Enrichment over iterated monoidal categories

Abstract

Joyal and Street note in their paper on braided monoidal categories [Braided tensor categories, Advances in Math. 102(1993) 20–78] that the 2–category $\mathcal{V}$–Cat of categories enriched over a braided monoidal category $\mathcal{V}$ is not itself braided in any way that is based upon the braiding of $\mathcal{V}$. The exception that they mention is the case in which $\mathcal{V}$ is symmetric, which leads to $\mathcal{V}$–Cat being symmetric as well. The symmetry in $\mathcal{V}$–Cat is based upon the symmetry of $\mathcal{V}$. The motivation behind this paper is in part to describe how these facts relating $\mathcal{V}$ and $\mathcal{V}$–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 [Iterated Monoidal Categories, Adv. Math. 176(2003) 277–349]. 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 $\mathcal{V}$ a $k$–fold monoidal category, $\mathcal{V}$–Cat becomes a $\left(k-1\right)$–fold monoidal $2$–category in a canonical way. In the next paper I indicate how this process may be iterated by enriching over $\mathcal{V}$–Cat, along the way defining the 3–category of categories enriched over $\mathcal{V}$–Cat. In future work I plan to make precise the $n$–dimensional case and to show how the group completion of the nerve of $\mathcal{V}$ is related to the loop space of the group completion of the nerve of $\mathcal{V}$–Cat.

This paper is an abridged version of ‘Enrichment as categorical delooping I: Enrichment over iterated monoidal categories’, math.CT/0304026.