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Enrichment over iterated monoidal categories | | Stefan Forcey
; | | Date: |
9 Mar 2004 | | Journal: | Algebraic and Geometric Topology 4 (2004) 95-119 | | Subject: | Category Theory; Algebraic Topology; Quantum Algebra MSC-class: 18D10, 18D20 | math.CT math.AT math.QA | | 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’ math.CT/0304026. | | Source: | arXiv, math.CT/0403152 | | Services: | Forum | Review | PDF | Favorites | |
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