A special class of tensor categories initiated by inverse braid monoids

Abstract

In this paper, we introduce the concept of a wide tensor category which is a special class of a tensor category initiated by the inverse braid monoids recently investigated by Easdown and Lavers [The Inverse Braid Monoid, Adv. in Math. 186 (2004) 438–455]. The inverse braid monoids IB n is an inverse monoid which behaves as the symmetric inverse semigroup so that the braid group B n can be regarded as the braids acting in the symmetric group. In this paper, the structure of inverse braid monoids is explained by using the language of categories. A partial algebra category, which is a subcategory of the representative category of a bialgebra, is given as an example of wide tensor categories. In addition, some elementary properties of wide tensor categories are given. The main result is to show that for every strongly wide tensor category C, a strict wide tensor category C str can be constructed and is wide tensor equivalent to C with a wide tensor equivalence F. As a generalization of the universality property of the braid category B , we also illustrate a wide tensor category through the discussion on the universality of the inverse braid category IB (see Theorem 3.3, 3.6 and Proposition 3.7).