categories enriched over a symmetric monoidal V. This construction is the strict analogue for V-functors in V-Cat of Brian Day’s probicategories for V-modules in V-Mod. Here I generalize the strict version to enriched n-categories for 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 αi. The symmetric case can easily be recovered. This paper proposes a recursive definition of V-n-categories and their morphisms. We show that for V k-fold monoidal the structure of a (k−n)-fold monoidal strict (n + 1)-category is possessed by V-n-Cat. This article is a completion of the work begun in [Forcey, 2003], and the initial sections duplicate the beginning of that paper.
Forcey, Stefan, "Vertically Iterated Classical Enrichment" (2006). Mathematical Sciences Faculty Research. 3.