Higher Enrichment: N-fold Operads and Enriched N-categories, Delooping and Weakening

The most familiar example of higher, or vertically iterated enrichment is that in the definition of strict n-category. We begin with strict n-categories based on a general symmetric monoidal category V. Motivation is offered through a comparison of the classical and extended versions of topological quantum field theory. A sequence of categorical types that filter the category of monoidal categories and monoidal functors has been given by Balteanu, Fiedorowicz, Schwanzl and Vogt. These subcategories of MonCat are called n-fold monoidal categories. A k-fold monoidal category is nfold monoidal for all n ≤ k, and a symmetric monoidal category is n-fold monoidal for all n. Operads and enriched categories were originally defined as enriched over a symmetric monoidal category V. The symmetry in V is required in order to describe the associativity axiom the operad composition must obey, to describe the associativity that must hold in the action of an operad on one of its algebras and to define the product of operads. For enriched categories the symmetry is required in order to describe the opposite of a V-category and the product of V-categories. We demonstrate how all this can be accomplished instead by use of the interchange transformations of an n-fold monoidal category. There are suggestive results about higher enrichment over n-fold categories. Briefly, it turns out that for V k-fold monoidal, V-n-Cat is a (k − n)-fold monoidal (n+ 1)-category. Since iterated monoidal categories have iterated loop space nerves, we suspect a delooping process. This is shown to be the case in the example of group torsors. Next we discuss how weakening higher enrichment may shed light on comparisons of n-category definitions. We pictorially describe the process of weakening and the appropriate morphisms using polytopes, and then define an operad whose action provides a concise equivalent description. Thanks to XY-pic for the diagrams. 2000 Mathematics Subject Classification: 18D10; 18D20.