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 –Cat of categories enriched over a braided monoidal category is not itself braided in any way that is based upon the braiding of . The exception that they mention is the case in which is symmetric, which leads to –Cat being symmetric as well. The symmetry in –Cat is based upon the symmetry of . The motivation behind this paper is in part to describe how these facts relating and –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 –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 –fold monoidal categories and their higher dimensional counterparts. The main result is that for a –fold monoidal category, –Cat becomes a –fold monoidal –category in a canonical way. In the next paper I indicate how this process may be iterated by enriching over –Cat, along the way defining the 3–category of categories enriched over –Cat. In future work I plan to make precise the –dimensional case and to show how the group completion of the nerve of is related to the loop space of the group completion of the nerve of –Cat.
This paper is an abridged version of ‘Enrichment as categorical delooping I: Enrichment over iterated monoidal categories’, math.CT/0304026.
"Enrichment over iterated monoidal categories." Algebr. Geom. Topol. 4 (1) 95 - 119, 2004. https://doi.org/10.2140/agt.2004.4.95