Traced monoidal categories BY ANDRE JOYAL

This paper introduces axioms for an abstract trace on a monoidal category. This trace can be interpreted in various contexts where it could alternatively be called contraction, feedback, Markov trace or braid closure. Each full submonoidal category of a tortile (or ribbon) monoidal category admits a canonical trace. We prove the structure theorem that every traced monoidal category arises in this way. Naively speaking, the construction is a glorification of the construction of the integers from the natural numbers. Less naively, the construct provides a left biadjoint to the forgetful 2-functor from the 2-category of tortile monoidal categories to the 2-category of traced monoidal categories, and we prove that the unit for this biadjunction is fully faithful. It should be kept in mind that the more familiar symmetric monoidal categories [ML] are obtained as special cases of balanced monoidal categories by taking the twist isomorphisms 6A:A^A to be identities. In the same way, compact closed categories [KL] are special tortile monoidal categories. In the diagrams for these special cases the reader may replace the ribbons by strings and ignore over and under crossings. Giulio Katis pointed out that an early attempt at the construction by the second author was too simplistic to be correct for the non-symmetric case. We shall describe here, for motivation, the meaning of our trace for linear functions between finite dimensional vector spaces. Consider a linear function / : F® U-> PF® U where U,V,W are vector spaces with bases (ut), (v}), {wk). The trace of/ with respect to U is the linear function t:V^»W given by