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Enrichment over iterated monoidal categories. (English) Zbl 1058.18003
A. Joyal and R. Street note [Adv. Math. 102, 20–78 (1993; Zbl 0817.18007)] that the 2-category \(\mathcal V\)-Cat of categories enriched over a braided monoidal category \(\mathcal V\) is not itself braided in any natural way based upon the braiding of \(\mathcal V\). On the other hand, in case \(\mathcal V\) is symmetric, the 2-category \(\mathcal V\)-Cat is itself symmetric in a canonical way.
The author shows that this phenomenon has a nice interpretation in terms of a categorical analogue of topological delooping. More precisely, following closely C. Balteanu, Z. Fiedorowicz, R. Schwänzl and R. Vogt [Adv. Math. 176, 277–349 (2003; Zbl 1030.18006)], the author introduces \(k\)-fold monoidal categories as a general setting to deal with monoidal, braided and symmetric categories. Namely, monoidal, braided and symmetric categories can be seen as \(1\)-fold, \(2\)-fold and \(\infty\)-fold monoidal categories, respectively, where an \(\infty\)-fold monoidal category is a category which is \(k\)-fold monoidal for any \(k\). The main result of the paper is that if \(\mathcal V\) is a \(k\)-fold monoidal category, then \(\mathcal V\)-Cat is naturally a \((k-1)\)-fold monoidal \(2\)-category, providing a clear explanation for the phenomenon noted by Joyal and Street.
In the final section of the paper, the author anticipates the main result of the sequel paper [S. Forcey, Theory Appl. Categ. 12, 299–325 (2004; Zbl 1056.18003)], namely that delooping by enrichment increases categorical dimension as it decreases monoidalness. In particular, \(n\) iterations of the process of categorical delooping by enrichment result in a strict \((n+1)\)-category.

MSC:
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18D20 Enriched categories (over closed or monoidal categories)
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