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
–Cat
of categories enriched over a braided monoidal category
is not itself braided in any way that is based upon the braiding of
.
The exception that they mention is the case in which
is symmetric,
which leads to –Cat
being symmetric as well. The symmetry in
–Cat is based upon
the symmetry of .
The motivation behind this paper is in part to describe how these facts relating
and
–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
–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
–fold
monoidal categories and their higher dimensional counterparts. The main result is that
for a
–fold monoidal
category, –Cat
becomes a –fold
monoidal –category
in a canonical way. In the next paper I indicate how this process may be iterated by enriching
over –Cat,
along the way defining the 3–category of categories enriched over
–Cat. In future work I plan to
make precise the –dimensional
case and to show how the group completion of the nerve of
is related to the loop space of the group completion of the nerve of
–Cat.
This paper is an abridged version of ‘Enrichment as categorical delooping I:
Enrichment over iterated monoidal categories’, math.CT/0304026.
