On the axiomatisability of the dual of compact ordered spaces

We provide a direct and elementary proof of the fact that the category of Nachbin's compact ordered spaces is dually equivalent to an Aleph_1-ary variety of algebras. Further, we show that Aleph_1 is a sharp bound: compact ordered spaces are not dually equivalent to any SP-class of finitary algebras.

Notation. Given morphisms f i : X → Y i for i ∈ {0, 1}, the unique morphism induced by the universal property of the product is f 0 , f 1 : X → Y 0 × Y 1 . Similarly, given morphisms g i : X i → Y with i ∈ {0, 1}, the coproduct map is g0 g1 : X 0 + X 1 → Y . For infinite coproducts, we use the notation i∈I X i . Epimorphisms are denoted by ։, while monomorphisms (resp. regular monomorphisms) by (resp. ֒→). We use the symbol for pre-orders, and for partial orders.

Varieties as categories
In this section we provide the background needed to state a well-known characterisation of those categories which are equivalent to some (quasi-)variety of algebras. See Theorem 2 below. Throughout, unless otherwise stated, (quasi-)varieties admit possibly infinitary function symbols in their signatures.
Recall that a category C is regular provided (i) it has finite limits, (ii) it admits coequalisers of kernel pairs, and (iii) regular epimorphisms in C are stable under pullbacks. For instance, varieties and quasi-varieties of algebras (with homomorphisms) are regular categories. Those regular categories in which there is a good correspondence between regular epimorphisms and equivalence relations are called exact. To give a precise definition, we recall the notion of equivalence relation in a category.
Let C be a category with finite limits, and A an object of C. An (internal) equivalence relation on A is a subobject p 0 , p 1 : R A × A satisfying the following properties: reflexivity: there exists a morphism d : A → R in C such that the following diagram commutes; then there is a morphism t : P → R such that the right-hand diagram commutes.
Definition 1. An equivalence relation p 0 , p 1 : R A × A is effective if it coincides with the kernel pair of the coequaliser of p 0 and p 1 . A regular category C is exact if every equivalence relation in C is effective.
For categories of algebras, the definition of equivalence relation given above coincides with the usual notion of congruence. Varieties of algebras are therefore exact categories, while the effective equivalence relations in quasi-varieties are the so-called relative congruences. We need one last piece of terminology to state the desired characterisation of (quasi-)varieties of algebras. Recall that an object G of a locally small category C is a regular generator if (i) for every set I the copower I G exists in C, and (ii) for every object A of C, the canonical morphism is a regular epimorphism. Further, G is regular projective if, for any morphism f : G → A and every regular epimorphism g : B → A, f factors through g. We can now state the following result.
Theorem 2. For any locally small category C, consider the following conditions: (1) C is regular with coequalisers for equivalence relations; (2) C has a regular projective regular generator G; (3) every equivalence relation in C is effective.
The category C is equivalent to a quasi-variety iff it satisfies 1 and 2, and it is equivalent to a variety iff it satisfies 1, 2 and 3.
Proof. The abstract characterisation of varieties and quasi-varieties has a long history in category theory, starting with the works of Lawvere, Isbell, Linton, Felscher and Duskin in the 1960s. We do not attempt here to provide an accurate historical account. For the case of quasi-varieties we refer the reader to [17,Theorem 1.8], and for varieties to [5,Theorem 4.4.5] or [23]. Further, we point out that the assumption that C be regular can be omitted provided C has all coequalisers, cf. [2, Theorem 3.6].

Compact ordered spaces and their dual variety
We collect here some basic facts about compact ordered spaces, first introduced by Nachbin [16]. In particular, we describe their limits and colimits. This will come handy in the following sections.
Definition 3. A compact ordered space (or compact pospace, for short) is a pair (X, ) where X is a compact space, and is a partial order on X which is closed in the product topology of X × X. We write KH for the category of compact pospaces and continuous monotone maps.
A basic example of compact pospace is the unit interval [0, 1] equipped with the Euclidean topology, and its usual total order. Note that, for any compact pospace (X, ), the opposite order op = {(x, y) | y x} is also closed in the product topology of X × X. The intersection ∩ op coincides with the diagonal This shows that there is a forgetful functor KH → KH, where KH denotes the category of compact Hausdorff spaces and continuous maps. On the other hand, there is also a functor ∆ : KH → KH sending a compact Hausdorff space X to the compact pospace (X, ∆ X ). It is readily seen that ∆ is left adjoint to the forgetful functor KH → KH. In symbols, We will see in a moment that KH admits all limits and colimits. By the adjunction in (1), limits in KH are computed in KH, whence in the category of sets. However, this is not the case for colimits. To circumvent this issue, we embed KH in a larger category where colimits admit a simpler description.

Definition 4.
A pre-ordered compact Hausdorff space is a pair (X, ) where X is a compact Hausdorff space, and is a pre-order on X which is closed in the product topology of X × X. We write KH for the category of pre-ordered compact Hausdorff spaces and continuous monotone maps.
Clearly, KH is a full subcategory of KH , and the adjunction in (1) lifts to an adjunction between KH and KH. Further, the forgetful functor KH → KH has, in addition to the left adjoint ∆, also a right adjoint. Write ∇ : KH → KH for the functor sending a compact Hausdorff space X to the pre-ordered compact Hausdorff space (X, ∇ X ), where ∇ X = X × X is the improper relation on X. It is immediate that ∇ is right adjoint to the forgetful functor KH → KH.
Given a pre-ordered compact Hausdorff space (X, ), we can consider the quotient of X with respect to the symmetrization of , that is the equivalence relation ∼ = ∩ op . The pre-order descends to a partial order on the quotient space X/∼, and the map ρ X : (X, ) ։ (X/∼, ) is continuous and monotone. The pair (X/∼, ) is readily seen to be a compact pospace. This assignment extends to a functor ρ : KH → KH , which is left adjoint to the inclusion KH → KH . In other words, KH is a reflective subcategory of KH .
The category KH is complete and cocomplete, hence so is its reflective subcategory KH , see [22, Example 2 and Corollary 2]. Since the forgetful functor KH → KH has a right adjoint, colimits in KH are computed in KH. In turn, the colimit of a diagram in KH can be obtained by first computing the colimit in KH , and then applying the reflector ρ. For more details, cf. the following remark.
Remark 5. Let X, Y be compact pospaces. Their coproduct in KH is the disjoint union of X and Y , with the coproduct topology and the coproduct pre-order. The latter is a compact pospace, whence it coincides with the coproduct of X and Y in KH . Next, we describe certain pushouts in KH .
Consider regular monomorphisms f 0 : X ֒→ Y 0 , f 1 : X ֒→ Y 1 in KH , and their pushout in the category KH , as displayed in the following diagram.
As a space, P is homeomorphic to the quotient of the coproduct space Y 0 + Y 1 with respect to the topological closure of the equivalence relation generated by the set Let i ∈ {0, 1}, and write i * = 1 − i. With this notation, the pre-order on P is given by The relation Θ is clearly reflexive, and it is transitive because f 0 and f 1 are order-embeddings by item 1 in Since the downward closure ↓D of any closed subset D of a compact pospace is again closed [16,Proposition 4], we conclude that Θ ′′ is also closed. Whence, Θ is a closed pre-order. It is not difficult to see that it is the smallest pre-order on P making λ 0 and λ 1 monotone. Finally, the pushout of f 0 along f 1 in KH is obtained by applying the reflector ρ : KH → KH to P . Proposition 6. The following statements hold: (1) the regular monomorphisms in KH are the continuous order-embeddings; (2) the epimorphisms in KH are the continuous monotone surjections; (3) the unit interval [0, 1] is a regular injective regular cogenerator in KH .
Corollary 7. The category KH is dually equivalent to a quasi-variety of algebras.
Proof. By Theorem 2, it is enough to show that (i) KH op is regular with coequalisers for equivalence relations, and (ii) it admits a regular projective regular generator G.
We already observed that KH is complete and cocomplete. Whence, so is KH op . To show that KH op is regular, it suffices to prove that regular monos, i.e. continuous order-embeddings, are stable under pushouts in KH . Pushouts in KH can be computed by first taking the pushout in KH , and then composing with the reflection map. Reasoning as in Remark 5, it is not difficult to see that the pushout of a continuous order-embedding in KH is again a continuous order-embedding. Further, composing with the reflection yields again a continuous order-embedding, i.e. a regular mono in KH . This proves (i). In turn, (ii) follows at once from item 3 in Proposition 6, by setting G = [0, 1].
The latter fact was already observed in [11] where, in addition, the authors provide a description of an ℵ 1 -ary quasi-variety dually equivalent to KH [11,Theorem 3.15]. Our main contribution consists in a direct proof of the following result: A proof of the previous theorem is provided in Sections 3-4. We conclude this section by observing that Theorem 8 implies that KH op is equivalent to an ℵ 1 -ary variety of algebras, that is a variety of algebras in a language consisting of function symbols of, at most, countably infinite arity.
Corollary 9. The category KH is dually equivalent to an ℵ 1 -ary variety of algebras.
Proof. By Corollary 7 we know that KH is dually equivalent to a quasi-variety of algebras. Theorems 2 and 8 entail that KH is in fact dually equivalent to a variety of algebras. Indeed, KH op is equivalent to the category of Eilenberg-Moore algebras for the monad induced by the adjunction

Equivalence co-relations on compact ordered spaces
In this section we provide a description of equivalence relations in the category KH op , which will then be exploited in the next section to prove that equivalence relations in KH op are effective.
To start with, we dualise the notion of subobject. Given a compact pospace X, a quotient object of X is a subobject of X in the category KH op . The poset of quotient objects of X is denoted by Q(X). Explicitly, Q(X) is the poset of (equivalence classes of) epimorphisms with domain X, where Remark. We warn the reader that our terminology is non-standard. By a quotient object we do not mean a regular epimorphism, but what may be called a co-subobject (not every epimorphism in KH is regular).
By definition, an equivalence relation on X in the opposite category KH op is a subobject of X × X (where the product is computed in KH op ) which is reflexive, symmetric and transitive. This corresponds to a quotient object q0 q1 : X + X ։ S of the compact pospace X + X satisfying the dual properties:

co-transitivity
A quotient object of X + X which satisfies the three properties above will be called an equivalence co-relation on X.
The key observation is that equivalence co-relations are more manageable than their duals, because quotient objects of X are in bijection with certain pre-orders on X.
Indeed, if f : (X, X ) ։ (Y, Y ) is an epimorphism in KH , then The monotonicity of f entails X ⊆ f . Further, recalling that epimorphisms in KH are precisely the continuous monotone surjections (see item 2 in Proposition 6), we see that f is closed in X × X because it coincides with the preimage of Y under the continuous map f × f : X × X → Y × Y . Let us denote by P(X) the poset of all closed pre-orders on X which extend X , ordered by reverse inclusion. By the previous discussion, there is a map Q(X) → P(X) sending f to f . This function is well-defined, as f does not depend on the choice of a representative in the equivalence class of f . Conversely, given a pre-order in P(X), consider its symmetrization ∼ = ∩ op . The space X/∼, equipped with the quotient topology, is compact. The direct image of under the quotient map is a partial order on X/∼, and it is closed because so is . Moreover, since X ⊆ , we get an epimorphism X ։ X/∼ in KH . Taking its equivalence class, we obtain an element of Q(X). The following fact follows easily.
Lemma 10. For every compact pospace X, the assignments induce an isomorphism between the posets P(X) and Q(X).
Remark 11. Assume f 1 : X → Y 1 and f 2 : X → Y 2 are surjective morphisms in KH . By Lemma 10 there exists g : In fact, it is not difficult to see that this is true even if f 2 is not surjective, as we can factor it as a surjective map followed by an injective one.
Recall from Remark 5 that the compact pospace X + X is isomorphic to the disjoint union equipped with the coproduct topology and the coproduct order.
For the rest of this section, we fix a quotient object q0 q1 : X + X ։ S of a compact pospace X. We write ( q 0 q 1 ) , or simply S , for the associated pre-order on X + X. We say that S is co-reflexive (co-symmetric, co-transitive) if so is q0 q1 . To improve readability, we write [(x, i)] instead of q0 q1 (x, i). Lemma 12. The following statements hold.
(1) The pre-order S is co-reflexive if, and only if, (x, i) S (y, j) entails x y.
Proof. (1) By definition, S is co-reflexive if, and only if, q0 q1 : X + X ։ S is above 1X 1X : X + X ։ X in the poset Q(X + X). By Lemma 10, this is equivalent to S ⊆ ( 1 X 1 X ) . Given (x, i), (y, j) ∈ X + X, we have (x, i) ( 1 X 1 X ) (y, j) ⇐⇒ x y. It follows that the pre-order S is co-reflexive if, and only if, (x, i) S (y, j) entails x y.
(2) Again, by definition, S is co-symmetric if and only if q0 q1 : X + X ։ S is above q1 q0 : X + X ։ S in Q(X + X). By Lemma 10, this happens exactly when S ⊆ ( q 1 q 0 ) . Given (x, i), (y, j) ∈ X + X, (x, i) ( q 1 q 0 ) (y, j) ⇐⇒ (x, i * ) S (y, j * ). Therefore, the pre-order S is co-symmetric if, and only if, (x, i) S (y, j) entails (x, i * ) S (y, j * ). Lemma 13. Assume the pre-order S is co-reflexive. Then it is co-transitive if, and only if, Proof. Recall that S is co-transitive if, and only if, given a pushout square in KH as in the left-hand diagram below, there is t : S → P making the right-hand diagram commute. By Remark 11, such a t exists precisely when, for We conclude that S is co-reflexive if, and only if, equation (2) holds whenever (x, i) S (y, j). In turn, this is equivalent to the condition in the statement of the lemma.
From Lemmas 12 and 13, we obtain the following characterisation of equivalence co-relations in KH .

Proposition 14.
The pre-order S is an equivalence co-relation on X if, and only if, and

Proof of Theorem 8
Assume q0 q1 : X + X ։ S is an equivalence co-relation on X. Dualising Definition 1, we say that q0 q1 is co-effective provided it coincides with the co-kernel pair of its equaliser. That is, provided the following is a pushout square in KH , where k is the equaliser of q 0 , q 1 : X ⇒ S in KH . Also, we say that the pre-order S is co-effective if so is the corresponding quotient object. By item 1 in Proposition 6, the space Y can be identified with a closed subset of X, equipped with the induced order and topology. Define the relation Y on X + X as follows: (4) (x, i) Y (y, j) ⇐⇒ (i = j and x y) or (i * = j and ∃z ∈ Y s.t. x z y).
Lemma 15. Y is the pre-order associated with the pushout of the inclusion Y ֒→ X along itself.
Proof. This is an immediate consequence of Remark 5.
For the next proposition, recall that ∼ S = S ∩ op S is the symmetrization of the pre-order S . Proposition 16. The equivalence co-relation S is co-effective if, and only if, Proof. Recall that the equivalence co-relation S is co-effective if and only if the diagram in (3) is a pushout in KH . In turn, by Lemma 15, this is equivalent to saying that S = Y . Since Y = {x ∈ X | (x, i) ∼ S (x, i * )}, (x, i) Y (y, j) ⇐⇒ ∃z ∈ X [x z y and (z, i) ∼ S (z, i * )].
Therefore, to settle the statement, it suffices to show that the inclusion Y ⊆ S is always satisfied.
Note that any equivalence co-relation on X satisfies (x, i) (y, i) if, and only if, x y. The leftto-right implication follows from item 1 in Lemma 12, while the right-to-left implication holds because extends the coproduct order of X + X. Whence, (x, i) Y (y, i) if, and only if, (x, i) S (y, i). Suppose now (x, i) Y (y, i * ), and let z ∈ Y satisfy x z y. We have where the two inequalities hold because S extends the partial order of X + X. Therefore, Y ⊆ S .
We can finally prove Theorem 8, stating that every equivalence relation in KH op is effective.
Proof of Theorem 8. Let (X, ) be a compact pospace, and an equivalence co-relation on X. In view of Proposition 16 it is enough to show that, whenever (x, i) (y, i * ), there is z ∈ X such that x z y and (z, i) ∼ (z, i * ).
Fix arbitrary x, y ∈ X and i ∈ {0, 1} satisfying (x, i) (y, i * ), and set The idea is to apply Zorn's Lemma to show that Ω has a maximal element z satisfying the desired properties.
First, note that Ω is non-empty because is co-transitive, cf. Lemma 13. We claim that every non-empty chain C ⊆ Ω admits an upper bound in Ω. Every directed set in a compact pospace has a supremum, which coincides with the topological limit of the set regarded as a net [10, Proposition VI.1.3]. Thus, C has a supremum s in X, which belongs to the topological closure C of C.
Claim. Ω is a closed subset of X.
Proof. The set Ω can be written as the intersection of the sets Hence, it is enough to show that Ω 1 , Ω 2 are closed in X. We show that Ω 1 is closed. The proof for Ω 2 is the same, mutatis mutandis. The set Ω 1 is the preimage, under the coproduct injection ι i * : X → X + X, of Since is a closed pre-order on X + X, the set ↑(x, i) is closed in X + X [16, Proposition 1]. Therefore, its preimage Ω 1 is closed in X.
The previous claim entails that s ∈ C ⊆ Ω, i.e. C has a supremum in Ω. Hence, every non-empty chain in Ω admits an upper bound. By Zorn's Lemma, Ω has a maximal element z. By co-reflexivity of (see item 1 in Lemma 12), (x, i) (z, i * ) and (z, i) (y, i * ) imply x z y. It remains to show (z, i) ∼ (z, i * ).
We saw that, for every compact pospace X and closed subset Y ⊆ X, there is a pre-order Y on X + X given as in (4). In fact, by Lemma 15, Y is the equivalence co-relation on X associated with the pushout of the inclusion Y ֒→ X along itself. Conversely, every equivalence co-relation on X yields a closed subset of X, namely Proof. The two maps are clearly monotone. For any closed subset Y ⊆ X, Moreover, it follows at once from Theorem 8 and Proposition 16 that, for any equivalence co-relation on X, ⊆ Φ( ) . For the converse inclusion, see the proof of Proposition 16.

Epilogue: negative axiomatisability results
In the previous sections we have given a direct proof of the fact that the category KH of compact ordered spaces is dually equivalent to an ℵ 1 -ary variety of algebras. One may wonder whether it is necessary to resort to infinitary operations. In this section we show that KH op is not equivalent to any SP-class of finitary algebras (i.e., one closed under subalgebras and Cartesian products), let alone a finitary (quasi-)variety.
Henceforth, we assume all categories under consideration are locally small. Recall that an object A of a category C is (Gabriel-Ulmer ) finitely presentable if the covariant hom-functor hom C (A, −) : C → Set preserves directed colimits. See [8,Definition 6.1] or [3,Definition 1.1]. Further, C is finitely accessible provided it has directed colimits, and there exists a set S of its objects such that (i) each object of S is finitely presentable, and (ii) each object of C is a directed colimit of objects in S. See [ Definition 18. A Priestley space is a compact pospace (X, ) which is totally order-disconnected, i.e. for all x, y with x y there is a clopen C ⊆ X which is an up-set for , and satisfies x ∈ C but y / ∈ C. Proof. It suffices to show that every object in F is a Priestley space. We claim that every finitely copresentable object in F (i.e. one which is finitely presentable when regarded as an object of F op ) is finite.
Let (X, ) be an arbitrary finitely copresentable object in F. Consider an epimorphism γ : Y ։ X in KH with Y a Priestley space. (E.g., let Y = β|X| be theČech-Stone compactification of the underlying set of X equipped with the discrete topology, and γ : (β|X|, =) → (X, ) the unique continuous extension of the identity function |X| → |X|). By Lemma 19, Y is the codirected limit in KH of finite posets {Y i } i∈I with the discrete topologies. Denote by α i : Y → Y i the i-th limit arrow. Since Y lies in F, and the full embedding F → KH reflects limits, Y is in fact the codirected limit of {Y i } i∈I in F.
The object X being finitely copresentable in F, there are j ∈ I and a morphism ϕ : Y j → X such that γ = ϕ • α j . The map γ is surjective, hence so is ϕ. This shows that X is finite, and thus the claim is settled. Since F op is finitely accessible, every object of F is the codirected limit of finitely copresentable objects. Using again the fact that the full embedding F → KH reflects limits, we deduce from Lemma 19 that every object of F is a Priestley space, as was to be shown. Finally, we have already observed that finitary varieties and finitary quasi-varieties are finitely accessible categories.
Corollary 21. KH op is not equivalent to any SP-class of finitary algebras.
Proof. By Theorem 8, every equivalence relation in KH op is effective. In turn, Banaschewski observed in [4] that every SP-class of finitary algebras in which every equivalence relation is effective is a variety of algebras. The statement then follows from Theorem 20.

Remark 22.
In a recent work, Lieberman, Rosický and Vasey [14] proved that the opposite of the category KH of compact Hausdorff spaces is not equivalent to any elementary class of structures, with morphisms all the homomorphisms. In fact, they show that there exists no faithful functor KH op → Set which preserves directed colimits. Since directed colimits in elementary classes are concrete [19], the preceding statement follows. This implies that KH op is not equivalent to any elementary class of structures. Indeed, note that the embedding ∆ : KH op → KH op (cf. equation (1)) preserves directed colimits. Hence, if there were a faithful functor F : KH op → Set preserving directed colimits, the composition F • ∆ : KH op → Set would also be a faithful functor preserving directed colimits, contradicting the aforementioned result. This shows that KH op cannot be equivalent to an elementary class of structures.