Tortile tensor categories

Abstract

A tortile tensor category is a braided tensor category in which every object A is equipped with a twist θ A : A≅ A and a compatible right dual (A ∗,d A,e A) . Given a category A we describe the free tortile tensor category F A on A by generators and relations. By observing that in any tortile tensor category, there are canonical isomorphisms (A⊗B) ∗≅B ∗⊗A ∗,I ∗≅I , and (non-canonical) isomorphisms A ∗∗≅A , we show that F A is equivalent to the simpler RF A consisting of the reduced objects and reduced maps of F A . This equivalence will later be used to show that F A is equivalent to the category T̃ ʃ A of double tangles labelled by A . To define T̃ ʃ A we first consider a double knot, which may be thought of as a (tame) link with two “parallel” components, or as the boundary of a ribbon in 3-space. A knot with the same diagram as one of these components is called its underlying knot. We associate to each double knot an integral quantity called its twist number, and show that this together with an underlying knot completely determine (up to equivalence) the double knot. Thus to give a double knot is to give an ordinary knot and an integer. A tangle is a disjoint union of knots and of directed paths connecting two points on+ ∂([0, 1] × P), where P is a Euclidean plane. A double tangle is a tangle with an integer attached to each of its connected components. Given a double tangle we label the points in its boundary by objects of A and its arcs by maps of A , and get what we call a double tangle labelled by A . Equivalence classes of these form a tortile tensor category T ̃ ʃ A . Because of the existence of an “inclusion” functor A → T ̃ ʃ A , there is by the freeness of F A a canonical strict tortile tensor functor F A → T ̃ ʃ A . Our main theorem asserts that this functor is an equivalence of tortile tensor categories, giving an explicit description of F A .