A note on the biadjunction between 2-categories of traced monoidal categories and tortile monoidal categories

We illustrate a minor error in the biadjointness result for 2-categories of traced monoidal categories and tortile monoidal categories stated by Joyal, Street and Verity. We also show that the biadjointness holds after suitably changing the denition of 2-cells. In the seminal paper "Traced Monoidal Categories" by Joyal, Street and Verity [4], it is claimed that the Int-construction gives a left biadjoint of the inclusion of the 2category TortMon of tortile monoidal categories, balanced strong monoidal functors and monoidal natural transformations in the 2-category TraMon of traced monoidal categories, traced strong monoidal functors and monoidal natural transformations [4, Proposition 5.2]. However, this statement is not correct. We shall give a simple counterexample below. Notation. We follow notations and conventions used in [4]. We write Int V for the tortile monoidal category obtained by the Int-construction on a traced monoidal category V, and N : V ! Int V for the canonical functor dened by N(X) = (X; I) and N(f) = f. Example 1. Let N = (N; 0; +; ) be the traced symmetric monoidal partially ordered set of natural numbers. Then the compact closed preordered set Int N is equivalent to the compact closed partially ordered set Z = (Z; 0; +; ; ) of integers. The biadjointness would imply that TraMon(N; Z) is equivalent to TortMon(Int N; Z), which in turn is equivalent to TortMon(Z; Z). However, some calculation shows that TraMon(N; Z) is isomorphic to the partially ordered set of natural numbers, while TortMon(Z; Z) is isomorphic to a discrete category with countably many objects. It is possible to recover the biadjointness, by introducing the 2-category TraMong of traced monoidal categories, traced strong monoidal functors and invertible monoidal natural transformations. Note that the 2-cells of TortMon are invertible because of the presence of duals [3, 5], and the inclusion of TortMon in TraMon factors through TraMong. Proposition 1. The inclusion of the 2-category TortMon in the 2-category TraMong has a left biadjoint with unit having component at a traced monoidal category V by N : V ! Int V.