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 definition 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 2-category 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 defined 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 . Proof. What we need to show is, for each traced monoidal category V and tortile monoidal category W , composition with N induces an equivalence of categories from 108 MASAHITO HASEGAWA AND SHIN-YA KATSUMATA TortMon(Int V,W) to TraMong(V,W). We prove that this induced functor is essentially surjective on objects, and is fully faithful. For showing that it is essentially surjective, the proof of [4, proposition 5·2 ] is sufficient. For a traced monoidal functor F: V → W , let K : Int V → W be the balanced strong monoidal functor sending (X,U ) to F X ⊗ (FU )∨ and f : (X,U ) → (Y, V ) to F X ⊗ (FU )∨ 1⊗η⊗1 −−−→ F X ⊗ FV ⊗ (FV )∨ ⊗ (FU )∨ F f ⊗1 −−−→FY ⊗ FU ⊗ (FV )∨ ⊗ (FU )∨ 1⊗c−1⊗1 −−−−−→ FY ⊗ (FV )∨ ⊗ FU ⊗ (FU )∨ 1⊗ε′ −→ FY ⊗ (FV )∨. That K is a balanced strong monoidal functor is shown exactly in the same manner as in the proof of [4, proposition 5·2]. Clearly K N F holds. For showing the full faithfulness, for an invertible monoidal natural transformation β: K N → K ′N with balanced strong monoidal functors K , K ′: Int V → W , let β: K → K ′ be the monoidal natural transformation whose (X,U )-component is given by K (X,U ) → K N X ⊗ (K NU )∨ βX⊗(β −1 U ) ∨ −−−−−→ K ′N X ⊗ (K ′NU )∨ → K ′(X,U ). That β is a monoidal natural transformation is verified by direct calculation. We have αN = α for a monoidal natural transformation α: K → K ′, as K (X,U ) αN (X,U ) −−−−−→ K ′(X,U ) = K (X,U ) → K N X ⊗ (K NU )∨ (αN )X⊗((αN ) −1 U ) ∨ −−−−−−−−−→ K ′N X ⊗ (K ′NU )∨ → K ′(X,U ) = K (X,U ) → K N X ⊗ (K NU )∨ αN X⊗(α −1 NU ) ∨ −−−−−−−→ K ′N X ⊗ (K ′NU )∨ → K ′(X,U ) = K (X,U ) → K N X ⊗ (K NU )∨ → K N X ⊗ (K ((NU )∨))∨∨ αN X⊗α∨∨ (NU )∨ −−−−−−−→ K ′N X ⊗ (K ′((NU )∨))∨∨ → K ′N X ⊗ (K ′NU )∨ → K ′(X,U ) = K (X,U ) → K N X ⊗ K ((NU )∨) αN X⊗α(NU )∨ −−−−−−−→ K ′N X ⊗ K ′((NU )∨) → K ′(X,U ) = K (X,U ) α(X,U ) −−−→ K ′(X,U ) where we have omitted some details on the structural isomorphisms. Note the isomorphism (X,U ) (X, I ) ⊗ (I,U ) = N X ⊗ (NU )∨; also note that, for a 2-cell α : K → K ′ in TortMon, its inverse α−1 : K ′ → K is given by (cf. [3, proposition 7·1], [5, corollary 2·2]) K ′C → (K ′(C∨))∨ (αC∨ ) ∨ −−−→ (K (C∨))∨ → K C. On the other hand, it is easy to see that βN = β holds. Hence the mapping α → αN is a bijection, and the functor induced by composition with N is full and faithful. Remark. This biadjointness result has been frequently quoted in the literature, often with no mention of 2-cells. However, there are some cases where the incorrect statement in [4] is inherited, with explicit mention of 2-cells. For example, in [2], the biadjunction is incorrectly stated for non-invertible 2-cells [2, section 5·1], although the technical development there does not depend on the choice of 2-cells and the error has no effect on the results. Another case is [1] in which the biadjointness of a variant of the Int-construction for linearly Biadjunction between traced categories and tortile categories 109 distributive categories is stated [1, proposition 27]; it contains the same problem as [4, proposition 5·2], and we expect that a similar change in the definition of 2-cells will make the claim correct. Again, this error has no effect on the other results in [1]. Acknowledgements. This research was partly supported by the Grant-in-Aid for Sci-entific Research (C) 20500010 (Hasegawa) and the Grant-in-Aid for Young Scientists (B)20700012 (Katsumata). REFERENCES[1] R. F. BLUTE, J. R. B. COCKETT and R. A. G. SEELY. Feedback for linearly distributive categories:traces and fixpoints. J. Pure Appl. Algebra 154 (2000), 27–69.[2] M. HASEGAWA, M. HOFMANN and G. D. PLOTKIN. Finite dimensional vector spaces are completefor traced symmetric monoidal categories. In Pillars of Computer Science: Essays Dedicated to Boris(Boaz) Trakhtenbrot on the Occasion of His 85th Birthday. Lecture Notes in Comput. Sci. vol. 4800(Springer-Verlag, 2008), pp. 367–385.[3] A. JOYAL and R. STREET. Braided tensor categories. Adv. Math. 102 (1993), 20–78.[4] A. JOYAL, R. STREET and D. VERITY. Traced monoidal categories. Math. Proc. Cam. Phil. Soc. 119(1996), 447–468.[5] M.-C. SHUM. Tortile tensor categories. J. Pure Appl. Algebra 93 (1994), 57–110.