Vertically Iterated Classical Enrichment

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, Schwänzl and Vogt but with the addition of making visible the coherent associators α. 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.