Theory and Applications of Categories, Vol. 28, No. 7, 2013, pp. 206–212.

TRACED ∗-AUTONOMOUS CATEGORIES ARE COMPACT CLOSED

TAMÁS HAJGATÓ AND MASAHITO HASEGAWA

Abstract. We show that any traced ∗-autonomous category is compact closed.

1. Introduction

Suppose that ℂ = (ℂ, I, ⊗, ⊸, ⊥, Tr) is a traced ∗-autonomous category; here we understand that a ∗-autonomous category is a symmetric monoidal closed category (ℂ, I, ⊗, ⊸) (we write A ⊸ B for the internal hom from A to B) equipped with a dualizing object ⊥ [Barr, 1979], and that the trace

Tr_{A,B}^{X} : ℂ(A ⊗ X, B ⊗ X) ⟶ ℂ(A, B)

is given on the symmetric monoidal structure in the sense of Joyal, Street and Verity [Joyal et al., 1996] (rather than the trace for linearly distributive categories with MIX by Blute, Cockett and Seely [Blute et al., 2000]).

In ℂ, we have the trace of the evaluation map

\frac{(X ⊸ (⊥ ⊗ X)) ⊗ X ⟶^{ev} ⊥ ⊗ X}{X ⊸ (⊥ ⊗ X) ⟶^{Tr^{X}ev} ⊥}

which gives rise to a morphism t_{X} : I ⟶ X ⊗ (X ⊸ I) via the isomorphism

(X ⊸ (⊥ ⊗ X)) ⊸ ⊥ ≃ X ⊗ (X ⊸ I).

It is then natural to ask if t_{X} satisfies the equations

X ⟶^{t_{X}⊗X} X ⊗ (X ⊸I) ⊗ X ⟶^{X⊗ev_{X,I}} X = id_{X}  (1)

and

X ⊸I ⟶^{(X⊸I)⊗t_{X}} (X ⊸I) ⊗ X ⊗ (X ⊸I) ⟶^{ev_{X,I}⊗(X⊸I)} X ⊸I = id_{X⊸I}  (2)

The first author was partly supported by the European Union and the European Regional Development Fund under the grant agreement TÁMOP-4.2.2/B-10/1-2010-0012. The second author was partly supported by the Grant-in-Aid for Scientific Research (C) 23500016.

Received by the editors 2013-02-21 and, in revised form, 2013-03-18.

Transmitted by Ross Street. Published on 2013-04-10.

2010 Mathematics Subject Classification: 18D10,18D15.

Key words and phrases: symmetric monoidal closed categories, traced monoidal categories, ∗- autonomous categories, compact closed categories.

\begin{matrix}c \\ ◯ \end{matrix} Tamás Hajgató and Masahito Hasegawa, 2013. Permission to copy for private use granted.

TRACED ∗-AUTONOMOUS CATEGORIES ARECOMPACT CLOSED  207

which mean that X ⊸ I is a (left) dual of X, hence ℂ is compact closed [Kelly and Laplaza, 1980]. Below we see that this is the case.

Before proceeding to the proof, let us explain how we came across this observation. In the sequel, we write f̂ : A ⟶ B ⊸ C for the transpose of f : A ⊗ B ⟶ C in a symmetric monoidal closed category. In a traced symmetric monoidal closed category, we have a family of morphisms

τ_{B}^{X} = Tr_{X⊸(B⊗X),B}^{X}(ev_{X,B⊗X}) : X ⊸ (B ⊗ X) ⟶ B.

It is easy to see that τ’s are sufficient to determine trace of any f : A ⊗ X ⟶ B ⊗ X as

Tr_{A,B}^{X}f = A \underrightarrow{f̂} X ⊸ (B ⊗ X) ⟶^{τ_{B}^{X}} B.

In the case of traced ∗-autonomous categories, we can further restrict our attention to τ’s with B = ⊥, from which τ_{B}^{X} for any B is recovered as

X ⊸ (B ⊗ X) ≃ X ⊸ ((B & ⊥) ⊗ X)
⟶^{X⊸δ} X ⊸ (B & (⊥ ⊗ X))
≃ B & (X ⊸ (⊥ ⊗ X))
⟶^{B ^{&}τ_{⊥}^{X}} B & ⊥
≃ B

where δ denotes linear distributivity [Cockett and Seely, 1997].

Therefore, giving τ_{⊥}^{X} : (X ⊸ (⊥ ⊗ X)) ⟶ ⊥ for each X is enough to determine a trace. With some efforts of spelling out how the trace can be recovered directly from τ_{⊥}^{X}’s, we noticed that τ_{⊥}^{X} actually determines the unit map t_{X} : I ⟶ X ⊗ (X ⊸ I) of the duality between X and X ⊸ I.

To make the proof short and readable, we use some basic results on (enriched) extraordinary natural transformations of Eilenberg and Kelly [Eilenberg and Kelly, 1966, Kelly, 1982], though it is also possible to derive the result by direct calculation from scratch.

2. A Characterization of Compact Closedness

There are many ways of characterizing compact closed categories as special symmetric monoidal closed categories [Day, 1977, Kelly and Laplaza, 1980]. In our development, the following characterization turns out to be useful:

2.1. Proposition. Suppose that ℂ is a symmetric monoidal closed category such that there is an extraordinary ℂ-natural transformation t_{X} : I ⟶ X ⊗ (X ⊸ I) with t_{I} invertible. Then ℂ is compact closed.

Proof. Recall that, for a symmetric monoidal closed category 𝕍, 𝕍-categories 𝔸, 𝔹, a 𝕍functor F : 𝔸^{op}⊗𝔸 → 𝔹 and an object B of 𝔹, a family of morphisms α_{X} : B ⟶ F (X, X) is said to be an extraordinary 𝕍-natural transformation [Kelly, 1982] when the following diagram commutes for all X, X′.

[figure: 𝔸(X, X′); F (X, −); F (−, X′); 𝔹(F (X, X), F (X, X′)); 𝔹(F (X′, X′), F (X, X′)); 𝔹(α_{X}, id); 𝔹(α_{X′}, id); 𝔹(B, F (X, X′))]

In the proposition, the assumption that t_{X} : I ⟶ X⊗(X ⊸I) is extraordinally ℂ-natural means that the following diagram commutes for all X and X′:

[figure: X ⊸X′; (X ⊸X′) ⊗ t_{X}; t_{X′} ⊗ (X ⊸X′); (X ⊸X′) ⊗ X ⊗ (X ⊸I); X′ ⊗ (X′ ⊸I) ⊗ (X ⊸X′); ev ⊗ (X ⊸ I); X′ ⊗ comp; X′ ⊗ (X ⊸I)]

By letting X be I in the diagram above, we see that (modulo some obvious simplifications)

X′ ⟶^{t_{X′}⊗X′} X′ ⊗ (X′ ⊸ I) ⊗ X′ ⟶^{X′⊗ev} X′

agrees with

X′ ⟶^{X′⊗t_{I}} X′ ⊗ I ⊗ (I ⊸ I) ⟶^{≃} X′.

Similarly, by letting X′ be I in the diagram, we have that

X ⊸ I ⟶^{(X⊸I)⊗t_{X}} (X ⊸ I) ⊗ X ⊗ (X ⊸ I) ⟶^{ev⊗(X⊸I)} X ⊸ I

agrees with

X ⊸ I ⟶^{t_{I}⊗(X⊸I)} I ⊗ (I ⊸ I) ⊗ (X ⊸ I) ⟶^{≃} X ⊸ I.

Hence

t′_{X} = I ⟶^{≃} I ⊗ (I ⊸ I) ⟶^{t_{I}^{−1}} I ⟶^{t_{X}} X ⊗ (X ⊸ I)

satisfies the equations (1) and (2) for making X ⊸ I a left dual of X.

Note that, in the proof above, t′_{X} agrees with t_{X} if t_{I} : I ⟶ I ⊗ (I ⊸ I) itself is the canonical isomorphism from I to I ⊗ (I ⊸ I).

2.2. Remark. In Proposition 2.1, the assumption that t_{I} is invertible cannot be dropped. For instance, consider the category ωCppo_{⊥} of pointed ω-complete partial orders and strict ω-continuous functions. ωCppo_{⊥} is symmetric monoidal closed, with Sierpinski space as the unit object, smash products as tensor and strict function spaces as internal hom. In ωCppo_{⊥}, there is an extraordinary ωCppo_{⊥}-natural transformation t_{X} : I ⟶ X ⊗ (X ⊸I) given by the constant functions returning the least element. However, t_{I} is not invertible, and ωCppo_{⊥} is not compact closed.

3. Proof of the Main Result

As in the introduction, let us define

τ_{B}^{X} = Tr_{X⊸(B⊗X),B}^{X}(ev_{X,B⊗X}) : X ⊸ (B ⊗ X) ⟶ B

in traced symmetric monoidal closed categories.

3.1. Lemma. In a traced symmetric monoidal closed category ℂ with an object B,

τ̂_{B}^{X} : I ⟶ (X ⊸ (B ⊗ X)) ⊸ B

is extraordinary ℂ-natural in X.

Proof. The extranaturality amounts to the commutativity of

[figure: (X ⊸X′) ⊗ (X′ ⊸(B ⊗ X)); id ⊗ τ̂_{B}^{X}; id ⊗ τ̂_{B}^{X′}; (X ⊸X′) ⊗ (X′ ⊸(B ⊗ X)) ⊗ ((X ⊸(B ⊗ X))⊸B); (X ⊸X′) ⊗ (X′ ⊸(B ⊗ X)) ⊗ ((X′ ⊸(B ⊗ X′))⊸B); comp ⊗ id; comp ⊗ id; (X ⊸(B ⊗ X)) ⊗ ((X ⊸(B ⊗ X))⊸B); (X′ ⊸(B ⊗ X′)) ⊗ ((X′ ⊸(B ⊗ X′))⊸B); ev; ev; B]

which is a consequence of the sliding property (dinaturality) of trace.

3.2. Lemma. In a ∗-autonomous category ℂ, the isomorphism

φ_{X,Y} : (X ⊸ (⊥ ⊗ Y)) ⊸ ⊥ ⟶^{≃} X ⊗ (Y ⊸ I)

given by

(X ⊸ (⊥ ⊗ Y)) ⊸ ⊥ ≃ (X ⊸ (((⊥ ⊗ Y) ⊸ ⊥) ⊸ ⊥)) ⊸ ⊥
≃ (X ⊸ ((Y ⊸ (⊥ ⊸ ⊥)) ⊸ ⊥)) ⊸ ⊥
≃ (X ⊸ ((Y ⊸ I) ⊸ ⊥)) ⊸ ⊥
≃ ((X ⊗ (Y ⊸ I)) ⊸ ⊥) ⊸ ⊥
≃ X ⊗ (Y ⊸ I)

is ℂ-natural in X and Y.

Proof. Each isomorphism involved in φ is ℂ-natural.

3.3. Lemma. [Eilenberg and Kelly, 1966, Kelly, 1982] Assume that 𝕍 is a symmetric monoidal closed category. Let 𝔸, 𝔹 be 𝕍-categories and G, H be 𝕍-functors of the form 𝔸^{op} ⊗𝔸 → 𝔹 and suppose that B is an object of 𝔹. If α_{X} : B → G(X, X) is extraordinary 𝕍-natural in X and β_{X,Y} : G(X, Y) → H(X, Y) is 𝕍-natural in X and Y, then

β_{X,X} ∘ α_{X} : B → H(X, X)

is extraordinary 𝕍-natural in X.

3.4. Corollary. In a traced ∗-autonomous category ℂ, t_{X} = φ_{X,X} ∘ τ̂_{⊥}^{X} : I ⟶ X ⊗ (X ⊸ I) is extraordinary ℂ-natural in X. □

3.5. Lemma. t_{I} : I ⟶ I ⊗ (I ⊸ I) agrees with the canonical isomorphism from I to I ⊗ (I ⊸ I).

Proof. A consequence of the vanishing property of trace.

Putting Proposition 2.1, Corollary 3.4 and Lemma 3.5 together, we obtain our main result.

3.6. Theorem. Any traced ∗-autonomous category is compact closed.

It is possible that a compact closed category is equipped with a dualizing object which is not isomorphic to the unit object (and par not isomorphic to tensor). For instance, the linearly ordered set of integers ℤ is compact closed with unit I = 0 and tensor X ⊗ Y = X + Y and duality X^{∗} = −X, while any element of ℤ serves as a dualizing object. (The same can be done for any partially ordered Abelian group regarded as a compact closed poset.)

Since a compact closed category has a unique trace (cf. [Hasegawa, 2009]), we have:

3.7. Theorem. To give a traced ∗-autonomous category is to give a compact closed category with a dualizing object.

Note that a dualizing object in a compact closed category is just an object ⊥ such that the unit morphism I ⟶ ⊥ ⊗ ⊥^{∗} is invertible, cf. the Abelian group example above.

4. On Linear Distributivity

In a compact closed category with a dualizing object ⊥, linear distributivity [Cockett and Seely, 1997] on the ∗-autonomous structure is invertible. To see this, recall that the linear distributivity δ : (A & B) ⊗ C ⟶ A & (B ⊗ C) in a ∗-autonomous category (regarded as a symmetric linearly distributive category with negation) amounts to the canonical morphism (A^{⊥} ⊸ B) ⊗ C ⟶ A^{⊥} ⊸ (B ⊗ C) which is just the associativity isomorphism ((A^{⊥})^{∗} ⊗ B) ⊗ C ≃ (A^{⊥})^{∗} ⊗ (B ⊗ C) in a compact closed category.

Conversely, a ∗-autonomous category with invertible linear distributivity is compact closed. We have

A ⊸ B ≃ A^{⊥} & B ≃ A^{⊥} & (I ⊗ B) ≃^{δ^{−1}} (A^{⊥} & I) ⊗ B ≃ (A ⊸ I) ⊗ B

In particular, the canonical map (A ⊸ I) ⊗ A ⟶ A ⊸ A is invertible, and it follows that the category is compact closed (cf. [Day, 1977]).

Together with Theorem 3.7, we have that the following three structures are essentially the same:

• a traced ∗-autonomous category,

• a compact closed category equipped with a dualizing object, and

• a ∗-autonomous category with invertible linear distributivity.

4.1. Remark. As noted in [Cockett and Seely, 1997], in a symmetric linearly distributive category with invertible linear distributivity and also equipped with a tensor-inverse of ⊥ (an object ⊥^{∗} such that there is an isomorphism I ≃ ⊥ ⊗ ⊥^{∗} subject to a coherence axiom), the par A & B is isomorphic to the “⊥-shifted tensor” A ⊗ ⊥^{∗} ⊗ B. This is the case for ∗-autonomous categories with invertible linear distributivity (equivalently: traced ∗-autonomous categories, or compact closed categories with a dualizing object), in which ⊥^{∗} = ⊥ ⊸ I serves as a tensor-inverse of ⊥ and we have A & B ≃ A ⊗ (⊥ ⊸ I) ⊗ B.

Acknowledgements. The second author is grateful to Naohiko Hoshino for helpful discussions on ∗-autonomous categories and traced monoidal categories.

References

[Barr, 1979] Barr, M. (1979) ∗-Autonomous Categories. Lecture Notes in Math. 752, Springer-Verlag.

[Blute et al., 2000] Blute, R.F., Cockett, J.R.B. and Seely, R.A.G. (2000) Feedback for linearly distributive categories: traces and fixpoints. J. Pure Appl. Algebra 154, 27– 69.

[Cockett and Seely, 1997] Cockett, J.R.B. and Seely, R.A.G. (1997) Weakly distributive categories. J. Pure Appl. Algebra 114, 133–173.

[Day, 1977] Day, B.J. (1977) Notes on compact closed categories. J. Austral. Math. Soc. (Series A) 24, 309–311.

[Eilenberg and Kelly, 1966] Eilenberg, S. and Kelly, G.M. (1966) A generalization of the functorial calculus. J. Algebra 3, 366–375.

[Hasegawa, 2009] Hasegawa, M. (2009) On traced monoidal closed categories. Math. Struct. in Comput. Science 19, 217–244.

[Joyal et al., 1996] Joyal, A., Street, R. and Verity, D. (1996) Traced monoidal categories. Math. Proc. Cambridge Phils. Soc. 119 (1996), 447–468.

[Kelly, 1982] Kelly, G.M. (1982) Basic Concepts of Enriched Category Theory. London Math. Soc. Lecture Note Ser. 64, Cambridge University Press. Reproduced in Reprints in Theory and Applications of Categories 10, 1–136 (2005).

[Kelly and Laplaza, 1980] Kelly, G.M. and Laplaza, M.L. (1980) Coherence for compact closed categories. J. Pure Appl. Algebra 19, 193–213.

Department of Computer Science University of Szeged Szeged, Hungary

Research Institute for Mathematical Sciences Kyoto University Kyoto, Japan

Email: hajgato@inf.u-szeged.hu
      hassei@kurims.kyoto-u.ac.jp

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