Operads in iterated monoidal categories

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. 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 a lar...

This paper gives an explicit description of the categorical operad whose algebras are precisely symmetric monoidal categories. This allows us to place the operad in a sequence of four, and therefore a sequence of four successively stricter concepts of symmetric monoidal category. A companion paper will use this operadic presentation to describe a vast array of underlying multicategories for a symmetric monoidal category.

We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and free iterated monoidal categories give rise to finite simplicial operads of the same homotopy type as the classical little cubes operads used to parametrize the higher H-space structure of iterated loop spaces. Iterated monoidal categories e...

In this paper, I introduce a new generalization of the concept of an operad, further generalizing the concept of an opetope introduced by Baez and Dolan, who used this for the definition of their version of non-strict $n$-categories. Opetopes arise from iterating a certain construction on operads called the $+$-construction, starting with monoids. The first step gives rise to plain operads, i.e. operads without symmetries. The permutation axiom in a symmetric operad, however,...

A general notion of operad is given, which includes as instances, the operads originally conceived to study loop spaces, as well as the higher operads that arise in the globular approach to higher dimensional algebra. In the framework of this paper, one can also describe symmetric and braided analogues of higher operads, likely to be important to the study of weakly symmetric, higher dimensional monoidal structures.

Joyal and Street note in their paper on braided monoidal categories [Braided tensor categories, Advances in Math. 102(1993) 20-78] that the 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception that they mention is the case in which V is symmetric, which leads to V-Cat being symmetric as well. The symmetry in V-Cat is based upon the symmetry of V. The motivation behind th...

Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term `(S,*)-multicategory', subject to certain conditions on S and *. Different choices of S and * give some of the existing notions of operad and multicategory. We then describe the `algebras' for an (S,*)-multicategory and, finally, present a tentative selection of further de...

The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n=2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the o...

The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well. This paper describes how these facts are related to a categorical analogue of topological delooping. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal ...

We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of $n$-monoidal categories and discuss several naturally occurring examples...