LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES

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1 Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES MARIA MANUEL CLEMENTINO AND IGNACIO LÓPEZ FRANCO Abstract: We give an account o lax orthogonal actorisation systems on orderenriched categories. Among them, we deine and characterise the kz-relective ones, in a way that mirrors the characterisation o relective orthogonal actorisation systems. We use simple monads to construct lax orthogonal actorisation systems, such as one on the category o T 0 topological spaces closely related to continuous lattices. Keywords: lax orthogonal actorization system, lax idempotent monad, orderenriched category, weak actorization system, relective actorization system, continuous lattice Math. Subject Classiication: Primary 18D05, 18A32. Secondary 55U Introduction Weak actorisation systems (wss) have been a eature o Homotopy Theory even beore Quillen s deinition o model categories and the recognition o their importance. Wss, whose deinition can be ound in 4.a, can be described as a pair o classes o morphisms pl, Rq that satisy three properties. First, each morphism o the category must be a composition o a morphism rom L ollowed by one o R (in a not necessarily unique way). Secondly, each r P R must have the right liting property with respect to each l P L; in other words, each commutative square, as displayed, has a (not necessarily unique) diagonal iller. l r (1.1) Received January 30, Research partially supported by Centro de Matemática da Universidade de Coimbra UID/MAT/00324/2013, unded by the Portuguese Government through FCT/MCTES and counded by the European Regional Development Fund through the Partnership Agreement PT2020. The second author was supported by a Research Fellowship o Gonville and Caius College, Cambridge, the Department o Mathematics and Mathematical Statistics o the University o Cambridge, PEDECIBA, and SNI ANII. 1

2 2 M M CLEMENTINO AND I LÓPEZ FRANCO Lastly, pl, Rq is, in a precise way, maximal. Each one o Quillen s model categories comes equipped with two ws (by deinition). Orthogonal actorisations systems (os) arose at the same time as wss and can be described as wss in which the diagonal iller (1.1) not only exists but it is unique. This makes the actorisation o a morphism as r l, with l P L and r P R, unique up to unique isomorphism. Two typical examples o oss are the actorisation o a unction as a surjection ollowed by an injection, and o a continuous map between topological spaces as a surjection ollowed by an embedding (ie an homeomorphism onto its image). When the ambient category has a terminal object, denoted by 1, there is a case o (1.1) o special interest, namely: l A 1 (1.2) I the unique morphism A Ñ 1 has the right (unique) liting property with respect to l, one says that A is injective with respect (resp., orthogonal to) l. Clearly each os pl, Rq gives rise to a class o objects that are orthogonal to each member o L: those objects A such that A Ñ 1 belongs to R. The extent to which pl, Rq is determined by this class o objects is the subject o study o [5]. The oss so determined are called relective. In addition to their widespread use in Homological Algebra, injective objects play a role in many other areas o Mathematics. For example, in the category o metric spaces and non-expansive maps, hyperconvex spaces are the objects injective with respect to the amily o isometries (see [2] and [15]). There are examples, as those introduced by D. Scott [29], o squares (1.2) where the diagonal iller is not unique but there exists a smallest one (with respect to an ordering between morphisms). The main example rom [29] consists o those topological spaces that arise rom endowing continuous lattices with the Scott topology. These spaces are characterised by their injectivity with respect to topological embeddings. In act, i l is a topological embedding and A is a continuous lattice in (1.2), there is a diagonal iller that is the smallest with respect to the (opposite o) the pointwise specialisation order (see 13 or more details). Another example comes rom complete lattices, which can be characterised as those posets that are injective with respect to embeddings o posets. As

3 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 3 in the previous example, in the situation (1.2) where A is a complete lattice and l is a poset embedding, there exists a smallest diagonal iller. Motivated by the above examples, one can generalise the existence o a smallest diagonal iller in the situation (1.2) to the situation o a commutative square (1.1). By doing so, one arrives to the notion o lax orthogonal actorisation system. The present paper gives an account, in the context o order-enriched categories, o lax orthogonal actorisation systems (los), a notion that sits between oss and wss. orthogonal lax orthogonal weak actorisation Ă actorisation Ă actorisation system system system Loss were introduced and studied in the context o 2-categories by the authors in [7]. We cover here some o the same material in the much simpler ramework o order-enriched categories and some completely new results on relective loss, as well as new examples (see below). In a los, the existence o a diagonal iller (1.1) is replaced by the existence o a smallest diagonal iller. More precisely, there is a diagonal iller d with the property that d ď d 1 or any other diagonal iller d 1. l d ď d 1 r Since morphisms between two objects in an order-enriched category orm a poset, the above property uniquely deines the smallest diagonal iller. There are, however, advantages in providing these diagonals by means o an algebraic structure, instead o postulating the existence o a smallest diagonal iller. This algebraic structure is provided by the algebraic weak actorisation systems (awss), introduced with a dierent name in [14] and slightly modiied in [13]; we use the deinition given in the latter. An aws on an order-enriched category C consists o a locally monotone comonad L and a locally monotone monad R on C 2 interrelated by axioms, and that deine a locally monotone unctorial actorisation R L. Inspired by the observation o [14] that oss correspond to awss whose monad and comonad are idempotent, we deined in [7] loss as awss whose monad and comonad are lax idempotent, or Kock-Zöberlein. We reprise this

4 4 M M CLEMENTINO AND I LÓPEZ FRANCO deinition in the context o order-enriched categories, which enables some simpliications. A undamental example o los on the order-enriched category o posets actors each morphism as a let adjoint right inverse (or lari) ollowed by a split opibration. This actorisation can be constructed on any order-enriched category with suicient (inite) limits, and plays a similar role or loss as the actorisation isomorphism morphism (that actors as 1 dompq ollowed by ) plays or oss ( 4.d). We introduce kz-relective loss as those loss pl, Rq that are determined by the restriction o the monad R on C 2 to C (here C is viewed as the ull subcategory o C 2 with objects o the orm A Ñ 1). We characterise kzrelective loss in a way that mirrors the characterisation in [5] o relective oss pl, Rq as those with the ollowing property: i g and g belong to L, then so does g ( 9). For example, the los o lari split opibration mentioned above will be relective with our deinition. Another contribution o [5] was the construction o relective oss rom the so-called simple relections. The morphisms inverted by them always orm a let class o an os. We introduce simple monads in the order-enriched context, as those satisying a certain property that allows us to build loss. Ater providing suicient conditions or a lax idempotent monad to be simple ( 11), we recover the example o topological spaces discussed above in this introduction as a consequence o the simplicity o a certain monad: the ilter monad, which associates to each topological space the space o ilters o its open subsets endowed with a natural topology ( 13). The algebras or the ilter monad are precisely the continuous lattices (with the Scott topology). The induced los on (T 0 ) topological spaces has an associated ws that was considered in [4]. We also provide easy-to-veriy conditions guaranteeing that a submonad o a simple lax idempotent monad enjoys these same properties ( 12). When applied to the ilter monad we obtain loss closely related to continuous Scott domains, stably compact spaces and sober spaces. Another example that we obtain rom a simple monad is a los on the order-enriched category o (skeletal) generalised metric spaces 14. The restriction o this los to the category o metric spaces yields an os whose let class o morphisms are the dense inclusions. Further examples are explored in [6] in a very general ramework that covers, or example, R. Lowen s approach spaces as well as the examples mentioned above.

5 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 5 An appendix A discusses part o the theory o loss that can be developed in the setting o locally presentable categories, where, under mild hypotheses, there is a relection between the category o accessible lax idempotent monads and the category o accessible loss. The appendix demands more knowledge o some parts o Category Theory. 2. Order-enriched categories and lax idempotent monads By an ordered set we shall mean what is usually called a poset, that is, a pair px, ďq where X is a set and ď is a relation on X that is relexive, transitive and antisymmetric. Ordered sets can be identiied with small categories with at most one morphism between any two objects and whose isomorphisms are identity morphisms. We denote by Ord the category o ordered sets and monotone maps (unctions that preserve ď). This is a cartesian closed category, with exponential Y X deined as the set o all order-morphisms X Ñ Y, and endowed with the pointwise order. A category enriched in Ord, or Ord-category, is a locally small category C whose hom-sets are equipped with an order structure, and whose composition preserves the inequality: i g ď g 1 then hg ď hg 1 and g ď g 1, whenever these compositions are deined. In other words, the composition unctions CpY, Zq ˆ CpX, Y q ÝÑ CpX, Zq are monotone maps. The category Ord o ordered sets can be regarded as a ull subcategory o the category o small categories Cat by regarding ordered sets as small categories, as mentioned above. This means that Ord-categories can be regarded as 2-categories, but we do not go to that level o generality. A locally monotone unctor F : C Ñ D, or Ord-unctor, between Ordcategories is an ordinary unctor between the underlying ordinary categories that is moreover monotone on homs; ie that each CpX, Y q Ñ DpF X, F Y q is a monotone map. The category o Ord-categories and Ord-unctors will be denoted by Ord-Cat. It is a cartesian closed category. Example 2.1. The category Ord has a canonical structure o an Ord-category Ord whose ordered sets are OrdpX, Y q Y X. Many other categories

6 6 M M CLEMENTINO AND I LÓPEZ FRANCO constructed rom Ord are Ord-enriched, such as the categories o joinsemilattices, complete lattices, distributive lattices, and Heyting algebras. Example 2.2. I X is a topological space, deine a preorder on X by x ď y i all the neighbourhoods o y are also neighbourhoods o x, or, equivalently, denoting by OX the topology o X, x P U whenever y P U or every U P OX; in other words, x ď y i y P txu. The opposite o this order is usually called the specialisation order and was introduced by D. Scott in [29]. The preorder px, ďq is an ordered set precisely when X is a t 0 space. Any continuous unction : X Ñ Y between topological spaces preserves the order ď. The category Top 0 o t 0 topological spaces and continuous maps can be endowed with an Ord-category structure i we deine, or any pair, g : X Ñ Y o continuous unctions, ď g i pxq ď gpxq or all x P X. 2.a. Full morphisms and locally ull unctors. Deinition 2.3. (1) A monotone map between ordered sets is ull i it relects inequalities; ie pxq ď pyq implies x ď y. (2) A locally monotone unctor F : A Ñ B between Ord-categories is locally ull i each monotone map F A,B : ApA, Bq ÝÑ BpF A, F Bq is ull. (3) A morphism g : X Ñ Y in an Ord-category C is ull i or each Z P C the monotone map is ull. CpZ, gq: CpZ, Xq ÝÑ CpZ, Y q Full morphisms are necessarily monomorphisms; or i : X Ñ Y is a ull monotone morphism o ordered sets and pxq pyq, then we have both x ď y and y ď x, so x y. Lemma 2.4. Suppose that F % U : B Ñ A is an adjunction o locally monotone unctors between Ord-categories, with unit η : 1 A ñ UF. Then F is locally ull i each component η A : A Ñ UF A is a ull morphism.

7 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 7 Proo : The naturality o η is expressed by the commutativity o the ollowing diagram. ApA, Bq F A,B BpF A, F Bq U F A,F B ApUF A, UF Bq Ap1,η B q Apη A,1q ApA, UF Bq I η B is ull, the diagonal morphism is ull and thereore F A,B must be ull too. 2.b. Order-enriched (co)limits. Limits. The category o ordered sets admits the construction o two-dimensional limits, which will be undamental or us. We denote by 2 the order with two elements 0 ď 1. I X is an ordered set, then the exponential X 2 is X 2 tpx, yq P X ˆ X : x ď yu Ď X ˆ X with the order inherited rom X ˆ X. We denote by d 0 and d 1 the two projections rom X 2 onto X. Slightly more involved is the comma-object o two order morphisms : X Ñ Z Ð Y : g Ó g tpx, yq P X ˆ Y : pxq ď gpyqu Ď X ˆ Y that can equally well be constructed rom Z 2 by taking the limit o the ollowing diagram. X ÝÑ Z d 0 ÐÝ Z 2 d ÝÑ 1 Z ÐÝ g Y The constructions o the previous paragraphs can be deined in any Ordcategory C. I X P C, then deine X 2 as an object equipped with two morphisms d 0 ď d 1 : X 2 Ñ X that induce isomorphisms o orders CpZ, X 2 q CpZ, Xq 2 or all Z P C, in the sense that, or each pair o morphisms 0 ď 1 : Z Ñ X, there exists a unique morphism h: Z Ñ X 2 such that 0 d 0 h and 1 d 1 h. Furthermore, i k : Z Ñ X 2 is another morphism, then the conjunction o d 0 h ď d 0 k and d 1 h ď d 1 k implies h ď k. This universal property guarantees that X 2 is unique up to canonical isomorphism. Similarly, given morphisms : X Ñ Z Ð Y : g in C, one can deine a comma-object Ó g in C as an object equipped with two morphisms d 0 and

8 8 M M CLEMENTINO AND I LÓPEZ FRANCO d 1 as shown X d 0 Ó g ď d 1 Y that induce an order-isomorphism Z CpW, Ó gq CpW, q Ó CpW, gq or all W P C. In other words, or each pair o morphisms h 0 : W Ñ X and h 1 : W Ñ Y such that h 0 ď g h 1, there exists a unique h: W Ñ Ó g satisying d 0 h h 0 and d 1 h h 1. Furthermore, i k : W Ñ Ó g is another morphism, then the conjunction o d 0 h ď d 0 k and d 1 h ď d 1 k implies h ď k. Colimits. Let D be an ordinary category. I D : D Ñ C is a unctor (ie a diagram in C), we say that an object C P C together with a natural transormation α X : DpXq Ñ C is a colimit o D i Cpα X, C 1 q: CpC, C 1 q ÝÑ CpDpXq, C 1 q is a limiting cone in the category Ord, or all C 1 P C. This is the same as saying that pc, αq is a limit o sets and the bijection CpC, C 1 q lim CpD, C 1 q is an isomorphism o posets. It is not hard to veriy that iltered colimits in Ord can be constructed in a completely analogous way to those in the category o sets. Furthermore, it can easily be veriied that iltered colimits commute, or distribute, over inite enriched limits in Ord, in the sense that the Ord-unctor lim: rf, Ords Ñ Ord preserves iltered colimits i F is inite. For example, the unctor p q 2 : Ord Ñ Ord preserves iltered colimits, as do pullbacks, and thereore comma-objects preserve colimits (since comma-objects can be constructed rom p q 2 and pullbacks). This phenomenon is part o the general theory o locally initely presentable enriched categories developed in [18]. 2.c. Adjunctions, extensions and litings. An adjunction in an Ordcategory C consists o two morphisms : X Ñ Y and g : Y Ñ X in opposite directions with inequalities 1 X ď g and g ď 1 Y. Such an adjunction is usually written % g. g

9 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 9 By the usual argument, adjoints are unique up to canonical isomorphism, which in our case, by the antisymmetry o the ordering, means that adjoints are unique. For, i % g and % g 1, then g 1 X g ď g 1 g ď g 1 1 Y g 1 and symmetrically, g 1 ď g. A notion related to adjunctions is that o a let extension. I j : X Ñ Y and : X Ñ Z are morphisms in the Ord-category C, we say that an inequality ď lan j j exhibits lan j : Y Ñ Z as a let extension o by j i, or any other g : Y Ñ Z that satisies ď g j, the inequality lan j ď g holds. X j ď Y Z g X j Y ď lan j ď This universal property makes lan j unique i it exists. When j has a right adjoint j, there always exists a let extension lan j, or any : the extension is given by lan j j. The notion dual to that o a let extension is called let liting. I j : X Ñ Y and : Z Ñ Y are morphisms in C, we say that an inequality ď j j as depicted exhibits j as a let liting o through j i, or any other morphism g, the inequality ď j g implies j ď g. g X Z ě j Y When j has a let adjoint j l, then j l is a let liting o through j. 2.d. Lax idempotent monads. Beore recalling the notion o order-enriched monad, let us remind the reader o the deinition o a monad on a category. A monad on a category A is a triple T pt, η, µq where T is an endounctor o A and η : 1 A ñ T ð T 2 : µ are natural transormations that satisy the associativity and unit axioms: g ě X j Z ě j Z Y g T 3 T µ T T η ηt T µt T 2 µ T 2 µ T 1 T 2 µ T 1

10 10 M M CLEMENTINO AND I LÓPEZ FRANCO An algebra or the monad T, or a T-algebra, is a pair pa, aq where a: T A Ñ A is a morphism in A that satisies two axioms: T 2 A T a T A µ A T A a A a A ηa T A a 1 A A morphism o T-algebras pa, aq Ñ pb, bq is a morphism : A Ñ B in A that satisies b T a. Algebras and their morphisms orm a category T-Alg, that comes equipped with a orgetul unctor into A. Let C be an Ord-category. An order-enriched monad, or Ord-monad, on C consists o a monad T pt, η, µq on the ordinary category C with the additional requirement that T be an Ord-unctor. When the context is clear, we will reer to Ord-monads simply as monads. Deinition 2.5. A monad T pt, η, µq on an Ord-category C is lax idempotent, or Kock Zöberlein, i it satisies any o the ollowing equivalent conditions. (1) T η µ ď 1. (2) 1 ď ηt µ. (3) For any T-algebra a: T A Ñ A, the inequality 1 T A ď η A a holds. (4) A morphism l : T A Ñ A deines a T-algebra structure pa, lq i and only i l % η A with l η A 1 A. (5) T η ď ηt. (6) For any pair o T-algebras pa, aq and pb, bq and all morphisms : A Ñ B in C, b T ď a holds. (7) For any T-algebra pa, aq and any morphism : X Ñ A in C, the equality a T η X exhibits a T as a let extension o along η X : X Ñ T X. The equivalences o the above conditions can be ound, in the more general case o 2-categories, in [20]. Morphisms satisying condition (6) are called lax morphisms o T-algebras, even or a monad T that is not lax idempotent; so condition (6) says that T is lax idempotent i any morphism in C between T-algebras is a lax morphism o T-algebras. Deinition 2.6. The notion o a lax idempotent comonad G pg, ε, δq is a dual one: G is a lax idempotent comonad on C i pg op, ε op, δ op q, the corresponding monad on C op, is lax idempotent. We only translate explicitly

11 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 11 condition (7) o Deinition 2.5: or any G-coalgebra a: A Ñ GA and any morphism : A Ñ X in C, the equality ε X G a exhibits G a as a let liting o through ε X (see 2.c or the deinition o let litings). Ga GX A ě ε X X Example 2.7. Given an ordered set X, denote by P pxq the set o down-closed subsets o X, ie the set o those subsets Y Ď X satisying px ď yq ^ py P Y q ñ x P Y ; the set P pxq is canonically ordered by the inclusion o subsets o X. We denote by η X : X Ñ P pxq the monotone unction η X : X ÝÑ P pxq x ÞÑ Óx ty P X : y ď xu. The assignment X ÞÑ P pxq can be extended to a unctor whose value on a monotone unction : X Ñ Y is P pxq ÝÑ P py q pzq ty P Y : pdx P Zqpy ď pxqqu Y xpz Ópxq. Observe that always has a right adjoint : P py q Ñ P pxq given by pzq tx P X : Dz P Z such that pxq ď zu. Clearly, ď g i ď g, so P is an Ord-unctor. It is well-known that X ÞÑ P pxq deines a monad on Ord, where P pxq is ordered by inclusion, with unit η and multiplication µ given by P 2 pxq ÝÑ P pxq `U Ď P pxq ÞÑ YtY P Uu Ď X. This Ord-monad on the Ord-category Ord is lax idempotent, since P η X pzq Y xpz ÓpÓxq Ď ÓZ η P pxq pzq. The Ord-category P-Alg is the category o complete lattices (posets with arbitrary suprema or joins) with morphisms those monotone maps that preserve arbitrary suprema. Example 2.8. I Top 0 is the category o t 0 topological spaces and Top 0 is the associated Ord-category, with ordering induced by the opposite o the specialisation order, as in Example 2.2, there is an endo-ord-unctor F : Top 0 Ñ Top 0 that sends X to the set F pxq o ilters o open sets o X, with topology generated by the subsets U # tϕ P F pxq : U P ϕu, or

12 12 M M CLEMENTINO AND I LÓPEZ FRANCO U P OX. This is in act the unctor part o the lax idempotent ilter monad on Top 0 that will be studied in Section 13. There is a well-known result about algebras or lax idempotent monads on Ord-categories (see [22] and [12]) that can be summarised by saying that algebras are closed under retracts. More precisely: Lemma 2.9. I T pt, η, µq is a lax idempotent monad on an Ord-category, the ollowing conditions on an object A are equivalent. (1) A admits a (unique) T-algebra structure (we simply say that A is a T-algebra). (2) η A : A Ñ T A has a right inverse. (3) A is a retract o T A. (4) A is a retract o a T-algebra. Given two monads S ps, ν, θq and T pt, η, µq on the category C we recall that a monad morphism ϕ : S Ñ T is a natural transormation such that, or every object X o C, the ollowing diagrams commute. SSX θ X SX ϕ SX Sϕ X T SX ST X ϕ X T ϕ X T T X ϕ T X µ X T X SX ν X X ϕ X η X T X (There is a more general notion o morphism between monads on dierent categories, which we will not need.) Lemma Let T and S be monads on an Ord-category. Then there is at most one monad morphism T Ñ S i T is lax idempotent. Proo : Suppose that ϕ X : T X Ñ SX are the components o a monad morphism. The morphism ψ X : T SX ϕ SX ÝÝÑ S 2 X µs X ÝÑ SX is a T-algebra structure on SX, and thereore it is uniquely deined as the let adjoint to the unit SX Ñ T SX. Thereore, ϕ X ψ X T pη S Xq is uniquely deined.

13 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES Orthogonal actorisations and simple relections, revisited In this section we revisit some o the material o Cassidy Hébert Kelly work on simple relections [5] rom a slightly dierent perspective, more amenable to generalisation. Suppose that T : A Ñ A is a relection, with unit η A : A Ñ T A, on the category A, which we assume to admit pullbacks. The corresponding relective subcategory will be denoted by T-Alg, as it consists o the algebras or the idempotent monad T associated to T, whose invertible multiplication we denote by µ: T 2 ñ T. We say that a morphism in A is a T -isomorphism, or is T -invertible, i T is an isomorphism. Each morphism : A Ñ B can be actorised through a pullback square, as displayed. R L A L K R η A q pb T A B ηb T B T (3.1) Remark 3.2. The actorisation R L is unctorial, in the sense that, i ph, kq: Ñ g is a morphism in the arrow category A 2, then there is a morphism Kph, kq: K Ñ Kg h k g ÞÝÑ h L K Kph,kq R k Kg Lg Rg yielding a unctor K : A 2 Ñ A. Remark 3.3. The assignment that sends a morphism ÞÑ L is part o an endounctor on A 2, given on morphisms by ph,kq ÝÝÑ g ÞÝÑ L ph,kph,kqq ÝÝÝÝÝÝÑ Lg.

14 14 M M CLEMENTINO AND I LÓPEZ FRANCO Furthermore, there is a natural transormation Φ: L ñ 1 with components Φ L R Remark 3.4. The assignment ÞÑ R underlies a monad on the arrow category A 2. Its unit and multiplication are given by Λ L R Π where the morphism π : KR Ñ K is the unique morphism into the pullback K such that R 2 q π µ dompq T q q R and R π RR. One o the contributions o [5] is to introduce a property on the relection T that guarantees that the actorisation R L is an orthogonal actorisation system (os): the property o being simple. Deinition 3.5. The relection T pt, ηq is simple i L is a T -isomorphism. As pointed out in [5], i T is simple then the actorisation R L deines an orthogonal actorisation system, with let class o morphisms that o T -isomorphisms. To say only a ew words about this act, any morphism o the orm T is orthogonal to T -isomorphisms, and so R, as a pullback o T, is also orthogonal to T -isomorphisms; together with the simplicity hypothesis that L be a T -isomorphism, we obtain an orthogonal actorisation. I we denote by F T : A Ñ T-Alg the let adjoint o the inclusion T-Alg Ă A, then we can consider the ull subcategory T-Iso Ă A 2 whose objects are those morphisms o A that are T -isomorphisms (equivalently, those morphisms such that F T pq is an isomorphism) as a pullback. T-Iso pb (3.6) A 2 pf T q 2 T-Alg 2 Lemma 3.7. The subcategory T-Iso ãñ A 2 is corelective i and only i the relection T is simple. In this case, the associated idempotent comonad is Iso π R

15 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 15 given by ÞÑ L and has counit L R Proo : I T is simple, we know that the T -isomorphisms are the let class o an orthogonal actorisation system, and thus corelective in A 2. To be more explicit, i pe, M q is an orthogonal actorisation system in A, and m e with e P E and m P M, then the morphism e m exhibits e as a corelection o into the ull subcategory o A 2 deined by E. Beore moving to proving the converse, we make the observation that, or any category B, the ull subcategory Iso Ă B 2 o isomorphisms is corelective (as well as relective) with corelection given by Υ : Ipq 1 dompq Ñ Ψ 1 dompq To prove the converse, suppose that the inclusion o T-Iso into A 2 is corelective, with corelection given by counits Ψ : G Ñ in A 2. Then the pullback diagram (3.6) can be rewritten in the ollowing orm, where the categories o coalgebras are those or the respective copointed endounctors Ψ: G ñ 1 and Υ: I ñ 1. pg, Ψq-Coalg pb pi, Υq-Coalg A 2 pf T q 2 T-Alg 2 It is well known that, in these circumstances, pg, Ψq is given by a pullback in the category o endounctors o A 2 G pu T q 2 IpF T q 2 Ψ 1 A 2 η 2 T 2 pu T q 2 ΥpF T q 2

16 16 M M CLEMENTINO AND I LÓPEZ FRANCO I we apply the domain unctor dom: A 2 Ñ A to this pullback, we obtain that dompψq can be taken to be the identity transormation, since dompu T Υ F T pqq is an identity morphism or any. I we apply the codomain unctor cod instead, we obtain a pullback square codpgq T dom cod Ψ T codpq ηcodpq T codpq (we have used that cod U T IpF T pqq U T codp1 dompf T pqqq T dompq). In other words, cod Ψ R and codpgq K as deined in diagram (3.1). From here it is straightorward to veriy that G L. Thereore L P T-Iso, which says that T is a simple relection, concluding the proo. The lemma proved above gives a characterisation o simple relections, so one could deine simple relections as those relections T on A such that the ull subcategory T-Iso Ă A 2 is corelective. The associated idempotent comonad on A 2 is given by ÞÑ L. 4. Lax orthogonal actorisations We now proceed to study lax orthogonal actorisation systems on Ordcategories. Beore that, we briely recall basic acts on algebraic weak actorisation systems. 4.a. Weak actorisation systems. This short section recalls the deinition o weak actorisation system, a notion that appeared as part o Quillen s deinition o model category [27]. We say that a morphism g has the right liting property with respect to another, and that has the let liting property with respect to g, i every time we have a commutative square as shown, there exists (a not necessarily unique) diagonal iller. A weak actorisation system (ws) in a category consists o two amilies o morphisms L and R such that: g

17 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 17 R consists o those morphisms with the right liting property with respect to each morphism o L. L consists o those morphisms with the let liting property with respect to each morphism o R. Each morphism in the category is equal to the composition o one element o L ollowed by one o R. 4.b. Algebraic weak actorisation systems. Algebraic weak actorisation systems (awss) where irst introduced by M. Grandis and W. Tholen in [14], with an extra distributivity condition later added by R. Garner in [13]. In this section we shall give the deinition o awss on order-enriched categories, which is the case we will need, even though the deinitions remain virtually unchanged. Deinition 4.1. An Ord-unctorial actorisation on an Ord-category C consists o a actorisation dom λ ùñ E ρ ùñ cod in the category o locally monotone unctors C 2 Ñ C o the natural transormation dom ñ cod with component at P C 2 equal to : dompq Ñ codpq. It is important that in this actorisation E should be a locally monotone unctor. As in the case o unctorial actorisations on ordinary categories, an Ordunctorial actorisation as the one described in the previous paragraph can be equivalently described as: A copointed endo-ord-unctor Φ: L ñ 1 C 2 on C 2 with dompφq 1. A pointed endo-ord-unctor Λ: 1 C 2 ñ R on C 2 with codpλq 1. The three descriptions o an Ord-unctorial actorisation are related by: dompλ q L λ codpφ q R ρ. (4.2) Deinition 4.3. An algebraic weak actorisation system, abbreviated aws, on an Ord-category C consists o a pair pl, Rq, where L pl, Φ, Σq is an Ordcomonad and R pr, Λ, Πq is an Ord-monad on C 2, such that pl, Φq and pr, Λq represent the same Ord-unctorial actorisation on C (ie, the equalities (4.2) hold), plus a distributivity condition that we proceed to explain. The unit axiom ΠpΛRq 1 o the monad R implies, since codpλq 1, that codpπq 1; dually dompσq 1, so these transormations have components

18 18 M M CLEMENTINO AND I LÓPEZ FRANCO that look like: Σ L σ L 2 and Π R 2 π R One can orm a transormation : LR ùñ RL K LR KR σ 1 π KL K RL The distributivity axiom requires to be a mixed distributive law between the comonad L and the monad R; this amounts to the commutativity o the ollowing diagrams. LR 2 LΠ LR R RLR R R 2 L RL ΠL LR RL ΣR RΣ (4.4) L 2 R L LRL L RL 2 (The two axioms o a mixed distributive law that involve the unit o the monad and the counit o the comonad automatically hold.) Example 4.5. Each os pe, M q on C gives rise (upon choosing an pe, M q- actorisation or each morphism) to an aws pl, Rq, where L is the idempotent comonad associated to the corelective subcategory E Ă C 2 and R is the idempotent monad associated to the relective inclusion M Ă C 2. Conversely, an aws pl, Rq with both L and R idempotent induces an os. This was irst shown in [14, Thm. 3.2], and [3, Prop. 3] urther shows that it suices that either L or R be idempotent. I pl, Rq is an aws on C, an L-coalgebra structure on and an R-algebra structure on g can be depicted by commutative squares p s L Rg g

19 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 19 and the (co)algebra axioms can be written in the ollowing way (where the morphisms σ and π g are those described in Deinition 4.3). R s 1 p Lg 1 Kp1, sq s σ s p Kpp, 1q p π g A morphism o L-coalgebras p, sq Ñ p 1, s 1 q is a morphism ph, kq: Ñ 1 in C 2 that is compatible with the coalgebra structures in the usual way: Kph, kq s s 1 k. Similarly, a morphism o R-algebras pg, pq Ñ pg 1, p 1 q is a morphism pu, vq: g Ñ g 1 such that p 1 Kpu, vq u p. With the obvious composition and identities we obtain categories L-Coalg and R-Alg, equipped with orgetul unctors into C 2. These are Ord-categories by stipulating that the ordering o morphisms o (co)algebras is inherited rom the ordering o morphisms in C 2 ; as a consequence, the orgetul unctors rom L-Coalg and R-Alg to C 2 become Ord-enriched. 4.c. Underlying wss. Each aws pl, Rq (enriched or not) has an underlying ws pl, Rq. The class L consists o all those morphisms that admit a structure o coalgebra over the copointed endounctor pl, Φq that underlies L; similarly, R consists o all those morphisms that admit a structure o an algebra over the pointed endounctor pr, Λq that underlies R. 4.d. Laris and awss. One o the most important examples o awss or us will be provided by the so-called laris. Deinition 4.6. A let adjoint right inverse, or lari, in an Ord-category is a morphism that is part o an adjunction % g with 1 g. In the same situation, we say that g is a right adjoint let inverse, or rali. Suppose given another adjunction 1 % g 1 with 1 g 1 1, and morphisms h and k as in the displayed diagram. X % g Y h k X 1 1 % Y 1 g 1

20 20 M M CLEMENTINO AND I LÓPEZ FRANCO We say that ph, kq is a morphism o laris Ñ 1, and that ph, kq is a morphism o ralis g Ñ g 1, i 1h k and g 1k hg. With the obvious notion o composition, laris and ralis orm categories that come equipped with orgetul unctors into C 2. Furthermore, i C is an Ord-category, there are Ord-categories LaripCq and RalipCq with objects and morphisms described above, and ordering between morphisms those o C 2. Example 4.7. Consider the ree (split) opibration monad M on Ord, given on : X Ñ Y by Mpq K Ó 1 codpq px, yq P X ˆ Y : pxq ď y ( M ÝÝÑ Y px, yq ÞÑ y with ordering inherited rom that o X ˆ Y. Furthermore, M is a locally monotone endounctor o Ord 2. The category M-Alg o algebras or this monad has objects the (split) opibrations, ie monotone unctions : X Ñ Y with a choice or each x P X and y P Y that satisy pxq ď y, o an x y P X such that: x ď x y, px y q y, and px ď x 1 q ^ ppx 1 q yq implies x y ď x 1. As an aside comment, we note that there is no dierence between the notions o an opibration and o a split opibration in Ord due to the antisymmetry property satisied by the orderings. Any monotone unction : X Ñ Y can be actorised as : X E ÝÑ K M ÝÝÑ Y where Epxq px, pxqq P Ó Y Ó 1 Y. This is in act part o an aws, as we proceed to show. As the unctorial actorisation is the one just described, the locally monotone endounctor E o Ord 2 has a copoint Φ p1 X, Mq: E Ñ. The monotone unction E : X Ñ Ó Y has a right adjoint r : Ó Y Ñ X, given by r px, yq x y. We can deine σ : K Ó Y ÝÑ KE E Ó K px, yq ÞÑ pr px, yq, px, yqq and morphisms Σ that orm the comultiplication o a comonad E pe, Φ, Σq. X X Σ : E ÝÑ E 2 E K σ E 2 KEpq The morphism MEpq: KE Ñ K is a let adjoint to σ, as can be easily veriied. Furthermore, Φ E % Σ, which means that the comonad E is lax

21 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 21 idempotent. The distributivity axiom o awss can be veriied by hand, or, alternatively, one can appeal to Theorem 7.2. We conclude with the observation that the endounctors E and M preserve iltered colimits; equivalently, the unctor K : Ord 2 Ñ Ord preserves iltered colimits. This is so because K is constructed by means o comma-objects and the comments at the end o 2.b. Example 4.8. Precisely the same construction can be carried out in any Ordcategory that admits comma-objects (see 2.b); or example, in any Ordcategory that admits cotensor products with 2 and pullbacks. The morphism M is a projection in the comma-object depicted. K M B The let part o the actorisation E : X Ñ K is the unique morphism deined by the conditions r ě X M E and r E 1 X. It is not hard to show that E % r. The endo-ord-unctor ÞÑ M is part o the ree (split) opibration monad on C. The endo-ord-unctor E is part o a comonad with counit Φ E p1, Mq: E Ñ and comultiplication Σ p1, σ q: E Ñ E 2 deined by r E σ r and ME σ 1 K. Lemma 4.9. Suppose that C is an Ord-category with comma-objects and pe, Mq the aws constructed in the previous example. I Φ E : E ñ 1 is the underlying copointed endounctor o the comonad E, then: (1) There is an isomorphism LaripCq E-Coalg over C 2. (2) The orgetul unctor is an isomorphism. B E-Coalg ÝÑ pe, Φ E q-coalg (4.10) Proo : This proo ollows a direction not suggested by the statement. We shall irst prove that there is an isomorphism between LaripCq and pe, Φ E q-coalg and then show that (4.10) is an isomorphism. The reason the lemma is stated in the present orm is that this orm extends to 2-categories [7].

22 22 M M CLEMENTINO AND I LÓPEZ FRANCO Suppose given a morphism in C 2 as depicted. A A E B s K (4.11) The morphism s: B Ñ K Ó B corresponds to a pair o morphisms r : B Ñ A and u: B Ñ B that satisy r ď u. The morphisms r and u are the composition o s with, respectively, the projections Ó B Ñ A and M : Ó B Ñ B. The commutativity o (4.11) translates into r 1 and u. Now suppose that (4.11) is a morphism o pe, Φ E q-coalgebras, ie that M s 1. By deinition o u, this is equivalent to saying that u 1. Thereore, to give an pe, Φ E q-coalgebra structure on is equivalent to giving a morphism r : B Ñ A such that r ď 1 and r 1. In other words, an pe, Φ E q- coalgebra structure on is the same as a lari structure on. To conclude the proo, we show that any pe, Φ E q-coalgebra structure p1, sq: Ñ E is an E-coalgebra, ie it satisies the coassociativity equality σ s Kp1, sq s. (4.12) The codomain o the morphisms at both sides o the equality is KE, so (4.12) holds precisely when it does ater composing with the projections ME : KE Ñ K and r E : KE Ñ X. One o these equalities is obvious, since ME σ s 1 s s s 1 s M s ME Kp1, sq s. The second equality holds by the ollowing string o equalities, the irst o which uses the deinition o σ and the last uses r E Kp1, sq r. r E σ s r s r E Kp1, sq s. This completes the proo o the lemma. 4.e. Lax orthogonal actorisation systems. Deinition An aws pl, Rq on an Ord-category C is a lax orthogonal actorisation system (abbreviated los) i either o the ollowing equivalent conditions holds: The comonad L is lax idempotent.

23 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 23 The monad R is lax idempotent. Beore proving the equivalence between the above properties we describe more explicitly what it means or pl, Rq to be lax orthogonal. According to our notation, the unit and multiplication o R and the counit and comultiplication o L are depicted as morphisms in C 2 as ollows. L Λ R R 2 π Π R L Φ R L Σ σ L 2 Then, pl, Rq is lax orthogonal i and only i any o the ollowing conditions hold (the equivalence o these conditions will be shown in Proposition 4.16): KpL, 1q π ď 1 1 ď LR π 1 ď σ RL σ Kp1, Rq ď 1. (4.14) In terms o R-algebras and L-coalgebras, the lax idempotency o pl, Rq is described as ollows. I p, sq is an L-coalgebra and pg, pq is an R-algebra, as displayed below, p,sq s L then the aws is lax orthogonal i and only i any o the ollowing two equivalent conditions hold, or all p, sq and pg, pq (again, the equivalence will be shown in Proposition 4.16): Rg p pg,pq 1 ď s R and 1 ď Lg p. (4.15) Proposition I pl, Rq is an aws on an Ord-category C, then L is lax idempotent i and only i R is lax idempotent. Proo : In this proo we use the ollowing general property o awss, whose details can be ound in [3, 2.8]. I pl, Rq is an aws on an (ordinary) category and, g are two composable morphisms each one o which carries an L-coalgebra structure, then their composition g carries a canonical L- coalgebra structure. We regard morphisms o the orm L as L-coalgebras with structure given by the comultiplication Σ p1, σ q: L Ñ L 2. Furthermore, we use the ollowing act, whose proo can be ound in [3, 3.1]: g

24 24 M M CLEMENTINO AND I LÓPEZ FRANCO the morphism p1, π q depicted is a morphism o L-coalgebras rom LR L to L. L A K LR KR π A K L Assuming that L is lax idempotent, we shall show that R is lax idempotent by exhibiting an inequality RΛ Π ď 1, where Λ and Π are the unit and multiplication o the monad. The converse, namely that L is lax idempotent i R is so, is not necessary to prove, as it ollows by a duality argument, more speciically, by taking the opposite Ord-category. Let : A Ñ B be a morphism o C, and consider the composition o the morphisms p1 A, π q: LR L Ñ L with LΛ pl, KpL, 1qq: L Ñ LR, as depicted. A A L K L K L LR (4.17) LR KR π K KpL,1q KR The composition o this diagram with the counit Φ R p1, R 2 q equals the morphism pl, R 2 q: LR L Ñ R, depicted on the right below, since A L K R 2 KpL, 1q π R π R 2. L K R (4.18) LR KR R2 B Since L is lax idempotent, the L-coalgebra morphism (4.17) is a let liting o (4.18) through Φ R (see Deinition 2.6).

25 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 25 On the other hand, the morphism in C 2 depicted below is also equal to (4.18) upon composition with the counit Φ R L A L K K LR (4.19) LR KR KR and by the universal property o litings we deduce that (4.17) is less or equal than (4.19), so KpL, 1q π ď 1 KR. It remains to prove that this deines an inequality in C 2 with identity codomain component; in other words, that the inequality becomes an equality upon composition with R 2. But this holds, since both sides become equal: concluding the proo. R 2 KpL, 1q π R π R 2, Example The aws pe, Mq o Example 4.7, or which M-algebras are opibrations and E-coalgebras are laris, is lax orthogonal. Indeed, the monad M is well-known to be lax idempotent. 4.. Categories o awss. There is a category AWFSpCq whose objects are awss on the Ord-category C. A morphism pl, Rq ÝÑ pl 1, R 1 q is a natural amily o morphisms ϕ that make the ollowing diagrams commute. L L 1 K ϕ K 1 R R 1 (4.21) Furthermore, the morphisms p1, ϕ q: L Ñ L 1 must orm a comonad morphism L Ñ L 1, and the morphisms pϕ, 1q: R Ñ R 1 must orm a monad morphism R Ñ R 1. There is a ull subcategory LOFSpCq o AWFSpCq consisting o the loss. Lemma LOFSpCq is a preorder.

26 26 M M CLEMENTINO AND I LÓPEZ FRANCO Proo : I the morphisms ϕ as in (4.21) orm a morphism rom pl, Rq to pl 1, R 1 q, then the morphisms pϕ, 1q: R Ñ R 1 deine a morphism o monads. There can only be one such, by Lemma Liting operations In this section we introduce kz liting operations and explain the motivation behind the deinition o lax orthogonal actorisation systems. Beore all that, we must say something about how liting operations work in relation to awss on Ord-categories. 5.a. Liting operations on Ord-categories. Suppose that U : A Ñ C 2 Ð B: V are locally monotone unctors between Ord-categories. A liting operation rom U to V can be described as a choice o a diagonal iller φ a,b ph, kq or each morphism ph, kq: Ua Ñ V b in C 2. Ua h φ a,b ph,kq k These diagonal illers must satisy a naturality condition with respect to morphisms in A and B. I α: a 1 Ñ a and β : b Ñ b 1 are morphisms in A and B respectively, then φ a1,b 1` dom V βhdom Uα, cod V βkcod Uα pdom V βqφa,b ph, kqpcod Uαq as depicted in the ollowing diagram. V b dom Uα h dom V β Ua 1 cod Uα Ua k V b cod V β V b 1 So ar, the deinition o liting operation is the one given in [13], but our categories are enriched in Ord and the unctors U and V are locally monotone, so we require that the diagonal iller satisies: i ph, kq and ph 1, k 1 q: Ua Ñ V b are commutative squares in C with ph, kq ď ph 1, k 1 q (ie h ď h 1 and k ď k 1 ) then φ a,b ph, kq ď φ a,b ph 1, k 1 q.

27 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 27 5.b. Liting operations rom Ord-unctorial actorisations. The idea o a unctorial actorisation dom ñ E ñ cod, as deined in Deinition 4.1, is that it induces a canonical liting operation between the orgetul Ordunctors U and V U : pl, Φq-Coalg ÝÑ C 2 ÐÝ pr, Λq-Alg: V. Here Φ: L ñ 1 C 2 and Λ: 1 C 2 ñ R are, respectively, the copointed endo- Ord-unctor and the pointed endo-ord-unctor on C 2 associated to the given Ord-unctorial actorisation. A coalgebra or pl, Φq can be depicted as the commutative square on the let below, while an algebra or pr, Λq is a commutative square on the right p, sq s L pg, pq satisying R s 1 and plg 1. Given a commutative square ph, kq: Ñ g, there is a canonical diagonal iller φ p,sq,pg,pq ph, kq p Eph, kq s. It is immediate to see that these diagonal illers orm a liting operation rom U to V. Remark 5.1. Even though an (Ord-)unctorial actorisation R L as the one discussed in the previous paragraphs yields a liting operation o pl, Φq-coalgebras against pr, Λq-algebras, there is no guarantee o being able to ind a diagonal iller or a commutative diagram o the orm L since L may not support an pl, Φq-coalgebra structure, and Rg may not support an pr, Λq-algebra structure. A natural way o endowing L and Rg with the corresponding structures is to require that pl, Φq extends to a comonad and pr, Λq extends to a monad; in this way, L is a (coree) coalgebra and Rg is a (ree) algebra. This one o the reasons or the orm that the deinition o aws takes (see Deinition 4.3). There is an useul act that is worth including at this point, and will be useul in the proo o Theorem 5.6. Rg Rg p g

28 28 M M CLEMENTINO AND I LÓPEZ FRANCO Lemma 5.2. For any aws pl, Rq, the diagonals φ L,R pl, Rq are identity morphisms. L 1 L R Proo : I we write the commutative square o the statement as a pasting o two commutative squares p1, Rq and pl, 1q, as displayed, we can easily compute the diagonal iller. L R R L φ L,R pl, Rq π KpL, Rqσ π KpL, 1qKp1, Rqσ Remark 5.3. As pointed out in [3, 2.5], the commutativity o the two diagrams (4.4) that express the act that : LR ñ RL is a mixed distributive law is equivalent to the requirement that the diagonal iller o the displayed square be σ π. K LR σ σ π KL KR π K 5.c. KZ liting operations. In the previous section we saw that each aws canonically induced a liting operation. It is logical to expect that liting operations that arise rom lax orthogonal awss carry extra structure. In this section we identiy this structure. Deinition 5.4. Suppose given a liting operation φ rom U : A Ñ C 2 to V : B Ñ C 2 on an Ord-category C as deined in 5.a. We say that φ is a kz-liting operation i, or all a P A, b P B and each commutative diagram as on the let, the inequality on the right holds. Ua d h k V b ùñ R RL φ a,b ph, kq ď d

29 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 29 In other words, the diagonal iller given by the liting operation φ is a lower bound o all possible diagonal illers. Example 5.5. Consider the Ord-unctor 0: 1 Ñ 2 that includes the terminal ordered set as the initial element o the ordered set 2 p0 ď 1q. There is a bijection between opibration structures on a morphism g : X Ñ Y in Ord and kz liting operations on g against the morphism 0. To see this, irst notice that a commutative square is equally well given by an element x P X and an element y P Y such that gpxq ď y. The existence o a diagonal iller is the existence o an element x y P X with x ď x y and gpx y q y. This diagonal iller is a lower bound i or any other x ď x with gp xq y there is an inequality x y ď x. The element x y is unique and the assignment px, yq ÞÑ x y deines a split opibration structure on g. Theorem 5.6. The ollowing conditions are equivalent or an aws pl, Rq on an Ord-category C. (1) The aws is a los. (2) The liting operation rom the orgetul unctor U : L-Coalg Ñ C 2 to the orgetul unctor V : R-Alg Ñ C 2 is a kz-liting operation. Proo : Assume that pl, Rq is lax orthogonal, p, sq is an L-coalgebra and pg, pq is an R-algebra. Given a diagonal iller d as depicted, we must show φ p,sq,pg,pq ph, kq ď d. Using the inequalities 1 ď s R and 1 ď Lg p rom (4.15), we obtain `φp,sq,pg,pq ph, kq p Kph, kq s ď d ô `Kph, kq ď Lg d R. d h k There is a morphism plgdr, kq: R Ñ Rg in C 2, as shown by the diagram below, which precomposed with the unit Λ pl, 1q: Ñ R o R equals X Y g g

30 30 M M CLEMENTINO AND I LÓPEZ FRANCO Λ g ph, kq plg h, kq: Ñ Rg. R R d k k Lg g Rg On the other hand, by the lax idempotency o R, we have that Kph, kq is a let extension o Λ g ph, kq along Λ, so there exists Kph, kq ď Lg d R, as desired. Conversely, assume that the liting operation φ induced by the aws is kz, and consider the commutative square LR 1 π LR R 2 LR R R 2 By Lemma 5.2, φ provides the diagonal iller φ LR,R2 plr, R 2 q 1, so we have an inequality 1 ď LR π as required. Theorem 5.7. Let pl, Rq be a los on an Ord-category C. Then, the ollowing statements about a morphism o C are equivalent: (1) has an (unique) R-algebra structure (we simply say that is an R-algebra). (2) is injective with respect to L-coalgebras, in the sense that any commutative square l with l P L-Coalg has a diagonal iller. (3) admits a (non-necessarily unique) pr, Λq-algebra structure. (4) is a retract in C 2 o an R-algebra. The ws that underlies pl, Rq has as let part those morphisms in the image o the orgetul unctor L-Coalg Ñ C 2 and as right part those morphisms in the image o the orgetul unctor R-Alg Ñ C 2.

31 LAX ORTHOGONAL FACTORISATIONS IN ORDERED STRUCTURES 31 Proo : We have seen in 5.b that (1) implies (2). To prove that (2) implies (3), consider the diagonal iller below, which shows that pp, 1q: R Ñ is is an pr, Λq-algebra structure. L p R The implications (3)ñ(4)ñ(1) are particular instances o part o Lemma 2.9, since R is lax idempotent. As mentioned in 4.c, the underlying ws pl, Rq o pl, Rq has as right class the algebras or the pointed endounctor pr, Λq. Then, P R (or, by duality, P L) precisely when is an R-algebra (an L-coalgebra). 6. Horizontally ordered double categories and loss 6.a. Horizontally ordered double categories. Double categories, introduced by C. Ehresmann [9], can be succinctly described as internal categories in the cartesian category o categories. They consist o an internal graph o categories and unctors G 1 Ñ G 0 (domain and codomain) with an identity unctor id: G 0 Ñ G 1 and a composition unctor G 1 ˆG0 G 1 Ñ G 1 that satisy the usual associativity and identity axioms. The morphisms o G 0 will be represented as horizontal arrows. The objects o G 1 have a domain and a codomain that are objects o G 0, and will be represented as vertical morphisms. Morphisms o G 1 will be represented as squares; or example a morphism α: x Ñ y in G 1 will be represented as x α Objects o G 1, ie vertical arrows, can be vertically composed, as well as squares as the one above. Deinition 6.1. A horizontally ordered double category is an internal category in the cartesian category Ord-Cat o Ord-categories and Ord-unctors. This means that in a horizontally ordered double category we can speak o inequalities between horizontal morphisms and between squares. A monotone double unctor between two horizontally ordered double categories is a y