Operads in iterated monoidal categories

TL;DR

This paper extends the concept of operads to $k$-fold monoidal categories, showing they can be defined and inherit structure in a generalized setting, with applications to specific examples.

Contribution

It generalizes the definition of operads to $k$-fold monoidal categories and analyzes how operad structures inherit and reduce monoidal levels.

Findings

Category of $n$-fold operads is a $(k-n)$-fold monoidal strict 2-category.

$n$-fold operads are automatically $(n-1)$-fold operads.

Classified operads in newly introduced simple $k$-fold monoidal categories.

Abstract

The structure of a $k$-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schw\"anzl and Vogt 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.