Injective objects and lax idempotent monads

Transcription

1 Master Project Injective objects and lax idempotent monads Author: Eiichi Piguet Supervised by: Dr. Gavin Jay Seal Autumn 2013

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3 i Acknowledgement I would like to express my deepest thanks to Dr. Gavin Jay Seal for giving me the chance to do this master project under his supervision, and for the choice of the subject. I am very grateful for the time that he spent with me during all the semester, in particular for the reading, comments, advice and help for this project. I would also like to thank my parents and my friends for their very warm support and encouragements, which allowed me to overcome difficult times.

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5 CONTENTS iii Contents Introduction 1 1 Basic categorical notions Monads and Kleisli triples The Eilenberg Moore category and the Kleisli category Distributive laws Lax idempotent monads Injective objects Definition and elementary properties Injective objects via algebras Example: Injective objects in OrdMon via Dn Example: Injective objects in Sup via Up Example: Injective objects in Top 0 via F and F Example: Injective objects in Sup via Fil Construction of lax idempotent monads Algebraically adjoint monads T-monoids Construction of a derived monad Properties of the derived monad Example: Injective objects in SLat via Ufin Example: Injective objects in Sup via U Example: Injective objects in OrdMon lax via PL Example: Injective objects in PrOrd via P Unsolved questions 39 References 40

6 1 BASIC CATEGORICAL NOTIONS 1 Introduction Given a category X and a class of morphisms M, characterizing the M-injective objects in X is, in general, far to be easy. However, Escardó proved in [8] that if T is a lax idempotent monad on an ordered category X and M is the class of T-embeddings, then the M-injective objects are exactly the T-algebras. This is a motivation to study more lax idempotent monads, and in particular, find a method to construct such monads. In [19], Seal introduced the notion of an order-adjoint monad T on Set, from which he defined the category of Kleisli monoids (or T-monoids) and Kleisli morphisms, denoted as Mon(Set T ). Briefly put, the theory presented in [19] describes the passage from an order-adjoint monad T on Set to a lax idempotent monad T on Mon(Set T ). In the first part of this project, we recall some basic notions of category theory, and give the definition and properties of a lax idempotent monad. The notions presented in this part will only be the necessary tools for the next two parts and come principally from [7]. The reader who wants more informations about category theory is also invited to read the excellent [4] (the second volume [5]) and [15]. A novice in category theory may rather be referred to [2], which is much easier to read. In the second section, we study, initially, the M-injective objects for an abstract class M. Despite the fact that, in the literature, a lot of results about injective objects are over monomorphisms, we will see that many elementary properties are still available in the abstract case. Secondly, we give the first main results of this project, which generalize the result of [8] mentioned above. The category X can, in fact, even be preordered, and we give more freedom in the choice of the class M. Moreover, the result is still true for any full subcategory of X, provided that the subcategory contains all the M-injective objects in X. Finally, in the last section, we will see that the theory presented in [19] over Set can be generalized, in a natural way, to an abstract category. Given a category X, we define the concept of an algebraically adjoint monad T on X. As in [19], we define the category of T-monoids and morphisms of T-monoids, denoted as Mon(X T ). The second main result of this project states that the category Mon(X T ) can be provided with a preordered category structure, and that there is a derived monad T on Mon(X T ) which is lax idempotent, and even separated. The main points of this project are presented as follows: basic categorical notions (Section 1); the M-injective objects are the T-algebras (Theorem ); construction of a derived monad T on Mon(X T ) (Section 3.3); the category X T is isomorphic to Mon(X T ) T (Theorem 3.4.1); the monad T on Mon(X T ) is separated lax idempotent (Theorem 3.4.4). Many examples appear at the end of Sections 2 and 3, and illustrate the theorems mentioned above. 1 Basic categorical notions As already mentioned in the introduction, the aim of this section is to remind the reader of some basic concepts of category theory and give the definition and properties of a lax idempotent monad. 1.1 Monads and Kleisli triples Definition A monad T = (T, µ, η) on a category X is given by a functor T : X X and two natural transformations, the multiplication µ : T T T and the unit η : 1 X T, satisfying the

7 1.1 Monads and Kleisli triples 2 multiplication law and the right and left unit laws: µ µt = µ T µ, µ ηt = 1 T = µ T η. Equivalently, these equalities mean that the diagrams T T T T µ T T T ηt T T T η T µt T T µ T µ 1 T T µ 1 T commute. A monad morphism (R, σ) : S T from a monad S = (S, ν, δ) on A to a monad T = (T, µ, η) on X is given by a functor R : X A together with a natural transformation σ : SR RT such that Rη = σ δr, Rµ σt Sσ = σ νr. In the case where A = X and R is the identity functor, one writes σ : S T rather than (1 X, σ) : S T. The next proposition tells us that any adjunction yields a monad. Proposition Let F : X be A : G an adjunction with unit η and counit ε. Then T = (GF, GεF, η) is a monad on X. Proof. Routine verification by using naturality of ε and triangular identities. Terminology The monad in Proposition is called the monad associated to the adjunction. Definition A Kleisli triple (T, ( ) T, η) on a category X consists of (1) a function T : Ob(X) Ob(X) sending X to T X, (2) an extension operation ( ) T sending a morphism f : X T Y to a morphism f T : T X T Y, (3) a morphism η X : X T X for each X Ob(X), subject to (g T f) T = g T f T, η T X = 1 T X, f T η X = f (1.1) for all X Ob(X), f : X T Y and g : Y T Z. The following proposition gives another way to characterize monads. Proposition A Kleisli triple (T, ( ) T, η) on X defines a monad T = (T, µ, η) on X by setting T f := (η Y f) T, µ X := (1 T X ) T for all morphisms f : X Y. Conversely, given a monad T = (T, µ, η) on X, one obtains a Kleisli triple (T, ( ) T, η) via f T := µ Y T f for all morphisms f : X T Y. Furthermore, these correspondences are mutually inverse. Proof. Routine verification by using (1.1).

8 1.2 The Eilenberg Moore category and the Kleisli category The Eilenberg Moore category and the Kleisli category Definition Given a monad T = (T, µ, η) on X, a T-algebra (or Eilenberg Moore algebra) is a pair (X, a), where X is an object of X, and the structure morphism a : T X X satisfies a T a = a µ X and 1 X = a η X. Equivalently, these equalities mean that the following diagrams commute: T T X T a T X X η X T X µ X T X a X a 1 X a X. In particular, (T X, µ X ) forms a T-algebra, the free T-algebra on X. A T-homomorphism f : (X, a) (Y, b) is a morphism f : X Y such that which is the same as commutativity of the diagram f a = b T f, T X T f T Y a X f Y. b The category of T-algebras and T-homomorphisms is denoted by X T, and is called the Eilenberg Moore category of T. The next proposition says that every monad arises from an adjunction. Proposition Let T = (T, µ, η) be a monad on a category X. There exists an adjunction F T : X X T : G T with unit η T : 1 X G T F T original monad T. and counit ε T : F T G T 1 X T, such that the associated monad gives back the Proof. We define the functor G T : X T X as the forgetful functor, and its left adjoint (routine verification) F T : X X T by X (T X, µ X ), (f : X Y ) (T f : T X T Y ) for any X Ob(X) and X-morphism f : X Y. The unit η T : 1 X G T F T = T of this adjunction is η, and the counit is described by its components as ε T (X,a) = a : (T X, µ X) (X, a) for all T-algebras (X, a). We can easily verify that the monad associated to F T : X gives X T : G T back the original T = (T, µ, η).

9 1.3 Distributive laws 4 Proposition If (T, ( ) T, η) is a Kleisli triple on a category X, then the T-algebras associated to the monad T are those pairs (X, a) with X Ob(X) and a : T X X an X-morphism such that f, g X(Y, T X) (a f = a g = a f T = a g T ) and a η X = 1 X. (1.2) Proof. Given f, g X(Y, T X) such that a f = a g, by using the bijective correspondence of Proposition 1.1.5, we get a f T = a µ X T f = a T a T f = a T (a f) = a T (a g) = a T a T g = a µ X T g = a g T. Conversely, assuming (1.2), one can show that (X, a) is a T-algebra. Indeed, since a (η X a) = a = a 1 T X, one obtains that a T a = a (η X a) T = a (1 T X ) T = a µ X. Whence (X, a) is a T-algebra. Definition Given a monad S = (S, ν, δ) on A and a monad T = (T, µ, η) on X, a functor R : X T A S is algebraic over a functor R : X A if it makes the diagram X T R A S G T X R A G S commute. Proposition Any monad morphism (R, σ) : S T from a monad S = (S, ν, δ) on A to a monad T = (T, µ, η) on X yields an algebraic functor R : X T A S over R via R(X, a) := (RX, Ra σ X ) on objects (and necessarily sends an X-morphism f to Rf). Conversely, every algebraic functor R : X T A S over R : X A induces a monad morphism (R, σ) via σ X := µ X SRη X, where µ X : SRT X RT X denotes the A-morphism defined by R(T X, µ X ) = (RT X, µ X ). Proof. See Exercise 3.H [7]. Definition The Kleisli category, denoted by X T, associated to the monad T = (T, µ, η) on X admits as objects the objects of X, and a morphism f : X Y in X T is simply an X-morphism f : X T Y. The Kleisli composition of f : X Y and g : Y Z in X T is defined via the composition in X as g f := µ Z T g f = g T f. The identity 1 X : X X in X T is the component η X : X T X of the unit η. 1.3 Distributive laws In all this section, we will fix two monads T = (T, µ, η) and S = (S, ν, δ) on a category X. Almost all the results are given without proofs. The reader who wants more details may read Section 3.8, Chapter II of

10 1.3 Distributive laws 5 [7]. Definition A distributive law of T over S is a natural transformation λ : T S ST making the following diagrams commute: T SS λs ST S Sλ SST T (1.3) T ν T S λ ST νt T S T δ λ δt ST. µs T T S T λ Sµ T ST λt ST T ηs S Sη Definition A lifting of the monad S on X through G T : X T X is a monad S = ( S, ν, δ) on X T such that G T S = SGT, G T ν = νg T, G T δ = δgt. (1.4) Remark The first condition may be used to identify the domains and codomains of the natural transformations in the last two equalities. The last two conditions state that the underlying X-morphisms of ν (X,a), δ (X,a) are ν X, δ X, respectively, for any T-algebra (X, a). Therefore, a lifting of S through G T : X T X consists of a functor S : X T X T making the diagram X T S X T G T X S X G T commute, and such that ν X : S S(X, a) S(X, a), δx : (X, a) S(X, a) are X T -morphisms for all (X, a) Ob(X T ). Definition One says that a monad ST = (ST, w, δ η), where δ η denotes the horizontal composition of δ and η, is a composite of S and T if the natural transformations Sη : S ST and δt : T ST are monad morphisms, and if the following diagram commutes: Sη δt ST 1 ST (1.5) ST ST w ST. The main result is that distributive laws, liftings and composite monads are equivalent concepts: Proposition For monads S and T on X, there is a bijective correspondence between: (i) distributive laws λ of T over S; (ii) liftings S of S through G T : X T X; (iii) monads ST = (ST, w, δ η) that are composites of S and T. Proof. We only sketch the proof. The idea is to prove the following implications: (iii) (i) (ii).

11 1.4 Lax idempotent monads 6 (i) = (ii): If λ : T S ST is a distributive law of T over S, then a lifting S : X T X T can be obtained via (X, a) (SX, Sa λ X ), (f : X Y ) (Sf : SX SY ). The fact that the components of ν : S S S and δ : 1 X T S are X T -morphisms follows from commutativity of Diagrams (1.3). (ii) = (iii): In turn, a lifting of S through G T : X T X gives rise the following composite adjunction G S (X T ) S F S X T G T X. F T Since ST = SG T F T = G T SF T = G T G S F S F T, the monad on X associated to this adjunction is where ST = (ST, w, δ η), w = G T G S ε S F S F T G T G S F S ε T G S F S F T = G T νf T G T Sε T SF T = νt SG T ε T SF T. (1.6) By combining (1.4) and (1.6), one can check the commutativity of Diagram (1.5). (iii) = (i): Finally, if a monad ST = (ST, w, δ η) on X is a composite of S = (S, ν, δ) and T = (T, µ, η), then the natural transformation λ : T S ST defined by becomes a distributive law of T over S. λ := w (δt Sη) Corollary The monads ST that are composites of S = (S, ν, δ) and T = (T, µ, η) are exactly those of the form ST = (ST, (ν µ) SλT, δ η), where λ is a distributive law of T over S. There are also monad morphisms Sη : S ST, δt : T ST. Proposition For a composite monad ST and the corresponding lifting S (given by Proposition 1.3.5), there is an isomorphism between (X T ) S and X ST. 1.4 Lax idempotent monads Definition A preordered category is a category X with each hom-class carrying a preorder (that is, a reflexive and transitive relation ), such that the composition maps X(X, Y ) X(Y, Z) X(X, Z) (f, g) g f, are monotone for all X, Y, Z Ob(X). Similarly, X is an ordered category if the same definition holds, but by replacing preorder by order. An adjunction f g : Y X in X is a pair of morphisms

12 1.4 Lax idempotent monads 7 f : X Y, the left adjoint, and g : Y X, the right adjoint, satisfying the inequalities 1 X g f and f g 1 Y. A functor T : X X is a 2-functor if it preserves the preorder on hom-classes. In other words: for all f, g X(X, Y ). f g = T f T g Remark If X is a preordered category, left and right adjoints are determined up to equivalence, while they are uniquely determined if X is an ordered category. Remark also that 2-functors preserve any adjunction. Proposition Let T = (T, µ, η) be a monad on a preordered category X, and assume that T is a 2-functor. Then the following conditions are equivalent: (i) T η X η T X (ii) T η X µ X (iii) µ X η T X for all X Ob(X); for all X Ob(X); for all X Ob(X). Proof. Let us fix X Ob(X). (i) = (ii): T η X µ X = µ T X T T η X µ T X T η T X = 1 T T X. (ii) = (iii): 1 T T X = T µ X T η T X µ T X η T T X T µ X η T T X = η T X µ X. (iii) = (i): T η X η T X µ X T η X = η T X. Definition A monad T = (T, µ, η) on a preordered category X is lax idempotent (or Kock Zöberlein) if T is a 2-functor and satisfies the condition of Proposition Moreover, if the preorder on X(X, T Y ) is an order for all X, Y Ob(X), then T is separated. If X is an ordered category, we obtain one more characterization of lax idempotent monads. Proposition Let T = (T, µ, η) be a monad on an ordered category X, and assume that T is a 2-functor. Then the following conditions are equivalent: (i) T is lax idempotent; (ii) for all X Ob(X), a morphism a : T X X is a structure of a T-algebra iff 1 T X η X a and a η X = 1 X. (1.7) Proof. (i) = (ii): Assume that a : T X X is a structure of a T-algebra. One gets 1 T X = T a T η X T a η T X = η X a, which shows that (1.7) is satisfied. Conversely, assume that (1.7) holds. The adjunctions T a T η X, a η X and µ X η T X yield a T a T η X η X and a µ X η T X η X. Since T η X η X = η T X η X, one gets that a T a = a µ X by Remark Thereby (ii) is satisfied.

13 1.4 Lax idempotent monads 8 (ii) = (i): Since µ X is a structure of a T-algebra for T X, one obtains µ X η T X by (1.7). The condition (iii) of Proposition is verified, whence T is lax idempotent. Remark Since adjoints are uniquely determined in an ordered category, if X is an ordered category, then every object has at most one structure map. Remark also that the hypotheses of Proposition can be weakened. Indeed, it is sufficient that T is a 2-functor and X preordered, but only that the preorder on X(T T X, X) is an order for all X Ob(X). Example Let X be an ordered set. For x X, let X x = x = {y X y x} be the down-set of x in X. The down-closure of A X is X A = A = x A x, and A is down-closed (or a down-set) if A = A. There is a fully faithful map : X Dn X = {A X A = A}, where the set of down-sets in X is ordered by inclusion. The down-set functor Dn : Ord Ord sends an ordered set X to Dn X and a monotone map f : X Y to Dn f : Dn X Dn Y defined by Dn f(a) = f(a) for all A Dn X. This functor with the union maps Dn X = : Dn Dn X Dn X (where A = A A A for all A Dn Dn X) and the down-set maps X : X Dn X form the down-set monad (see [7]) Dn = (Dn, Dn, ) on Ord (which is an ordered category with the pointwise order). As Dn X is left adjoint to the down-set map Dn X and the down-set functor is a 2-functor, the monad Dn is lax idempotent. For X Ob(Ord), we write X op for the same set equipped with the opposite order. By setting Up X := (Dn (X op )) op, X := ( X op) op, Up X := ( Dn (X op ) )op, one can define the up-set monad Up = (Up, Up, ) on Ord, which is colax idempotent, as it is lax idempotent with the hom-sets equipped with their dual order. Proposition Let T = (T, µ, η) and S = (S, ν, δ) be two monads on an ordered category X. Assume that λ : T S ST is a distributive law of T over S. If S is lax idempotent, then the corresponding lifting S is also lax idempotent. Proof. Since S is lax idempotent, we have Sδ X δ SX for all X Ob(X). Recall (Proposition 1.3.5) that the functor S : X T X T is given by (X, a) (SX, Sa λ X ) and (f : X Y ) (Sf : SX SY ) for all T-algebras (X, a) and X T -morphisms f : (X, a) (Y, b). Let us fix a T-algebra (X, a). By Remark 1.3.3, one gets that S δ (X,a) = Sδ X δ SX = δ S(X,a). Finally, since S is a 2-functor, we have immediately, by definition of S, that S is also a 2-functor. Consequently, S is lax idempotent over X T.

14 2 INJECTIVE OBJECTS 9 2 Injective objects The aim of this section is to generalize a result given in [8], which briefly says that under some assumptions the injective objects can be given in terms of T-algebras. 2.1 Definition and elementary properties Definition Let X be a category and M a class of morphisms in X. An object X Ob(X) is M-injective if for any morphisms f : Y X and j : Y Z with j M, there exists a morphism f : Z X such that the following diagram commutes: Y j Z f f X. We denote by M-Inj(X) the class of all M-injective objects in X. Examples The following examples come almost from [1]. (1) In Set, if M consists of all injections (i.e. monomorphisms), then the M-injective objects are precisely the non-empty sets. (2) In R Mod, if M consists of all injections (i.e. monomorphisms), then the M-injective objects are precisely the injective modules. (3) In Ab, if M consists of all injections (i.e. monomorphisms), then the M-injective objects are precisely the divisible abelian groups. (4) In Top, if M consists of all embeddings of closed subspaces of normal spaces, then [0, 1] and R are M-injective objects (c.f. Tietze Extension Theorem). (5) In Top, if M consists of the single embedding j : {0, 1} [0, 1], then the M-injective objects are the pathwise connected spaces. Let us see now some elementary properties of M-injective objects. Proposition Let X be a category. Every terminal object is M-injective. Proof. Immediate from the definition of a terminal object. Proposition Let X be a category. A product of M-injective objects, when it exists, is again an M-injective object. Proof. Consider the following diagram: Y j X f g g i i I P i p i P i. Let P i Ob(X) be M-injective objects (with i I), j : Y X a morphism in M, and f : Y i I P i an arbitrary morphism. Assume that the p i s are the projections. Since each P i is M-injective, there exists a g i such that p i f = g i j for all i I. By definition of the product, one gets a unique

15 2.2 Injective objects via algebras 10 g : X i I P i such that p i g = g i for all i I. Thus, p i g j = g i j = p i f for all i I. Unicity of the induced morphism implies that g j = f. Proposition Let X be a category. A retract of an M-injective object is also M-injective. Proof. Consider the following diagram: Y i f f P j i r X g r g R. Let P be an M-injective object, j : Y X a morphism in M, f : Y R an arbitrary morphism, where R is a retract of P. So, there exists two morphisms r : P R and i : R P such that r i = 1 R. Since P is M-injective, one gets a morphism g : X P such that g j = i f. Thus, r g j = r i f = f, and therefore, R is M-injective. Corollary Let X be a category with products and a zero object. If (P i ) i I is a family of objects of X, then the following conditions are equivalent: (i) i I P i is M-injective; (ii) P i is M-injective for all i I. Proof. For a fixed index j I, define f j : P j P i by f j = 1 Pi if i = j and f j = 0 (the zero morphism) if i j. This induces a unique morphism s j : P j i I P i such that p j s j = 1 Pj for i = j and p i s j = 0 for i j, where the p j s are the projections. In particular, each P j is a retract of i I P i. If each P j is M-injective for all j I, then i I P i is also M-injective by Proposition If i I P i is M-injective, then each P j is M-injective for all j I by Proposition Injective objects via algebras The following definition generalizes that given in [8] for a preordered category. Definition Let T = (T, µ, η) be a monad on a preordered category X. A morphism f : X Y is a T-embedding if T f : T X T Y admits a right adjoint g : T Y T X such that g T f = 1 T X. Let us now give two others definitions. Definition Let T = (T, µ, η) be a monad on a preordered category X. A morphism f : X Y is lax T-unital if T f : T X T Y admits a right adjoint g : T Y T X such that η X g η Y f. If the last inequality is an equality, then f is T-unital. Remark Under the same assumptions as the definitions above, we obtain the following implications: T-embedding = T-unital = lax T-unital. Indeed, the second implication is immediate from definitions. For the first implication, take a T-embedding f : X Y and a right adjoint g : T Y T X of T f such that g T f = 1 T X. Using the naturality of η, we get that η X = g T f η X = g η Y f, which shows that f is T-unital.

16 2.2 Injective objects via algebras 11 Notations Let T = (T, µ, η) be a monad on a preordered category X. From now on, we will write E := {η X } X Ob(X), U for the class of T-unitals, and L for the class of lax T-unitals. Moreover, if A is a subcategory of X and M is a class of morphisms in X, then M A := M Mor(A). Finally, if X is an ordered category, if it exists, the unique right adjoint of a morphism f : X Y will be denoted by f. Lemma Let T = (T, µ, η) be a lax idempotent monad on a preordered category X. Then E U. In particular, E L. Proof. Let us fix X Ob(X). Since T is lax idempotent, one has T η X µ X. Furthermore, by definition of a monad, one also has that µ X T η X = 1 T X. This shows that all the elements of E are T-embeddings. By Remark 2.2.3, we obtain E U. Example Let Dn = (Dn, Dn, ) be the down-set monad on Ord (Example 1.4.7), and take the ordinal number 2 = {0, 1} Ob(Ord). The identity map 1 2 : 2 2 is Dn-unital, but 1 2 / E. This shows that E U. Lemma Let T = (T, µ, η) be a separated lax idempotent monad on a preordered category X. For Y Ob(X), if η Y is a section, then the preorder on X(X, Y ) becomes an order for all X Ob(X). Proof. By definition of a section, there is a morphism b : T Y Y such that b η Y = 1 Y. Take f, g X(X, Y ) such that f g and g f. This implies that η Y f η Y g and η Y g η Y f. Since T is separated, the preorder on X(X, T Y ) is an order, and thus one has that η Y f = η Y g. Composing the last equality by b on the left, one obtains that f = g. Corollary Let T = (T, µ, η) be a separated lax idempotent monad on a preordered category X. If (Y, b) is a T-algebra, then the preorder on X(X, Y ) becomes an order for all X Ob(X). Proof. Since b η Y = 1 Y, Lemma yields the result. We now give the first main result (generalization of a result given in [8]) of this project, which says that, under some assumptions, injective objects can be given in terms of T-algebras. Proposition Let T = (T, µ, η) be a separated lax idempotent monad on a preordered category X and M a class of morphisms such that E M L. Then there is a one-to-one correspondence that sends X M-Inj(X) to a T-algebra (X, a): M-Inj(X) = Ob(X T ). Proof. Let X M-Inj(X). Since E M, there exists a morphism a : T X X making the following diagram commute: X η X T X 1 X In other words, a η X = 1 X. Moreover, using the condition (i) of Proposition 1.4.3, one has a X. η X a = T a η T X T a T η X = T (a η X ) = 1 T X. Thus the conditions (1.7) of Proposition are satisfied. Since by Lemma the preorder on X(T T X, X) is an order, the second part of Remark implies that a : T X X is a structure map, and therefore X is a T-algebra.

17 2.2 Injective objects via algebras 12 Conversely, let (X, a) be a T-algebra and take any morphism j : Y Z with j M and any morphism f : Y X. Since M L, j is lax T-unital. So the right adjoint (T j) : T Z T Y satisfies η Y (T j) η Z j. The idea (which is inspired from [8]) is to define f : Z X by as illustrated by the following diagram: f := a T f (T j) η Z, T Y (T j) η Z Z T Z T f T X a X. f Since j is lax T-unital, one has f j = a T f (T j) η Z j a T f η Y = a η X f = f, and on the other hand, since T j (T j), one obtains f j = a T f (T j) η Z j = a T f (T j) T j η Y a T f η Y = a η X f = f. By Corollary 2.2.8, we obtain that f j = f. This proves that X is an M-injective object. Finally, the bijection is a consequence of Remark Theorem Let T = (T, µ, η) be a separated lax idempotent monad on a preordered category X and M a class of morphisms such that E M L. Let A be a full subcategory of X such that M-Inj(X) Ob(A). Then there is a one-to-one correspondence that sends X M A -Inj(A) to a T- algebra (X, a): M A -Inj(A) = Ob(X T ). Proof. Let X M A -Inj(A). As (T X, µ X ) forms a T-algebra on X, Proposition implies that T X is an M-injective object in X. Since M-Inj(X) Ob(A), T X Ob(A) (in particular, the monad T restricts to A). Since A is a full subcategory of X, the same reasoning as in the proof of Proposition shows that X is a T-algebra. Conversely, let (X, a) be a T-algebra and take any A-morphism j : Y Z with j M A and any A-morphism f : Y X. Since M A L, j is lax T-unital. So the right adjoint (T j) : T Z T Y satisfies η Y (T j) η Z j. As A is a full subcategory of X, g is an A-morphism, whence the morphism f : Z X given by f := a T f (T j) η Z is well-defined. The end of the proof is similar to the proof of Proposition Again, the bijection is a consequence of Remark The next result, which characterize the T-algebras in terms of retracts, comes originally from [12], but it also can be proved by using Proposition Corollary Let T = (T, µ, η) be a separated lax idempotent monad on a preordered category X. The following statements are equivalent for any X Ob(X): (i) X is a T-algebra; (ii) X is a retract of a free T-algebra; (iii) X is a retract of a T-algebra.

18 2.3 Example: Injective objects in OrdMon via Dn 13 Proof. (i) = (ii): Let (X, a) be a T-algebra. Since a η X = 1 X, a is a retraction, thus X is a retract of T X. (ii) = (iii): Obvious. (iii) = (i): Consider the following diagram: Z j C i f f i X Y. f r g Let Y be a T-algebra and X a retract of Y. Therefore, there are morphisms i : X Y and r : Y X such that r i = 1 X. Take a morphism j : Z C with j M and a morphism f : Z X. Since Y is a T-algebra, Proposition implies that Y is M-injective. So there exists a morphism g : C Y such that g j = i f. If we take f := r g, one obtains that f j = r g j = r i f = f, which shows that X is an M-injective object. By Proposition 2.2.9, X is a T-algebra. Notation Assume that T is a monad on a preordered category X, and that A, B are two classes of morphisms in X such that A B. Then, any class M of morphisms with A M B will be denoted by M T A,B. Finally, the following result will be illustrated by examples in the upcoming parts. Corollary Let T = (T, µ, η) and S = (S, ν, δ) be two monads on a preordered category X. Assume that S is separated lax idempotent and that λ : T S ST is a distributive law of T over S. If ST is the corresponding composite monad on X, and S is the corresponding lifting monad of S to X T, then there is a bijection M S E,L-Inj(X T ) = Ob(X ST ). Proof. Since S is separated lax idempotent (Proposition 1.4.8), one obtains, by Propositions and 1.3.7, the following sequence of bijections: M S E,L-Inj(X T ) = Ob((X T ) S ) = Ob(X ST ). 2.3 Example: Injective objects in OrdMon via Dn Before starting to see our first example, we give some definitions. For more details, see [3]. Definition An ordered monoid (or po-monoid) is a monoid M together with a partial order such that, when the binary operation is denoted as a tensor, x y = z x z y and x z y z for all x, y, z M. A monotone map f : M N is a homomorphism of ordered monoids if f(m m ) = f(m) f(m ) and f(1) = 1 (2.1)

19 2.3 Example: Injective objects in OrdMon via Dn 14 for all m, m M. Furthermore, f : M N is an embedding of ordered monoids if it is injective, and for all m 1,..., m n, m M, one has f(m 1 ) f(m n ) f(m) = m 1 m n m. The category of ordered monoids with homomorphisms of ordered monoids is denoted by OrdMon. Remark The category OrdMon can also be defined as the category of monoids and homomorphisms of monoids in Ord, seen as a monoidal category. Definition A quantale V is a complete lattice which carries a monoid structure such that, when the binary operation is denoted as a tensor, a ( ) : V V, ( ) b : V V are sup-maps for all a, b V. Hence the tensor distributes over suprema: a i I b i = i I(a b i ), A homomorphism of quantales f : V W is a map satisfying i I a i b = i I (a i b). f(a b) = f(a) f(b), f(1) = 1, f( A) = f(a) for all a, b V and A V. The category of quantales with homomorphisms of quantales is denoted by Qnt. Let G : OrdMon Ord be the forgetful functor. We can define a left adjoint F : Ord OrdMon by sending an ordered set (X, ) to (F X, ), where F X is the free monoid over X and is an order on F X defined by: [x 1,..., x n ] [y 1,..., y m ] n = m and x i y i i = 1,..., n for all [x 1,..., x n ], [y 1,..., y m ] F X, and sends a monotone map f : X Y to F f : F X F Y defined by F f([x 1,..., x n ]) := [f(x 1 ),..., f(x n )] for all [x 1,..., x n ] F X. Defining the unit η : 1 Ord GF and the counit ε : F G 1 OrdMon by η X : X GF X and ε M : F GM M x [x] [m 1,..., m n ] m 1 m n for all X Ob(Ord) and M Ob(OrdMon), one can easily check that F : Ord. OrdMon : G Let L = (L, m, e) be the monad associated to this adjunction. One calls L the list monad on Ord. We get the following results. Proposition There is an isomorphism that commutes with the forgetful functors to Ord: Proof. Define a functor K : OrdMon Ord L by Ord L = OrdMon. M (GM, Gε M ) and (f : M N) (Gf : GM GN)

20 2.3 Example: Injective objects in OrdMon via Dn 15 for all M Ob(OrdMon) and f : M N in OrdMon. The inverse H : Ord L OrdMon can be defined on objects by (X, a) (X, a ) for any L-algebra (X, a), and where a : X X X (x, y) a([x, y]) for all x, y X, and sends any morphism to the same underlying morphism. Moreover, we define the neutral element by 1 X := a([ ]). where [ ] denotes the empty word. The verifications are left to the reader. Proposition Let L = (L, m, e) be the list monad on Ord and Dn = (Dn, Dn, ) the down-set monad on Ord. There is a distributive law λ : LDn DnL of L over Dn given by λ X ([A 1,..., A n ]) = {[a 1,..., a n ] a i A i i = 1,..., n} for all X Ob(Ord) and [A 1,..., A n ] LDn X. Proof. Routine verification (but a bit technical). Proposition If DnL is the composite monad corresponding to the distributive law given in Proposition 2.3.5, then Ord DnL = Qnt. Proof. Define a functor S : Qnt Ord DnL identically on morphisms and on objects by (X,, ) (X, a, ) for all (X,, ) Ob(Qnt), and where a, : Dn LX X is defined by a, (A) := { n a i [a 1,..., a n ] A} i=1 for all A Dn LX. The inverse R : Ord DnL Qnt can be defined identically on morphisms and on objects by (X, a) (X, a, a ) for any DnL-algebra (X, a), where a : X X X and a : P X X are defined by a : X X X and a : P X X (x, y) a( LX [x, y]) Y a( LX {[y] y Y }) for all x, y X and Y X. Again, all the verifications are left to the reader. Proposition Let Dn be the corresponding lifting of the distributive law given in Proposition on Ord L. If f : X Y is a Dn-embedding, then f is an embedding of ordered monoids. Proof. Let f : X Y be a Dn-embedding. Since the functor Dn is defined as Dn on L-algebras by Remark 1.3.3, let us write Dn instead of Dn. The condition (Dn f) Dn f = 1 Dn X implies that Dn f is

21 2.4 Example: Injective objects in Sup via Up 16 a section, thus Dn f is injective. Let us fix x, y X, x y. Then x y. Since Dn f is injective, the naturality of implies that f(x) = Dn f( x) Dn f( y) = f(y). Thus f(x) f(y), which shows that f is injective. Fix x 1,..., x n, x X and assume that f(x 1 ) f(x n ) f(x). Since f is a morphism of OrdMon, we get the following implications: f(x 1 ) f(x n ) f(x) = f(x 1 x n ) f(x) = f(x 1 x n ) f(x) = Dn f( x 1 x n ) Dn f( x) (naturality of ) = x 1 x n x ((Dn f) Dn f = 1 Dn X ) = x 1 x n x. Therefore, f is an embedding of ordered monoids. Since Dn is lax idempotent (Example and Proposition 1.4.8), Corollary implies the following bijections: M Dn E,L-Inj(OrdMon) = Ob(Ord DnL ) = Ob(Qnt). Therefore, the ME,L Dn -injective objects in OrdMon are exactly the quantales. Remark Lambek et al. [13] proved that the injective objects over embeddings of ordered monoids are exactly the quantales, but with a slightly different definition of homomorphism of ordered monoids. They consider the lax condition f(m) f(m ) f(m m ) instead of f(m m ) = f(m) f(m ) in (2.1). We tried to prove that an embedding of ordered monoids is a Dn-embedding, but we did not succeed. This is a possible reason that Lambeck et al. took the lax definition for homomorphisms of ordered monoids given in [13]. 2.4 Example: Injective objects in Sup via Up Recall that Sup denotes the category of complete lattices with sup-maps. As presented in [7], completeness of an ordered set X is characterized by the existence of an adjunction X X : X Dn X. In the same spirit, we give the following definition (see [7] or [18] for more information). Definition A complete lattice X is completely distributive (or constructively completely distributive) if the left adjoint X has itself a left adjoint X: X X : Dn X X. The category of completely distributive lattices with maps that preserve simultaneously all suprema and infima is denoted as Ccd. Definition A morphism f : X Y in Sup is a sup-embedding if it is injective, and for all x, y X, one has f(x) f(y) = x y. Proposition There is an isomorphism that commutes with the forgetful functors to Ord: Ord Dn = Sup.

22 2.5 Example: Injective objects in Top 0 via F and F + 17 Proof. If (X, a) is a Dn-algebra (X is thus an ordered set), then we can define the suprema of any A X by a A := a( A). Conversely, if X is a complete lattice, then one can define a structure map a X : Dn X X by a X (A) := A for all A Dn X. We can now define two functors F : Ord Dn commute with the forgetful functors to Ord, by Sup and G : Sup Ord Dn, that F : Ord Dn Sup and G : Sup Ord Dn (X, a) (X, a ) and it can be easily checked that they are mutual inverses. X (X, a X), Proposition Let Up = (Up, Up, ) be the up-set monad on Ord and Dn = (Dn, Dn, ) the down-set monad on Ord. There is a distributive law λ : DnUp UpDn of Dn over Up. Proof. See [16, Proposition 5.5]. Theorem If UpDn is the corresponding composite monad of the distributive law given in Proposition 2.4.4, then Ord UpDn = Ccd op. Proof. See [16, Theorem 6.2]. Recall (Example 1.4.7) that Up becomes a lax idempotent monad on Ord if we define the order on hom-sets of Ord by the dual pointwise order, that is: f g f(x) g(x) x X for all monotone maps f, g : X Y. In this case, Ũp also becomes a lax idempotent monad on Sup. Therefore, Corollary implies the following bijections: MŨp E,L -Inj(Sup) = Ob(Ord UpDn ) = Ob(Ccd op ) = Ob(Ccd). This shows that the MŨp E,L-injective objects in Sup are exactly the completely distributive lattices. Remark The last result is not really surprising, since we already know that the injective objects in Sup over embeddings (in the sense of [1]) are exactly the completely distributive lattices. Moreover, as in Example 2.3, an Ũp-embedding is a sup-map, but we do not know about the converse. 2.5 Example: Injective objects in Top 0 via F and F + The following example comes from [9]. Definition A complete lattice X is continuous if the restriction of the supremum map X : Dn X X to the set Idl X of ideals in X has a left adjoint X : X X : Idl X X

23 2.6 Example: Injective objects in Sup via Fil 18 (compare with Definition 2.4.1). The category of continuous lattices and inf-maps that preserve updirected suprema is denoted by Cnt. Escardó and Flagg [9] defined the filter monad F = (F, µ, η) on the category Top 0 of T 0 -topological spaces as follows. Given a T 0 -space X, one denotes its lattice of open sets by OX. The elements of F X are the filters in OX. We put on F X the topology, which is in fact the Scott topology (see [10]), generated by the sets U = {x F X U x} with U OX. These sets form a base since U U = (U U ) for all U, U OX. Given a continuous map f : X Y, one defines F f : F X F Y by F f(x) = {V OY f 1 (V ) x} for all x F X. We also define the natural transformations η : 1 Top0 F and µ : F F F by η X (x) = {U OX x U}, µ X (X) = {U OX U X} for all X Ob(Top 0 ), x X and X F F X. These definitions make F = (F, µ, η) into a monad. The category Top 0 becomes an ordered category with the following order. For continuous maps f, g : X Y, f g f 1 (V ) g 1 (V ) for all V OY. (2.2) With this definition, F becomes a lax idempotent monad on Top 0. Note that we took the reverse inclusion in (2.2), since the definition of lax idempotent monad presented in [9] is the dual of Definition Furthermore, it is proved in op. cit. that the Eilenberg Moore category is isomorphic to the category of continuous lattices (endowed with the Scott topology) and that the F-embeddings are precisely the subspace embeddings. If M is the class of F-embeddings, then Proposition implies that M-Inj(Top 0 ) = Ob(Top F 0 ). Thus we obtain a well-known result in topology, which says that the injective objects over subspace embeddings in Top 0 are exactly the continuous lattices endowed with the Scott topology (see [10]). Escardó and Flagg [9] also treated the case for the proper filter monad F + on Top 0, defined in a similar way as F, but by taking, for all X Ob(Top 0 ), the set of proper filters on OX instead of F X. By a similar argument as above, they proved that the F + -algebras are exactly the Scott domains (see [10]) endowed with the Scott topology, and that the F + -embeddings are precisely the dense subspace embeddings. As before, if M is the class of F + -embeddings, then we obtain by Proposition that M-Inj(Top 0 ) = Ob(Top F+ 0 ). This shows that the injective objects over dense subspace embeddings are exactly the Scott domains endowed with the Scott topology (this result was originally proved by Wyler [20]). 2.6 Example: Injective objects in Sup via Fil We give the dual notion of Definition For more informations, see [7]. Definition A complete lattice X is cocontinuous if the restriction of the infimum map X :

24 3 CONSTRUCTION OF LAX IDEMPOTENT MONADS 19 Up X X to the set Fil X of filters in X has a right adjoint X X : X Fil X. The category of cocontinuous lattices and sup-maps which preserve down-directed infima is denoted by Cnt co. The function Fil defined on ordered sets is the object part of a 2-functor Fil : Ord Ord that is the restriction of the up-set 2-functor Up to filters: for a monotone map f : X Y, one has Fil f(a) = x A f(x) for all filters A X. The up-set map of an ordered set X corestricts to a monotone map X : X Fil X, and union yields the infimum map Fil X : Fil Fil X Fil X. X : Definition The monad Fil = (Fil, Fil, ) is called the ordered-filter monad on Ord. Proposition Let Dn = (Dn, Dn, ) be the down-set monad on Ord and Fil = (Fil, Fil, ) the ordered-filter monad on Ord. There is a distributive law λ : DnFil FilDn of Dn over Fil given by for all X Ob(Ord) and x DnFil X. Proof. See Exercise 4.A [7]. λ X (x ) = {A Dn X B x (A B )} Proposition If FilDn is the corresponding composite monad of the distributive law given in Proposition 2.6.3, then Ord FilDn = Cnt co. Proof. See [7, Proposition 4.4.1]. Proposition If Ord is endowed with the dual pointwise order, then the ordered-filter monad Fil = (Fil, Fil, ) becomes lax idempotent. Proof. Immediate, since Up = (Up, Up, ) becomes lax idempotent with the dual pointwise order (Example 1.4.7) and that Fil is a restriction of Up. Therefore, if Ord is ordered with the dual pointwise order, then the hypotheses of Corollary are satisfied, and thus we obtain the following bijections: M Fil E,L-Inj(Sup) = Ob(Ord FilDn ) = Ob(Cnt co ). This shows that the ME,L Fil -injective objects in Sup are exactly the cocontinuous lattices. 3 Construction of lax idempotent monads In this section, we will see a generalization of the theory presented in [19], which gives an explicit construction of a lax idempotent monad. Some results are directly inspired from [19].

25 3.1 Algebraically adjoint monads Algebraically adjoint monads Definition A monad T = (T, µ, η) on a category X is algebraically adjoint if for all X, Y Ob(X), the hom-class X(X, T Y ) admits an order such that X T becomes an ordered category, T a admits a right adjoint (T a) for any algebra structure a : T X X, and each component µ Y of the monad multiplication has a right adjoint µ Y, such that µ Y preserves the order on the right: f g = µ Y f µ Y g for all f, g : X T Y. The following lemma describes the compositions which preserve the order. Lemma If T = (T, µ, η) is an algebraically adjoint monad on a category X, then: (i) the extension operation ( ) T preserves the order on the hom-classes. In other words, f g = f T g T for all f, g : X T Y ; (ii) f g = µ Y f µ Y g for all f, g : X T T Y ; (iii) f g = T h f T h g for all f, g : X T Y and h : Y Z; (iv) f g = f h g h for all f, g : Y T Z and h : X Y. Proof. (i) Take f, g : X T Y such that f g. By definition of an algebraically adjoint monad, one has f T 1 T X g T 1 T X, thus f T g T. (ii) Take f, g : X T T Y such that f g. Then, (1 T Y ) T f (1 T Y ) T g, which is equivalent to µ Y f µ Y g. (iii) Take f, g : X T Y such that f g and h : Y Z. Then, f g = (η Z h) T f (η Z h) T g µ Z T η Z T h f µ Z T η Z T h g T h f T h g. (iv) Take f, g : Y T Z such that f g and h : X Y. Then, f g = f T η Y h g T η Y h µ Z T f η Y h µ Z T g η Y h µ Z η T Z f h µ Z η T Z g h f h g h.

26 3.2 T-monoids 21 Lemma Let T = (T, µ, η) be an algebraically adjoint monad on a category X. Assume that f : T Y T Z is a morphism such that the right adjoint f : T Z T Y exists, and that f preserves the order on the right: g h = f g f h for all g, h : X T Y. Then f is a retraction if and only if f f = 1 T Z. Proof. On one hand, if f f = 1 T Z, then f is a retraction by definition. On the other hand, assume that there is a morphism g : T Z T Y such that f g = 1 T Z. Since 1 T Y f f, if we compose this inequality with g on the right, then we get g f by Lemma Finally, using that f preserves the order, we get 1 T Z = f g f f. Since f f 1 T Z, we can conclude that f f = 1 T Z. Lemma If T = (T, µ, η) is an algebraically adjoint monad on a category X and a : T X X an algebra structure, then for the morphism a : X T X defined by a := µ X (T a) η X, one has 1 T X a a and a a = 1 X. (3.1) Moreover, a morphism which satisfies (3.1) is unique. Proof. On one hand, T a T η X = 1 T X implies that T a (T a) = 1 T X by Lemma 3.1.3, and a a = a µ X (T a) η X = a T a (T a) η X = a η X = 1 X. On the other hand, using Lemma and naturality of η, we have that 1 T X = µ X η T X µ X (T a) T a η T X = µ X (T a) η X a = a a. For the last claim, assume that α : X T X is a morphism such that 1 T X α a and a α = 1 X. Then, by Lemma 3.1.2, we obtain which implies that α = a. α a a α = a and a α a a = α, Remark As a consequence of Lemma 3.1.4, we obtain µ X = µ X for all X Ob(X). Proposition Let T = (T, µ, η) be an algebraically adjoint monad on a category X. Then for all f, g : X T Y, one has f g µ Y f µ Y g. Proof. Since T is algebraically adjoint, if f g, then µ Y f µ Y g. Conversely, since µ Y µ Y = 1 T Y (Remark 3.1.5) and µ Y preserves the order on the right (Lemma 3.1.2), if µ Y f µ Y g, then f g. From now, we will no more mention when we use the points (ii) (iv) of Lemma T-monoids Definition Given an algebraically adjoint monad T = (T, µ, η) on a category X, a T-monoid (or Kleisli monoid) is a pair (X, α), where X Ob(X) and α : X T X is a reflexive and transitive morphism: η X α, α T α α. (3.2)

27 3.2 T-monoids 22 A morphism of T-monoids f : (X, α) (Y, β) is a morphism f : X Y with T f α β f, and f composes with another morphism of T-monoids g : (Y, β) (Z, γ) as in X. The category of T-monoids and morphisms of T-monoids is denoted by Mon(X T ). If X Ob(X) and (Y, β) Ob(Mon(X T )), then the hom-class X(X, Y ) can be equipped with a preorder, the Kleisli preorder induced by β : Y T Y, as follows. For all f, g : X Y, f K g β f β g. (3.3) Remark In our context, it will be crucial to remark that in the presence of the reflexivity condition in (3.2), transitivity may be expressed as an equality. Indeed, α = µ X η T X α = µ X T α η X µ X T α α = α T α α. Examples Let T = (T, µ, η) be an algebraically adjoint monad on a category X. (1) Since one has η X η X and η T X η X = η X, the pair (X, η X ) forms a T-monoid, called the discrete T-monoid. (2) Using the naturality of η, any morphism f : X Y yields a morphism of T-monoids f : (X, η X ) (Y, η Y ). Given any T-monoid (X, α), the identity 1 X : X X also defines a morphism of T-monoids 1 X : (X, η X ) (X, α), since T 1 X η X = η X α = α 1 X. Similarly, the structure morphism α also defines a morphism of T-monoids α : (X, α) (T X, µ X), because the condition µ X T α α = α T α α implies T α α µ X α by adjunction. (3) Proposition implies that (T X, µ X ) is a T-monoid for any X Ob(X). Since 1 T T X µ X µ X and µ X T µ X = µ X µ T X for all X Ob(X), one gets T µ X µ T X µ X µ X T µ X µ T X = µ X µ X. Thus, µ X : (T T X, µ T X ) (T X, µ X ) is a morphism of T-monoids. (4) If P = (P,, { }) denotes the powerset monad (see [7] or [15]), and if Set(X, P Y ) admits the pointwise order (P Y ordered by set-inclusion) for all X, Y Ob(Set), then Mon(Set P ) = PrOrd. Indeed, a map α : X P X can be identified with a relation on X, and the reflexivity and transitivity conditions make this relation into a preorder. Therefore, α is the down-set map X : X P X. A map f : X Y is a morphism of Mon(Set P ) if and only if it preserves the relations, that is, if and only if f is a monotone map. (5) If F = (F, µ, η) denotes the filter monad on Set (see [19]), and if for any X, Y Ob(Set) the hom-set Set(X, F Y ) admits the pointwise order, but with F Y ordered by reverse set-inclusion,

28 3.2 T-monoids 23 then Mon(Set F ) = Top. Indeed, α : X F X associates to each point x X a filter α(x) F X. All the elements of this filter contain x by reflexivity of α, and transitivity is translated as the axiom required of a family of filters (α(x)) x X to form the set of neighbourhoods of the topology it determines ([11, Proposition 1.21]). A map f : X Y is a morphism of Mon(Set F ) if and only if the image of the neighbourhood α(x) is finer than the neighbourhood β(f(x)) of f(x), that is, if and only if f is continuous. Proposition Let T = (T, µ, η) be an algebraically adjoint monad on a category X. There is a functor Q : X T Mon(X T ) that sends (X, a) to (X, a ) for any T-algebra (X, a). In particular, any algebra structure a : (T X, µ X ) (X, a ) is a morphism of T-monoids. Proof. Since the Eilenberg Moore conditions imply a η X = 1 X and a µ X = a T a, one has η X a and µ X T (a ) a (a a T a) T (a ) a = a. Similarly, a T-homomorphism f : (X, a) (Y, b) yields a morphism of T-monoids f : (X, a ) (Y, b ), as the condition b T f = f a implies T f a b f. The last claim follows from the fact that a T a = a µ X for any algebra structure a : T X X. Lemma For a T-monoid (Y, β), one has for any morphisms f, g : X Y. η Y f β g β f β g Proof. Since η Y β, one observes that for f, g : X Y, if β f β g, then Conversely, if η Y f β g, then η Y f β f β g. β f = µ Y η T Y β f = µ Y T β η Y f µ Y T β β g = β T β g = β g. Proposition Any morphism of T-monoids h : (Y, β) (Z, γ) preserves the Kleisli preorder. In other words, β f β g = γ h f γ h g for any morphisms f, g : X Y. Proof. Assume that f, g : X Y are such that β f β g. One has η Z h f = T h η Y f T h β f T h β g γ h g. By Lemma 3.2.5, the last inequality is equivalent to γ h f γ h g. Corollary Endowed with the Kleisli preorder, Mon(X T ) becomes a preordered category.