Theory and Applications of Categories, Vol. 26, No. 7, 2012, pp. 194–203.

NOTE ON STAR-AUTONOMOUS COMONADS

CRAIG PASTRO

Abstract. We develop an alternative approach to star-autonomous comonads via linearly distributive categories. It is shown that in the autonomous case the notions of star-autonomous comonad and Hopf comonad coincide.

1. Introduction

Given a linearly distributive category C, this note determines what structure is required of a comonad G on C so that C^{G}, the category of Eilenberg-Moore coalgebras of G, is again a linearly distributive category. Furthermore, if C is equipped with negations (and is hence a star-autonomous category), the structure required to lift the negations to C^{G} is determined as well. This latter is equivalent to lifting star-autonomy and it is shown that the notion presented is equivalent to a star-autonomous comonad [PS09]. As a consequence of the presentation given here, it may be easily seen that any star-autonomous comonad on an autonomous category is a Hopf monad [BV07].

2. Lifting linear distributivity

Suppose C is a monoidal category and G : C → C is a comonad on C. Recall that C^{G}, the category of (Eilenberg-Moore) coalgebras of G, is monoidal if and only if G is a monoidal comonad [M02]. In this section we are interested in the structure required to lift linear distributivity to the category of coalgebras.

A linearly distributive category C is a category equipped with two monoidal structures (C, ⋆, I) and (C, ⋄, J),^{1} and two compatibility natural transformations (called “linear distributions”)

∂_{l} : A ⋆ (B ⋄ C) → (A ⋆ B) ⋄ C
∂_{r} : (B ⋄ C) ⋆ A → B ⋄ (C ⋆ A),

satisfying a large number of coherence diagrams [CS97].

Partially supported by Grant-in-Aid for Scientific Research Wakate Kenkyu (B) No. 22740015.

Received by the editors 2011-07-26 and, in revised form, 2012-03-25.

Transmitted by Richard Blute. Published on 2012-04-02.

2000 Mathematics Subject Classification: 18C15, 18C20, 18D10.

Key words and phrases: Star-autonomous, autonomous, rigid, linearly distributive, comonad, Hopf.

\begin{matrix}c \\ ◯ \end{matrix} Craig Pastro, 2012. Permission to copy for private use granted.

1For simplicity we assume that the monoidal structures are strict, although this is not necessary.

Furthermore, in their original paper [CS97] the tensor products ⋆ and ⋄ are respectively denoted by ⊗ and ⊕, and called tensor and par, emphasizing their connection to linear logic.

Suppose G = (G, δ, ϵ) is a comonad on a linearly distributive category C which is a monoidal comonad on C with respect to both ⋆ and ⋄, with structure maps (G, ϕ, ϕ_{0}) and (G, ψ, ψ_{0}) respectively. If, for G-coalgebras A, B, and C, the comonad G satisfies

[figure: GA ⋆ (GB ⋄ GC); GA ⋆ G(B ⋄ C); G(A ⋆ (B ⋄ C)); (GA ⋆ GB) ⋄ GC; G(A ⋆ B) ⋄ GC; G((A ⋆ B) ⋄ C),; 1⋆ψ; ϕ; ϕ⋄1; ψ; ∂_{l}; ∂_{l}]  (1)

it may be seen that the morphism ∂_{l} becomes a G-coalgebra morphism. If G satisfies a similar axiom for ∂_{r}, i.e.,

[figure: (GB ⋄ GC) ⋆ GA; G(B ⋄ C) ⋆ GA; G((B ⋄ C) ⋆ A); GB ⋄ (GC ⋆ GA); GB ⋄ G(C ⋆ A); G(B ⋄ (C ⋆ A)),; ψ⋆1; ϕ; 1⋄ϕ; ψ; ∂_{r}; ∂_{r}]  (2)

then ∂_{r} also becomes a G-coalgebra morphism. Thus,

2.1. Proposition. Given a linearly distributive category C and a comonad G : C → C satisfying axioms (1) and (2), the category C^{G} is a linearly distributive category.

2.2. Example. Let C be a symmetric linearly distributive category and (B, μ, η, δ, ϵ) a bialgebra in C with respect to ⋄. That is, the structure morphisms are given as

μ : B ⋄ B → B  δ : B → B ⋄ B
η : J → B  ϵ : B → J.

Then, G = B ⋄ − is a comonad and is monoidal with respect to both ⋄ and ⋆. The latter via I ≅ J ⋄ I ⟶^{η⋄1} B ⋄ I, and the following,

(B ⋄ U) ⋆ (B ⋄ V) ⟶^{∂_{r}} B ⋄ (U ⋆ (B ⋄ V))
⟶^{1 ⋄ (1 ⋆ c)} B ⋄ (U ⋆ (V ⋄ B))
⟶^{1 ⋄ ∂_{l}} B ⋄ ((U ⋆ V) ⋄ B)
⟶^{1 ⋄ c} B ⋄ (B ⋄ (U ⋆ V))
⟶^{≅} (B ⋄ B) ⋄ (U ⋆ V)
⟶^{μ ⋆ 1} B ⋄ (U ⋆ V).

Rather large diagrams, which we leave to the faith of the reader, prove that B ⋄ − satisfies (1) and (2), so that C^{B} = Comod_{C}(B), the category of comodules of B, is a linearly distributive category.

3. Lifting negations

Suppose now that C is a linearly distributive category equipped with negations S and S′ (corresponding to ^{⊥}(−) and (−)^{⊥} in [CS97]). That is, functors S, S′ : C^{op} → C together with the following (dinatural) evaluation and coevaluation morphisms

\begin{matrix}SA ⋆ A ⟶^{e_{A}} J & A ⋆ S′A ⟶^{e′_{A}} J \\ I ⟶^{n_{A}} A ⋄ SA & I ⟶^{n′_{A}} S′A ⋄ A, \end{matrix}  (3)

satisfying the four evident “triangle identities”. One such is

(A ≅ I ⋆ A ⟶^{n⋆1} (A ⋄ SA) ⋆ A ⟶^{∂_{r}} A ⋄ (SA ⋆ A) ⟶^{1⋄e} A ⋄ J ≅ A) = 1_{A}.

If C is equipped with such negations we say simply that C is a linearly distributive category with negations.

We are interested to lift negations to C^{G}. This means we must ensure that the “negation” functors S, S′ : C^{op} → C lift to functors (C^{G})^{op} → C^{G}, and the evaluation and coevaluation morphisms are in C^{G}, i.e., are G-coalgebra morphisms.

The following is essentially known from [S72].

3.1. Proposition. A (contravariant) functor S : C^{op} → C may be lifted to a functor S̃: (C^{G})^{op} → C^{G} such that the diagram

[figure: (C^{G})^{op}; C^{G}; C^{op}; C,; S̃; U; U; S]

(in which U : C^{G} → C is the underlying functor) commutes, if and only if there is a natural transformation

ν : S → GSG

satisfying the following two axioms

[figure: S; GSG; SG; ν; ϵ_{SG}; Sϵ; S; GSG; G^{2}SG; GSG; G^{2}SG^{2}.; ν; δ_{SG}; ν; Gν_{G}; G^{2}Sδ]  (4)

This may be viewed as a distributive law of a contravariant functor over a comonad [S72]. In this case, we say that S may be lifted to C^{G}, and a functor S̃ : (C^{G})^{op} → C^{G} is defined as

S̃(A, γ) = (SA, SA ⟶^{ν} GSGA ⟶^{GSγ} GSA)  S̃(f) = Sf.

(To see the reverse direction, suppose (A, γ) is a coalgebra and S̃ is a functor C^{G} → C^{G}, so that S̃A = (SA, γ̃) is again a coalgebra. Define

ν := SA \underrightarrow{γ̃} GSA ⟶^{GSϵ_{A}} GSGA,

which may be seen to satisfy the axioms in (4).) We will usually let the context differentiate between S and S̃ and simply write S in both cases.

Now, suppose S and S′ may be lifted to C^{G}, that is, they are equipped respectively with natural transformations

ν : S → GSG and ν′ : S′ → GS′G,

satisfying (4). It remains to lift the evaluation and coevaluation morphisms (3). Consider the following axioms.

[figure: SA ⋆ GA; SA ⋆ A; J; GSGA ⋆ G^{2}A; G(SGA ⋆ GA); GJ; ν⋆δ; ϕ; Ge_{GA}; 1⋆ϵ; e_{A}; ψ_{0}]  (5)

[figure: I; GA ⋄ SGA; GA ⋄ GSG^{2}A; G(A ⋄ SG^{2}A); G(A ⋄ SGA); GI; G(A ⋄ SA); n; 1⋄ν; ϕ; G(1⋄Sδ); ϕ_{0}; Gn; G(1⋄Sϵ)]  (6)

[figure: GA ⋆ S′A; A ⋆ S′A; J; G^{2}A ⋆ GS′GA; G(GA ⋆ S′GA); GJ; δ⋆ν′; ϕ; Ge′_{GA}; ϵ⋆1; e′_{A}; ψ_{0}]  (7)

[figure: I; S′GA ⋄ GA; GS′G^{2}A ⋄ GA; G(S′G^{2}A ⋄ A); G(S′GA ⋄ A); GI; G(S′A ⋄ A); n′; ν′⋄1; ϕ; G(S′δ⋄1); ϕ_{0}; Gn′; G(S′ϵ⋄1)]  (8)

3.2. Proposition. Suppose C is a linearly distributive category with negation, G is a monoidal comonad satisfying axioms (1) and (2) (so that C^{G} is linearly distributive), and that S and S′ may be lifted to C^{G}. Then, G satisfies axioms (5), (6), (7), and (8) if and only if C^{G} is a linearly distributive category with negation.

Proof. Suppose (A, γ) is a G-coalgebra. We start by proving that axiom (5) holds if and only if e : SA ⋆ A → J is a G-coalgebra morphism. The following diagram proves the “only if” direction,

[figure: SA ⋆ A; GSGA ⋆ GA; G(SGA ⋆ A); G(SA ⋆ A); SA ⋆ GA; GSGA ⋆ G^{2}A; G(SGA ⋆ GA); GJ,; SA ⋆ A; J; (5); 1; ν⋆γ; ϕ; G(Sγ⋆1); 1⋆γ; 1⋆Gγ; G(1⋆γ); Ge; ν⋆δ; ϕ; Ge; 1⋆ϵ; e; ψ_{0}]

and this next diagram the “if” direction

[figure: SA ⋆ GA; GSGA ⋆ G^{2}A; SA ⋆ A; SGA ⋆ GA; GSG^{2}A ⋆ G^{2}A; GSGA ⋆ G^{2}A; G(SGA ⋆ GA); J; GJ,; 1⋆ϵ; Sϵ⋆1; ν⋆δ; GSGϵ⋆1; 1; ϕ; e; e_{GA}; ν⋆δ; GSδ⋆1; ϕ; Ge_{GA}; ψ_{0}]

where the bottom square commutes as e_{GA} is a G-coalgebra morphism.

Next we prove that axiom (6) holds if and only if n : I → A ⋄ SA is a G-coalgebra morphism. The “only if” direction is given by

[figure: I; A ⋄ SA; GA ⋄ SGA; GA ⋄ GSG^{2}A; G(A ⋄ SG^{2}A); G(A ⋄ SGA); GI; GA ⋄ SA; GA ⋄ GSGA; G(A ⋄ SGA); G(A ⋄ SA),; G(A ⋄ SA); (6); ϕ_{0}; Gn; G(1⋄Sϵ); G(1⋄Sγ); n; 1⋄ν; ϕ; G(1⋄Sδ); n; γ⋄1; 1⋄ν; ϕ; G(1⋄Sγ); 1⋄Sγ; 1⋄GSGγ; G(1⋄SGγ); 1]

and the “if” direction by

[figure: I; GA ⋄ SGA; G^{2}A ⋄ GSG^{2}A; G(GA ⋄ SG^{2}A); G(GA ⋄ SGA); G(A ⋄ SA); GI; GA ⋄ GSG^{2}A; G(A ⋄ SG^{2}A); G(A ⋄ SGA),; ϕ_{0}; n_{GA}; Gn_{GA}; Gn; δ⋄ν; 1⋄ν; ψ; Gϵ⋄1; G(1⋄Sδ); G(ϵ⋄1); G(ϵ⋄1); G(1⋄Sϵ); ψ; G(1⋄Sδ)]

where the top square commutes as n_{GA} is a G-coalgebra morphism.

The remaining two axioms are proved similarly.

4. Star-autonomous comonads

Suppose C = (C, ⊗, I, S, S′) is a star-autonomous category. A star-autonomous comonad G : C → C is a comonad satisfying axioms (described below) so that C^{G} becomes a starautonomous category [PS09]. In this section we show that comonads as in Proposition 3.2 and star-autonomous comonads coincide.

We recall the definition of star-autonomous comonad [PS09], but, as it suits our needs better here, we present a more symmetric version. First recall that a star-autonomous category C = (C, ⊗, I, S, S′) may be defined as a monoidal category (C, ⊗, I) equipped with an adjoint equivalence

S ⊣ S′ : C^{op} ⟶^{≃} C

such that

C(A ⊗ B, SC) ≅ C(A, S(B ⊗ C)),  (9)

natural in A, B, C ∈ C. The functor S is called the left star operation and S′ the right star operation.

By the Yoneda lemma, the isomorphism in (9) determines, and is determined by, the two following “evaluation” morphisms:

e = e_{A,B} : S(A ⊗ B) ⊗ A → SB and e′ = e′_{B,A} : B ⊗ S′(A ⊗ B) → S′A.

4.1. Definition. A star-autonomous comonad on a star-autonomous category C is a monoidal comonad G : C → C equipped with

ν : S → GSG  and  ν′ : S′ → GS′G,

satisfying (4) (i.e., S and S′ may be lifted to C^{G}), and this data must be such that the following four diagrams commute.

[figure: SS′G; G; GSGS′G; GSS′; ≅; ≅; ν; GSν′; S′SG; G; GS′GSG; GS′S; ≅; ≅; ν′; GSν]

[figure: S(A ⊗ B) ⊗ GA; S(A ⊗ B) ⊗ A; SB; GSG(A ⊗ B) ⊗ G^{2}A; GSGB; G(SG(A ⊗ B) ⊗ GA); G(S(GA ⊗ GB) ⊗ GA); 1⊗ϵ; e_{A,B}; ν; ν⊗δ; ϕ; G(Sϕ⊗1); Ge_{GA,GB}]

[figure: GB ⊗ S′(A ⊗ B); B ⊗ S′(A ⊗ B); S′A; G^{2}B ⊗ GS′G(A ⊗ B); GS′GA; G(GB ⊗ S′G(A ⊗ B)); G(GB ⊗ S′(GA ⊗ GB)); ϵ⊗1; e′_{B,A}; ν′; δ⊗ν′; ϕ; G(1⊗S′ϕ); Ge′_{GB,GA}]

The first two diagrams above ensure that the unit and the counit of the adjoint equivalence S ⊣ S′ lifts to C^{G}, while the latter two diagrams above respectively ensure that e and e′ are G-coalgebra morphisms, so that the isomorphism (9) also lifts to C^{G}.

We wish to show that star-autonomous comonads and comonads as in Proposition 3.2 coincide. It should not be surprising considering the following theorem.

4.2. Theorem. [CS97, Theorem 4.5] The notions of linearly distributive categories with negation and star-autonomous categories coincide.

Given a star-autonomous category (C, ⊗, I_{⊗}, S, S′), identifying ⋆ := ⊗ (and the monoidal unit I := I_{⋆} = I_{⊗}) and defining

A ⋄ B := S′(SB ⋆ SA) ≅ S(S′B ⋆ S′A) J := SI ≅ S′I  (10)

results in a linearly distributive category with negations (C, ⋆, I, ⋄, J, S, S′) [CS97]. In [CS97], they consider the symmetric case, but the correspondence between linearly distributive categories with negation and star-autonomous categories holds in the noncommutative case as well.

Given Theorem 4.2, Proposition 3.2 says that if C is star-autonomous, and G is such a comonad, then C^{G} is star-autonomous. We now compare the two definitions.

Suppose now that G is a comonad on a linearly distributive category C, as in Proposition 3.2. We wish to show that it is a star-autonomous comonad. Rather than proving the axioms, it is simpler to show directly that the morphisms under consideration are G-coalgebra morphisms. To this end, the unit and the counit of the adjoint equivalence S ⊣ S′ are defined respectively by the composites

A ≅ I ⋆ A ⟶^{n′_{SA}⋆1} (S′SA ⋄ SA) ⋆ A ⟶^{∂_{r}} S′SA ⋄ (SA ⋆ A) ⟶^{1⋄n} S′SA ⋄ J ≅ S′SA

and

S′SA ≅ I ⋆ S′SA ⟶^{n_{A}⋆1} (A ⋄ SA) ⋆ S′SA ⟶^{∂_{r}} A ⋄ (SA ⋆ S′SA) ⟶^{1⋄e′_{SA}} A ⋄ J ≅ A,

and the evaluation morphisms e_{A,B} and e′_{B,A} respectively by the composites

[figure: S(A ⋆ B) ⋆ A; (S(A ⋆ B) ⋆ A) ⋆ I; (S(A ⋆ B) ⋆ A) ⋆ (B ⋄ SB); (S(A ⋆ B) ⋆ A ⋆ B) ⋄ SB,; J ⋄ SB; SB; ≅; 1⋆n; ∂_{l}; e_{A⋆B}⋄1; ≅; e_{A,B}]

and

[figure: B ⋆ S′(A ⋆ B); I ⋆ (B ⋆ S′(A ⋆ B)); (S′A ⋄ A) ⋆ (B ⋆ S′(A ⋆ B)); S′A ⋄ (A ⋆ B ⋆ S′(A ⋆ B)).; S′A ⋄ J; SB; ≅; n′⋆1; ∂_{r}; 1⋄e′_{A⋆B}; ≅; e′_{B,A}]

In the situation of Proposition 3.2, we see that all four of these morphisms are given as composites of G-coalgebra morphisms, and thus, are G-coalgebra morphisms themselves. Therefore, G is a star-autonomous comonad.

In the other direction suppose G is a star-autonomous comonad on a star-autonomous category C. It is similar to show that it is a comonad satisfying the requirements of Proposition 3.2. Using the identifications in (10), the two linear distributions are defined as the following composites.

[figure: A ⋆ (B ⋄ C); A ⊗ S′(SC ⊗ SB); A ⊗ S′(SC ⊗ S(A ⊗ B) ⊗ A); S′(SC ⊗ S(A ⊗ B)); (A ⋆ B) ⋄ C; ∂_{l}; ≅; 1⊗S′(1⊗e); e′; ≅; (B ⋄ C) ⋆ A; S(S′C ⊗ S′B) ⊗ A; S(A ⊗ S′(C ⊗ A) ⊗ S′B) ⊗ A); S(S′(C ⊗ A) ⊗ S′B); B ⋄ (C ⋆ A); ∂_{r}; ≅; S(e′⊗1)⊗1; e; ≅]

The evaluation maps e_{A} and e′_{A} are defined as e_{A,I} and e′_{A,I}, and the coevaluation maps n_{A} and n′_{A} as

n_{A} = (I ≅ SS′I ⟶^{S(e′_{A,I})} S(A ⊗ S′A) = A ⋄ SA)

n′_{A} = (I ≅ S′SI ⟶^{S′(e_{A,I})} S′(SA ⊗ A) = S′A ⋄ A)

Again, each morphism is a G-coalgebra morphism, or composite thereof, and therefore is itself a G-coalgebra morphism.

Thus, both notions coincide, and we will simply call either a star-autonomous comonad, and let context differentiate the axiomatization.

4.3. Example. Any Hopf algebra H in a star-autonomous category C gives rise to a star-autonomous comonad H ⊗ − : C → C. See [PS09, p. 3515] for details.

4.4. Example. If C is a symmetric closed monoidal category with finite products, then we may apply an instance of the Chu construction [B79] to produce a star-autonomous category Chu(C, 1). The category C fully faithfully embeds into Chu(C, 1),

C ↪ Chu(C, 1)

and this functor is strong symmetric monoidal. Thus, any Hopf algebra in C becomes a Hopf algebra in Chu(C, 1), and thus, an example of a star-autonomous comonad.

5. The compact case ⋆ = ⋄

If C is a linearly distributive category with negation for which ⋆ = ⋄ (and thus, I = J), then C is an autonomous (= rigid) category. The functor S provides left duals, while S′ provides right duals. It is not hard to see that, in this case, any star-autonomous monad G (after turning arrows around) is a Hopf monad [BV07]. Set ⋆ = ⋄ and I = J and dualize axioms (5), (6), (7), and (8). They correspond in [BV07] to axioms (23), (22), (21), and (20) respectively. (In their notation ^{∨}(−) = S and (−)^{∨} = S′.) Therefore, we have:

5.1. Proposition. Star-autonomous monads on autonomous categories are Hopf monads.

Acknowledgements: I would like to thank Robin Cockett and Masahito Hasegawa for valuable discussions, and an anonymous referee for valuable suggestions.

References

[B79] Michael Barr. ∗-Autonomous categories, Volume 752 of Lecture Notes in Mathematics. Springer, Berlin, 1979. With an appendix by Po Hsiang Chu.

[BV07] Alain Bruguières and Alexis Virelizier. Hopf monads, Advances in Mathematics 215 no. 2 (2007) 679–733.

[CS97] J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories, Journal of Pure and Applied Algebra 114 (1997) 133–173. Corrected version available from the second authors webpage.

[M02] I. Moerdijk. Monads on tensor categories, Journal of Pure and Applied Algebra 168 (2002) 189–208.

[PS09] Craig Pastro and Ross Street. Closed categories, star-autonomy, and monoidal comonads, Journal of Algebra 321 no. 11 (2009) 3494–3520.

[S72] Ross Street. The formal theory of monads, Journal of Pure and Applied Algebra 2 no. 2 (1972) 149–168.

Department of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Email: craig@math.kyushu-u.ac.jp

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