Combinatoric N-fold Categories and N-fold Operads

Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal categories and monoidal functors is given by Balteanu, Fiedorowicz, Schwänzl and Vogt in [2]. These subcategories of MonCat have objects that are called n-fold monoidal categories. A k– fold monoidal category is n-fold monoidal for all n ≤ k, and a symmetric monoidal category is n-fold monoidal for all n. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce the lower branches of an extended family tree of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras in an iterated monoidal category. It is seen that the interchanges in an iterated monoidal category are the natural requirement for expressing operad associativity. The definition is developed from the starting point of iterated monoids in a category of collections. Since monoids are special cases of enriched categories this allows us to describe the iterated monoidal and higher dimensional categorical structure of iterated operads. We show that for V k-fold monoidal the structure of a (k−n)-fold monoidal strict n-category is possessed by the category of n-fold operads in V. We discuss examples of these operads that live in the previously described categories. Finally we describe the algebras of n-fold V-operads and their products.