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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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