Algebraic and Geometric Topology 4 (2004), paper no. 7, pages 95-119.

Enrichment over iterated monoidal categories

Stefan Forcey

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 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' arXiv:math.CT/0304026.

Keywords. Loop spaces, enriched categories, n-categories, iterated monoidal categories

AMS subject classification. Primary: 18D10. Secondary: 18D20.

DOI: 10.2140/agt.2004.4.95

E-print: arXiv:math.CT/0403152

Submitted: 29 September 2003. (Revised: 1 March 2004.) Accepted: 4 March 2004. Published: 6 March 2004.

Notes on file formats

Stefan Forcey
Department of Mathematics, Virginia Tech
460 McBryde Hall, Blacksburg, VA 24060, USA

AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to