Algebraic and Geometric Topology 4 (2004),
paper no. 7, pages 95-119.
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
Loop spaces, enriched categories, n-categories, iterated monoidal categories
AMS subject classification.
Submitted: 29 September 2003.
(Revised: 1 March 2004.)
Accepted: 4 March 2004.
Published: 6 March 2004.
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