Theory and Applications of Categories, Vol. 24, No. 2, 2010, pp. 25–38.

THE FROBENIUS RELATIONS MEET LINEAR DISTRIBUTIVITY

J.M. EGGER

Abstract. The notion of Frobenius algebra originally arose in ring theory, but it is a fairly easy observation that this notion can be extended to arbitrary monoidal categories. But, is this really the correct level of generalisation?

For example, when studying Frobenius algebras in the ∗-autonomous category Sup, the standard concept using only the usual tensor product is less interesting than a similar one in which both the usual tensor product and its de Morgan dual (par) are used.

Thus we maintain that the notion of linear-distributive category (which has both a tensor and a par, but is nevertheless more general than the notion of monoidal category) provides the correct framework in which to interpret the concept of Frobenius algebra.

1. Introduction

We recall that a linearly distributive category is defined to be a category equipped with two tensor products, here denoted ∩^{×} and ∪^{×}, which are related by a pair of (generally non-invertible) natural transformations

[figure: x ∩^{×} (y ∪^{×} z); κ^{(ℓℓ)}; (x ∩^{×} y) ∪^{×} z; (x ∪^{×} y) ∩^{×} z; κ^{(rr)}; x ∪^{×} (y ∩^{×} z)]

which are required to satisfy a large number of coherence conditions.

The reader is referred to [Cockett and Seely, 1997] for the full definition, as well as for proofs of all statements in this section. The definition of linear functor between linearly distributive categories can be found in [Cockett and Seely, 1999]. Note that symmetries are not included in the definition of linearly distributive category; we use the word planar to mean not necessarily symmetric.

1.1. Example. Every (planar) ∗-autonomous category (K, ∩^{×}, e, {− \atop ∘}, ∘−, d) has an underlying linearly distributive category, in which the second tensor product is defined as the de Morgan dual of the first.

x ∪^{×} y := ^{∗}(y^{∗} ∩^{×} x^{∗})

Here, as usual, x^{∗} is an abbreviation for x {− \atop ∘} d, and ^{∗}x is an abbreviation for d ∘− x.

Research partially supported by NSERC. Received by the editors 2007-02-22 and, in revised form, 2010-02-15. Transmitted by Ross Street. Published on 2010-02-18. 2000 Mathematics Subject Classification: 03F52,18D10,18D15. Key words and phrases: Frobenius algebras, linear distributive categories. \begin{matrix}c \\ ◯ \end{matrix} J.M. Egger, 2010. Permission to copy for private use granted.

(Note that ^{∗}(y^{∗} ∩^{×} x^{∗}) ≅ (^{∗}y ∩^{×} ^{∗}x)^{∗} holds in any (planar) ∗-autonomous category, so the asymmetry present in the definition above is only one of appearance.)

In particular, a boolean algebra may be viewed as a linearly distributive category with

∩^{×} = ∧ and ∪^{×} = ∨.

One of the mount profound and useful observations made in [Cockett and Seely, 1997] is that what one might call duality theorems in a (planar) ∗-autonomous category can be characterised in terms of its underlying linearly distributive structure.

1.2. Theorem. Let (K, ∩^{×}, e, {− \atop ∘}, ∘−, d) be a (planar) ∗-autonomous category, and suppose that we are given a pair of arrows

[figure: x; θ; ^{∗}y.; φ]

Let \xymatrix{x ∩^{×} y \ar[r]^{γ} & d} denote the transpose of θ, and τ the composite

[figure: e; ^{⌜}φ^{⌝}; ^{∗}y {− \atop ∘} x; ∼; y ∪^{×} x.]

Then θ and φ are inverse to each other if and only if the following diagrams commute.

[figure: x ∩^{×} e; ι ∩^{×} τ; υ^{(r)}; (lti_{1}); x ∩^{×} (y ∪^{×} x); κ^{(ℓℓ)}; (x ∩^{×} y) ∪^{×} x; γ ∪^{×} ι; x; _{υ^{(ℓ)}_{−1}}; d ∪^{×} x]

[figure: e ∩^{×} y; τ ∩^{×} ι; υ^{(ℓ)}; (lti_{2}); (y ∪^{×} x) ∩^{×} y; κ^{(rr)}; y ∪^{×} (x ∩^{×} y); ι ∪^{×} γ; y; _{υ^{(r)}_{−1}}; y ∪^{×} d]

(Here we abuse notation by using the same symbol to denote both of the left-unit isomorphisms; similarly, the right-unit isomorphisms.)

1.3. Definitions.

1. A pair of arrows in a linearly distributive category (K, ∩^{×}, e, ∪^{×}, d)

[figure: x ∩^{×} y; γ; d; e; τ; y ∪^{×} x]

which satisfies (lti_{1}) and (lti_{2}) is called a linear adjoint.

2. A pair of linear adjoints

[figure: x ∩^{×} y; γ_{1}; d; e; τ_{1}; y ∪^{×} x; y ∩^{×} x; γ_{2}; d; e; τ_{2}; x ∪^{×} y]

is called a cyclic linear adjoint.

1.4. Remark. It is to be emphasised that any monoidal category (K, ⊗, i) can be regarded as a linearly distributive category by choosing ∩^{×} = ⊗ = ∪^{×}, e = i = d, κ^{(rr)} = α, and κ^{(ℓℓ)} = α^{−1}.

In this sense, linearly distributive categories are actually more general than monoidal categories and when, as here, duality theory is at the centre of one’s attention, this is often a useful point of view: that monoidal categories are merely degenerate linearly distributive categories.

Note that in the strict degenerate case, the diagrams defining linear adjoint reduce to the more usual triangle identities.

1.5. Lemma. In an arbitrary linearly distributive category, each half of a linear adjoint determines the other in the sense that, if

[figure: x ∩^{×} y; γ_{1}; d; e; τ_{1}; y ∪^{×} x; x ∩^{×} y; γ_{2}; d; e; τ_{2}; y ∪^{×} x]

are linear adjoints, then (γ_{1} = γ_{2}) ⟺ (τ_{1} = τ_{2}).

2. Linear points and Frobenius monoids

Throughout the next two sections (K, ∩^{×}, e, ∪^{×}, d) shall denote an arbitrary linearly distributive category.

It shall be our convention that: the term monoid refers to a monoid in (K, ∩^{×}, e); the term comonoid refers to a comonoid in (K, ∪^{×}, d); and, the term monoid/comonoid pair refers to a monoid and a comonoid (in the senses above) which have the same underlying object.

2.1. Lemma. Let M = (m, μ, η) be a monoid, (x, σ^{(ℓ)}) a left M-object, and (y, σ^{(r)}) a right M-object; then the maps

[figure: m ∩^{×} (x ∪^{×} y); κ^{(ℓℓ)}; (m ∩^{×} x) ∪^{×} y; σ^{(ℓ)} ∪^{×} ι; x ∪^{×} y; (x ∪^{×} y) ∩^{×} m; κ^{(rr)}; x ∪^{×} (y ∩^{×} m); ι ∪^{×} σ^{(r)}; x ∪^{×} y]

define a two-sided action of M on x ∪^{×} y.

Proof. We shall prove only the left/right-compatibility diagram, sometimes also known as middle associativity, as an illustration.

[figure: (m ∩^{×} (x ∪^{×} y)) ∩^{×} m; α; κ^{(ℓℓ)} ∩^{×} ι; (coh); m ∩^{×} ((x ∪^{×} y) ∩^{×} m); ι ∩^{×} κ^{(rr)}; ((m ∩^{×} x) ∪^{×} y) ∩^{×} m; (σ^{(ℓ)} ∪^{×} ι) ∩^{×} ι; κ^{(rr)}; m ∩^{×} (x ∪^{×} (y ∩^{×} m)); κ^{(ℓℓ)}; ι ∩^{×} (ι ∪^{×} σ^{(r)}); (x ∪^{×} y) ∩^{×} m; (nat); κ^{(rr)}; (m ∩^{×} x) ∪^{×} (y ∩^{×} m); _{σ^{(ℓ)} ∪^{×} σ^{(r)}}; σ^{(ℓ)} ∪^{×} ι; ι ∪^{×} σ^{(r)}; m ∩^{×} (x ∪^{×} y); (nat); κ^{(ℓℓ)}; x ∪^{×} (y ∩^{×} m); ι ∪^{×} σ^{(r)}; (m ∩^{×} x) ∪^{×} y; σ^{(ℓ)} ∪^{×} ι; x ∩^{×} y]

The remainder is left as an exercise.

2.2. Remark. In (the linearly distributive category arising from) a (planar) ∗-autonomous category (K, ∩^{×}, e, {− \atop ∘}, ∘−, d), we have canonical isomorphisms x^{∗} {− \atop ∘} y ≅ x ∪^{×} y ≅ x ∘− ^{∗}y. In this case, the left (right) m-action we have just defined on x ∪^{×} y coincides the usual one on x ∘− ^{∗}y (respectively, x^{∗} {− \atop ∘} y). But in an arbitrary (left- and right-) closed monoidal category, there would seem to be no way of expressing the idea that these left- and rightactions combine to form a two-sided action.

For the most part, we shall be interested in only the following cases of Lemma 2.1, and their duals: if M = (m, μ, η) is a monoid, then of course m carries a canonical two-sided M-action; and so any two-sided action on x induces canonical two-sided actions on both

x ∪^{×} m and m ∪^{×} x.

This observation is necessary to make the following definition type correctly.

2.3. Definitions.

1. A cyclic nuclear monoid in (K, ∩^{×}, e, ∪^{×}, d) consists of:

(a) a monoid M = (m, μ, η); (b) a comonoid G = (g, δ, ε);

(c) a two-sided M-action (σ^{(ℓ)}, σ^{(r)}) on g; and

(d) a two-sided G-coaction (θ^{(ℓ)}, θ^{(r)}) on m;

such that:

(e) \xymatrix{m \ar[r]^{θ^{(ℓ)}} & g ∪^{×} m} and \xymatrix{m \ar[r]^{θ^{(r)}} & m ∪^{×} g} are M-equivariant; and

(f) \xymatrix{m ∩^{×} g \ar[r]^{σ^{(ℓ)}} & g} and \xymatrix{g ∩^{×} m \ar[r]^{σ^{(r)}} & g} are G-coequivariant.

2. A Frobenius monoid in (K, ∩^{×}, e, ∪^{×}, d) is a cyclic nuclear monoid with m = g, σ^{(ℓ)} = μ = σ^{(r)}, and θ^{(ℓ)} = δ = θ^{(r)}.^{1}

2.4. Theorem. A cyclic nuclear monoid in (K, ∩^{×}, e, ∪^{×}, d) is the same thing as a linear point of (K, ∩^{×}, e, ∪^{×}, d)—i.e., a linear functor 1 ⟶ (K, ∩^{×}, e, ∪^{×}, d).

A Frobenius monoid is the same thing as a monoid/comonoid pair (m, μ, η, δ, ε) which satisfy the linear Frobenius relations depicted below.

[figure: m ∩^{×} (m ∪^{×} m); κ^{(ℓℓ)}; (lfr_{1}); (m ∩^{×} m) ∪^{×} m; μ ∪^{×} ι; m ∩^{×} m; ι ∩^{×} δ; μ; δ ∩^{×} ι; m; δ; m ∪^{×} m; (m ∪^{×} m) ∩^{×} m; κ^{(rr)}; (lfr_{2}); m ∪^{×} (m ∩^{×} m); ι ∪^{×} μ]

Proof. It follows directly from the definition of linear functor that a linear point consists of: a monoid M = (m, μ, η), a comonoid G = (g, δ, ε), and maps σ^{(ℓ)}, σ^{(r)}, θ^{(ℓ)}, θ^{(r)} whose sources and targets match those given in the definition of cyclic nuclear monoid. Therefore all that remains to show is that the eighteen coherence conditions required of σ^{(ℓ)}, σ^{(r)}, θ^{(ℓ)}, θ^{(r)} by the definition of linear functor are (collectively) equivalent to the conditions placed on them by the definition of cyclic nuclear monoid.

In fact, Definition 2.3 also requires that σ^{(ℓ)}, σ^{(r)}, θ^{(ℓ)} and θ^{(r)} satisfy a total of eighteen diagrams:

1. five diagrams for σ^{(ℓ)}, σ^{(r)} being an M-action;

2. five diagrams for θ^{(ℓ)}, θ^{(r)} being a G-coaction;

3. four diagrams for θ^{(ℓ)}, θ^{(r)} being M-equivariant; and

1I have two suggestions for replacing the phrase Frobenius monoid with something shorter: Frobenium and ambimonoid—indeed, I should like to propose the term ambimonoidal as a synonym for linearly distributive so that we can speak of ambimonoids in an ambimonoidal category.

4. four diagrams for σ^{(ℓ)}, σ^{(r)} being G-coequivariant.

We claim that the two sets of conditions are not merely equivalent, but actually equal.

To demonstrate this claim, we need merely identify the five symmetry-based groupings which appear in [Cockett and Seely, 1999, Definition 1]. These are as follows:

1. the four (co)unit diagrams;

2. the four (co)associativity diagrams;

3. the two left/right-compatibility diagrams;

4. the four same-parity (co)equivariance diagrams—i.e., the left equivariance of θ^{(ℓ)}, the right equivariance of θ^{(r)}, the left coequivariance of σ^{(ℓ)} and the right coequivariance of σ^{(r)};

5. the remaining four (co)equivariance diagrams.

(What distinguishes the same-parity (co)equivariance diagrams is that, in each case, they contain an alternating ternary string: either mgm or gmg—for an example, see below.)

Turning to the case of Frobenius monoids, we see that the first ten diagrams (according to either grouping) are trivial; the remaining eight diagrams collapse to two.

Consider, for example, the M-equivariance of θ^{(ℓ)}. In general, this states that the diagrams

[figure: m ∩^{×} m; μ; ι ∩^{×} θ^{(ℓ)}; m ∩^{×} (g ∪^{×} m); κ^{(ℓℓ)}; m ∩^{×} m; μ; θ^{(ℓ)} ∩^{×} ι; (g ∪^{×} m) ∩^{×} m; κ^{(rr)}; (m ∩^{×} g) ∪^{×} m; σ^{(ℓ)} ∪^{×} ι; g ∪^{×} (m ∩^{×} m); ι ∪^{×} μ; m; θ^{(ℓ)}; g ∪^{×} m; m; θ^{(ℓ)}; g ∪^{×} m]

commute—but, if g = m, σ^{(ℓ)} = μ and θ^{(ℓ)} = δ, then these reduce to (lfr_{1}) and (lfr_{2}), respectively.

Each of the three other pairs of diagrams—the G-coequivariance of σ^{(ℓ)}, that of σ^{(r)}, and the M-equivariance of θ^{(r)}—do the same.

(Note that each pair contains one diagram with a κ^{(ℓℓ)} and one with a κ^{(rr)}—the former reduces to (lfr_{1}) and the latter to (lfr_{2}).)

2.5. Scholium. For a monoid/comonoid pair (m, μ, η, δ, ε) in (K, ∩^{×}, e, ∪^{×}, d), the following are equivalent:

1. m ⟶^{δ} m ∪^{×} m is left (m, μ, η)-equivariant;

2. m ⟶^{δ} m ∪^{×} m is right (m, μ, η)-equivariant;

3. m ∩^{×} m ⟶^{μ} m is left (m, δ, ε)-coequivariant;

4. m ∩^{×} m ⟶^{μ} m is right (m, δ, ε)-coequivariant;

5. (m, μ, η, δ, ε) is a Frobenius monoid.

In the degenerate case, this fact is well-known and appears, for example, in [Kock, 2004].

3. Duality

In the degenerate case, Frobenius monoids are usually studied in connection with duality theory, so there should be little surprise that the same is true in the general case. Indeed, it follows from general results in [Cockett and Seely, 1999] and [Cockett et al., 2000] that cyclic nuclear monoids and Frobenius monoids must be self-dual. Here we formulate and prove the same result with somewhat more precision.

3.1. Theorem. If ((m, μ, η), (g, δ, ε), (σ^{(ℓ)}, σ^{(r)}), (θ^{(ℓ)}, θ^{(r)})) form a cyclic nuclear monoid, then the maps

[figure: e; η; m; θ^{(ℓ)}; g ∪^{×} m; m ∩^{×} g; σ^{(ℓ)}; g; ε; d; e; η; m; θ^{(r)}; m ∪^{×} g; g ∩^{×} m; σ^{(r)}; g; ε; d]

form a cyclic linear adjoint.

In particular, if (m, μ, η, δ, ε) is a Frobenius monoid, then

[figure: e; η; m; δ; m ∪^{×} m; m ∩^{×} m; μ; m; ε; d]

form a linear adjoint.

Proof. Each of the four same-parity (co)equivariance diagrams is used once, together with appropriate (co)unit axioms, to obtain one of the four necessary diagrams.

For instance, the left-equivariance of θ^{(ℓ)} (see proof of Theorem 2.4 above) can be combined with its left counit axiom and the right unit axiom for μ to produce (lti_{1}) for η ; θ^{(ℓ)} and σ^{(ℓ)} ; ε.

[figure: m ∩^{×} e; ι ∩^{×} η; υ^{(r)}; m ∩^{×} m; μ; ι ∩^{×} θ^{(ℓ)}; m ∩^{×} (g ∪^{×} m); κ^{(ℓℓ)}; (m ∩^{×} g) ∪^{×} m; σ^{(ℓ)} ∪^{×} ι; m; θ^{(ℓ)}; _{υ^{(ℓ)}_{−1}}; g ∪^{×} m; ε ∪^{×} ι; d ∪^{×} m]

3.2. Scholium. By examining the proof of Theorem 3.1, we can derive the following formula for θ^{(ℓ)}:

[figure: m ∩^{×} e; ι ∩^{×} (η ; θ); m ∩^{×} (g ∪^{×} m); κ^{(ℓℓ)}; (m ∩^{×} g) ∪^{×} m; σ^{(ℓ)} ∪^{×} ι; m; θ^{(ℓ)}; _{υ^{(r)}_{−1}}; g ∪^{×} m]

—but recall from Lemma 1.5, that the composite η ; θ^{(ℓ)} is uniquely determined by σ^{(ℓ)} ; ε.

Thus, the structure of a cyclic nuclear monoid is overdetermined. In particular, θ^{(ℓ)} is determined, indirectly, by σ^{(ℓ)} and ε.

In the case of a Frobenius monoid in a degenerate linearly distributive category, this is a celebrated result: the map δ = θ^{(ℓ)} is determined by μ = σ^{(ℓ)} and ε—see, for example, [Kock, 2004].

3.3. Remark. It is interesting to note that only four of the eight (co)equivariance conditions are actually used in the proof of Theorem 3.1.

Also interesting is the fact that each linear adjoint is constructed using only one half of the action/coaction structure. This suggests that there exists a more general notion than that of cyclic nuclear monoid, for which non-cyclic linear adjoints can be constructed.

The latter idea will be more fully explored in a subsequent paper.

4. Frobenius quantales

Throughout the next two sections: E shall denote an arbitrary elementary topos; 1 ⟶^{⊤} 2 its subobject classifier; ℙ the power-E-object monad, (2^{( )}, ⋃, {}); and Ě = Sup(E) the category of ℙ-algebras—equivalently, the category of (internally) cocomplete ordered Eobjects and (internal) sup-homomorphisms. Henceforth, we shall speak of the objects of E as if they were sets, and therefore also drop the modifiers internal and internally.

As demonstrated in [Joyal and Tierney, 1984], Ě can be given a symmetric ∗-autonomous structure, and hence also a (symmetric) linearly distributive structure. (We shall postpone the definition of symmetric linearly distributive category until the next section, as we will not need this extra structure until then.) We shall continue to write ∩^{×} and ∪^{×} for the two tensor products on Ě, but we shall abuse notation, slightly, by denoting their units by 2 and 2^{op} respectively. (Each of these carries a canonical ℙ-algebra structure.) Following the convention laid out in Example 1.1, x^{∗} does not denote x^{op} but rather x {− \atop ∘} 2^{op}; we write _{⌞}−_{⌟} for the canonical isomorphism \xymatrix{x^{op} \ar[r] & x^{∗} = ^{∗}x} .

We write |−| for the forgetful functor Ě ⟶ Ord(E), ∩^{×} also for the canonical map

[figure: |x| × |y|; ∩^{×}; |x ∩^{×} y|]

—hence (|−| , ∩^{×}, ⊤) is the forgetful monoidal functor (Ě, ∩^{×}, 2) ⟶ (Ord(E), ×, 1).

A quantale is a monoid in (Ě, ∩^{×}, 2). (We shall only consider unital quantales in the current paper.) Given a quantale (q, μ, η), we write (|q| , &, η) for the corresponding monoid in (Ord(E), ×, 1); in particular, & denotes the composite

[figure: |q| × |q|; ∩^{×}; |q ∩^{×} q|; |μ|; |q| .]

We write → and ← for the left- and right-closed structures on (|q| , &, η), respectively.

We recall that a quantale equipped with a cyclic dualising element is commonly called a Girard quantale, [Rosenthal, 1990]. (By a (cyclic) dualising element for (q, μ, η), one actually means a (cyclic) dualising element for (|q| , &, η, →, ←).)

Surprisingly, there does not seem to be a standard (short) name for a quantale equipped with an arbitrary dualising element; but this proves to be fortunate, in light of the following theorem.

4.1. Theorem. A Frobenius monoid in (Ě, ∩^{×}, 2, ∪^{×}, 2^{op}) amounts to a quantale equipped with a dualising element; equivalently, a ∗-autonomous cocomplete poset.

Proof. Suppose that (q, μ, η, δ, ε) is a Frobenius monoid in (Ě, ∩^{×}, 2, ∪^{×}, 2^{op}), and let ϶ = sup ker ε (so that ε = _{⌞}϶_{⌟}). We will abbreviate ( ) → ϶ and ϶ ← ( ) to ( )^{⊥} and ^{⊥}( ) respectively.

By Theorem 3.1

[figure: 2; η; q; δ; q ∪^{×} q; q ∩^{×} q; μ; q; ε; 2^{op}]

form a linear adjoint. Hence, by Theorem 1.2, the transpose of the latter composite

[figure: q; (μ ; ε)^{r}; 2^{op} ∘− q = ^{∗}q]

is invertible.

But it is easy to check that this transpose equals the map

α ↦ _{⌞}α^{⊥}_{⌟}

since we have

⌞α^{⊥}_{⌟}(β) = ⊥ ⟺ β ≤ α^{⊥} = α → ϶
⟺ μ(α ∩^{×} β) = α & β ≤ ϶
⟺ ε(μ(α ∩^{×} β)) = ⊥.

(This may not look contructive, but it is: we’re just using the universal properties of ∩^{×} and of 2. Here, ⊥ means ⊥_{2^{op}}—i.e., ⊤_{2}.)

Moreover, this suffices to demonstrate that ϶ is a dualising element, since for posets we always have (^{⊥}(( )^{⊥}))^{⊥} = ( )^{⊥}.

The previous arguments are reversible in the sense that if we start with a dualising element ϶, then we can define ε = _{⌞}϶_{⌟}, and the composite \xymatrix{q ∩^{×} q \ar[r]^{μ} & q \ar[r]^{ε} & 2^{op}} defines half of a linear adjoint. According to Scholium 3.2, there is then a unique δ making (q, μ, η, δ, ε) into a Frobenius monoid in (Ě, ∩^{×}, 2, ∪^{×}, 2^{op}).

We therefore propose that a quantale equipped with a (not necessarily) dualising element should be called a Frobenius quantale. We shall also use the term Frobenius locale for a Frobenius quantale whose underlying quantale is a locale; this amounts to a complete boolean algebra.

By way of comparison, note that a Frobenius monoid in (Ě, ∩^{×}, 2, ∩^{×}, 2) whose underlying quantale is a locale amounts to a power-object, [Pedicchio and Wood, 1999]. Thus, in a boolean topos, E, Frobenius quantales are more general than Frobenius monoids in (Ě, ∩^{×}, 2, ∩^{×}, 2); but, in non-boolean toposes, neither concept is contained in the other.

5. Canonical cyclicity

We recall that, by definition, a linearly distributive category (K, ∩^{×}, e, ∪^{×}, d) is called symmetric if both ∩^{×} and ∪^{×} are equipped with a symmetry, and if these symmetries (which shall both be denoted χ) satisfy one extra coherence condition:

[figure: x ∩^{×} (y ∪^{×} z); κ^{(ℓℓ)}; ι ∩^{×} χ; x ∩^{×} (z ∪^{×} y); χ; (z ∪^{×} y) ∩^{×} x; κ^{(rr)}; (x ∩^{×} y) ∪^{×} z; χ; z ∪^{×} (x ∩^{×} y); ι ∪^{×} χ; z ∪^{×} (y ∩^{×} x).]

The following lemma should be fairly obvious, but its proof does utilise this extra coherence condition, demonstrating the latter’s worth.

5.1. Lemma. If (K, ∩^{×}, e, ∪^{×}, d) is a symmetric linearly distributive category, and

[figure: x ∩^{×} y; γ; d; e; τ; y ∪^{×} x]

is a linear adjoint in (K, ∩^{×}, e, ∪^{×}, d), then

[figure: y ∩^{×} x; χ; x ∩^{×} y; γ; d; e; τ; y ∪^{×} x; χ; x ∪^{×} y]

is also a linear adjoint in (K, ∩^{×}, e, ∪^{×}, d).

Briefly, every linear adjoint in (K, ∩^{×}, e, ∪^{×}, d) gives rises to a cyclic linear adjoint.

Proof. We can obtain (lti_{1}) as follows:

[figure: y ∩^{×} e; ι ∩^{×} τ; χ; υ^{(r)}; y ∩^{×} (y ∪^{×} x); ι ∩^{×} χ; χ; y ∩^{×} (x ∪^{×} y); κ^{(ℓℓ)}; e ∩^{×} y; τ ∩^{×} ι; υ^{(ℓ)}; (y ∪^{×} x) ∩^{×} y; κ^{(rr)}; y ∪^{×} (y ∩^{×} x); ι ∪^{×} χ; χ; (y ∩^{×} x) ∪^{×} y; χ ∪^{×} ι; y ∪^{×} (x ∩^{×} y); ι ∪^{×} γ; (x ∩^{×} y) ∪^{×} y; γ ∪^{×} ι; y; _{υ^{(r)}_{−1}}; _{υ^{(ℓ)}_{−1}}; y ∪^{×} d; χ; d ∪^{×} y]

and (lti_{2}) by a symmetric argument.

What is, perhaps, not obvious is whether every cyclic linear adjoint in (K, ∩^{×}, e, ∪^{×}, d) arises in this way. As we shall see later in this section, the answer turns out to be no.

5.2. Definitions.

1. A cyclic linear adjoint in (K, ∩^{×}, e, ∪^{×}, d) is called canonically cyclic if the diagram

[figure: x ∩^{×} y; γ_{1}; χ; d; e; τ_{1}; y ∪^{×} x; χ; y ∩^{×} x; γ_{2}; d; e; τ_{2}; x ∪^{×} y.]

commutes.

2. A cyclic nuclear monoid in (K, ∩^{×}, e, ∪^{×}, d) is called canonically cyclic if the cyclic linear adjoint described in Theorem 3.1 is so—i.e., if the diagram

[figure: e; η; m; θ^{(ℓ)}; g ∪^{×} m; χ; m ∩^{×} g; χ; σ^{(ℓ)}; g; ε; d; e; η; m; θ^{(r)}; m ∪^{×} g; g ∩^{×} m; σ^{(r)}; g; ε; d]

commutes.

3. A Girard monoid in (K, ∩^{×}, e, ∪^{×}, d) is a Frobenius monoid whose underlying cyclic nuclear monoid is canonically cyclic—i.e., if it satisfies

[figure: e; η; m; δ; m ∪^{×} m; χ; m ∩^{×} m; μ; χ; m; ε; d; e; η; m; δ; m ∪^{×} m; m ∩^{×} m; μ; m; ε; d]

5.3. Remark. As a result of Lemmata 1.5 and 5.1, it suffices to check only one of each pair of diagrams in the definitions above.

From an abstract point of view, the significance of canonically cyclic linear adjoints has to do with the fact that in a symmetric ∗-autonomous category (K, ∩^{×}, e, {− \atop ∘}, ∘−, d), we have a canonical natural isomorphism \xymatrix{x^{∗} \ar[r]^{ω} & ^{∗}x} defined as the transpose of the composite

[figure: x^{∗} ∩^{×} x; χ; x ∩^{×} x^{∗}; ε^{(r)}; d.]

(Note that, for us, the term ∗-autonomous category denotes a (left- and right-) closed monoidal category equipped with a dualising object; and that the term symmetric ∗- autonomous category denotes a ∗-autonomous category equipped with a symmetry. Thus we do not assume in general that x {− \atop ∘} y = y ∘− x, although this is what frequently occurs in practice.)

5.4. Theorem. Let (K, ∩^{×}, e, {− \atop ∘}, ∘−, d) be a symmetric ∗-autonomous category. Then a cyclic linear adjoint in (K, ∩^{×}, e, ∪^{×}, d)

[figure: x ∩^{×} y; γ_{1}; d; e; τ_{1}; y ∪^{×} x; y ∩^{×} x; γ_{2}; d; e; τ_{2}; x ∪^{×} y]

is canonically cyclic if and only if the diagram

[figure: x; γ_{2}^{ℓ}; γ_{1}^{r}; y^{∗}; ω; ^{∗}y]

commutes.

Proof. It suffices to show that the composite

[figure: x ∩^{×} y; γ_{2}^{ℓ} ∩^{×} ι; y^{∗} ∩^{×} y; ω ∩^{×} ι; ^{∗}y ∩^{×} y; ε^{(ℓ)}; d]

equals γ_{1}.

But the definition of ω amounts to

[figure: y^{∗} ∩^{×} y; ω ∩^{×} ι; χ; ^{∗}y ∩^{×} y; ε^{(ℓ)}; y ∩^{×} y^{∗}; ε^{(r)}; d]

and combining this with a naturality square, we obtain

[figure: x ∩^{×} y; γ_{2}^{ℓ} ∩^{×} ι; χ; y^{∗} ∩^{×} y; ω ∩^{×} ι; χ; ^{∗}y ∩^{×} y; ε^{(ℓ)}; y ∩^{×} x; ι ∩^{×} γ_{2}^{ℓ}; γ_{2}; y ∩^{×} y^{∗}; ε^{(ℓ)}; d.]

5.5. Remark. The previous theorem is the main reason we have concentrated on symmetries rather than braidings.

In a braided ∗-autonomous category, we have at least two isomorphisms \xymatrix{x^{∗} \ar[r]^{ω} & ^{∗}x} which need not be equal; therefore neither is genuinely canonical.

Readers familiar with the classical theory of Frobenius algebras (i.e., Frobenius monoids in the degenerate linearly distributive category (Vec, ⊗, k, ⊗, k)) will have already recognised that not every Frobenius monoid is a Girard monoid, and hence that not every cyclic linear adjoint is canonically cyclic. (Girard monoids in (Vec, ⊗, k, ⊗, k) go by the uninspired, and potentially misleading, name symmetric Frobenius algebras.)

For everyone else we offer the following class of counter-examples.

5.6. Corollary. A Girard monoid in (Ě, ∩^{×}, 2, ∪^{×}, 2^{op}) is the same thing as a Girard quantale.

Proof. By the previous theorem, a Girard monoid in (Ě, ∩^{×}, 2, ∪^{×}, 2^{op}) is the same thing as a Frobenius quantale (q, μ, η, δ, ε) such that the diagram

[figure: q; (μ ; ε)^{r}; ^{∗}q; q; (μ ; ε)^{ℓ}; q^{∗}]

commutes.

But we have already shown, in the proof of Theorem 4.1, that the top map equals α ↦ _{⌞}α^{⊥}_{⌟}; by a symmetric argument, the bottom map equals α ↦ _{⌞}^{⊥}α_{⌟}.

Acknowledgements

The author would like to thank: Rick Blute, Robin Cockett and Robert Seely, for much encouragement in developing the material of sections 3 and 4; and David Kruml, for long and interesting conversations which, among other things, provided the impetus for proving the results of section 5.

References

[Cockett et al., 2000] Cockett, J. R. B., Koslowski, J., and Seely, R. A. G. (2000). Introduction to linear bicategories. Math. Structures Comput. Sci., 10(2):165–203. The Lambek Festschrift: mathematical structures in computer science (Montreal, QC, 1997).

[Cockett and Seely, 1997] Cockett, J. R. B. and Seely, R. A. G. (1997). Weakly distributive categories. J. Pure Appl. Algebra, 114(2):133–173.

[Cockett and Seely, 1999] Cockett, J. R. B. and Seely, R. A. G. (1999). Linearly distributive functors. J. Pure Appl. Algebra, 143(1-3):155–203. Special volume on the occasion of the 60th birthday of Professor Michael Barr (Montreal, QC, 1997).

[Joyal and Tierney, 1984] Joyal, A. and Tierney, M. (1984). An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc., 51(309):vii+71.

[Kock, 2004] Kock, J. (2004). Frobenius algebras and 2D topological quantum field theories, volume 59 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge.

[Pedicchio and Wood, 1999] Pedicchio, M. C. and Wood, R. J. (1999). Groupoidal completely distributive lattices. J. Pure Appl. Algebra, 143(1-3):339–350. Special volume on the occasion of the 60th birthday of Professor Michael Barr (Montreal, QC, 1997).

[Rosenthal, 1990] Rosenthal, K. I. (1990). Quantales and their applications, volume 234 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow.

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