#### Volume 4, issue 1 (2004)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Enrichment over iterated monoidal categories

### Stefan Forcey

Algebraic & Geometric Topology 4 (2004) 95–119
 arXiv: math.CT/0403152
##### 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 $\mathsc{V}$–Cat of categories enriched over a braided monoidal category $\mathsc{V}$ is not itself braided in any way that is based upon the braiding of $\mathsc{V}$. The exception that they mention is the case in which $\mathsc{V}$ is symmetric, which leads to $\mathsc{V}$–Cat being symmetric as well. The symmetry in $\mathsc{V}$–Cat is based upon the symmetry of $\mathsc{V}$. The motivation behind this paper is in part to describe how these facts relating $\mathsc{V}$ and $\mathsc{V}$–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 $\mathsc{V}$ a $k$–fold monoidal category, $\mathsc{V}$–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 $\mathsc{V}$–Cat, along the way defining the 3–category of categories enriched over $\mathsc{V}$–Cat. In future work I plan to make precise the $n$–dimensional case and to show how the group completion of the nerve of $\mathsc{V}$ is related to the loop space of the group completion of the nerve of $\mathsc{V}$–Cat.

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

##### Keywords
loop spaces, enriched categories, $n$–categories, iterated monoidal categories
Primary: 18D10
Secondary: 18D20
##### Publication
Received: 29 September 2003
Revised: 1 March 2004
Accepted: 4 March 2004
Published: 6 March 2004
##### Authors
 Stefan Forcey Department of Mathematics Virginia Tech 460 McBryde Hall Blacksburg VA 24060 USA