Enrichment over iterated monoidal categories

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 $V$ –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 [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 $V$ 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 defining 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.

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

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Keywords

loop spaces, enriched categories, $n$–categories, iterated monoidal categories

Mathematical Subject Classification 2000

Primary: 18D10

Secondary: 18D20

References

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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

Volume 4, issue 1 (2004)

Stefan Forcey

Abstract