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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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##### References:
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