Iterated Monoidal Categories

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 encompass, as a special case, the notion of braided tensor categories, as used in the theory of quantum groups.Comment: 55 pages, 3 PostScript figure

Enrichment over iterated monoidal categories

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 this paper is in part to describe how these facts relating V and V-Cat are in turn related to a categorical analogue of topological delooping. To do so I need to pass to a more general setting than braided and symmetric categories -- in fact the k-fold monoidal categories of Balteanu et al in [Iterated Monoidal Categories, Adv. Math. 176(2003) 277-349]. 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 objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k-fold monoidal category, V-Cat becomes a (k-1)-fold monoidal 2-category in a canonical way. In the next paper I indicate how this process may be iterated by enriching over V-Cat, along the way defining the 3-category of categories enriched over V-Cat. In future work I plan to make precise the n-dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V-Cat. This paper is an abridged version of `Enrichment as categorical delooping I' math.CT/0304026.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-7.abs.htm

Enrichment As Categorical Delooping I: Enrichment Over Iterated Monoidal Categories

Joyal and Street note in their paper on braided monoidal categories [10] 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. What is meant by “based upon” here will be made more clear in the present paper. 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 this paper is in part to describe how these facts relating V and V–Cat are in turn related to a categorical analogue of topological delooping first mentioned by Baez and Dolan in [1]. To do so I need to pass to a more general setting than braided and symmetric categories – in fact the k–fold monoidal categories of Balteanu et al in [3]. 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 objects, including k–fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k–fold monoidal category, V–Cat becomes a (k − 1)–fold monoidal 2–category in a canonical way. I indicate how this process may be iterated by enriching over V–Cat, along the way defining the 3–category of categories enriched over V–Cat. In the next paper I hope to make precise the n–dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V–Cat

Classification of braids which give rise to interchange

It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2-category V-Cat of enriched categories and functors over V, a monoidal bicategory V-Mod of enriched categories and modules, a category of operads in V and a 2-fold monoidal category structure on V. We will begin by focusing our exposition on the first and last in this list due to their ability to shed light on a new question. We ask, given a braiding on V, what non-equal structures of a given kind in the list exist which are based upon the braiding. For instance, what non-equal monoidal structures are available on V-Cat, or what non-equal operad structures are available which base their associative structure on the braiding in V. We demonstrate alternative underlying braids that result in an infinite family of associative structures. The external and internal associativity diagrams in the axioms of a 2-fold monoidal category will provide us with several obstructions that can prevent a braid from underlying an associative structure.Comment: Previous title: Equivalence of associative structures over a braiding. This journal ready version adds proof details involving the free braided category with dual