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Operads in iterated monoidal categories
#
Operads in iterated monoidal categories

##
Stefan Forcey, Jacob Siehler and E. Seth Sowers

The structure of a $k$-fold monoidal category as introduced by
Balteanu, Fiedorowicz, Schw\"anzl and Vogt in \cite{Balt} can be
seen as a weaker structure than a symmetric or even braided monoidal
category. In this paper we show that it is still sufficient to
permit a good definition of ($n$-fold) operads in a $k$-fold monoidal
category which generalizes the definition of operads in a braided
category. Furthermore, the inheritance of structure by the category
of operads is actually an inheritance of iterated monoidal structure,
decremented by at least two iterations. We prove that the category of $n$-fold operads in a
$k$-fold monoidal category is itself a $(k-n)$-fold monoidal, strict
$2$-category, and show
that $n$-fold operads are automatically $(n-1)$-fold operads.
We also introduce a family of simple examples of
$k$-fold monoidal categories and classify operads in the example
categories.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 1, pp. 1-43