Dear all, Recently we have noticed that there is an error in the paper "Traced Monoidal Categories" by Joyal, Street and Verity, published in 1996. It is about the bidjointness of the Int-construction and, to our best knowledge, never pointed out before. We think that it is of some interest for quite a few people, as this biadjointness is widely known and frequently quoted in the literature, especially in various work in theoretical computer science. In the paper by JSV, it is claimed that (in Proposition 5.2) the Int-construction gives a left biadjoint of the inclusion of the 2-category TortMon (of tortile monoidal categories, balanced monoidal functors and monoidal natural transformations) in the 2-category TraMon (of traced monoidal categories, traced strong monoidal functors and monoidal natural transformations). However, we found that the statement is not quite correct. There is a simple counterexample: --------------------------------------------------------------------- Counterexample: Let N=(N,0,+,\leq) be the traced partially ordered set of natural numbers. Then Int(N) is equivalent to the compact closed partially ordered set Z=(Z,0,+,-,\leq) of integers. The biadjointness would say 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. --------------------------------------------------------------------- JSV's proof was almost perfect, except the very last three lines where some details on 2-cells were missing/wrong. We think that the easiest (and possibly the only) way to recover the biadjointness is to modify the definition of TraMon as the 2-category of traced monoidal categories, traced strong monoidal functors and *invertible* monoidal natural transformations - this is a natural choice, as the 2-cells of TortMon are invertible because of the presence of duals - then all seem to work well (and Ross agreed in his reply to our message). In practice, as far as we can see, this seems to be a relatively harmless error, as most uses of the Int-construction in the literature do not depend on the details on 2-cells (they often do not mention 2-cells at all). However, there are some papers explicitly mentioning 2-cells and thus inheriting the incorrect statement from the JSV paper. (Unfortunately, it is the case for a paper by one [Hasegawa] of us ...) Best, Masahito Hasegawa Shin-ya Katsumata -- Masahito Hasegawa <hassei@kurims.kyoto-u.ac.jp> Research Institute for Mathematical Sciences, Kyoto University