Enrichment Over Iterated Monoidal Categories

Joyal and Street note in their paper on braided monoidal categories [9] that the 2{category V{Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V . The exception that they mention is the case in which V is symmetric, which leads to V {Cat being symmetric as well. The symmetry in V {Cat is based upon the symmetry of V . The motivation behind this paper is in part to describe how these facts relating V and V{Cat are in turn related to a categorical analogue of topological delooping. To do so I need to pass to a more general setting than braided and symmetric categories | in fact the k{fold monoidal categories of Balteanu et al in [2]. It seems that the analogy of loop spaces is a good guide for how to dene the concept of enrichment over various types of monoidal objects, including k{fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k{fold monoidal category, V{Cat becomes a (k - 1){fold monoidal 2{category in a canonical way. In the next paper I indicate how this process may be iterated by enriching over V{Cat, along the way dening the 3{category of categories enriched over V{Cat. In future work I plan to make precise the n{dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V{Cat.