EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 6 June 2010. For the current production of this journal, please refer to http://www.springer.com/mathematics/geometry/journal/40062.

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