The Exact Completion for Regular Categories enriched in Posets
aa r X i v : . [ m a t h . C T ] J a n AN EXACT COMPLETION FOR REGULAR CATEGORIESENRICHED IN POSETS
VASILEIOS ARAVANTINOS-SOTIROPOULOS
Abstract.
We construct an exact completion for regular categories enrichedin the cartesian closed category
Pos of partially ordered sets and monotonefunctions by employing a suitable calculus of relations. We then character-ize the embedding of any regular category into its completion and use thisto obtain examples of concrete categories which arise as such completions.In particular, we prove that the exact completion in this enriched sense ofboth the categories of Stone and Priestley spaces is the category of compactordered spaces of L. Nachbin. Finally, we consider the relationship betweenthe enriched exact completion and categories of internal posets in ordinarycategories. Introduction
The notions of regularity and (Barr-)exactness have been fundamental in Cat-egory Theory for quite some time. Exactness was introduced by Barr [2] in 1970and motivated by a result of Tierney which essentially exhibited the notion as thenon-additive part of the definition of abelian category. From another perspective,it is the basic property which is common to both abelian categories and elementarytoposes. Regularity is a weaker property which can be viewed as the requirementthat the category affords a good calculus of internal relations. Alternatively, fromthe perspective of Categorical Logic, regular categories are those which correspondto the fragment of first-order Logic on the operations ∧ , ⊤ , ∃ .In this paper we look at these notions in an enriched setting. More precisely,we work with versions of them that apply to categories enriched over the carte-sian closed category Pos of partially ordered sets and monotone functions. Ourmain motivation comes from the paper [12] by A. Kurz and J. Velebil, where
Pos -enriched regularity and exactness were first explicitly considered. The authorsemploy these notions to obtain categorical characterizations of (quasi-)varieties ofordered algebras in the sense of Bloom & Wright [3], very much along the linesof the corresponding characterizations for ordinary (quasi-)varieties of UniversalAlgebra. Broadly speaking, varieties turn out to be the exact categories possessinga “nice” generator, while quasivarieties can be characterized in a similar fashion byreplacing exactness with the weaker regularity.Recall here that ordered algebras in the sense of [3] are algebras over some sig-nature Σ which consist of a poset X together with a monotone map [ σ ] : X n → X ,for each specified n -ary operation σ . A homomorphism of such algebras is a mono-tone map which preserves the operations. Then a variety in this context is definedas a class of ordered algebras satisfying formal inequalities s ≤ t , where s, t areΣ-terms. A quasivariety is a class defined by more general formal implications V i ∈ I ( s i ≤ t i ) = ⇒ s ≤ t , where again the s i , t i , s, t are Σ-terms.The categories OrdGrp and
OrdMon of ordered semigroups and ordered monoidsrespectively are both examples of varieties which play an important role in thetheory of automata. More generally, any quasivariety of ordinary algebras gives rise to a quasivariety of ordered algebras defined by the same axioms. A differentexample of quasivariety is given by torsion-free monoids
OrdMon t . f . , i.e. the orderedmonoids ( M, + , ≤ ) satisfying the implication nx ≤ ny = ⇒ x ≤ y for all n ∈ N ∗ . Afurther source of examples is furnished by ordinary varieties whose axioms containthose of semi-lattices, since they can be equipped with the equationally definableorder x ≤ y ⇐⇒ x ∨ y = y . Yet more examples of quasivarieties are given by the Kleene algebras of Logic.While (quasi-)varieties of ordered algebras are a central source of examples of
Pos -enriched categories, there are other interesting examples that will appear inthe present paper. For one, we have the category S - Pos [6] of monotone actions ofan ordered monoid S on a poset and monotone equivariant maps between them.Furthermore, there are categories of ordered topological spaces, such as Priestleyspaces or the compact ordered spaces of Nachbin. These are all examples of cat-egories which are either themselves monadic over Pos or reflective in a monadiccategory.The thread of this paper can be seen as one continuation of the ideas developedin [12] and is in part suggested by the authors at the end of the latter paper. Ourmain contribution is a construction of the exact completion of a regular category for regular
Pos -categories which employs a suitably enriched version of the calculusof relations . We then identify varieties of ordered algebras which occur as suchcompletions of corresponding (quasi-)varieties of ordered or unordered algebras.Furthermore, we prove that the exact completion of the category of
Priestley spaces is precisely the category of
Nachbin spaces . This provides an ordered version of thefolklore result which identifies the category of compact Hausdorff spaces as theexact completion (in the ordinary sense) of the regular category of Stone spaces.In fact, it will follow by the same token that the exact completion of Stone spacesin the enriched sense is also the category of Nachbin spaces.1.1.
Organization of the paper.
In section 2 we collect some preliminaries in-volving regularity for categories enriched over
Pos , mostly for the convenience ofthe reader. There is only one original contribution here, Proposition 2.15, whichprovides a simplification of the definition of regularity that was presented in [12].More precisely, we prove that one of the defining conditions is a consequence of theother three and can thus be omitted.Section 3 discusses the main aspects of the calculus of relations which is availablein any regular category. The main result in this section is Theorem 3.7, whichidentifies the morphisms of a regular category as the left adjoints in a suitablebicategory of relations.Section 4 is represents the crux of the paper and is where we construct theexact completion of a regular category. After the initial definition, we prove ina sequence of steps that our proposed construction indeed satisfies the requiredproperties, culminating in Theorem 4.19. The arguments here make extensive useof the calculus of relations relying on the previous section.In section 5 we characterize the embedding of a regular category into its comple-tion. This is subsequently used to obtain examples of categories which arise as exactcompletions of one or more of their regular subcategories. In particular, we showthat the category of Nachbin spaces is exact and can obtained as the completion ofeither the category of Priestley or Stone spaces.Finally, in section 6 we examine the relationship between the process of exactcompletion and that of taking internal posets in an ordinary category. We provethat, in a suitable sense, these two commute.
N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 3 Preliminaries on Regularity
In this section we collect some preliminaries concerning the notion of regularityfor categories enriched over the cartesian closed category
Pos of posets and order-preserving functions, as defined by Kurz and Velebil in [12]. After recalling somebasic facts about finite limits, we reexamine the definition of regularity and observethat one of the conditions therein is in fact redundant.Throughout the paper by ‘a category C ’ we shall always mean a category thatis enriched over the cartesian closed category Pos of partially ordered sets andmonotone functions. Explicitly, this means that C is a category such that eachHom C ( X, Y ) is equipped with a partial order relation and such that compositionof morphisms is order-preserving in each variable. If we wish to refer to categoriesin the usual non-enriched sense we will always use the adjective ‘ordinary’.A functor F : C → D will always mean a
Pos -functor, i.e. an ordinary functorthat furthermore preserves the order of morphisms.Similarly, whenever we speak of limits or colimits in a category C , these willalways mean weighted (co-)limits (also called indexed (co-)limits in [9]). We knowfrom [9] that completeness of a category C , i.e. the existence of all small weightedlimits, is equivalent to the existence in C of all small conical limits and all powers.The former of these can in turn be constructed via products and equalizers, so that C is complete iff it possesses products, equalizers and powers. Recall here that thepower of an object X ∈ C to a poset P is an object X P ∈ C for which there existsa natural isomorphismHom C (C , X P ) ∼ = Hom Pos (P , Hom C (C , X))When the base of enrichment is locally finitely presentable as a monoidal category[11], as in our case with
Pos , there is also a useful notion of finite weighted limit. Inparticular, we have that C is finitely complete iff it has finite products, equalizersand finite powers. By finite power here we mean a power object X P where P is afinitely presentable object in Pos , i.e. a finite poset.We begin by recalling some basic notions, most of which can also be foundin [12]. First, the notion of monomorphism that is more appropriate in the orderedcontext and which will form part of the factorization system leading to the notionof regularity for
Pos -categories.
A morphism m : X → Y in a category C is called an ff -morphism (or representably fully faithful , or an order-monomorphism ) if for every Z ∈ C themonotone map C ( Z, m ) : C ( Z, X ) → C ( Z, Y ) in
Pos also reflects the order.We shall use the terms ‘ ff -morphism’ and ‘order-monomorphism’ interchangeablythroughout the paper. Furthermore, we will use the term ‘order-epimorphism’ forthe dual notion.Explicitly, m : X → Y is an ff -morphism when for every f, g : Z → X theimplication mf ≤ mg = ⇒ f ≤ g holds. In Pos , m is an ff -morphism preciselywhen it is an order-embedding, i.e. a map which preserves and reflects the order.Any such map is of course a monomorphism, but the converse is not true.The shift from monomorphisms to ff -morphisms is also essentially the differencebetween the notion of conical (weighted) limit in a category C and the ordinarylimit of the same type in the underlying ordinary category C . For example, con-sider two objects X, Y ∈ C . Then a diagram
X X × Y π X o o π Y / / Y is a productdiagram if the usual unique factorization property is satisfied, along with the fol-lowing additional condition: given any two morphisms u, v : Z → X × Y , the pairof inequalities π X u ≤ π X v and π Y u ≤ π Y v together imply that u ≤ v . In otherwords, the pair of projections π X , π Y must be jointly order -monic, rather than just VASILEIOS ARAVANTINOS-SOTIROPOULOS jointly monic. This stems from the fact that the universal property is a naturalisomorphism Hom(
Z, X × Y ) ∼ = Hom( Z, X ) × Hom(
Z, Y ) in
Pos , rather than in
Set .A similar observation applies to colimits in C .Let us record below a few basic properties of ff -morphisms familiar for monomor-phisms in an ordinary category. Consider morphisms f : X → Y and g : Y → Z in a category C .Then: (1) If f, g are ff -morphisms, then so is gf . (2) If gf is an ff -morphism, then so is f . (3) ff -morphisms are stable under pullback.Proof. Perhaps only item (3) needs some details, so consider the following pullbacksquare in C where f is an ff -morphism and assume that u, v : A → P are such that qu ≤ qv . P p / / q (cid:15) (cid:15) X f (cid:15) (cid:15) Y g / / Z We then have gqu ≤ gqv = ⇒ f pu ≤ f pv = ⇒ pu ≤ pv , since f is an ff -morphism.Then, because we have both pu ≤ pv and qu ≤ qv , we conclude by the limitproperty of the pullback that u ≤ v . (cid:3) We recall next two particular examples of weighted limits which are not conicaland which play an important role in the context of
Pos -categories. See also [12].The comma object of an ordered pair of morphisms ( f, g ) with common codomainis a square
C YX Z c c ≤ gf such that f c ≤ gc and which is universal with this property, the latter meaningprecisely the following two properties:(1) Given u : W → X and u : W → Y in C such that f u ≤ gu , there existsa u : W → C ∈ C such that c u = u and c u = u .(2) The pair ( c , c ) is jointly order-monic.Note in particular that the factorization given in (1) will be unique by (2). We willusually denote the comma object C by f /g . In Pos , the comma object is given by f /g = { ( x, y ) ∈ X × Y | f ( x ) ≤ g ( y ) } with the order induced from the product.The inserter of an ordered pair ( f, g ) of parallel morphisms X Y fg is amorphism e : E → X ∈ C such that f e ≤ ge and universal in the following sense:(1) If h : Z → X ∈ C is such that f h ≤ gh , then there exists a u : Z → A suchthat eu = h .(2) e is an ff -morphism.Again, note that the factorization posited in (1) is unique by property (2). In Pos ,the inserter is precisely E = { x ∈ X | f ( x ) ≤ g ( x ) } with the order induced from X .It will be convenient for us to have an alternative way of constructing all finiteweighted limits using inserters along with some conical limits. This is then thecontent of the following proposition, which should be well-known and in any case N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 5 follows from more general facts about 2-categorical limits. We nevertheless includea proof for the sake of completeness.
A category C is finitely complete iff it has finite products andinserters.Proof. Suppose that C has finite products and inserters. To have all finite conicallimits it suffices to construct equalizers. So consider any parallel pair of morphisms X f / / g / / Y in C . Let m : M → X be the inserter of the pair ( f, g ) and then let n : E → M be the inserter of ( gm, f m ). We claim that now e := mn : E → X isthe desired equalizer. Indeed, note first that gmn ≤ f mn and also f m ≤ gm = ⇒ f mn ≤ gmn , so that f mn = gmn . Then suppose that h : Z → X is such that f h = gh . Since in particular f h ≤ gh , there exists a unique u : Z → M suchthat mu = h . Now u is such that gmu = gh ≤ f h = f mu , so we have a unique v : Z → E such that nv = u . So ev = mnv = mu = h . Finally, it is clear that e isorder-monic, since both m, n are so.Second, we need to construct finite powers, so let P be a finite poset and con-sider any X ∈ C . Consider the product Q a ∈ P X (i.e. the ordinary power) andfor every pair of elements a, b ∈ P with a ≤ b form the inserter of ( π a , π b ), say E ab / / e ab / / Q a ∈ P X π a / / π b / / X in C . Set E := Q a ≤ b E ab . We claim the E is the power X P .To show this, consider any family of morphisms ( f a : C → X ) a ∈ P with a ≤ b = ⇒ f a ≤ f b , i.e. a homomorphism P → Hom C ( C, X ) in
Pos . There is thena unique f : C → Q a ∈ P X such that π a f = f a for all a ∈ P . Now for each pair ofelements a, b ∈ P with a ≤ b we have f a ≤ f b , which is to say π a f ≤ π b f . Hence,there is a unique u ab : C → E ab with e ab u ab = f . This in turn induces a unique u : C → E such that π ab u = u ab whenever a, b ∈ P with a ≤ b .Finally, let’s show that this assignment is order-preserving and order reflecting.So consider another family ( g a : C → X ) a ∈ P , with g : C → Q a ∈ P X and v : C → E corresponding to f and u as defined above for ( f a ) a ∈ P . If ( f a ) a ∈ P ≤ ( g a ) a ∈ P , then f a ≤ g a for all a ∈ P , i.e. π a f ≤ π a g for all a ∈ P and hence f ≤ g by theuniversal property of the product. This in turn means that whenever a ≤ b wehave e ab u ab = f ≤ g = e ab v ab and so u ab ≤ v ab , hence u ≤ v . It is clear that theseimplications can also be reversed. (cid:3) If a category C has comma objects, then in particular for any morphism f : X → Y ∈ C we can form the comma of the pair ( f, f ). This comma measures the extentto which f fails to be an ff -morphism and ultimately connects with the notions ofregularity and exactness to be introduced shortly. Given any morphism f : X → Y in a category C , the commaobject f /f is called the kernel congruence of f . For any morphism f : X → Y with kernel congruence f /f f / / f / / X in a category C , the following are equivalent: (1) f is an ff -morphism. (2) f ≤ f . (3) The canonical morphism ι f : 1 X / X → f /f is an iso. VASILEIOS ARAVANTINOS-SOTIROPOULOS
Proof. If f is an ff -morphism, then f f ≤ f f = ⇒ f ≤ f .If f ≤ f , then 1 X f ≤ X f implies that ( f , f ) must factor through 1 X / X via a morphism which is then easily seen to be inverse to 1 X / X → f /f .Finally, assume that f /f ∼ = 1 X / X and let u , u : Z → X be such that f u ≤ f u . Then ( u , u ) factors through f /f and so through 1 X / X . But the lattermeans precisely that u ≤ u . (cid:3) As we have mentioned already, the class of ff -morphisms will be the ‘mono part’of a factorization system for regular categories. The other class of morphisms istaken to be the class of morphisms orthogonal to all ff -morphisms, as has to bethe case in any orthogonal factorization system. Let us first recall the definition oforthogonality in this enriched context. Given morphisms e : A → B and m : X → Y in a category C , wesay that e is left orthogonal to m and write e ⊥ m if the square C ( B, X ) −◦ e / / m ◦− (cid:15) (cid:15) C ( A, X ) m ◦− (cid:15) (cid:15) C ( B, Y ) −◦ e / / C ( A, Y )is a pullback in
Pos .To make things more explicit, the statement e ⊥ m means two things:(1) The usual diagonal fill-in property.(2) Given two commutative squares A e / / u (cid:15) (cid:15) B v (cid:15) (cid:15) d ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ X m / / Y A e / / u (cid:15) (cid:15) B v (cid:15) (cid:15) d ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ X m / / Y in C with u ≤ u and v ≤ v , the diagonal fill-ins must also satisfy d ≤ d .Then, as in [12] we introduce the following class of morphisms, which in somesense are the Pos -enriched analogue of strong epimorphisms for ordinary categories.
A morphism e : A → B is called and so -morphism (or surjectiveon objects ) if e ⊥ m for every ff -morphism m : X → Y ∈ C . We note here that, in the special case of checking a condition e ⊥ m with m an ff -morphism, the 2-dimensional part of the definition of orthogonality(property 2 above) follows for free. Indeed, in the notation of the above diagrams,the fact that md = v ≤ v = md then implies d ≤ d .The so -morphisms will be the ‘epi part’ of the factorization system on regularcategories. Indeed, given the existence of some limits, every so -morphism is anorder-epimorphism. If C has inserters, then every so -morphism in C is an order-epimorphism.Proof. Let e : A → B be an so -morphism and consider f, g : B → C such that f e ≤ ge . Let m : M → B be the inserter of ( f, g ). Then there exists a unique u : A → M such that mu = e . But we have e ⊥ m and so we obtain a v : B → M such that ve = u , mv = 1 B . Now m is both a split epi and a mono, hence an isoand so f m ≤ gm = ⇒ f ≤ g . (cid:3) The notion of regularity for ordinary categories, as is well known, can be definedeither in terms of strong epimorphisms or in terms of regular epimorphisms. Infact, one can argue that a significant part of the power of regularity is that itforces these two classes of epimorphisms to coincide. From another perspective,
N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 7 the regular epimorphisms are the categorical notion of quotient that is usuallyappropriate in ordinary categories. For
Pos -categories the corresponding notion isthat of coinserter , which is dual to the notion of inserter defined earlier. Intuitively,whereas taking a coequalizer corresponds to adding new equalities, constructing acoinserter should be thought of as adding new inequalities .Now, just as every regular epimorphism is always strong, so too we have thefollowing.
Every coinserter is an so -morphism.Proof. Consider the following commutative diagram, where e is assumed to be thecoinserter of ( f , f ) and m is an ff -morphism. C f / / f / / A e / / u (cid:15) (cid:15) B v (cid:15) (cid:15) X m / / Y Now we have muf = vef ≤ vef = muf , so by virtue of m being an ff -morphismwe get uf ≤ uf . Then by the coinserter property there exists a unique d : B → X such that de = u . It follows also that mu = v because e is an order-epi. (cid:3) A morphism q : X → Y is called an effective (epi-)morphism ifit is the coinserter of some pair of parallel morphisms.The notion of regularity of a category is essentially an exactness property relatingkernel congruences and coinserters. There are however some exactness propertiesinvolving these notions that always hold, regardless of regularity. This is the contentof the following proposition, which again should be compared to the correspondingfacts involving kernel pairs and coequalizers in an ordinary category (see [4]). (1) If an effective epimorphism has a kernel congruence,then it must be the coinserter of that kernel congruence. (2)
If a kernel congruence has a coinserter, then it is also the kernel congruenceof that coinserter.Proof. (1) Suppose X f / / f / / Y q / / Q is a coinserter diagram and assume q has a kernel congruence q/q q / / q / / Y in C . Then qf ≤ qf = ⇒ ( ∃ ! u : X → q/q ) q u = f , q u = f , by the universal property of kernel congru-ence.Now let g : Y → Z be such that gq ≤ gq . Then gq u ≤ gq u = ⇒ gf ≤ gf and so ( ∃ ! v : Q → Z ) vq = g . Finally, q is already an order-epiby virtue of being a coinserter of some pair of morphisms.(2) Suppose that R r / / r / / X q / / Q is a coinserter diagram and that ( r , r )is the kernel congruence of some f : X → Y . Then f r ≤ f r implies theexistence of a unique u : Q → Y such that uq = f .Now let a, b : A → X be such that qa ≤ qb . Then also uqa ≤ uqb , i.e. f a ≤ f b and so ( ∃ ! v : A → R ) r v = a, r v = b . (cid:3) VASILEIOS ARAVANTINOS-SOTIROPOULOS
We now come to the definition of regularity for
Pos -enriched categories, as pre-sented by Kurz and Velebil in [12]. We label it as ‘provisional’ in the context ofthis paper for reasons that will be justified shortly. (provisional) A category C is regular if it satisfies the following:(R1) C has all finite (weighted) limits.(R2) C has ( so , ff )-factorizations.(R3) so -morphisms are stable under pullback in C .(R4) Every so -morphism is effective in C .A main feature of this definition is that it posits the existence of a stable ( so , ff )factorization system in C . The authors go on to state that the ‘gist of the definition’is property (R4), i.e. the assumption that so -morphisms and effective morphismscoincide, which essentially states that it is equivalent to require the stable factoriza-tion system to be (effective, ff ). However, we shall show that condition (R4) in factfollows from the first three conditions, much like in the case of ordinary regularityone can state the definition equivalently either in terms of regular epimorphismsor strong epimorphisms. In fact, the proof is essentially a direct adaptation of thecorresponding one in the ordinary context (see [4]).Before making good on our claim, we need a preparatory result on the pastingof a pullback square with a comma square. This is well-known from the realmof 2-categories, but we include its proof for the sake of making this paper moreself-contained. Consider the following diagram in a category C , where the right-hand square is a comma square and the left-hand square commutes. P p / / p (cid:15) (cid:15) Q q / / q (cid:15) (cid:15) ≤ Z g (cid:15) (cid:15) X ′ x / / X f / / Y Then the outer rectangle is a comma square iff the left-hand square is a pullback.Proof.
Assume first that the left-hand square is a pullback. Note first that by ourassumptions on the diagram we have f xp = f q p ≤ gq p . Now suppose that u : A → X ′ and v : A → Z are such that f xu ≤ gv . Since the right-hand square isa comma, ( ∃ ! w : A → Q ) q w = xu, q w = v . The first of these equalities by virtueof the pullback property gives that ( ∃ ! z : A → P ) p z = u, p z = w . Then we have q p w = q w = v as well.Finally, assume that z, z ′ : A → P are such that p z ≤ p z ′ and q p z ≤ q p z ′ .Then we also have xp z ≤ xp z ′ = ⇒ q p z ≤ q p z ′ , so that by the universalproperty of the comma square we get p z ≤ p z ′ . The latter inequality togetherwith p z ≤ p z ′ yield z ≤ z ′ by the universal property of the pullback.Conversely, assume that the outer square is a comma and let u : A → X ′ and v : A → Q be such that xu = q v . Then we have f xu = f q v ≤ gq v , so the outerrectangle being a comma says that ( ∃ ! w : A → P ) p w = u, q p w = q v . But sincealso q p w = xp w = xu = q v and q , q are jointly monic, we obtain also that p w = v . Finally, it is clear that p , p are jointly order-monic because p , q p areso. (cid:3) Now we can prove that condition (R4) in the definition of regularity is superflu-ous. The proof that follows is almost identical to that of Proposition 2.2.2 in [4],concerning ordinary regularity, where we replace some uses of the familiar lemmaon pasting of pullback squares with Lemma 2.14.
N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 9 If C is a category satisfying conditions (R1),(R2) and (R3) ofDefinition 2.13, then it also satisfies (R4).Proof. Let f : X → Y be an so -morphism and consider its kernel congruence f /f k / / k / / X , which exists in C by (R1). We want to show that f is the coinserterof ( k , k ).So let g : X → Z ∈ C be such that gk ≤ gk . We can consider then theinduced morphism h f, g i : X → Y × Z . By (R2) we can factor this morphism as an so -morphism followed by an ff -morphism, say X p / / / / I / / i / / Y × Z . We thenform the following diagram in C , where we begin by forming the bottom right-handsquare as a comma square and then the remaining three squares are pullbacks. P v / / v (cid:15) (cid:15) P x / / u (cid:15) (cid:15) X p (cid:15) (cid:15) (cid:15) (cid:15) P u / / x (cid:15) (cid:15) C c / / c (cid:15) (cid:15) ≤ I π Y i (cid:15) (cid:15) X p / / / / I π Y i / / Y By an application of Lemma 2.14 and its order-dual as well as the usual pullbackgluing lemma we deduce that the big outer square resulting from the pasting of allfour smaller ones is also a comma square. Then, since π Y ip = f , we have P ∼ = f /f and we can assume that x v = k and x v = k . Also, observe that by (R3) wehave that u , u and then also v , v are so -morphisms.Now we want to show that π Y i is an iso. Since ( π Y i ) p = f is an so -morphism,we already know that π Y i is an so -morphism as well. Thus, it suffices to show thatit is also an ff -morphism, which is equivalent to showing c ≤ c . Since u v = u v is an so -morphism, so in particular an order-epi, the latter inequality is equivalentto having c u v ≤ c u v . This is in turn equivalent to ic u v ≤ ic u v , because i is an ff -morphism. To prove this last inequality we now observe the following: π Y ic u v = π Y ipx v = f x v = f k ≤ f k = f x v = π Y ipx v = π Y ic u v π Z ic u v = π Z ipx v = gx v = gk ≤ gk = gx v = π Z ipx v = π Z ic u v Then the universal property of the product yields the desired inequality.Finally, we now have a morphism π Z i ( π Y i ) − : Y → Z such that π Z i ( π Y i ) − f = π Z i ( π Y i ) − π Y ip = π Z ip = g . Furthermore, we know already that f is an order-epibecause it is an so -morphism by assumption. (cid:3) Thus, we can officially strike condition (R4) from the definition of regularity andhenceforth adopt the following more economical one.
A category C will be called regular if it satisfies the following:(R1) C has all finite (weighted) limits.(R2) C has ( so , ff )-factorizations.(R3) so -morphisms are stable under pullback in C .Similarly, we can now furthermore establish another equivalent characterizationof regularity in terms of the existence of quotients for kernel congruences. A finitely complete category C is regular iff the following hold: (1) Every kernel congruence in C has a coinserter. (2) Effective morphisms are stable under pullback in C .0 VASILEIOS ARAVANTINOS-SOTIROPOULOS
A category C will be called regular if it satisfies the following:(R1) C has all finite (weighted) limits.(R2) C has ( so , ff )-factorizations.(R3) so -morphisms are stable under pullback in C .Similarly, we can now furthermore establish another equivalent characterizationof regularity in terms of the existence of quotients for kernel congruences. A finitely complete category C is regular iff the following hold: (1) Every kernel congruence in C has a coinserter. (2) Effective morphisms are stable under pullback in C .0 VASILEIOS ARAVANTINOS-SOTIROPOULOS Proof. If C is regular, then it is easy to see that it satisfies the two conditions aboveby definition and be an appeal to part (2.) of Proposition 2.12.Conversely, let us assume that C satisfies the two conditions in the statement.Consider any f : X → Y ∈ C , its kernel congruence f /f f / / f / / X and the coin-serter q : X → Q in C of the latter, which exists by condition 1. Since f f ≤ f f ,there exists a unique m : Q → Y such that f = mq . It now suffices to show that m is an ff -morphism.For this, consider the kernel congruence m/m m / / m / / Q and form the followingdiagram where the bottom right-hand square is a comma and the other three arepullbacks. P v / / v (cid:15) (cid:15) P x / / u (cid:15) (cid:15) X q (cid:15) (cid:15) (cid:15) (cid:15) P u / / x (cid:15) (cid:15) m/m m / / m (cid:15) (cid:15) ≤ Q m (cid:15) (cid:15) X q / / / / Q m / / Y Similarly to the proof of Proposition 2.15, we have P ∼ = f /f and we can assumethat x v = f and x v = f . Furthermore, by the assumed stability of effectivemorphisms under pullback, we can deduce that u v = u v is an order-epi. Nowwe have that m u v = qx v = qf ≤ qf = qx v = m u v whence we deduce that m ≤ m and so that m is an ff -morphism. (cid:3) To end this section, let us list a few examples of regular categories. For moredetails on most of these one can consult [12]. (1)
Pos is regular as a
Pos -category. So is any enriched presheafcategory [ C op , Pos ] for C a small category.(2) Any ordinary regular category C is also regular in the Pos -enriched sensewhen equipped with the discrete order on its Hom-sets. Indeed, in thiscase ff -morphisms coincide with monomorphisms and so -morphisms withstrong epimorphisms. Note that Pos is an example of a category which isnot regular in the ordinary sense, but is regular as an enriched category.(3)
Quasivarieties of ordered algebras in the sense of Bloom and Wright [3] areregular categories [12]. As particular examples here we have the categories
OrdMon of ordered monoids,
OrdSGrp of ordered semi-groups,
OrdCMon ofcommutative ordered monoids and
OrdMon of ordered monoids with theneutral element 0 of the monoid operation as the minimum element for theorder. These are all in fact varieties. An example of a quasivariety whichis not a variety is the category OrdCMon t . f . of torsion-free commutativemonoids, i.e. those ordered commutative monoids ( M, + , ≤ ) satisfying theimplication nx ≤ ny = ⇒ x ≤ y for all x, y ∈ M and n ∈ N .(4) The categories Nach of Nachbin spaces (or compact ordered spaces) and
Pries of Priestley spaces with continuous order-preserving functions in bothinstances are examples of regular categories. We shall have more to say onthese in section 5.
N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 11 (5) If C is monadic over Pos for a monad T : Pos → Pos which preserves so -morphisms (i.e. surjections), then C is regular. An example of this kind isgiven by the category S - Pos of S -posets for any ordered monoid S (see [6]).The objects in the latter category are monoid actions S × X → X on aposet X which are monotone in both variables, while the morphisms arethe monotone equivariant functions.3. Calculus of Relations and Exactness
Recall that in an ordinary regular category C , the existence of a stable (regularepi,mono) factorization system allows for a well-behaved calculus of relations in C .More precisely, the existence of the factorization system allows one to define thecomposition of two internal relations and then the stability of regular epimorphismsunder pullback is precisely equivalent to the associativity of this composition. Es-sentially the same facts hold also in our Pos -enriched setting.If E is a regular category, then by a relation in E we shall mean an order-subobject R X × Y , i.e. a subobject of a product represented by an ff-morphism. We shallwrite R : X Y to denote that R is a relation from X to Y in E . The factorizationsystem ( so , ff ) in E and the stability of so-morphisms under pullback allow us tohave a well-defined composition of relations in E : given R : X Y and S : Y Z in C , the relation S ◦ R : X Z is defined by first constructing the pullback squarebelow T t (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ t (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ R r ~ ~ ⑦⑦⑦⑦⑦⑦⑦ r (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ S s (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ s (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ X Y Z and then taking the (so,ff) factorization of h r t , s t i : T → X × Z , say T q / / / / M / / h m ,m i / / X × Z In other words, in our notation S ◦ R is the relation represented by the ff-morphism h m , m i .This leads to the locally posetal bicategory (a.k.a Pos -category) Rel( E ), whoseobjects are those of E and whose morphisms are the relations R : X Y in E . The identity morphism on the object X in Rel( E ) is the diagonal relation h X , X i : ∆ X X × X and composition of morphisms is given by composition ofrelations in E .If we forget about the 2-dimensional nature of the properties that define thetwo classes of morphisms in the factorization system ( so , ff ), then the structureand basic properties of Rel( E ) are essentially the calculus of relations relative toa stable factorization system , as explicated by Meisen [15], Richter [18], Kelly [10]and others. Thus, we shall feel free to take for granted many of the basic factsconcerning the structure of Rel( E ) without proving them here. As an exceptionto this rule, we include the proof of the following lemma because it describes away in which one can argue about relations in a regular category using generalizedelements which will be particularly useful to us in subsequent proofs. Recall herethat, given a relation R : X Y and generalized elements x : A → X , y : A → Y in E , we write ( x, y ) ∈ A R to indicate that h x, y i : A → X × Y factors through h r , r i . Let R : X Y and S : Y Z be relations in a regular category E and consider any generalized elements x : P → X and z : P → Z . Then ( x, z ) ∈ P S ◦ R iff there exists an effective epimorphism q : Q ։ P and a generalizedelement y : Q → Y such that ( xq, y ) ∈ Q R and ( y, zq ) ∈ Q S .Proof. Consider the diagram below, where the square is a pullback. T t (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ t (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ R r ~ ~ ⑦⑦⑦⑦⑦⑦⑦ r (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ S s (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ s (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ X Y Z
Then S ◦ R is given by the image factorization h r t , s t i = T e / / / / I / / h i ,i i / / X × Z .Assume first that ( x, z ) ∈ P S ◦ R , i.e. there exists a morphism u : P → I suchthat h i , i i u = h x, z i . We then form the pullback square below. Q v (cid:15) (cid:15) q / / / / P u (cid:15) (cid:15) T e / / / / I Note that q is an effective epi because e is such. Now set y := r t v = s t v : Q → Y . We then have h xq, y i = h i uq, r t v i = h i ev, r t v i = h r t v, r t v i = h r , r i t v and h y, zq i = h s t v, i uq i = h s t v, i ev i = h s t v, s t v i = h s , s i t v ,so that ( xq, y ) ∈ Q R and ( y, zq ) ∈ Q S .Conversely, assume that ( xq, y ) ∈ Q R and ( y, zq ) ∈ Q S for some y : Q → Y and effective epi q : Q ։ P . This means that there exist morphisms u : Q → R and v : Q → S such that h r , r i u = h xq, y i and h s , s i v = h y, zq i . Since r u = y = s v , there exists a unique w : Q → T such that t w = u and t w = v . Then h i , i i ew = h r t , s t i w = h r u, s v i = h xq, zq i , so that ( xq, zq ) ∈ Q SR . Since q is an effective epi, we can conclude that also ( x, z ) ∈ P SR . (cid:3) The above lemma can actually be used to prove many of the fundamental prop-erties of Rel( E ). In general, Rel( E ) is a tabular allegory with a unit (see e.g. [8], [7]),where the anti-involution ( − ) ◦ : Rel( E ) op → Rel( E ) is given by taking the opposite relation. In particular, we have that Freyd’s Modular Law holds in Rel( E ), i.e. QP ∩ S ⊆ Q ( P ∩ Q ◦ S )for any relations P : X Y , Q : Y Z and S : X Z in E .Every morphism f : X → Y ∈ E defines a relation X Y represented by theff-morphism h X , f i : X → X × Y , which we call its graph and denote by the sameletter. This assignment defines a faithful ordinary functor E → Rel( E ) on theunderlying ordinary category of E which is the identity on objects. Furthermore,in Rel( E ) we have an adjunction f ⊣ f ◦ , which means that the inclusions f ◦ f ⊇ ∆ X and f f ◦ ⊆ ∆ Y hold. We say then that the morphisms of E are maps inthe bicategory Rel( E ). We should perhaps stress here that taking the graph of amorphism does not define a functor E →
Rel( E ) because the order of morphismsis not preserved. In fact, since Rel( E ) is an allegory, the modular law forces anyinclusion f ⊆ g for morphisms f, g : X → Y to be an equality (see e.g. Lemma3.2.3 in [8]). N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 13
While Rel( E ) is a very useful category in terms of performing calculations withrelations in E , it does not really capture the enriched nature of E . As we havementioned earlier, Rel( E ) in some sense only involves an ordinary category with astable factorization system. In order to also capture the Pos -enriched aspects of aregular category E we will also need to work in a different bicategory of relations.In terms of our goals in this paper, this need is related to our desire to identify themorphisms of E as the maps in a certain bicategory of relations, thus generalizing afamiliar fact for ordinary regular categories. Indeed, while certainly any morphism f : X → Y ∈ E has as its right adjoint in Rel( E ) the opposite relation f ◦ , this isnot a complete characterization of (graphs of) morphisms. As can be deduced byarguments essentially contained in [10], being a map in Rel( E ) is a weaker propertythan being the graph of a morphism in E .Another reason for moving to a different bicategory of relations is dictated bythe form of congruences in this enriched setting and the way in which exactness isdefined. This will become apparent a little bit later. A relation R : X Y in the regular category E is called weakening or weakening-closed if, whenever x, x ′ : A → X and y, y ′ : A → Y are generalizedelements in E with x ′ ≤ x and y ≤ y ′ , the following implication holds x ′ ≤ x ∧ ( x, y ) ∈ A R ∧ y ≤ y ′ = ⇒ ( x ′ , y ′ ) ∈ A R For any given object X ∈ E , there is a weakening-closed relation I X : X X given by the comma I X := 1 X / X . Then it is easy to see that a relation R : X Y is weakening-closed iff we have R = I Y RI X in Rel( E ). In particular, the relations I X act as identity elements for composition of weakening-closed relations. Thus,we can define a bicategory Rel w ( E ) where the objects are again those of E but nowthe morphisms are the weakening-closed relations. Perhaps we should note here that, although every morphism ofRel w ( E ) is also a morphism in Rel( E ) and composition in both categories is thesame, this is not a functorial inclusion Rel w ( E ) ֒ → Rel( E ) because identity mor-phisms are not preserved.Now to any given morphism f : X → Y ∈ E we can canonically associate twoweakening-closed relations f ∗ : X Y and f ∗ : Y X via the following commas: f ∗ := f / Y and f ∗ := 1 Y /f . We sometimes call f ∗ and f ∗ the hypergraph and hypograph of f respectively. In terms of generalized elements x : A → X and y : A → Y in E we have ( x, y ) ∈ A f ∗ ⇐⇒ f x ≤ y and ( y, x ) ∈ A f ∗ ⇐⇒ y ≤ f x .The following are then easy to see, for example by arguing with generalizedelements. Let E be a regular category. Then for any f : X → Y and g : Y → Z in E we have (1) ( gf ) ∗ = g ∗ f ∗ . (2) ( gf ) ∗ = f ∗ g ∗ . It is also easy to see that, for any f, g : X → Y ∈ E , we have f ≤ g ⇐⇒ g ∗ ⊆ f ∗ ⇐⇒ f ∗ ⊆ g ∗ We thus have two fully order-faithful functors ( − ) ∗ : E co ֒ → Rel w ( E ) and ( − ) ∗ : E op ֒ → Rel w ( E ), where E co denotes the “order-dual” category, i.e. the categoryobtained from E by reversing the order on morphisms. Similarly, arguing withgeneralized elements we easily deduce the following. For any f : X → Y in a regular category E we have f ∗ f ∗ = f /f asrelations in E . In particular, we see that f : X → Y is an ff-morphism in E iff f ∗ f ∗ = I X inRel w ( E ). Similarly, a pair of morphisms Y X Z f g is jointly ff preciselywhen f ∗ f ∗ ∩ g ∗ g ∗ = I X .Now just as the graph of every morphism f : X → Y induces an adjunction f ⊣ f ◦ in Rel( E ), so do the hypergraph and hypograph of that morphism form anadjunction in the bicategory Rel w ( E ). For any f : X → Y in a regular category E , there is an adjunction f ∗ ⊣ f ∗ in Rel w ( E ) .Proof. We saw above that f ∗ f ∗ is precisely the kernel congruence of f , so clearlywe have I X ⊆ f ∗ f ∗ . To form the composition f ∗ f ∗ we consider the diagram below,where the top square is a pullback and the other two are commas. Q q (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ q (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ f ∗ s ~ ~ ⑦⑦⑦⑦⑦⑦⑦ s ❆❆❆❆❆❆❆ f ∗ r ~ ~ ⑦⑦⑦⑦⑦⑦⑦ r (cid:31) (cid:31) ❅❅❅❅❅❅❅ Y ❆❆❆❆❆❆❆❆ ❆❆❆❆❆❆❆❆ ≤ X f ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ f ❆❆❆❆❆❆❆❆ ≤ Y ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ Y Y
Now by definition we have that f ∗ f ∗ is given by the image of h s q , r q i . But s q ≤ f s q = f r q ≤ r q , so that h s q , r q i factors through I Y = 1 Y / Y .This yields the inclusion f ∗ f ∗ ⊆ I Y . (cid:3) In other words, every morphism f : X → Y in a regular category E is a mapin Rel w ( E ) via its hypergraph f ∗ . Our first result in this paper is that in anyregular category E this is now indeed a complete characterization of morphisms,i.e. every map in Rel w ( E ) is of the form f ∗ for a (necessarily unique) morphism f : X → Y ∈ E . If φ : X Y ∈ Rel w ( E ) has a right adjoint in Rel w ( E ) , then thereexists a morphism f : X → Y ∈ E such that φ = f ∗ .Proof. Let ψ : Y X ∈ Rel w ( E ) be the right adjoint, so that we have I X ⊆ ψφ and φψ ⊆ I Y . Suppose also that φ and ψ are represented respectively by the ff-morphisms h φ , φ i : T X × Y and h ψ , ψ i : T ′ Y × X . We next form thepullback square below, so that h φ u, φ u i = h ψ u ′ , ψ u ′ i : S X × Y represents therelation φ ∩ ψ ◦ : X Y ∈ Rel( E ). We first want to show that φ u = ψ u ′ is an iso,in which case we will have φ ∩ ψ ◦ = f for the morphism f := φ u ( φ u ) − : X → Y . S / / u ′ / / (cid:15) (cid:15) u (cid:15) (cid:15) T ′ (cid:15) (cid:15) h ψ ,ψ i (cid:15) (cid:15) T / / h φ ,φ i / / X × Y First of all, since I X ⊆ ψφ , we have (1 X , X ) ∈ X ψφ and hence there exist aneffective epi e : P ։ X and a y : P → Y such that ( e, y ) ∈ P φ and ( y, e ) ∈ P ψ . Then we have ( e, y ) ∈ P φ ∩ ψ ◦ and so there exists a v : P → S such that h φ u, φ u i v = h e, y i . In particular, since φ uv = e is an effective epi, we deducethat φ u is an effective epi as well. N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 15
Now it suffices to show that φ u is also an ff-morphism. So let a, b : A → S besuch that φ ua ≤ φ ub . Then ( φ ua, φ ub ) ∈ A I X = ⇒ ( φ ua, φ ub ) ∈ A ψφ , sothere is an effective epi e : P ։ A and a z : P → Y such that ( φ uae, z ) ∈ P φ and( z, φ ube ) ∈ P ψ . Since we also clearly have( φ uae, φ uae ) = ( ψ u ′ ae, ψ u ′ ae ) ∈ ψ, we get ( φ uae, z ) ∈ P φψ = ⇒ ( φ uae, z ) ∈ P I Y = ⇒ φ uae ≤ z .Similarly, since ( φ ube, φ ube ) ∈ P φ and ( z, φ ube ) ∈ ψ , we have ( z, φ ube ) ∈ P φψ = ⇒ ( z, φ ube ) ∈ P I Y = ⇒ z ≤ φ ube . Thus, φ uae ≤ z ≤ φ ube = ⇒ φ uae ≤ φ ube = ⇒ φ ua ≤ φ ub , the latter implication because e is an effectiveepimorphism. Since also φ ua ≤ φ ub , we obtain a ≤ b because h φ u, φ u i is anff-morphism.Finally, we now claim that φ = f ∗ . To see this, observe first that f ∗ = I Y f = I Y ( φ ∩ ψ ◦ ) ⊆ I Y φ = φ , where the last equality follows because φ is weakening-closed. Thus, f ∗ ⊆ φ . But by an application of the modular law in Rel( E ) we alsohave ψf = ψ ( φ ∩ ψ ◦ ) ⊇ ψφ ∩ ∆ X ⊇ I X ∩ ∆ X = ∆ X and so that φψf ⊇ φ . Then f ∗ = I Y f ⊇ φψf ⊇ φ and so we conclude that f ∗ = φ . (cid:3) Next, let us comment here on how the calculus of relations in a regular category E can be used to express various limit properties therein. For example, the statementthat a pair of morphisms X Z Y f g represents a given relation R : X Y is equivalent to the following two equalities between relations:(1) R = gf ◦ .(2) f ∗ f ∗ ∩ g ∗ g ∗ = I Z .Note here that we cannot replace condition (1.) by R = g ∗ f ∗ , even if R isweakening-closed. Similarly, we cannot replace (2.) by f ◦ f ∩ g ◦ g = I Z , because thelatter means that ( f, g ) are only jointly monic instead of jointly order-monic.Based on the above observation, now consider the statement that a commutativesquare P YX Z p p gf is a pullback. It is easy to see that this is equivalent to the following pair ofequalities:(1) g ◦ f = p p ◦ .(2) p ∗ p ∗ ∩ p ∗ p ∗ = I P .Similarly, the statement that the square P YX Z p p ≤ gf is a comma is equivalent to:(1) g ∗ f ∗ = p p ◦ .(2) p ∗ p ∗ ∩ p ∗ p ∗ = I P .Now we turn to discussing exactness for Pos -categories. First, let us recall thedefinition of congruence relation from [12]. This can be seen as the ordered analogue of equivalence relations in an ordinary category. It is also a special case of a moregeneral notion of congruence for 2-categories.
Let X be an object of the regular category E . A congruence on X is a relation E : X X ∈ Rel w ( E ) which is reflexive and transitive.We say that the congruence E is effective if there exists a morphism f : X → Y ∈ E such that E = f /f .Equivalently, we can say that E : X X is a congruence if it is a transitiverelation such that E ⊇ I X . In essence, a congruence is a pre-order relation on X which is compatible with the canonical order relation on X , the latter expressed bythe requirement that it is weakening-closed. We think of a congruence as imposingadditional inequalities on X , just as an equivalence relation corresponds to the ideaof imposing new equalities.With the notion of congruence in hand, we are lead naturally to the notion of(Barr-)exactness for Pos -categories as considered by Kurz-Velebil in [12].
A regular category E is called exact if every congruence in E iseffective. (1) Pos is exact and so is any presheaf category [ C , Pos ] forany small category C .(2) The locally discrete category Set is an example of a category which is regularbut not exact (see [12]).(3) Generalizing the case of
Pos , any variety of ordered algebras , always in thesense of Bloom & Wright [3], is an exact category. Particular examples hereare furnished by the categories
OrdSGrp , OrdMon , OrdMon and OrdCMon .On the other hand, the quasivariety
OrdCMon t . f . is not exact.(4) There are also examples of exact categories which are not varieties, but aremonadic over Pos . One such, which will appear in more detail in section 5,is the category
Nach of Nachbin spaces. Another is given by the category S - Pos for any ordered monoid S .A congruence in a regular category E , being a transitive relation, is an idempo-tent when considered as a morphism in either of Rel( E ) and Rel w ( E ). When it ismoreover effective, then it actually is a split idempotent in the latter bicategory.Indeed, if E = f /f for some f : X → Y , then we can assume that f is actually thecoinserter of E by Proposition 2.12. Then we have f ∗ f ∗ = f /f = E and f ∗ f ∗ = I Y .The next proposition shows that this splitting actually characterizes effective con-gruences. This is analogous to a familiar fact for ordinary regular categories, wherean equivalence relation splits in the bicategory of relations iff it occurs as a kernelpair. Let E : X X be a congruence in the regular category E .Then E is effective iff it splits as an idempotent in Rel w ( E ) .Proof. Suppose that E = ψφ for some φ : X Y and ψ : Y X in Rel w ( E ) with φψ = I Y . Since E is reflexive and weakening-closed, we have ψφ = E ⊇ I X . Sincealso trivially φψ ⊆ I Y , we have φ ⊣ ψ in Rel w ( E ) and so by Theorem 3.7 we havethat φ = f ∗ and ψ = f ∗ for some morphism f : X → Y ∈ E . Thus, we obtain E = ψφ = f ∗ f ∗ = f /f . (cid:3) In the following section we will embark on the goal of constructing the exactcompletion E ex/reg of a regular category E by a process of splitting idempotents ina bicategory of relations and then taking maps in the resulting bicategory. This isessentially an attempt to mimic the construction of the ordinary exact completionof a regular category, as initially described by Lawvere in [13] and then with more N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 17 details for example in [19], [7], [8], motivated by the combination of the two resultscontained in Theorem 3.7 and Proposition 3.11.However, even if the reader is not familiar with the construction of the splittingof idempotents, the next proposition can serve as a type of heuristic for comingup with the definition of morphisms in the completion E ex/reg . It will also beof practical use a little bit later. Recall here that in a regular category we say E e / / e / / X p / / / / P is an exact sequence if ( e , e ) is the kernel congruence of p and also p is the coinserter of ( e , e ). In terms of the calculus of relations, exactnessof the sequence is equivalent to the equalities p ∗ p ∗ = E and p ∗ p ∗ = I P . Let E / / / / X p / / / / P and F / / / / Y q / / / / Q be exact se-quences in the regular category E . Then there is an order-reversing bijection betweenthe following: (1) Morphisms P → Q in E . (2) Relations R ∗ : X Y ∈ Rel w ( E ) for which there exists another relation R ∗ : Y X ∈ Rel w ( E ) such that the following are satisfied: • F R ∗ E = R ∗ and ER ∗ F = R ∗ . • R ∗ R ∗ ⊇ E and R ∗ R ∗ ⊆ F .Proof. Consider first a morphism r : P → Q ∈ E . Set R ∗ := q ∗ r ∗ p ∗ and R ∗ := p ∗ r ∗ q ∗ . We have F R ∗ E = q ∗ q ∗ q ∗ r ∗ p ∗ p ∗ p ∗ = q ∗ r ∗ p ∗ = R ∗ and ER ∗ F = p ∗ p ∗ p ∗ r ∗ q ∗ q ∗ q ∗ = p ∗ r ∗ q ∗ = R ∗ . In addition, R ∗ R ∗ = p ∗ r ∗ q ∗ q ∗ r ∗ p ∗ = p ∗ r ∗ I Q r ∗ p ∗ = p ∗ r ∗ r ∗ p ∗ ⊇ p ∗ p ∗ = E and R ∗ R ∗ = q ∗ r ∗ p ∗ p ∗ r ∗ q ∗ ⊆ q ∗ r ∗ r ∗ q ∗ ⊆ q ∗ q ∗ = F .Conversely, consider a relation R ∗ as in (2.). Set φ := q ∗ R ∗ p ∗ and then also ψ := p ∗ R ∗ q ∗ . Then for these weakening-closed relations we have ψφ = p ∗ R ∗ q ∗ q ∗ R ∗ p ∗ ⊇ p ∗ R ∗ R ∗ p ∗ ⊇ p ∗ Ep ∗ = I P φψ = q ∗ R ∗ p ∗ p ∗ R ∗ q ∗ = q ∗ R ∗ ER ∗ q ∗ = q ∗ R ∗ R ∗ q ∗ ⊆ q ∗ F q ∗ = I Q Thus, by Theorem 3.7 we have φ = r ∗ for a (unique) morphism r : P → Q .The fact that these two assignments are inverse to each other is expressed by thetwo equalities q ∗ q ∗ r ∗ p ∗ p ∗ = r ∗ and q ∗ q ∗ R ∗ p ∗ p ∗ = F R ∗ E = R ∗ . (cid:3) Before ending this section, let us make a couple more observations on the as-signment R ∗ r from the last proposition. Note that, as the notation suggestsand the above proof exhibits, the relation R ∗ corresponds to the hypergraph r ∗ of a morphism r : P → Q . Specifically, r is uniquely determined from R ∗ by theequality r ∗ = q ∗ R ∗ p ∗ . We would like to also record here a relation R that in somesense corresponds directly to the (graph of the) morphism r .Given R ∗ : X Y as in Proposition 3.12, set R := R ∗ ∩ ( R ∗ ) ◦ : X Y . Thenwe claim that r = qRp ◦ . To see this, observe first that qRp ◦ is also a map in Rel( E )because ( qRp ◦ ) ◦ qRp ◦ = pR ◦ q ◦ qRp ◦ ⊇ pR ◦ Rp ◦ ⊇ p ( E ∩ E ◦ ) p ◦ = ∆ P ( qRp ◦ )( qRp ◦ ) ◦ = qRp ◦ pR ◦ q ◦ = qR ( E ∩ E ◦ ) R ◦ q ◦ = qRR ◦ q ◦ ⊆ q ( F ∩ F ◦ ) q ◦ = ∆ Q Here we used the fact that in an exact sequence E / / / / X p / / / / P we have E ∩ E ◦ = p ∗ p ∗ ∩ ( p ∗ p ∗ ) ◦ = p ◦ p , i.e. E ∩ E ◦ is precisely the kernel pair of p . Inaddition, the inclusions R ◦ R ⊇ E and RR ◦ ⊆ F were used, the first of which isclear and the second follows by the modular law (see the next section for details).Now, finally, we have r = r ∗ ∩ ( r ∗ ) ◦ = q ∗ R ∗ p ∗ ∩ ( q ∗ ) ◦ ( R ∗ ) ◦ ( p ∗ ) ◦ ⊇ q ( R ∗ ∩ ( R ∗ ) ◦ ) p ◦ = qRp ◦ . But since we have an inclusion between two maps in the allegory Rel( E ), thesemaps must be equal. Hence, we conclude that r = qRp ◦ .4. Exact Completion
In this section we come to the heart of this paper, which is the construction ofthe exact completion of a regular
Pos -category E .The main idea is to try to perform a construction that mimics one of the waysin which one can define the exact completion of an ordinary regular category C .Let us thus quickly recall this construction, as originally suggested by Lawverein [13]. We will also very much be drawing inspiration from the presentation ofSucci-Cruciani [19].Given a regular ordinary category C , one first performs a splitting of idempotents in the bicategory of relations Rel( C ). More precisely, one splits the class of equiv-alence relations, which are indeed idempotent as morphisms in Rel( C ). This stepyields a bicategory which, a posteriori, is identified as the bicategory of relationsRel( C ex / reg ) of the completion. The second step is then to identify the completionitself, which can be done by taking the category of maps in the bicategory producedby the first step.We note that the idea for this construction of the exact completion can be tracedback to the following two observations, valid in any ordinary regular category C :(1) An equivalence relation E on an object X ∈ C is effective iff it splits as anidempotent in Rel( C ).(2) The morphisms f : X → Y ∈ C are precisely the maps in Rel( C ).Accordingly, our hope to perform a Pos -enriched version of this constructionhinges on the validity of enriched versions of the two observations above, as con-tained respectively in Proposition 3.11 and Theorem 3.7. Hence, in our context, weshould first look at Rel w ( E ), split the idempotents therein which are congruencesin E , then finally take the category of maps in the resulting bicategory.The fact that we need to work with Rel w ( E ) rather than Rel( E ) already presentssome issues. As we have mentioned earlier, Rel( E ) has the structure of an allegoryand it is this fact that facilitates many computations. Furthermore, the theory ofallegories is well developed and in fact there is a precise correspondence betweenordinary regular and exact categories on the one hand and certain classes of alle-gories on the other (see e.g. [8], [7]). On the contrary, the structure of Rel w ( E ) isnot as rich. Fundamentally, the process of taking the opposite of a relation doesnot restrict to Rel w ( E ). Thus, in our quest to construct the Pos -enriched exactcompletion as indicated above, we cannot simply rely on the general theory of al-legories. While this creates a complication, at the same time it is in some senseto be expected. Indeed, allegories are in some aspects too simple for our enrichedcontext. For example, the only inclusions between maps in an allegory are equali-ties and it is precisely this fact that does not allow us to recover the order relationon morphisms from Rel( E ).Motivated by the above, we embark towards our goal by first defining a cate-gory Q w ( E ) by splitting the idempotents in Rel w ( E ) which are congruences in E .Explicitly, Q w ( E ) is defined as follows: • Objects of Q w ( E ) are pairs ( X, E ), where X is an object of E and E : X X is a congruence relation in E . • Morphisms Φ : (
X, E ) → ( Y, F ) in Q w ( E ) are (weakening-closed) relationsΦ : X Y in E such that Φ E = Φ = F Φ or equivalently Φ = F Φ E . N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 19
Composition in Q w ( E ) is composition of relations in E , while the identity morphismon ( X, E ) ∈ Q w ( E ) is the relation E itself. The morphisms are locally ordered byinclusion and it is clear that Q w ( E ) also has binary infima of morphisms given byintersection of relations in E . Then we define a category E ex/reg by taking the mapsin Q w ( E ). E ex/reg := Map(Q w ( E )).Explicitly, E ex/reg has the same objects as Q w ( E ) and as morphisms those R ∗ :( X, E ) → ( Y, F ) ∈ Q w ( E ) for which there exists an R ∗ : ( Y, F ) → ( X, E ) ∈ Q w ( E )such that R ∗ R ∗ ⊇ E and R ∗ R ∗ ⊆ F .Note that E ex/reg is not merely an ordinary category, but can be made into alegitimate Pos-category by defining for any R ∗ , S ∗ : ( X, E ) → ( Y, F ) in E ex/reg R ∗ ≤ S ∗ : ⇔ R ∗ ⊇ S ∗ the inclusion on the right-hand side being that of relations in E . This is clearly apartial order relation on Homs that is preserved by composition. Observe further-more that we can equivalently define the order R ∗ ≤ S ∗ by requiring R ∗ ⊆ S ∗ forthe right adjoints.We also have a canonical functor Γ : E → E ex/reg defined by mapping an object X ∈ E to ( X, I X ) and a morphism f : X → Y ∈ E to its hypergraph f ∗ : X Y considered as a morphism ( X, I X ) → ( Y, I Y ). Note that Γ is order-preserving andreflecting by definition of the order on morphisms in E ex/reg . We will consistently denote morphisms of E ex/reg by a capital letterwith a lower asterisk and their right adjoint in Q w ( E ) by the same letter with anupper asterisk. This notation represents our intuition that R ∗ is the hyper-graphof the morphism R and in some sense we are working towards making this a precisestatement. In particular, we will denote the identity morphism ( X, E ) → ( X, E )by 1 ( X,E ) ∗ , where as relations in E we have 1 ( X,E ) ∗ = E and 1 ∗ ( X,E ) = E .Our goal now in this section is to show that the category E ex/reg as definedabove is the exact completion of E as a regular category. The category Q w ( E ) willbe seen in the end to be precisely the category of weakening-closed relations in E ex/reg . Accordingly, the proofs of the various statements about E ex/reg later on inthis section will be motivated by the description of the various limit and exactnessproperties in terms of the calculus of relations in a regular category. However,we know that such a description cannot be achieved with only weakening-closedrelations. Thus, it will be convenient to construct also at the same time what willturn out to be the bicategory of all relations in E ex/reg .This leads us to define another bicategory Q ( E ) as follows: • Objects of Q ( E ) are again pairs ( X, E ), where E : X X is a congruencerelation in E . • Morphisms (
X, E ) → ( Y, F ) in Q ( E ) are relations Φ : X Y in E suchthat Φ( E ∩ E ◦ ) = Φ = ( F ∩ F ◦ )Φ or equivalently ( F ∩ F ◦ )Φ( E ∩ E ◦ ) = Φ.The composition of morphisms is that of relations in E while the identity on( X, E ) ∈ Q ( E ) is E ∩ E ◦ .In other words, Q ( E ) is the locally ordered bicategory obtained from Rel( E ) bysplitting those idempotents of the form E ∩ E ◦ for a congruence E in E . Noticethat idempotents of this form are equivalence relations in E and in particular aresymmetric. It then follows (see for example Theorem 3.3.4 in [8]) that Q ( E ) is anallegory, where the opposite of a morphism is given by taking the opposite relationin E .0 VASILEIOS ARAVANTINOS-SOTIROPOULOS
X, E ) → ( Y, F ) in Q ( E ) are relations Φ : X Y in E suchthat Φ( E ∩ E ◦ ) = Φ = ( F ∩ F ◦ )Φ or equivalently ( F ∩ F ◦ )Φ( E ∩ E ◦ ) = Φ.The composition of morphisms is that of relations in E while the identity on( X, E ) ∈ Q ( E ) is E ∩ E ◦ .In other words, Q ( E ) is the locally ordered bicategory obtained from Rel( E ) bysplitting those idempotents of the form E ∩ E ◦ for a congruence E in E . Noticethat idempotents of this form are equivalence relations in E and in particular aresymmetric. It then follows (see for example Theorem 3.3.4 in [8]) that Q ( E ) is anallegory, where the opposite of a morphism is given by taking the opposite relationin E .0 VASILEIOS ARAVANTINOS-SOTIROPOULOS Now we make some important observations regarding the connection betweenmorphisms of Q w ( E ) and Q ( E ) and between maps in these two bicategories. If thereader keeps in mind the intuition that these two categories should respectively beRel w ( E ex / reg ) and Rel( E ex / reg ), then these observations are to be expected.First, note that every morphism Φ : ( X, E ) → ( Y, F ) in Q w ( E ) can also beconsidered as a morphism in Q ( E ), sinceΦ = ∆ Y Φ∆ X ⊆ ( F ∩ F ◦ )Φ( E ∩ E ◦ ) ⊆ F Φ E = ΦHowever, it is important to note as well that this assignment is not functorial as itdoes not preserve the identity morphisms.Second, to any map in Q w ( E ), i.e. to any morphism of E ex/reg , we can associatein a natural way a map in Q ( E ) as follows. Consider any R ∗ : ( X, E ) → ( Y, F ) ∈ E ex/reg and define the relation gr ( R ∗ ) := R ∗ ∩ ( R ∗ ) ◦ : X Y in E . Then gr ( R ∗ ) is a map ( X, E ) → ( Y, F ) in Q ( E ) .Proof. First, it is easy to see that ( F ∩ F ◦ ) gr ( R ∗ )( E ∩ E ◦ ) = gr ( R ∗ ). Indeed, wehave gr ( R ∗ ) = ∆ Y gr ( R ∗ )∆ X ⊆ ( F ∩ F ◦ ) gr ( R ∗ )( E ∩ E ◦ ) ⊆ ( F gr ( R ∗ ) E ) ∩ ( F ◦ gr ( R ∗ ) E ◦ ) ⊆⊆ F R ∗ E ∩ F ◦ ( R ∗ ) ◦ E ◦ = R ∗ ∩ ( R ∗ ) ◦ = gr ( R ∗ )Second, to see that gr ( R ∗ ) is indeed a map in Q ( E ) we argue as follows: gr ( R ∗ ) gr ( R ∗ ) ◦ = ( R ∗ ∩ ( R ∗ ) ◦ )(( R ∗ ) ◦ ∩ R ∗ ) ⊆ R ∗ R ∗ ∩ ( R ∗ ) ◦ ( R ∗ ) ◦ ⊆ F ∩ F ◦ gr ( R ∗ ) ◦ gr ( R ∗ ) = ( R ◦∗ ∩ R ∗ )( R ∗ ∩ ( R ∗ ) ◦ ) = ( R ◦∗ ∩ R ∗ )(( R ∗ ∩ ( R ∗ ) ◦ ) ∩ R ∗ )= ( R ◦∗ ∩ R ∗ )(( R ◦∗ ∩ R ∗ ) ◦ ( E ∩ E ◦ ) ∩ R ∗ ) ⊇ ( E ∩ E ◦ ) ∩ ( R ◦∗ ∩ R ∗ ) R ∗ = ( E ∩ E ◦ ) ∩ ( E ◦ R ◦∗ ∩ R ∗ ) R ∗ ⊇ ( E ∩ E ◦ ) ∩ E ◦ ∩ R ∗ R ∗ ⊇ ( E ∩ E ◦ ) ∩ E ◦ ∩ E = E ∩ E ◦ where for establishing the first and second inclusions we used the modular law inRel( E ). (cid:3) Given a morphism R ∗ : ( X, E ) → ( Y, F ) ∈ E ex/reg , we call the relation gr ( R ∗ )defined above the graph of R ∗ . Observe furthermore that gr ( R ∗ ) satisfies the fol-lowing two basic equalities: F ◦ gr ( R ∗ ) = R ∗ gr ( R ∗ ) ◦ ◦ F = R ∗ Indeed, on one hand clearly F ◦ gr ( R ∗ ) ⊆ F R ∗ = R ∗ . On the other hand we have F ◦ gr ( R ∗ ) ⊇ R ∗ R ∗ gr ( R ∗ ) = R ∗ R ∗ ( R ∗ ∩ ( R ∗ ) ◦ ) ⊇ R ∗ ( R ∗ R ∗ ∩ E ◦ ) ⊇ R ∗ ( E ∩ E ◦ ) ⊇ R ∗ The second equality follows in a similar fashion.In fact, these two equalities characterize gr ( R ∗ ) in the following sense: if Φ :( X, E ) → ( Y, F ) is a map in Q ( E ) with F Φ = R ∗ and Φ ◦ F = R ∗ , then Φ = gr ( R ∗ ).Indeed,Φ ⊆ ( F ∩ F ◦ )Φ ⊆ F Φ ∩ F ◦ Φ = F Φ ∩ (Φ ◦ F ) ◦ = R ∗ ∩ ( R ∗ ) ◦ = gr ( R ∗ )and so Φ = gr ( R ∗ ) because Q ( E ) is an allegory and hence the inclusion of maps isdiscrete.Finally, the assignment R ∗ gr ( R ∗ ) is functorial. Given R ∗ : ( X, E ) → ( Y, F )and S ∗ : ( Y, F ) → ( Z, G ) we have G ( gr ( S ∗ ) gr ( R ∗ )) = ( G gr ( S ∗ )) gr ( R ∗ ) = S ∗ gr ( R ∗ ) = S ∗ F gr ( R ∗ ) = S ∗ R ∗ ( gr ( S ∗ ) gr ( R ∗ )) ◦ G = gr ( R ∗ ) ◦ gr ( S ∗ ) ◦ G = gr ( R ∗ ) ◦ S ∗ = gr ( R ∗ ) ◦ F S ∗ = R ∗ S ∗ , N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 21 so we conclude by the above observation that gr ( S ∗ ) gr ( R ∗ ) = gr ( S ∗ R ∗ ). Also, gr (1 ( X,E ) ∗ ) = 1 ( X,E ) ∗ ∩ (1 ∗ ( X,E ) ) ◦ = E ∩ E ◦ and the latter is an identity morphismin Map(Q( E )). (on notation) Henceforth, to ease the notation, we shall often denotethe graph gr ( R ∗ ) of a morphism R ∗ : ( X, E ) → ( Y, E ) ∈ E ex/reg simply by R , i.e. wewill just drop the lower asterisk. This shall not present too much risk for confusionas R ∗ will always have appeared before R .We now begin our work on proving that E ex/reg is indeed the desired exactcompletion of the regular category E . This is broken down into a sequence of morebite-sized pieces. First, we establish a fundamental result asserting the existenceof certain canonical representations for morphisms in the bicategories Q w ( E ) and Q ( E ). Following this, we repeatedly employ this representation to establish step bystep that E ex/reg has the desired finite limit and exactness properties making it anexact category. The arguments here are essentially motivated by the descriptionof these properties in terms of the calculus of relations. Finally, we show that wehave indeed constructed the exact completion by establishing the relevant universalproperty.To begin with, we record a small lemma concerning jointly order-monic pairs ofmorphisms in E ex/reg . If ( Y, F ) (
X, E ) R ∗ o o S ∗ / / ( Z, G ) is a pair of morphisms in E ex/reg such that R ∗ R ∗ ∩ S ∗ S ∗ = E , then this pair is jointly order-monic in E ex/reg .Proof. Assume that R ∗ R ∗ ∩ S ∗ S ∗ = E and let H ∗ , K ∗ : ( A, T ) → ( X, E ) ∈ E ex/reg be such that R ∗ H ∗ ≤ R ∗ K ∗ and S ∗ H ∗ ≤ S ∗ K ∗ i.e. R ∗ H ∗ ⊇ R ∗ K ∗ and S ∗ H ∗ ⊇ S ∗ K ∗ . Then we have K ∗ H ∗ = EK ∗ H ∗ = ( R ∗ R ∗ ∩ S ∗ S ∗ ) K ∗ H ∗ ⊆ R ∗ R ∗ K ∗ H ∗ ∩ S ∗ S ∗ K ∗ H ∗ ⊆ R ∗ R ∗ H ∗ H ∗ ∩ S ∗ S ∗ H ∗ H ∗ ⊆ R ∗ R ∗ ∩ S ∗ S ∗ , hence K ∗ H ∗ ⊆ E . But recall that by definition of morphisms we have an adjunction H ∗ ⊣ H ∗ in Q w ( E ). Thus, K ∗ H ∗ ⊆ E ⇐⇒ K ∗ ⊆ H ∗ , so that we obtain H ∗ ≤ K ∗ . (cid:3) The result that follows will be of central importance in establishing all the desiredproperties of E ex/reg throughout the remainder of this section. It says that anymorphism of Q ( E ) (hence also of Q w ( E )) can be expressed in a suitable way viamorphisms of E ex/reg . This should be compared to the fact that in any regularcategory C every relation R : X Y can be written as R = gf ◦ , where f : Z → X and g : Z → Y are morphisms in C with f ∗ f ∗ ∩ g ∗ g ∗ = I Z . Actually, our goal is toshow in the end that it is precisely this fact, since we will prove that Q ( E ) is exactlythe bicategory of relations of E ex/reg , while Q w ( E ) will be that of weakening-closedrelations. Let
Φ : (
X, E ) → ( Y, F ) be a morphism of Q ( E ) . Then thereexists a pair of morphisms ( X, E ) (
Z, T ) R ∗ o o R ∗ / / ( Y, F ) in E ex/reg such that (1) Φ = R R ◦ . (2) R ∗ R ∗ ∩ R ∗ R ∗ = T .Moreover, any pair ( R ∗ , R ∗ ) with these two properties has the following universalproperty: Given any morphisms ( X, E ) (
C, G ) S ∗ o o S ∗ / / ( Y, F ) in E ex/reg such that S S ◦ ⊆ Φ , there exists a unique morphism H ∗ : ( C, G ) → ( Z, T ) ∈ E ex/reg with R ∗ H ∗ = S ∗ and R ∗ H ∗ = S ∗ .Proof. Suppose that Φ is represented by the ff-morphism h r , r i : Z X × Y in E .We set T := r ◦ Er ∩ r ◦ F r and R ∗ := Er , R ∗ := F r . The relations r ◦ Er and r ◦ F r are inverse images along r , r respectively of the congruences E, F , henceare themselves congruences. Thus, so is their intersection T .Also, we claim that that R ∗ and R ∗ as defined above are morphisms( X, E ) (
Z, T ) R ∗ o o R ∗ / / ( Y, F ) in E ex/reg . Let’s check this for R ∗ : first, wehave ER ∗ = EEr = Er = R ∗ . Furthermore, R ∗ T = Er ( r ◦ Er ∩ r ◦ F r ) ⊆ Er r ◦ Er ⊆ E ∆ X Er = EEr = Er = R ∗ , hence R ∗ T = R ∗ . So R ∗ is at least a morphism in Q w ( E ). To show that it isactually a map, define R ∗ := r ◦ E . We then similarly have T R ∗ = ( r ◦ Er ∩ r ◦ F r ) r ◦ E ⊆ r ◦ Er r ◦ E ⊆ r ◦ EE = r ◦ E = R ∗ = ⇒ T R ∗ = R ∗ and R ∗ E = r ◦ EE = r ◦ E = R ∗ . And finally, R ∗ R ∗ = r ◦ EEr = r ◦ Er ⊇ T and R ∗ R ∗ = Er r ◦ E ⊆ EE = E .Similarly, it follows that R ∗ is a morphism ( Z, T ) → ( Y, F ) ∈ E ex/reg whoseright adjoint in Q w ( E ) is R ∗ := r ◦ F .Just by the definitions, we have R ∗ R ∗ ∩ R ∗ R ∗ = r ◦ EEr ∩ r ◦ F F r = r ◦ Er ∩ r ◦ F r = T. In addition, R = R ∗ ∩ ( R ∗ ) ◦ = Er ∩ ( r ◦ E ) ◦ = Er ∩ E ◦ r = ( E ∩ E ◦ ) r andsimilarly R = ( F ∩ F ◦ ) r , so that R R ◦ = ( F ∩ F ◦ ) r r ◦ ( E ∩ E ◦ ) = ( F ∩ F ◦ )Φ( E ∩ E ◦ ) = Φ . where the last equality holds because Φ ∈ Q ( E ).Next, we have to prove the stated universality property, so let( X, E ) (
C, G ) S ∗ o o S ∗ / / ( Y, F ) in E ex/reg be such that S S ◦ ⊆ Φ.Set H ∗ := R ∗ S ∗ ∩ R ∗ S ∗ and H ∗ := S ∗ R ∗ ∩ S ∗ R ∗ . Since, both H ∗ , H ∗ arebinary intersections of compositions of morphisms in Q w ( E ), it is immediate thatthey are both themselves morphisms in that bicategory. We will show that H ∗ ismoreover a map with H ∗ as its right adjoint.First of all, we have that H ∗ H ∗ ⊆ R ∗ S ∗ S ∗ R ∗ ∩ R ∗ S ∗ S ∗ R ∗ ⊆ R ∗ ER ∗ ∩ R ∗ F R ∗ = R ∗ R ∗ ∩ R ∗ R ∗ = T For the other inclusion we argue as follows: H ∗ H ∗ = ( S ∗ R ∗ ∩ S ∗ R ∗ )( R ∗ S ∗ ∩ R ∗ S ∗ ) ⊇ ( S ◦ R ∩ S ◦ R )( R ◦ S ∩ R ◦ S )= ( S ◦ R ∩ S ◦ R )(( R ◦ S ∩ R ◦ S )( G ∩ G ◦ ) ∩ R ◦ S ) ⊇ ( G ∩ G ◦ ) ∩ ( S ◦ R ∩ S ◦ R ) R ◦ S = ( G ∩ G ◦ ) ∩ (( G ∩ G ◦ ) S ◦ R ∩ S ◦ R ) R ◦ S ⊇ ( G ∩ G ◦ ) ∩ ( G ∩ G ◦ ) ∩ S ◦ R R ◦ S where we used the modular law in E twice. But now, using adjunction propertiesin Q ( E ) together with the assumption that S S ◦ ⊆ Φ, we observe that S S ◦ ⊆ Φ = R R ◦ = ⇒ S ⊆ R R ◦ S = ⇒ G ∩ G ◦ ⊆ S ◦ R R ◦ S , from which we deduce H ∗ H ∗ ⊇ ( G ∩ G ◦ ) ∩ S ◦ R R ◦ S = G ∩ G ◦ . Then, finally, H ∗ H ∗ G ⊇ ( G ∩ G ◦ ) G = ⇒ H ∗ H ∗ ⊇ G .Thus, H ∗ is indeed a morphism in E ex/reg . Furthermore, R ∗ H ∗ = R ∗ ( R ∗ S ∗ ∩ R ∗ S ∗ ) ⊆ R ∗ R ∗ S ∗ ⊆ ES ∗ = S ∗ N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 23 R ∗ H ∗ = R ∗ ( R ∗ S ∗ ∩ R ∗ S ∗ ) ⊇ R ( R ◦ S ∩ R ◦ S ) ⊇ S ∩ R R ◦ S = S where for the second line of inclusions we used the modular law together with thefact that S S ◦ ⊆ Φ = R R ◦ = ⇒ S S ◦ ⊆ R R ◦ = ⇒ S ⊆ R R ◦ S . Now ER ∗ H ∗ ⊇ ES = ⇒ R ∗ H ∗ ⊇ S ∗ . Thus, we have R ∗ H ∗ = S ∗ and similarlyone gets R ∗ H ∗ = S ∗ . Finally, uniqueness is clear because, as we’ve proved in theprevious lemma, the equality R ∗ R ∗ ∩ R ∗ R ∗ = T implies that R ∗ , R ∗ are jointlyorder-monic in E ex/reg . (cid:3) A pair of morphisms ( R ∗ , R ∗ ) in E ex/reg with the properties in Proposition 4.6will be called a tabulation of Φ : ( X, E ) → ( Y, F ) ∈ Q ( E ). This terminology isborrowed from the theory of allegories. Note incidentally that the latter theorycould have been directly applied to Q ( E ), but this approach would not work forus. The reason for that is that we can not identify the morphisms of the would becompletion E ex/reg as maps in this allegory, but rather only in Q w ( E ). Thus, weneed both of these bicategories at the same time: Q w ( E ) to identify the morphismsof E ex/reg and Q ( E ) to express the existence of tabulations and perform calculationsmore freely.Furthermore, since as we’ve noted earlier every morphism Φ : ( X, E ) → ( Y, F ) ∈ Q w ( E ) can be considered as a morphism in Q ( E ), we also have tabulations formorphisms of Q w ( E ). In this case the inclusion S S ◦ ⊆ Φ in the universal propertyof tabulations is equivalent to S ∗ S ∗ ⊆ Φ. Indeed, S S ◦ ⊆ Φ = ⇒ F S S ◦ E ⊆ F Φ E = ⇒ S ∗ S ∗ ⊆ Φ. Similarly, for the tabulation ( R ∗ , R ∗ ) we have Φ = R R ◦ = R ∗ R ∗ .As we’ve already claimed, the existence of tabulations will be a fundamentaltool for establishing results about E ex/reg . As a first example of this, we can nowcompletely characterize what it means for a pair of morphisms to be jointly order-monic in E ex/reg . A pair of morphisms ( Y, F ) (
X, E ) R ∗ o o S ∗ / / ( Z, G ) is jointlyorder-monic in E ex/reg iff R ∗ R ∗ ∩ S ∗ S ∗ = E .Proof. We’ve already proven sufficiency earlier, so assume conversely that R ∗ , S ∗ arejointly order-monic. By the proposition above, there exists a tabulation ( X, E ) (
A, T ) U ∗ o o V ∗ / / ( X, E )for the morphism R ∗ R ∗ ∩ S ∗ S ∗ ∈ Q w ( E ). Then we have V ∗ U ∗ = R ∗ R ∗ ∩ S ∗ S ∗ ⊆ R ∗ R ∗ = ⇒ R ∗ V ∗ U ∗ ⊆ R ∗ = ⇒ R ∗ V ∗ ⊆ R ∗ U ∗ and similarly we obtain S ∗ V ∗ ⊆ S ∗ U ∗ . Thus, we have R ∗ V ∗ ≥ R ∗ U ∗ and S ∗ V ∗ ≥ S ∗ U ∗ and hence V ∗ ≥ U ∗ ⇐⇒ V ∗ ⊆ U ∗ . But now we have R ∗ R ∗ ∩ S ∗ S ∗ = V ∗ U ∗ ⊆ U ∗ U ∗ ⊆ E Since the reverse inclusion always holds, we conclude that R ∗ R ∗ ∩ S ∗ S ∗ = E . (cid:3) Before beginning to prove the basic finite limit and exactness properties of E ex/reg we will need some information on ff and so-morphisms therein. Let R ∗ : ( X, E ) → ( Y, F ) be a morphism in E ex/reg . Then: (1) R ∗ is an ff-morphism in E ex/reg iff R ∗ R ∗ = E . (2) R ∗ is an iso iff R ∗ R ∗ = E and R ∗ R ∗ = F . (3) If R ∗ R ∗ = F , then R ∗ is an so-morphism in E ex/reg . Proof. (1) Suppose first that R ∗ R ∗ = E and let U ∗ , V ∗ : ( Z, G ) → ( X, E )be such that R ∗ U ∗ ≤ R ∗ V ∗ . Then R ∗ U ∗ ⊇ R ∗ V ∗ = ⇒ R ∗ R ∗ U ∗ ⊇ R ∗ R ∗ V ∗ = ⇒ EU ∗ ⊇ EV ∗ = ⇒ U ∗ ⊇ V ∗ = ⇒ U ∗ ≤ V ∗ .Conversely, assume that R ∗ is an ff-morphism. Consider a tabulationof R ∗ R ∗ : ( X, E ) → ( X, E ) ∈ Q w ( E ), say R ∗ R ∗ = V U ◦ = V ∗ U ∗ where( X, E ) (
A, T ) U ∗ o o V ∗ / / ( X, E ) . Then we have that R ∗ V ∗ ⊆ R ∗ V ∗ U ∗ U ∗ = R ∗ R ∗ R ∗ U ∗ = R ∗ U ∗ = ⇒ R ∗ V ∗ ≥ R ∗ U ∗ Thus, we get V ∗ ≥ U ∗ , i.e. V ∗ ⊆ U ∗ . Now R ∗ R ∗ = V ∗ U ∗ ⊆ U ∗ U ∗ ⊆ E .(2) Clear.(3) Consider a commutative square in E ex/reg as below, where M ∗ is an ff-morphism. ( X, E ) R ∗ / / V ∗ (cid:15) (cid:15) ( Y, F ) S ∗ (cid:15) (cid:15) ( Z, G ) / / M ∗ / / ( W, H )By (1.), we know that M ∗ M ∗ = G . Set P ∗ := V ∗ R ∗ . First, we claimthat P ∗ is a morphism ( Y, F ) → ( Z, G ) in E ex/reg with P ∗ = R ∗ V ∗ .Indeed, we have P ∗ P ∗ = R ∗ V ∗ V ∗ R ∗ ⊇ R ∗ R ∗ = F . Observe also that P ∗ = M ∗ S ∗ , because M ∗ V ∗ = S ∗ R ∗ = ⇒ M ∗ M ∗ V ∗ = M ∗ S ∗ R ∗ = ⇒ V ∗ = M ∗ S ∗ R ∗ = ⇒ V ∗ R ∗ = M ∗ S ∗ R ∗ R ∗ = M ∗ S ∗ . Then we can argue that P ∗ P ∗ = M ∗ S ∗ R ∗ V ∗ = M ∗ M ∗ V ∗ V ∗ = V ∗ V ∗ ⊆ G .Finally, clearly P ∗ R ∗ = M ∗ S ∗ R ∗ = M ∗ M ∗ V ∗ = V ∗ and M ∗ P ∗ = M ∗ V ∗ R ∗ = S ∗ R ∗ R ∗ = S ∗ . (cid:3) E ex/reg has finite limits and Γ :
E → E ex/reg preserves them.Proof.
Let’s first construct the inserter of a pair (
X, E ) R ∗ / / S ∗ / / ( Y, F ) . To this end,consider a tabulation (
X, E ) (
A, T ) Φ ∗ o o Φ ∗ / / ( X, E ) of S ∗ R ∗ ∩ ( E ∩ E ◦ ) as amorphism ( X, E ) → ( X, E ) in Q ( E ). Then we observe thatΦ = Φ ( T ∩ T ◦ ) = Φ (Φ ◦ Φ ∩ Φ ◦ Φ ) ⊆ Φ Φ ◦ Φ = ( S ∗ R ∗ ∩ E ∩ E ◦ )Φ ⊆ ( E ∩ E ◦ )Φ = Φ and hence Φ = Φ because the inclusion of maps in Q ( E ) is discrete. So wehave Φ ∗ = Φ ∗ , which we henceforth denote simply by Φ ∗ . To prove that Φ ∗ :( A, T ) → ( X, E ) is the inserter of ( R ∗ , S ∗ ) it now suffices, due to the universalproperty of tabulations, to show that for every H ∗ : ( Z, G ) → ( X, E ) we have R ∗ H ∗ ≤ S ∗ H ∗ ⇐⇒ HH ◦ ⊆ S ∗ R ∗ ∩ E ∩ E ◦ . Indeed, we have R ∗ H ∗ ≤ S ∗ H ∗ ⇐⇒ S ∗ H ∗ ⊆ R ∗ H ∗ ⇐⇒ H ∗ ⊆ S ∗ R ∗ H ∗ ⇐⇒ H ∗ H ∗ ⊆ S ∗ R ∗ ⇐⇒ HH ◦ ⊆ S ∗ R ∗ ⇐⇒ HH ◦ ⊆ S ∗ R ∗ ∩ E ∩ E ◦ Next, let us construct the product of a pair of objects (
X, E ) and (
Y, F ) in E ex/reg . Observe that the maximal relation X Y given by the product is clearlya morphism ( X, E ) → ( Y, F ) in Q ( E ). Therefore, by Proposition 4.6 it has atabulation. In fact, looking back at the proof of the latter proposition we caneasily see that the tabulation thus constructed is given by the pair of morphisms( X, E ) ( X × Y, E × F ) ( Y, F ) Π ( X,E ) ∗ Π ( Y,F ) ∗ , where Π ( X,E ) ∗ = Eπ X N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 25 and Π ( Y,F ) ∗ = F π Y . In any case, the universal property of tabulations gives pre-cisely the universal property of a product diagram in E ex/reg .Finally, it is easy to check that Γ1 is a terminal object for E ex/reg . (cid:3) Even though the above proposition tells us that E ex/reg inherits all finite weightedlimits from E , we shall need some more specific information as well, namely on theconstruction of comma squares and pullbacks.Consider any morphisms R ∗ : ( X, E ) → ( Z, G ) and S ∗ : ( Y, F ) → ( Z, G ) in E ex/reg . First, we construct the comma square below:( W, T ) P ∗ (cid:15) (cid:15) P ∗ / / ≤ ( Y, F ) S ∗ (cid:15) (cid:15) ( X, E ) R ∗ / / ( Z, G )For this, we take ( P ∗ , P ∗ ) to be a tabulation of S ∗ R ∗ : ( X, E ) → ( Y, F ) ∈ Q w ( E ).To prove that this square is indeed a comma, it suffices to prove that, given any U ∗ : ( A, H ) → ( X, E ) and V ∗ : ( A, H ) → ( Y, F ), we have V ∗ U ∗ ⊆ S ∗ R ∗ ⇐⇒ R ∗ U ∗ ≤ S ∗ V ∗ . But indeed, using properties of adjunctions we have R ∗ U ∗ ≤ S ∗ V ∗ ⇐⇒ S ∗ V ∗ ⊆ R ∗ U ∗ ⇐⇒ V ∗ ⊆ S ∗ R ∗ U ∗ ⇐⇒ V ∗ U ∗ ⊆ S ∗ R ∗ For pullbacks let us take now ( P ∗ , P ∗ ) to be a tabulation of the relation S ◦ R ∈ Q ( E ). Then we claim that the following square is a pullback.( W, T ) P ∗ (cid:15) (cid:15) P ∗ / / ( Y, F ) S ∗ (cid:15) (cid:15) ( X, E ) R ∗ / / ( Z, G )Indeed, in this case we have for any U ∗ : ( A, H ) → ( X, E ) and V ∗ : ( A, H ) → ( Y, F )that R ∗ U ∗ = S ∗ V ∗ ⇐⇒ RU = SV ⇐⇒ SV ⊆ RU ⇐⇒ SV U ◦ ⊆ R ⇐⇒ V U ◦ ⊆ S ◦ R Thus, the universal property of the pullback is identified with that of the tabulation.Next, we prove that E ex/reg admits the required factorization system. E ex/reg has (so,ff)-factorizations.Proof. Consider a morphism R ∗ : ( X, E ) → ( Y, F ) ∈ E ex/reg . Then RR ◦ ∈ Q ( E )and so it admits a tabulation ( Y, F ) (
Z, G ) S ∗ o o S ∗ / / ( Y, F ) . Since tautologi-cally R ∗ is such that RR ◦ ⊆ S S ◦ , there exists a unique Q ∗ : ( X, E ) → ( Z, G ) ∈E ex/reg such that S ∗ Q ∗ = R ∗ = S ∗ Q ∗ .Now observe that in Q ( E ) we have S ⊆ S S ◦ S = RR ◦ S ⊆ S and so wededuce that S = S and hence S ∗ = S ∗ . We denote this morphism now simplyby S ∗ . Then we have S ∗ S ∗ = G by the tabulation property and this tells us that S ∗ is an ff-morphism. It suffices now to show that Q ∗ Q ∗ = G , so that Q ∗ will bean so-morphism. For this we argue as follows: SQQ ◦ S ◦ = RR ◦ = SS ◦ = ⇒ S ◦ SQQ ◦ S ◦ S = S ◦ SS ◦ S = ⇒ ( G ∩ G ◦ ) QQ ◦ ( G ∩ G ◦ ) = G ∩ G ◦ = ⇒ QQ ◦ = G ∩ G ◦ Then Q ∗ Q ∗ = GQQ ◦ G = G ( G ∩ G ◦ ) G = G . (cid:3) For a morphism R ∗ : ( X, E ) → ( Y, F ) ∈ E ex/reg we have R ∗ R ∗ = F ⇐⇒ RR ◦ = F ∩ F ◦ . We showed the “ ⇐ = ”direction in the course of the aboveproof. For the converse, assume that R ∗ R ∗ = F and argue as follows: RR ◦ = R ( R ◦ ( F ∩ F ◦ ) ∩ R ∗ ) ⊇ ( F ∩ F ◦ ) ∩ RR ∗ = ( F ∩ F ◦ ) ∩ ( R ∗ ∩ ( R ∗ ) ◦ ) R ∗ ⊇ ( F ∩ F ◦ ) ∩ R ∗ R ∗ ∩ F ◦ = ( F ∩ F ◦ ) ∩ F ∩ F ◦ = F ∩ F ◦ A morphism R ∗ : ( X, E ) → ( Y, F ) is an so-morphism in E ex/reg iff R ∗ R ∗ = F iff RR ◦ = F ∩ F ◦ . With this equational characterization of so-morphisms in hand, we are now in aposition to prove their stability under pullback. so-morphisms are stable under pullback in E ex/reg .Proof. Consider the following pullback square in E ex/reg where we assume that R ∗ is an so-morphism, so that we have R ∗ R ∗ = G or equivalently RR ◦ = G ∩ G ◦ .( W, T ) Q ∗ / / P ∗ (cid:15) (cid:15) ( Y, F ) S ∗ (cid:15) (cid:15) ( X, E ) R ∗ / / / / ( Z, G )By the construction of pullbacks we know that ( P ∗ , Q ∗ ) is a tabulation of S ◦ R .Thus, we have QQ ◦ = Q ( T ∩ T ◦ ) Q ◦ = Q ( P ◦ P ∩ Q ◦ Q ) Q ◦ ⊇ ( QP ◦ P ∩ Q ) Q ◦ ⊇ QP ◦ P Q ◦ ∩ ( F ∩ F ◦ ) = S ◦ RR ◦ S ∩ ( F ∩ F ◦ )= S ◦ ( G ∩ G ◦ ) S ∩ ( F ∩ F ◦ ) = S ◦ S ∩ ( F ∩ F ◦ ) = F ∩ F ◦ Hence, QQ ◦ = F ∩ F ◦ ⇐⇒ Q ∗ Q ∗ = F and so Q ∗ is an so-morphism. (cid:3) Putting together what we have proved so far, we have the following. E ex/reg is a regular category and Γ :
E → E ex/reg is a fully order-faithful regular functor.
We next would like to prove that E ex/reg is exact. To accomplish this we firstmake good on promises made much earlier. Namely, we identify Q w ( E ) as thebicategory of weakening-closed relations in E ex/reg and, before that, Q ( E ) as thebicategory of all relations in E ex/reg . There is an equivalence
Rel( E ex / reg ) ≃ Q( E ) .Proof. We will define a functor F : Rel( E ex / reg ) → Q( E ) by letting it be the identityon objects and mapping a relation represented by any jointly order-monic pair( X, E ) (
Z, T ) R ∗ o o R ∗ / / ( Y, F ) in E ex/reg to the morphism R R ◦ : ( X, E ) → ( Y, F ) ∈ Q ( E ).To show that this assignment is functorial, consider first the diagonal relation onthe object ( X, E ) in Rel( E ex / reg ), i.e. the relation represented by the jointly order-monic pair ( X, E ) (
X, E ) ( X,E ) ∗ o o ( X,E ) ∗ / / ( X, E ) . Then the image of this relationunder F is 1 ( X,E ) ◦ ( X,E ) = ( E ∩ E ◦ )( E ∩ E ◦ ) ◦ = E ∩ E ◦ and so F preserves identitymorphisms.Next, we consider two relations R , S in E ex/reg represented by the pairs ( X, E ) (
A, T ) R ∗ o o R ∗ / / ( Y, F )and (
Y, F ) (
B, T ′ ) S ∗ o o S ∗ / / ( Z, G ) respectively. To calculate the compositionof these two relations we form the following pullback square in E ex/reg N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 27 ( C, Ω) Π ∗ / / Π ∗ (cid:15) (cid:15) ( B, T ′ ) S ∗ (cid:15) (cid:15) ( A, T ) R ∗ / / ( Y, F )and then the image factorization of h R ∗ Π ∗ , S ∗ Π ∗ i , say( C, Ω) Q ∗ / / / / ( D, Θ) / / h U ∗ ,U ∗ i / / ( X, E ) × ( Z, G )By construction of pullbacks in E ex/reg we know that Π Π ◦ = S ◦ R . Also, bydefinition F maps the composition of the two relations to F ( S R ) = U U ◦ . Butnow we have that U U ◦ = U QQ ◦ U ◦ = S Π Π ◦ R ◦ = S S ◦ R R ◦ = F ( S ) F ( R )Finally, the fact that F preserves the order of morphisms and is fully (order-)faithful is precisely the existence of tabulations proved in Proposition 4.6. Thus, F is an equivalence of bicategories. (cid:3) There is an equivalence
Rel w ( E ex / reg ) ≃ Q w ( E ) .Proof. We define a functor F : Rel w ( E ex / reg ) → Q w ( E ) exactly as in the proof ofthe previous proposition. Then the main observation to make here is the follow-ing: a relation R : ( X, E ) ( Y, F ) represented by the jointly order-monic pair(
X, E ) (
Z, T ) R ∗ o o R ∗ / / ( Y, F ) in E ex/reg is weakening-closed iff R ∗ R ∗ = R R ◦ as relations in E .Recall that R is a weakening-closed relation precisely if I ( Y,F ) R I ( X,E ) = R inRel( E ex / reg ). To compute the composition I ( Y,F ) R I ( X,E ) one has to form the follow-ing diagram in E ex/reg where the top square is a pullback and the bottom two arecommas and then take the image factorization of the morphism h U ∗ W ∗ , V ∗ W ∗ i .( C, X ) W ∗ z z ✉✉✉✉✉✉✉✉✉ W ∗ $ $ ■■■■■■■■■ ( A, Φ) U ∗ z z ✉✉✉✉✉✉✉✉✉ U ∗ $ $ ■■■■■■■■■ ( B, Ψ) V ∗ z z ✉✉✉✉✉✉✉✉✉ V ∗ $ $ ■■■■■■■■■ ( X, E ) ( X,E ) ∗ $ $ ■■■■■■■■■ ≤ ( Z, T ) R ∗ z z ✉✉✉✉✉✉✉✉✉ R ∗ $ $ ■■■■■■■■■ ≤ ( Y, F ) ( Y,F ) ∗ z z ✉✉✉✉✉✉✉✉✉ ( X, E ) (
Y, F )Note that by the various limit constructions in E ex/reg we know that we must have R ∗ = U U ◦ , R ∗ = V V ◦ and V ◦ U = W W ◦ .If I ( Y,F ) R I ( X,E ) = R , then there is a factorization in E ex/reg ( C, X ) Q ∗ / / / / ( Z, T ) / / h R ∗ ,R ∗ i / / ( X, E ) × ( Y, F )with Q ∗ an so-morphism, so that QQ ◦ = T ∩ T ◦ . Then we have that R ∗ R ∗ = V V ◦ U U ◦ = V W W ◦ U ◦ = R QQ ◦ R ◦ = R R ◦
08 VASILEIOS ARAVANTINOS-SOTIROPOULOS
Conversely, assume that R ∗ R ∗ = R R ◦ and let ( X, E ) (
C, G ) S ∗ o o U ∗ o o S ∗ / / V ∗ / / ( Y, F )be such that ( S ∗ , S ∗ ) factors through ( R ∗ , R ∗ ) and U ∗ ≤ S ∗ and S ∗ ≤ V ∗ . Thenwe respectively have S S ∗ ⊆ R ∗ R ∗ and U ∗ ⊆ S ∗ and V ∗ ⊇ S ∗ . Hence, V ∗ U ∗ ⊆ S ∗ S ∗ ⊆ R ∗ R ∗ = R R ◦ and so ( U ∗ , V ∗ ) must also factor through ( R ∗ , R ∗ ) by theuniversal property of tabulations.With this observation in hand, one can run the same proof as in the previousproposition to show that F : Rel w ( E ex / reg ) → Q w ( E ) thus defined is an equivalence.The only point of minor difference is in the proof that F preserves identity mor-phisms. For this, just recall that the identity on an object ( X, E ) in Rel w ( E ex / reg )is the relation given by the following comma square( A, T ) R ∗ / / R ∗ (cid:15) (cid:15) ≤ ( X, E ) ( X,E ) ∗ (cid:15) (cid:15) ( X, E ) ( X,E ) ∗ / / ( X, E )so that by construction of commas we have that R R ◦ = 1 ∗ ( X,E ) ( X,E ) ∗ = EE = E . (cid:3) Now from this last proposition one can immediately deduce that E ex/reg is indeedan exact category. The category E ex/reg is exact.Proof. Using the equivalence Rel w ( E ex / reg ) ≃ Q w ( E ), we see that a congruence R on an object ( X, E ) ∈ E ex/reg corresponds precisely to a congruence R on the object X ∈ E with R ⊇ E . Then by construction we have that any such congruence splitsas an idempotent in Q w ( E ) ≃ Rel w ( E ex / reg ). (cid:3) It remains to prove that E ex/reg , or more precisely Γ : E → E ex/reg satisfies therequired universal property. Before doing this, we observe in the proposition thatfollows that every object of E ex/reg appears as a quotient of a congruence comingfrom E in a canonical way. For every object ( X, E ) ∈ E ex/reg there exists an exact se-quence Γ E Γ e / / Γ e / / Γ X E / / / / ( X, E ) , where h e , e i : E X × X is an ff-morphismrepresenting the congruence E in E .Proof. Observe that E indeed defines a morphism E ∗ : Γ X → ( X, E ) in E ex/reg which is in fact effective because EI X = E = EE , E ∗ E ∗ = EE = E ⊇ I X , E ∗ E ∗ = EE = E . Now it suffices to show that the square below is a comma squarein E ex/reg . Γ E Γ e / / Γ e (cid:15) (cid:15) ≤ Γ X E (cid:15) (cid:15) (cid:15) (cid:15) Γ X E / / / / ( X, E )But by the construction of comma squares in E ex/reg this is equivalent to having(Γ e ) ∗ Γ e ∩ (Γ e ) ∗ Γ e = I E ⇐⇒ e ∗ e ∩ e ∗ e = I E and Γ e (Γ e ) ◦ = E ∗ E ∗ ⇐⇒ e e ◦ = EE as relations in E , both of which hold. (cid:3) N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 29
Now we at last come to the proof that E ex/reg satisfies the universal property thatexhibits it as the exact completion of the regular category E . Before proceeding,let us make a couple of quick observations that will be used in the course of ourcalculations in the proof that follows below.Consider an exact fork E / / / / X p / / / / P in the regular category E . Recallthat in the calculus of relations this means that E = p ∗ p ∗ and p ∗ p ∗ = I P , where thesecond equality can equivalently be replaced by pp ◦ = ∆ P . In addition, the kernelpair p ◦ p of p can be written as p ∗ p ∗ ∩ ( p ∗ p ∗ ) ◦ and hence we also have p ◦ p = E ∩ E ◦ We now observe that the following equalities must hold: • pE = p ∗ . • Ep ◦ = p ∗ . • p ∗ Ep ∗ = I P .Indeed, for the first of these we have pE ⊆ p ∗ E = p ∗ p ∗ p ∗ = p ∗ and also pE = pp ∗ p ∗ ⊇ pp ◦ p ∗ = ∆ P p ∗ = p ∗ . The second one follows similarly. Finally, for the lastone we have p ∗ Ep ∗ = p ∗ p ∗ p ∗ p ∗ = I P I P = I P . Let F : E → F be a regular functor with F an exact category.Then there is a unique (up to iso) regular functor F : E ex/reg → F such that F ◦ Γ = F .Proof. If F is to be regular, then by the previous proposition we must define it onany object ( X, E ) ∈ E ex/reg as the following coinserter in F F ( E ) F e / / F e / / F X p ( X,E ) / / / / F ( X, E )which exists because regular functors preserve congruences and F is exact.Also, given any morphism R ∗ : ( X, E ) → ( Y, G ) ∈ E ex/reg , we have relations F ( R ∗ ) : F X
F Y and F ( R ∗ ) : F Y
F X in F satisfying the following equations F ( G ) F ( R ∗ ) F ( E ) = F ( GR ∗ E ) = F ( R ∗ ) F ( E ) F ( R ∗ ) F ( G ) = F ( ER ∗ G ) = F ( R ∗ ) F ( R ∗ ) F ( R ∗ ) = F ( R ∗ R ∗ ) ⊇ F ( E ) F ( R ∗ ) F ( R ∗ ) = F ( R ∗ R ∗ ) ⊆ F ( G )where we used the fact that regular functors preserve the compositions and in-clusions of relations. Thus, by Proposition 3.12 we can define F ( R ∗ ) to be theuniquely associated morphism between quotients F ( X, E ) → F ( Y, G ). More ex-plicitly, F ( R ∗ ) is the morphism uniquely determined by the equality F ( R ∗ ) ∗ = p ( Y,G ) ∗ F ( R ∗ ) p ∗ ( X,E ) in Rel( F ). It is immediate that this defines a functor E ex/reg →C and clearly F ◦ Γ ∼ = F . Note that by the discussion following Proposition 3.12 wealso have F ( R ∗ ) = p ( Y,G ) ( F ( R ∗ ) ∩ F ( R ∗ ) ◦ ) p ◦ ( X,E ) = p ( Y,G ) F ( R ) p ◦ ( X,E ) as relationsin F .Now let’s show that F preserves finite limits. First, it is clear that it preservesthe terminal object Γ1, since F Γ1 = F F preserves the terminal object.Second, the preservation of binary products follows from the fact that exactsequences in any regular category are stable under binary products.It suffices then to prove the preservation of inserters. So suppose that ( A, T ) (
X, E ) (
Y, G ) Φ ∗ R ∗ S ∗ is an inserter diagram in E ex/reg . Recall from Proposition 4.9 that by constructionof inserters in E ex/reg this means that ΦΦ ◦ = S ∗ R ∗ ∩ ( E ∩ E ◦ ) and Φ ∗ Φ ∗ = T as0 VASILEIOS ARAVANTINOS-SOTIROPOULOS
Y, G ) Φ ∗ R ∗ S ∗ is an inserter diagram in E ex/reg . Recall from Proposition 4.9 that by constructionof inserters in E ex/reg this means that ΦΦ ◦ = S ∗ R ∗ ∩ ( E ∩ E ◦ ) and Φ ∗ Φ ∗ = T as0 VASILEIOS ARAVANTINOS-SOTIROPOULOS relations in E . Then first of all we have F (Φ ∗ ) ∗ F (Φ ∗ ) ∗ = p ( A,T ) ∗ F (Φ ∗ ) p ∗ ( X,E ) p ( X,E ) ∗ F (Φ ∗ ) p ∗ ( A,T ) = p ( A,T ) ∗ F (Φ ∗ ) F ( E ) F (Φ ∗ ) p ∗ ( A,T ) = p ( A,T ) ∗ F (Φ ∗ E Φ ∗ ) p ∗ ( A,T ) = p ( A,T ) ∗ F (Φ ∗ Φ ∗ ) p ∗ ( A,T ) = p ( A,T ) ∗ F ( T ) p ∗ ( A,T ) = I F ( A,T ) which tells us that F (Φ ∗ ) is an ff-morphism in F . Second, we have the followingsequence of calculations: F ( S ∗ ) ∗ F ( R ∗ ) ∗ ∩ ∆ F ( X,E ) == p ( X,E ) p ◦ ( X,E ) ( F ( S ∗ ) ∗ F ( R ∗ ) ∗ ∩ ∆ F ( X,E ) ) p ( X,E ) p ◦ ( X,E ) = p ( X,E ) p ◦ ( X,E ) ( p ( X,E ) ∗ F ( S ∗ ) p ∗ ( Y,G ) p ( Y,G ) ∗ F ( R ∗ ) p ∗ ( X,E ) ∩ ∆ F ( X,E ) ) p ( X,E ) p ◦ ( X,E ) = p ( X,E ) p ◦ ( X,E ) ( p ( X,E ) F ( E ) F ( S ∗ ) F ( G ) F ( R ∗ ) F ( E ) p ◦ ( X,E ) ∩ ∆ F ( X,E ) ) p ( X,E ) p ◦ ( X,E ) = p ( X,E ) p ◦ ( X,E ) ( p ( X,E ) F ( ES ∗ GR ∗ E ) p ◦ ( X,E ) ∩ ∆ F ( X,E ) ) p ( X,E ) p ◦ ( X,E ) = p ( X,E ) p ◦ ( X,E ) ( p ( X,E ) F ( S ∗ R ∗ ) p ◦ ( X,E ) ∩ ∆ F ( X,E ) ) p ( X,E ) p ◦ ( X,E ) = p ( X,E ) [ p ◦ ( X,E ) p ( X,E ) F ( S ∗ R ∗ ) p ◦ ( X,E ) p ( X,E ) ∩ p ◦ ( X,E ) p ( X,E ) ] p ◦ ( X,E ) = p ( X,E ) [ F ( E ∩ E ◦ ) F ( S ∗ R ∗ ) F ( E ∩ E ◦ ) ∩ F ( E ∩ E ◦ )] p ◦ ( X,E ) = p ( X,E ) ( F ( S ∗ R ∗ ) ∩ F ( E ∩ E ◦ )) p ◦ ( X,E ) = p ( X,E ) F ( S ∗ R ∗ ∩ E ∩ E ◦ ) p ◦ ( X,E ) = p ( X,E ) F (ΦΦ ◦ ) p ◦ ( X,E ) = p ( X,E ) F (Φ) F ( T ∩ T ◦ ) F (Φ ◦ ) p ◦ ( X,E ) = p ( X,E ) F (Φ) p ◦ ( A,T ) p ( A,T ) F (Φ ◦ ) p ◦ ( X,E ) = F (Φ ∗ ) F (Φ ∗ ) ◦ These tell us that F ( A, T ) F ( X, E ) F ( Y, G ) F (Φ ∗ ) F ( R ∗ ) F ( S ∗ ) is an inserter dia-gram in F .Next, consider an so -morphism R ∗ : ( X, E ) ։ ( Y, G ) ∈ E ex/reg . This means that R ∗ R ∗ = G and then we have F ( R ∗ ) ∗ F ( R ∗ ) ∗ = p ( Y,G ) ∗ F ( R ∗ ) p ∗ ( X,E ) p ( X,E ) ∗ F ( R ∗ ) p ∗ ( Y,G ) = p ( Y,G ) ∗ F ( R ∗ ) F ( E ) F ( R ∗ ) p ∗ ( Y,G ) = p ( Y,G ) ∗ F ( R ∗ R ∗ ) p ∗ ( Y,G ) = p ( Y,G ) ∗ F ( G ) p ∗ ( Y,G ) = I F ( Y,G ) from which we obtain that F ( R ∗ ) is an so -morphism in F . Thus, we have provedthat F is a regular functor.Finally, for any regular functor H : E ex/reg → F , for every object ( X, E ) ∈E ex/reg we must have an exact sequence H Γ E / / / / H Γ X / / / / H ( X, E ) in F .If H Γ ∼ = F , this forces H ∼ = F . (cid:3) A Characterization of the Exact Completion and Priestley Spaces
Having established the universal property of the exact completion, in this sectionwe present a result which identifies the situation in which an exact category is theexact completion of a given regular category E . More precisely, we will characterize N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 31 the canonical functor Γ :
E → E ex/reg as the unique up to equivalence functorfrom E into an exact category which satisfies some simple properties. This will inturn allow us to easily deduce some examples of categories which arise as exactcompletions of some familiar regular subcategory.The main example that we aim to cover here involves the category of Priestley spaces. Indeed, the latter is regular as a
Pos -category and we prove that its exactcompletion is the category of compact ordered spaces (or
Nachbin spaces). Thisprovides an ordered version of the folklore result which identifies the category ofcompact Hausdorff spaces as the exact completion (in the ordinary sense) of theregular category of
Stone spaces (see e.g. [1]).But first, we need some preliminaries. We will say that a functor F : C → D is order-faithful if, for every f, g : X → Y ∈ C we have F f ≤ F g = ⇒ f ≤ g . Inother words, F is order-faithful if for every X, Y ∈ C the morphism Hom C (X , Y) → Hom D (FX , FY) is an ff-morphism in
Pos . Note in particular that such a functoris faithful in the ordinary sense or, in more appropriate language, the underlyingfunctor between ordinary categories is faithful. In fact, if C has inserters which arepreserved by F : C → D , then the two notions coincide.The crux of the work now consists of establishing that certain properties of aregular functor F : E → F into an exact category F translate to correspondingproperties of the induced F : E ex/reg → F . Let F : E → F be a regular functor with F an exact category. If F is fully order-faithful, then F : E ex/reg → F is order-faithful.Proof. Let R ∗ , S ∗ : ( X, E ) → ( Y, G ) ∈ E ex/reg be such that F ( R ∗ ) ≤ F ( S ∗ ) in F .Then using the definition of F we have F ( R ∗ ) ∗ ⊇ F ( S ∗ ) ∗ = ⇒ p ∗ ( Y,G ) F ( R ∗ ) ∗ p ( X,E ) ∗ ⊇ p ∗ ( Y,G ) F ( S ∗ ) ∗ p ( X,E ) ∗ = ⇒ p ∗ ( Y,G ) p ( Y,G ) ∗ F ( R ∗ ) p ∗ ( X,E ) p ( X,E ) ∗ ⊇⊇ p ∗ ( Y,G ) p ( Y,G ) ∗ F ( S ∗ ) p ∗ ( X,E ) p ( X,E ) ∗ = ⇒ F ( G ) F ( R ∗ ) F ( E ) ⊇ F ( G ) F ( S ∗ ) F ( E )= ⇒ F ( R ∗ ) ⊇ F ( S ∗ )But since F is fully (order-) faithful, it reflects inclusions of subobjects and hencewe obtain R ∗ ⊇ S ∗ , i.e. R ∗ ≤ S ∗ in E ex/reg . (cid:3) We introduce some further properties of functors that will be of interest. Ourchoice of terminology follows the literature of Categorical Logic (e.g. [14]).
A functor F : C → D is called covering if, for every object Y ∈ D ,one can find an object X ∈ C and an effective epimorphism F X ։ Y .We say that F is full on subobjects if, for every ff-morphism B F X in D ,there exists an ff-morphism A X in C such that F A ∼ = B in Sub D (FX).The following basic observation (even for ordinary categories) seems to not haveappeared explicitly in the literature. Since we will need it below, we give its easyproof. Let F : C → D be a regular functor between regular categories. If F is full and covering, then it is full on subobjects.Proof. Consider an ff-morphism v : D F Y in D . Since F is covering, there existssome so-morphism q : F X ։ D in D . Now consider the composite F X q / / / / D / / v / / F Y .Since F is full, there is a morphism f : X → Y in C such that F f = vq .Since C is regular, we can factor f through its image, say f = X p / / / / I / / u / / Y .But F is a regular functor, so F X
F p / / / / F I / / F u / / F Y is the (so,ff) factorization of F f = vq . By uniqueness of such factorizations in the regular category D , wededuce that F I ∼ = D as subobjects of F Y . (cid:3) Let F : E → F be a regular functor with F an exact category.If F is fully order-faithful and covering, then F : E ex/reg → F is fully order-faithfuland covering.Proof. We saw earlier that F being fully order-faithful implies that F is order-faithful. Furthermore, it is immediate that F being covering implies the sameproperty for F , since we have F Γ ∼ = F .Now consider any morphism g : F ( X, E ) → F ( Y, G ) ∈ F . Let S ∗ : F X
F Y and S ∗ : F Y
F X denote the relations corresponding to this morphism via thebijection of Proposition 3.12, i.e. the relations S ∗ = p ∗ ( Y,G ) g ∗ p ( X,E ) ∗ and S ∗ = p ∗ ( X,E ) g ∗ p ( Y,G ) ∗ . Now since F is a full and covering regular functor, we know bythe previous lemma that it is also full on subobjects and so there exist relations R ∗ : X Y and R ∗ : Y X in E such that F ( R ∗ ) = S ∗ and F ( R ∗ ) = S ∗ .Furthermore, we have the following: F ( GR ∗ E ) = F ( G ) F ( R ∗ ) F ( E ) = F ( G ) S ∗ F ( E ) = S ∗ = F ( R ∗ ) F ( R ∗ R ∗ ) = F ( R ∗ ) F ( R ∗ ) = S ∗ S ∗ ⊇ F ( E ) F ( R ∗ R ∗ ) = F ( R ∗ ) F ( R ∗ ) = S ∗ S ∗ ⊆ F ( G )But now because F is fully (order-) faithful it reflects inclusions of subobjects.Thus, we deduce that GR ∗ E = R ∗ , R ∗ R ∗ ⊇ E and R ∗ R ∗ ⊆ G , so that R ∗ is amorphism ( X, E ) → ( Y, G ) ∈ E ex/reg .Finally, we have by definition of the functor F that F ( R ∗ ) ∗ = p ( Y,G ) ∗ F ( R ∗ ) p ∗ ( X,E ) = p ( Y,G ) ∗ S ∗ p ∗ ( X,E ) = p ( Y,G ) ∗ p ∗ ( Y,G ) g ∗ p ( X,E ) ∗ p ∗ ( X,E ) = g ∗ and hence F ( R ∗ ) = g . (cid:3) The final ingredient we need is the
Pos -enriched analogue of Lemma 1.4.9 from[14].
Let F : E → F be a regular functor between regular categories wheremoreover E is exact. If F is fully order-faithful and covering, then it is an equiva-lence.Proof. It suffices to show that F is essentially surjective on objects. So let Y ∈ F .By the assumption that F is covering, we can find a coinserter q : F X ։ Y in F for some object X ∈ E . Consider the kernel congruence q/q q / / q / / F X in F .Since F is a covering and full regular functor, we know that it must be also full onsubobjects. In particular, there is a relation E e / / e / / X in E such that F ( E ) = q/q as relations in F .Now because F is order faithful, F ( E ) = q/q being a congruence in F impliesthat E is a congruence on X in E . Since E is assumed to be exact, E has a coinserter,say p : X → P , and is the kernel congruence of that coinserter. Now by regularityof the functor F we have that F p : F X ։ F P is the coinserter of F ( E ) = q/q .Since q : F X → Y is also a coinserter of q/q , we deduce that there exists an iso F P ∼ = Y . (cid:3) Now putting everything together we have proved the following result.
Let F : E → F be a regular functor with F an exact category. Then F : E ex/reg → F is an equivalence iff F is fully order-faithful and covering. N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 33
In other words, if have found a regular functor F : E → F into an exact category F and F is fully faithful and covering, then we have identified F as E ex/reg . Thiscan be immediately applied to produce some examples of categories which are exactcompletions of regular categories. (1) Consider Set viewed as
Pos -category with discrete hom-sets. We have seen that it is regular but not exact. The discrete posetfunctor D : Set → Pos is clearly fully faithful and regular. It is also triviallycovering, since for any poset ( X, ≤ ) we have an order-preserving surjection( X, =) ։ ( X, ≤ ). Thus, by Theorem 5.6 we have that Set ex/reg ≃ Pos .(2) Consider the category
OrdCMon of ordered commutative monoids, which isa variety of ordered algebras in the sense of Bloom & Wright [3] and henceis an exact category [12]. Recall that an ordered commutative monoid M is torsion-free if it satisfies nx ≤ ny = ⇒ x ≤ y for any x, y ∈ M and n ∈ N ∗ . Thus, the full subcategory on such monoids OrdCMon t.f. is aregular category since it is an ordered quasivariety. Then by Theorem 5.6we see that (
OrdCMon t.f. ) ex/reg ≃ OrdCMon . Indeed, every ordered monoidadmits a surjective homomorphisms from a free one and clearly every freeordered monoids is torsion-free.(3) The ordinary category
Mon of monoids is regular, hence also regular asa locally discrete
Pos -category. The inclusion functor
Mon ֒ → OrdMon isregular and any ordered monoid ( M, ≤ ) admits a surjective homomorphism( M, =) ։ ( M, ≤ ). It follows then from Theorem 5.6 that Mon ex/reg ≃ OrdMon .(4) It is easy to see that in the above example there is nothing special aboutthe variety of monoids. Indeed, any ordinary quasivariety gives rise toa corresponding quasivariety of ordered algebras defined by the same setof axioms. The ordinary (unordered) version sits inside the ordered oneas the discrete ordered algebras and as such is a regular subcategory. Itfollows then that its exact completion qua
Pos -category yields preciselythe corresponding ordered quasivariety. Thus, for example, in the case ofsemigroups we similarly have
SGrp ex/reg ≃ OrdSGrp .The examples presented so far of exact completions have all been varieties ofordered algebras which appear as completions of certain corresponding quasiva-rieties. However, the main example we would like to present in this section isorder-topological in nature and involves the category of Priestley spaces. Let usthus first recall some terminology.We will say that a triple (
X, τ, ≤ ) with τ a topology and ≤ a partial order relationon the set X is an ordered topological space .A compact ordered space is an ordered topological space ( X, τ, ≤ ) such that( X, τ ) is compact and ≤ is closed as a subspace of X × X . This class of spaces wasintroduced and developed by L. Nachbin in [16] as an ordered analogue of compactHausdorff spaces, and so we will also call these Nachbin spaces. Together with thecontinuous order-preserving functions between them they form a category which wedenote by
Nach . Note that, under the assumption of compactness, the conditionthat the order relation be closed in the product space X × X means the following:whenever x, y ∈ X with x (cid:2) y , there exist a open upper set U and an open lowerset V such that x ∈ U , y ∈ V and U ∩ V = ∅ .Inside Nach sits the very interesting full subcategory
Pries of Priestley spaces.This is a class of ordered topological spaces introduced by H. A. Priestley [17] inorder to provide an extension of Stone duality to distributive lattices. In otherwords there is an equivalence of categories
DLat op ≃ Pries , where
DLat denotes thecategory of (bounded) distributive lattices and lattice homomorphisms.
Recall then than an ordered topological space (
X, τ, ≤ ) is a Priestley space if(
X, τ ) is compact and the following is satisfied: whenever x (cid:2) y , there exists aclopen upper set U such that x ∈ U and y / ∈ U . Ordered spaces satisfying thelatter condition are often called totally order-separated . It is immediate that everyPriestley space is indeed a Nachbin space. It is furthermore clear that the underlyingtopological space of a Priestley space is a Stone space. In fact, the category Stone ofStone spaces is embedded in
Pries as the full subcategory on the objects for whichthe order relation is discrete.
The category
Nach is exact, while the category
Pries is regular.Proof.
Observe that the coinserter
E X X/E ∩ E ◦ q of any internalcongruence E X × X in Nach is constructed by equipping the set
X/E ∩ E ◦ withthe quotient topology and the induced order relation by the pre-order E . Since E is closed in X × X , so is the equivalence relation E ∩ E ◦ and so at the level ofspaces we know that the quotient will be a compact Hausdorff space. It is then aNachbin space because the order relation is by definition equal to ( q × q )[ E ] andthe map q × q is closed.It now follows that the effective epimorphisms in Nach are precisely the contin-uous monotone surjections. Indeed, if f : X → Y ∈ Nach is surjective then on thelevel of spaces it is a continuous surjection between compact Hausdorff spaces andhence it is a quotient map. This means that the induced ¯ f : X/R → Y is a home-omorphism, where R = { ( x, x ′ ) | f ( x ) = f ( x ′ ) } = E ∩ E ◦ , for E := f /f . But ¯ f alsopreserves and reflects the order because by definition we have ¯ f ([ x ]) ≤ ¯ f ([ x ′ ]) ⇐⇒ f ( x ) ≤ f ( x ′ ) ⇐⇒ ( x, x ′ ) ∈ E ⇐⇒ [ x ] ≤ [ x ′ ]. Thus, f is the coinserter of itskernel congruence.This shows that Nach is regular, since the continuous monotone surjections areclearly stable under pullback. To see that
Pries is regular, it suffices to observe thatthe latter is closed under finite limits and subobjects in
Nach .Finally, consider any internal congruence E X × X in Nach and construct itscoinserter q as we did above. It is then immediate by the construction that E = q/q and so we have proved that Nach is exact. (cid:3)
We can now deduce an ordered version of the folklore result which identifies theexact completion of
Stone as the category of compact Hausdorff spaces. The latterseems to have first appeared in print in [1] where a similar argument involving theordinary exact completion was invoked.
Pries ex/reg ≃ Nach ≃ Stone ex/reg .Proof.
The inclusions
Stone ֒ → Pries ֒ → Nach are both regular functors. Further-more, if X ∈ Nach , then X is in particular a compact Hausdorff space and so admitsa continuous surjection β ( X ) ։ X from a Stone space β ( X ), the latter being theStone-Cech compactification of the discrete set X . Equipping β ( X ) with the equal-ity relation this becomes a continuous monotone surjection in Nach . Thus, bothinclusion functors are also covering and the result follows from Theorem 5.6. (cid:3)
Before ending this section, let us record a small observation that generalizes someof the examples we have seen so far. To this effect, recall that some varieties ofordered algebras were described as exact completions of certain ordinary varietieswhich appeared as the objects with discrete order relation. For example, we had
Mon ex/reg ≃ OrdMon for the category of ordered monoids. Similarly, in the contextof the above corollary we could have included the equivalence
CHaus ex/reg ≃ Nach ,where
CHaus denotes the locally discrete category of compact Hausdorff spaces.
N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 35
More generally now, consider any regular category E and define an object X ∈ E to be discrete if for every f, g : A → X ∈ E we have that f ≤ g = ⇒ f = g . Ifwe denote by Dis ( E ) the full subcategory on the discrete objects, then it is plainthat Dis ( E ) is a locally discrete category which is closed under finite limits andsubobjects in E . Thus, Dis ( E ) is a regular category as well. We will say that E has enough discrete objects if for every object X ∈ E there exists an so -morphism D ։ X in E with D ∈ Dis ( E ). By another application of Theorem 5.6 we nowdeduce the following: Let E be an exact category with enough discrete objects. Then, E ≃
Dis ( E ) ex/reg . Internal Posets and Exact Completion
In this final section we consider the process of taking internal posets in an ordi-nary category and how the ordinary and enriched notions of regularity and exactnessare related through said process. Furthermore, we prove a type of commutationbetween this construction of internal posets and that of exact completion.To begin with, suppose that C is any finitely complete ordinary category. Wecan then define a category Ord ( C ) as follows: • Objects: are pairs ( X, ≤ X ), where X is an object of C and ≤ X X × X isa partial order relation in C . • Morphisms: A morphism f : ( X, ≤ X ) → ( Y, ≤ Y ) ∈ Ord ( C ) is a morphism f : X → Y ∈ C such that f ( ≤ X ) ⊆≤ Y .The condition f ( ≤ X ) ⊆≤ Y means that there is a commutative diagram in C of theform ≤ X ≤ Y X × X Y × Y f × f Composition of morphisms and identities are those of C .Furthermore, given morphisms f, f ′ : ( X, ≤ X ) → ( Y, ≤ Y ) ∈ Ord ( C ), we define f ≤ f ′ to mean that there exists a commutative diagram X ≤ Y Y × Y h f,g i Now it is easy to see that this order relation on morphisms of
Ord ( C ) is preservedby composition. For example, if f, f ′ : ( X, ≤ X ) → ( Y, ≤ Y ) ∈ Ord ( C ) with f ≤ f ′ and g : ( Y, ≤ Y ) → ( Z, ≤ Z ), we have gf ≤ gf ′ by pasting the following commutativediagrams X ≤ Y ≤ Z Y × Y Z × Z h f,f ′ i g × g Thus,
Ord ( C ) is enriched in Pos . Our first observation below is that finite com-pleteness of the ordinary category C implies the existence of all finite weighted limitsin Ord ( C ). If C is finitely complete, then Ord ( C ) has finite weighted limits.Proof. It is easy to see that ( X, ≤ X ) ( X × Y, ≤ X × ≤ Y ) ( Y, ≤ Y ) π X π Y is a product diagram for every ( X, ≤ X ) , ( Y, ≤ Y ) ∈ Ord ( C ). Let us show how to construct the inserter of a pair of morphisms ( X, ≤ X ) ( Y, ≤ Y ) fg . For this,form the following pullback square in C E ≤ Y X Y × Y e e ′ h f,g i Let ≤ E be the restriction of ≤ X to the subobject E X , i.e. ≤ E = ( E × E ) ∩ ≤ X as subobjects of X × X . It is easy to see that ≤ E is itself an internal partial orderrelation on E ∈ C so that we have a morphism e : ( X, ≤ E ) ∈ Ord ( C ). Also, bycommutativity of the pullback square above we have f e ≤ ge .Now let h : ( Z, ≤ Z ) → ( X, ≤ X ) be such that f h ≤ gh . This means that h f h, gh i = h f, g i h factors through ≤ Y Y × Y , say via u : Z →≤ Y , so thenby the pullback property there exists a unique v : Z → E satisfying ev = h , e ′ v = u .Finally, e is an ff -morphism in Ord ( C ) by definition of ≤ E . Indeed, for any( Z, ≤ Z ) ( E, ≤ E ) hh ′ , the inequality eh ≤ eh ′ means that ( e × e ) h h, h ′ i factorsthrough ≤ X , which implies h h, h ′ i factors through ≤ E = ( E × E ) ∩ ≤ X , i.e. that h ≤ h ′ . (cid:3) Since it will be needed later, let us also record here how to construct commasquares ( C, ≤ C ) ( Y, ≤ Y )( X, ≤ X ) ( Z, ≤ Z ) c c ≤ gf in Ord ( C ). This is accomplished by constructing the following pullback square in C C ≤ Z X × Y Z × Z h c ,c i f × g and then setting ≤ C := ( C × C ) ∩ ( ≤ X × ≤ Y ).Before moving on, let us also discuss ff -morphisms in Ord ( C ). We saw in thecourse of the previous proof that an m : ( X, ≤ X ) → ( Y, ≤ Y ) ∈ Ord ( C ) with m : X → Y ∈ C monic and ≤ X = ( X × X ) ∩ ≤ Y is an ff -morphism in Ord ( C ). Itis in fact not too hard to see that this completely characterizes ff -morphisms inOrd( C ). Indeed, assume that m : ( X, ≤ X ) → ( Y, ≤ Y ) ∈ Ord ( C ) is an ff -morphism.If f, g : Z → X ∈ C are such that mf = mg , then we can also consider them asmorphisms f, g : ( Z, ∆ Z ) → ( X, ≤ X ) in Ord ( C ) and hence deduce that f = g . Thisproves m must be a monomorphism in C . Now arguing with generalized elementsone can easily deduce that ≤ X is indeed the restriction of ≤ Y along m : X Y . If f : ( X, ≤ X ) → ( Y, ≤ Y ) ∈ Ord ( C ) is such that the underlying f : X → Y is a strong epimorphism in C , then f is an so -morphism in Ord ( C ) . N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 37
Proof.
Consider the following commutative square in
Ord ( C ), where we assume m : ( M, ≤ M ) → ( Z, ≤ Z ) is an ff -morphism.( X, ≤ X ) ( Y, ≤ Y )( M, ≤ M ) ( Z, ≤ Z ) fh gm In particular, by the preceding discussion we know that m is monic in C and soby the property of f as a strong epimorphism in the latter category we deduce theexistence of a u : Y → M such that uf = h and mu = g . The fact that u is actuallya morphism in Ord ( C ) follows because mu = g is such a morphism and because m being an ff -morphism also means that ≤ M = ( M × M ) ∩ ≤ Z . (cid:3) We can now prove that ordinary regularity of C implies (enriched) regularity for Ord ( C ). Note that this result is in some sense a special case of Proposition 62 in [5],where the authors prove a form of 2-categorical regularity for the 2-category Cat ( E )of internal categories in a regular (1-)category E . Nevertheless, we include a proofhere in order to make the paper more self-contained. If C is an ordinary regular category, then Ord ( C ) is regular.Proof. Consider any f : ( X, ≤ X ) → ( Y, ≤ Y ) ∈ Ord ( C ) and let X M Y p m be the (regular epi,mono) factorization of f in C . Then by what we have alreadyestablished earlier, upon setting ≤ M := ( M × M ) ∩ ≤ Y , we obtain an ( so , ff ) factor-ization ( X, ≤ X ) ( M, ≤ M ) ( Y, ≤ Y ) p m in Ord ( C ).Now the existence of these factorizations together with the previous lemma implythat any f : ( X, ≤ X ) → ( Y, ≤ Y ) ∈ Ord ( C ) is an so -morphism in Ord ( C ) iff f : X → Y is a regular(=strong) epi in C . Since pullbacks in Ord ( C ) are constructed bysimply taking the pullback of the underlying morphisms in C , pullback-stability ofregular epimorphisms in C implies pullback-stability of so -morphisms in Ord ( C ). (cid:3) Similarly, ordinary exactness of C implies Pos -enriched exactness of
Ord ( C ).Again, this is a special case of Proposition 63 in [5]. If C is an ordinary exact category, then Ord ( C ) is exact.Proof. Suppose that ( E ≤ E ) ( X, ≤ X ) × ( X, ≤ X ) h e ,e i is a congruence on ( X, ≤ X ) ∈ Ord ( C ). Then we have a monomorphism E X × X ∈ C , i.e. a relation E on X in C . Reflexivity and transitivity of the congruence in Ord ( C ) imply the sameproperties for the relation E : X X in C . Then R := E ∩ E ◦ is an equivalencerelation on X in C .Since C is exact, there exists an exact sequence R X Q q in C ,which means that q is the coequalizer of R and R is the kernel pair of q . Let ≤ Q := q ( E ) be the image of E along q in C . Note that in the calculus of relation inthe ordinary regular category C we can write q ( E ) = qEq ◦ , while exactness of thesequence is equivalent to q ◦ q = R and qq ◦ = ∆ Q .Now observe that ≤ Q is indeed an internal partial order relation in C . It isreflexive because E is so and q is a regular epimorphism. For transitivity we argueas follows ≤ Q ◦ ≤ Q = qEq ◦ qEq ◦ = qEREq ◦ = qE ( E ∩ E ◦ ) Eq ◦ = qEq ◦ = ≤ Q Finally, for anti-symmetry we have ≤ Q ∩ ≤ ◦ Q = qq ◦ ( ≤ Q ∩ ≤ ◦ Q ) qq ◦ = q ( q ◦ ≤ q q ∩ q ◦ ≤ ◦ Q q ) q ◦ = q ( E ∩ E ◦ ) q ◦ = qq ◦ qq ◦ = ∆ Q Now we claim that ( E, ≤ E ) ( X, ≤ X ) e e is the kernel congruence of q :( X, ≤ X ) → ( Q, ≤ Q ) in Ord ( C ). So let g , g : ( Z, ≤ Z ) → ( X, ≤ X ) be such that qg ≤ qg . In terms of generalized elements in C this means that ( qg , qg ) ∈ Z ≤ Q ,which in turn is equivalent to ( g , g ) ∈ Z q − ( ≤ Q ). But now observe that q − ( ≤ Q ) = q − ( q ( E )) = q ◦ qEq ◦ q = RER = ( E ∩ E ◦ ) E ( E ∩ E ◦ ) = E Thus, ( g , g ) ∈ Z E as desired. (cid:3) In particular, if C is an ordinary regular category and C oex/reg is its ordinaryexact completion, then Ord ( C oex/reg ) is an exact Pos -category. In the remainderof this paper we want to prove that the latter category is equivalent to the exactcompletion in the enriched sense of both
Ord ( C ) and C itself.Consider a regular functor F : C → D between ordinary regular categories.Since F preserves internal partial order relations, we have an induced functor Ord ( F ) : Ord ( C ) → Ord ( D ) defined on objects by ( X, ≤ X ) ( F X, F ( ≤ X )). Bythe construction of finite weighted limits in Ord ( C ), the fact that F preserves finitelimits implies the same for the enriched functor Ord ( F ). Similarly, F preservingregular epimorphisms translates to the fact that Ord ( F ) preserves so -morphisms.Thus, Ord ( F ) is a regular functor between regular Pos -categories.We now turn to discussing how the properties of F being fully faithful andcovering translate to properties of Ord ( F ). Let F : C → D be an ordinary regular functor which is fully faithful.Then
Ord ( F ) : Ord ( C ) → Ord ( D ) is fully order-faithful.Proof. It is clear that
Ord ( F ) is faithful, since its action on morphisms is that of F itself. Since Ord ( C ) has finite limits and Ord ( F ) preserves them, this is equivalentto order-faithfulness.Now consider any h : ( F X, F ( ≤ X )) → ( F Y, F ( ≤ Y )). By fullness of F , thereexists an f : X → Y ∈ C such that F f = h . It suffices then to show that f isorder-preserving. For this, observe that F f = h being a morphism in Ord ( D ) meansthat there is a commutative diagram in D of the form F ( ≤ X ) F ( ≤ Y ) F ( X × X ) ∼ = F X × F X F Y × F Y ∼ = F ( Y × Y ) F f × F f
By full faithfulness of F this is then reflected to a commutative diagram in C whichexhibits f as a morphism ( X, ≤ X ) → ( Y, ≤ Y ) in Ord ( C ). (cid:3) Let F : C → D be an ordinary regular functor which is fully faithfuland covering. Then
Ord ( F ) : Ord ( C ) → Ord ( D ) is covering.Proof. Consider any object ( Y, ≤ Y ) ∈ Ord ( D ). Since F is covering, we can find aregular epimorphism q : F X ։ Y in D . Set ≤ F X := q − ( ≤ Y ). Then ≤ F X is aninternal partial order relation in D .Now because F is a full and covering regular functor, it is full on subobjects.Thus, there is a relation ≤ X X × X in C such that F ( ≤ X ) = ≤ F X . In addition,full faithfulness of F implies that the properties making ≤ F X a partial order arereflected to C . Hence ≤ X is an internal partial order relation on X in C .Finally, q : Ord ( F )( X, ≤ X ) ։ ( Y, ≤ Y ) is an so -morphism in Ord ( D ) because itsunderlying q : F X ։ Y is a regular epimorphism in D . (cid:3) Putting everything together we now obtain the main result of this section.
N EXACT COMPLETION FOR REGULAR CATEGORIES ENRICHED IN POSETS 39
For any regular ordinary category C there is an equivalence of Pos -categories
Ord ( C oex/reg ) ≃ Ord ( C ) ex/reg ≃ C ex/reg , where C oex/reg denotes theexact completion of C as an ordinary category.Proof. The ordinary regular functor Γ :
C → C oex/reg is fully faithful and covering.By the preceding lemmas we have then that
Ord (Γ) :
Ord ( C ) → Ord ( C oex/reg )satisfies the same properties in the enriched sense. Furthermore, the category Ord ( C oex/reg ) is exact by Proposition 6.4 and thus from Theorem 5.6 we deducethat Ord ( C oex/reg ) ≃ Ord ( C ) ex/reg .Similarly, the composite functor C →
Ord ( C ) → Ord ( C oex/reg ) is regular, fullyfaithful and covering, being a composition of two functors satisfying these proper-ties. Again by Theorem 5.6 we conclude that C ex/reg ≃ Ord ( C oex/reg ). (cid:3) References [1] V. Aravantinos-Sotiropoulos, P. Karazeris,
A property of effectivization and its uses in Cate-gorical Logic , Theory and Applications of Categories, Vol. 32, 2017, No. 22, pp. 769-779.[2] M. Barr,
Exact Categories , Lecture Notes in Mathematics, 236 Springer (1970) 1-120.[3] S. L. Bloom, J. B. Wright,
P-varieties - A signature independent characterization of varietiesof ordered algebras , Journal of Pure and Applied Algebra Handbook of Categorical Algebra vol. 2 , Cambridge University Press.[5] J. Bourke, R. Garner,
Two-dimensional regularity and exactness , Journal of Pure and AppliedAlgebra 218 (2014) 1436-1371.[6] S. Bulman-Fleming, M. Mahmoudi,
The Category of S-Posets , Semigroup Forum Vol. 71(2005) 443-461.[7] P. J. Freyd, A. Scedrov,
Categories, allegories , Mathematical Library Vol 39, North-Holland(1990).[8] P.T. Johnstone,
Sketches of an Elephant: A Topos Theory Compendium , Oxford Logic Guides[9] G. M. Kelly,
Basic Concepts of Enriched Category Theory , London Math. Soc. Lec. NoteSeries 64, Cambridge Univ. Press 1982.[10] G. M. Kelly,
A Note on Relations relative to a Factorization System , Proceedings of theconference “Category Theory ’90”held in Como, Italy, July 22-28 1990.[11] G. M. Kelly,
Structures defined by finite limits in the enriched context , Cah. Top. Geom. DiffCategoriques, tome 23, no1 (1982), p. 3-42[12] A. Kurz, J. Velebil,
Quasivarieties and varieties of ordered algebras: regularity and exactness ,Math. Struct. in Comp. Science (2017), vol. 27, pp. 1153-1194.[13] F.W. Lawvere,
Teoria delle categorie sopra un topos di base , lecture notes from Perugia(1972-73).[14] M. Makkai, G. E. Reyes,
First Order Categorical Logic , LNM 611, Springer-Verlag, Berlin1977[15] J. Meisen,
On bicategories of relations and pullback spans , Communications in Algebra1(1974), 377-401.[16] L. Nachbin,
Topology and Oder , Van Nostrand, Princeton, NJ, 1965.[17] H. A. Priestley,
Ordered topological spaces and the representation of distributive lattices ,Proc. London Math.Soc.3, 3 (1972), 507.[18] G. Richter,
Mal’cev conditions for categories , Categorical Topology, Proc. Conference Toledo,Ohio 1983 (Heldermann Verlag, Berlin, 1984), 453-469.[19] R. Succi Cruciani,
La teoria delle relazioni nello studio di categorie regolari e di categorieesatte , Riv. Mat. Univ. Parma (4) 1 (1975), 143-1580 VASILEIOS ARAVANTINOS-SOTIROPOULOS