Enrichment as Categorical Delooping I: Enrichment Over Iterated Monoidal Categories

Joyal and Street note in their paper on braided monoidal categories [10] 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. What is meant by “based upon” here will be made more clear in the present paper. 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 first mentioned by Baez and Dolan in [1]. 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 [3]. It seems that the analogy of loop spaces is a good guide for how to define 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. I indicate how this process may be iterated by enriching over V–Cat, along the way defining the 3–category of categories enriched over V–Cat. In the next paper I hope 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.