aa r X i v : . [ m a t h . C T ] J un Injective Hulls of ( L , V ) -Categories Eros Martinelli ∗ Abstract
In this communication some results obtained in [ Rum16 ] are generalized to categories enriched in acommutative quantale V . Using these results is it shown that every ( L , V ) -category admits an injectivehull. A connection between Isbell-duality and the construction of injective hulls made in [ Ban73 ] is made. In their article [ LBKR12 ] , Lambek et al. studied injective objects and injective hulls in the category ofordered monoids with sub-multiplicative monotones maps. They proved that quantales are the injectiveobjects and that every ordered monoid admits an injective hull. This injective hull is obtained as the fixedpoints of a quantic nuclei on the quantales of downclosed subsets.In [ Rum16 ] the problem is studied further by considering the so called Quantum B-Algebras , orderdedsets equipped with two implications that mimic the residuals in an quantale. It was shown that quantalesare injective in the category formed by them by taking as morphisms oplax homomorphisms. Moreover, itwas proven that every Quantum B-Algebra admits an injective hull.In this paper we show how the two results can be put under a common roof and generalized to therealm of enriched categories. This is done by considering ( L , V ) -categories, a particular example of ( T , V ) -categories, obtained by taking the list monad L . In this way both ordered monoids with submultiplicativemonotone maps and quantum B-algebras with oplax homomorphisms, or more generally their enrichedcounterparts, become full sub-categories of ( L , V ) - Cat : the category formed by taking ( L , V ) -categoriesalong with ( L , V ) -functor (which is a notion that is able to capture the both the submultiplicativiness andthe oplaxness).For such categories, Clementino, Hofmann and Tholen, have shown how many constructions comingfrom enriched category theory: [ Hof11, CH09b ] ((relative) cocompleteness and cocompletion), [ Hof14 ] (completeness), could be developed in the more general context of ( T , V ) -categories. In particular in [ Hof11 ] is it shown, generalizing results regarding topological spaces ( [ Esc97, Esc98 ] ), how injectiveobjects could be characterized as algebras for a Koch-Zöberlein monad which is a generalization of thepreasheaf monad.Using this characterization of injectives as algebras for a monad, by pure categorical arguments is pos-sible to generalize Theorem 4.1 of [ LBKR12 ] and proving that quantales (their enriched counterparts) areinjective also in ( L , V ) - Cat . Since quantales can be seen both as ordered monoids and as quantum B-algebras, we can recover the characterization of injectives contained in [ LBKR12 ] and in [ Rum16 ] . ∗ This work is financed by the ERDF – European Regional Development Fund through the Operational Programme for Competi-tiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT -Fundação para a Ciência e a Tecnologia, within project POCI-01-0145-FEDER-030947, and project UID / MAT / / / BD / / [ Rum16 ] wewill prove that, for a full-subcategory of ( L , V ) - Cat a sufficient and necessary condition for its objects toadmit an injective hulls is that quantales are objects of such categories.We would like to stress the fact that results of this paper are not a mere exercise in fuzzyfication , thatis to say, a rephrase of the ones in [ Rum16 ] for a general V . In order to be able to generalize them, onneeds a more general theory of colimits, which is provided by the notion of ( L , V ) -colimits we are going tointroduce. This provides a nice an elegant categorical way to shred light on some universal constructionsthat in the ordered case are somehow "hidden".The structure of the paper is the following: • The first section is devoted to introduce some background material on V -categories and to introducethe notion of ( L , V ) -categories. We will show how many constructions coming from enriched cate-gory theory (distributors, colimits, colimits complections etc.) can be developed in the more generalcontext of ( L , V ) -categories. • The second section is, as the title suggests, an intermezzo . We recall how injectives and injective hullsare built in the category of ordered sets and monotone maps between them. We show how the sameideas apply to the enriched case, for a general commutative quantale V . The aim of the section is tointroduce some ideas that later will be applied in the more general context of ( L , V ) -categories. • The third section forms the core of the paper. Promonoidal categories are introduced and theirrelation with ( L , V ) -categories is explained. Quantum B-algebras are introduced as representablespromonoidal categories, mirroring in a "dual" way the case of monoidal’s ones. A characterizationof injectives in ( L , V ) - Cat is given from which will follow the ones for promonoidal categories andquantum B-algebras. Using some ideas from the previous sections we will prove that every quantumB-algebras admits an injective hull, which is constructed by defining a quantic nuclei on the categoryof ( L , V ) -presheaves. • The fourth section is devoted to characterize those sub-categories of ( L , L ) - Cat for which injectivehulls exist. Using ideas from the previous section we will prove that a necessary and sufficient con-dition is given by having quantales as objects. • The final section sketch a connection between Isbell-dually in the ordered case and the constructionof injective hulls for topological spaces contained in [ Ban73 ] . Acknowledgement.
I am grateful to D. Hofmann for valuable discussions about the content of the paperand to I. Stubbe for the valuable suggestions he gave me during his visit in Aveiro. ( L , V ) - Cat and V - Cat
Enriched categories are a generalization of ordinary categories, in which the hom-sets take values in a cos-mos , a symmetric-closed monoidal category V . The standard reference about them is [ Kel82 ] .In this paper we are going to consider only the case in which V is a commutative quantale , that is to say,a commutative monoid in the category of sup-lattices and suprema preserving monotone maps. Althoughthis might seen not so useful, Lawvere, in his seminal paper [ Law73 ] , showed how taking enrichment in acommutative quantale is not only useful as a toy model in which some general constructions become easierto understand (due to the absence of coherent conditions), but that they are worthy to study as their own,since they are able to capture some important mathematical structures like metric spaces. ( L , V ) -categories are a special case of the more generals ( T , V ) -categories, where the list monad L isconsidered. They could also be seen as the ordered version of multicategories ( [ Lei04 ] for an account on2hem). The basic idea is that, instead of having arrows with just a single domain, we allow them to haveas domain a list of objects.In the following section we are going to recall / introduce some basics notions of such categories. Ourpoint of view is slightly different from the more "standard" one contained in [ Kel82 ] . Our point of viewwill be more "relational": following [ BCSW83, CT03 ] , we will introduce the quantaloid of V -relations anddefining V -categories starting from there. This might seen as an overkilling, but it will be clear in the sec-tion related to ( L , V ) -categories, how this allows us to smoothly introduce some concepts (as distributors,presheaves and colimits) in the ( L , V ) -case.The section is structured as follows: • The first two subsections are devoted to introduce V -categories, V -distributors and (co)-limits, pro-viding a lot of examples of their use and explaining how they generalize some (possibly) more familiarconcepts in order theory. • In following two subsections we will introduce ( L , V ) -categories and by mimic what is done for V -categories we will introduce ( L , V ) -distributors and ( L , V ) -colimits, explaining how they are able tocapture in more abstract and categorical way some constructions in order theory. • In the last subsection we will introduce the correspondent enriched version of quantales, a conceptthat plays a central role through the whole paper.
As we’ve already remarked before, we are going to consider only categories enriched in a commutativequantale:
Definition 2.1.
A commutative quantale ( V , ⊗ , k ) is complete lattice endowed with a (commutative) mul-tiplication preserving suprema in each variable ⊗ : V × V → V for which k ∈ V is a neutral element. Remark 2.2.
By the adjoint functor theorem applied to order sets, it follows that −⊗ = admits a rightadjoint (in each variable) denoted by: [ − , =] and called "internal hom". Examples 2.3.
1. The two-elements boolean algebra 2 = {
0, 1 } with ∧ as multiplication and ⇒ asinternal hom is a quantale.2. More generally, every frame F becomes a quantale with the multiplication given by ∧ . In particular k : = ⊤ .3. ([ + ∞ ]) op (with the reverse natural order) with max as multiplication is a quantale. The internalhom is given by the "truncated sum" defined as v ⊖ u : = max ( v − u , 0 ) .4. Consider the set: ∆ : = { ψ : [ + ∞ ] → [
0, 1 ] | for all α ∈ [ + ∞ ] : ψ ( α ) = _ β<α ψ ( β ) } ,of distribution functions; with the pointwise order it becomes a complete order set. For all ψ , φ ∈ ∆ , α ∈ [ + ∞ ] , define the following multiplication: ψ ⊗ φ ( α ) : = _ β + γ<α ψ ( β ) ∗ φ ( γ ) ,Where ∗ is the ordinary multiplication on [
0, 1 ] . It can be shown that ( ∆ , ⊗ , k ) is quantale, where k ( α ) = α > k ( ) : =
0. 3s we stated in the introduction of this section, we are going to present V -categories from a more "rela-tional" point of view. The first step is to define the so called quantaloid of V -relation , which is the enrichedgeneralization of the quantaloid Rel of 2 valued relations. For an account of quantaloids, we refer to [ Stu14 ] for a brief overview and to [ Stu05 ] for a more depth description.The quantaloid V - Rel is the quantaloid whose objects are sets, and an arrow r : X −7−→ Y is given by afunction: r : X × Y → Y .The composition of r : X → Y , s : Y → Z is given as "matrix multiplication" and is defined pointwise as: s · r ( x , y ) : = _ y ∈ Y r ( x , y ) ⊗ s ( y , z ) ;the identity arrow Id : X −7−→ X is defined as:Id ( x , x ) : = ¨ ⊥ if x = x k if x = x The complete order on V - Rel ( X , Y ) is the one induced (pointwise) by V , i.e: r ≤ r ′ in V - Rel ( X , Y ) iff r ( x , y ) ≤ r ′ ( x , y ) in V for all x , y ∈ X , Y . (1) Remark 2.4.
Notice that the order is complete because V is. Since multiplication of V preserves supremain both variables and because suprema commute with suprema, one has: ( _ i r i ) · ( _ j s j ) = _ i , j r i · s j .Proving that V - Rel is a quantaloid.
Remark 2.5.
Notice that in the case in which V =
2, 2-
Rel is the quantaloid of relations, and the "matrixmultiplication" defined previously becomes the "classical" relational composition.
Remark 2.6.
Note that (1) is equivalent to: k ≤ ^ x , y ∈ X , Y [ r ( x , y ) , s ( x , y )] ;indeed, fixing x , y ∈ X , Y we have: r ( x , y ) ≤ r ′ ( x , y ) iff r ( x , y ) ⊗ k ≤ r ′ ( x , y ) iff k ≤ [ r ( x , y ) , s ( x , y )] . Remark 2.7.
Notice that every function f : X → Y could be seen as a V -relation in a straightforward wayas follows: f ( x , y ) : = ¨ ⊥ if f ( x ) = yk if f ( x ) = y The identity in V - Rel ( X , X ) is an example of this construction.We have also an involution ( − ) ◦ : V - Rel op → V - Rel defined as r ◦ ( y , x ) : = r ( x , y ) , which satisfies: ( X ) ◦ = X , ( s · r ) ◦ = r ◦ · s ◦ ( r ◦ ) ◦ = r . Definition 2.8. A V -category ( X , a ) is a pair, where X is a set and a : X −7−→ X is a V -relation that satisfies: • I d ≤ a ; 4 a · a ≤ a . Remark 2.9.
In the paper, when the V -structure is clear from the context, we’ll denote a V -category ( X , a ) simply as X . Definition 2.10.
Given two V -categories ( X , a ) , ( Y , b ) , a V - functor f : ( X , a ) → ( Y , b ) is a function betweenthe two underlying sets such that: a ≤ f ◦ · b · f ,which pointwise means, for all x , y ∈ X : a ( x , y ) ≤ b ( f ( x ) , f ( y )) .If the equality holds, we call it fully-faithful . Examples 2.11.
1. In the case in which V =
2, as already stated before, a 2-category is an ordered setand a 2-functor is a monotone map.2. Categories enriched in the quantale ([ + ∞ ]) op , as first recognized by Lawvere in his seminal paper [ Law73 ] , are generalized metric spaces and ([ + ∞ ]) op -functors between them are non-expansivemaps.3. Categories enriched in ∆ are called probabilistic metric spaces as first recognized in [ Fla97 ] .4. The quantale V could be viewed as a V -category with the V -structure given by its internal hom [ − , =] .5. By using the involution in V - Rel , for every V -category ( X , a ) one can define its opposite category X op = ( X , a ◦ ) .6. Given two V -categories ( X , a ) , ( Y , b ) one can define the V -category formed by all V -functors f : ( X , a ) → ( Y , b ) , denoted by ([ X , Y ] , [ X , Y ]( − , =)) , by defining the following V -structure: [ X , Y ]( f , g ) : = ^ x ∈ X b ( f ( x ) , g ( x )) .In particular we have two very important V -categories: D ( X ) : = [ X op , V ] , the category of presheaves, U ( X ) : = [ X , V ] op , the category of co-presheaves.Notice that they are the generalization (for a general V ) of the classical downward (upward) closetsubsets construction, which correspond to the case in which V = V -category ( X , a ) , there are two V -functors, called the Yoneda and the co-Yoneda embedding: y X : ( X , a ) D ( X ) , x a ( − , x ) , λ X : ( X , a ) U ( X ) , x a ( x , =) .The adjective "embedding" is justified by the fact that one can prove that they are both fully-faithful.Moreover, it can be proved that: U ( X )[ λ X ( x ) , g ] = g ( x ) , D ( X )[ y X ( x ) , g ] = g ( x ) .The last result is known as the (co)-Yoneda Lemma. This justifies the adjective "embedding", sincefrom (co)-Yoneda Lemma it immediately follows that both are fully-faithful.Again, it is worth noticing that the two functors generalize the familiar monotone maps: x x , x x ,which associate to an element the set of elements which are below (above) it.5. Given two V -categories ( X , a ) , ( Y , b ) we can define their tensor product: X ⊠ Y = ( X × Y , a ⊗ b ) ,In particular one has: X ⊠ K ≃ X (where K denotes the one-point V -category ( k ) ).In this way we can define V - Cat as the category whose object are V -categories and whose arrows are V -functors among them; moreover, V - Cat becomes an order enriched category, by defining, for two V -functor f , g : ( X , a ) → ( Y , b ) : f ≤ g iff k ≤ ^ x ∈ X b ( f ( x ) , g ( x )) .Moreover, one can show that, with the tensor product previously defined, V - Cat becomes a closed monoidalcategory, since one can show that for three V -categories ( X , a ) , ( Y , b ) , ( Z , c ) one has: V - Cat ( X ⊠ Y , Z ) ≃ V - Cat ( X , [ Y , Z ]) ≃ V - Cat ( Y , [ X , Z ]) . Distributors were introduced by Bénabou in [ Bén73 ] and since them they played an important role in cat-egory theory. They could be seen as generalizations of ideal relations from order theory. That is to say,subsets of the cartesian product of two ordered set X , Y which are upward closed in X and downwardclosed in Y .Weighted (co)-limits encompass the classical notion of (co)-limits coming from "ordinary" category the-ory, by admitting a "weight" given by a distributor. In the case of ordered sets, due to the fact that everydistributor is a (particular) subset, the weight "disappear" and becomes the "set of condition" over whichsuprema / infima are taken. Definition 2.12.
Given two V -categories ( X , a ) , ( Y , b ) , a V - distributor (or simply a distributor) j : ( X , a ) −7−→ ( Y , b ) is a V -relation between them such that: • j · a ≤ j ; • b · j ≤ j .Since the composite of two distributors is again a distributor, we can define the quantaloid V - Dist inthe same way as we defined V - Rel . In V - Dist ( X , X ) the V -structure a plays the role of the identity, sincefor every j : X −7−→ Y , one has: b · j = j · a = j . Remark 2.13.
By juggling with the definition of distributor, one can show that distributors between two V -categories ( X , a ) , ( Y , b ) are in bijective correspondence with V -functors between X op ⊠ Y and V . if it easyto prove that this correspondence is functorial and that gives and equivalence of order sets: V - Dist ( X , Y ) ≃ V - Cat ( X op ⊠ Y , V ) ≃ V - Cat ( Y , D ( X )) .In particular, to every V -distributor j : X −7−→ Y we can associate its mate : ð j ñ : Y → D ( X ) , y j ( − , y ) .Given a V -functor f : ( X , a ) → ( Y , b ) , we can define two arrows in V - Rel : • f ∗ : X −7−→ Y , f ∗ ( x , y ) : = b ( f ( x ) , y ) ; • f ∗ : Y −7−→ X , f ∗ ( y , x ) : = b ( y , f ( x )) . 6ne has: Lemma 2.14.
The V -relations f ∗ and f ∗ are both distributors, moreover f ∗ ⊣ f ∗ in V - Dist .In this way we have two bifunctors: ( − ) ∗ : V - Cat co → V - Dist , ( − ) ∗ : V - Cat → V - Dist . Theorem 2.15. ( − ) ∗ : V -
Cat co → V -
Dist defines a proarrow equipment [ Woo82, Woo85 ] on V - Cat , thatis to say: • ( − ) ∗ is locally fully-faithful; • For all V -functor f : ( X , a ) → ( Y , b ) , there exists a right adjoint to f ∗ in V - Dist . Proof.
For the first point we have to show that: f ≤ g in V - Cat iff g ∗ ≤ f ∗ in V - Dist . ⇒ ) . Fix an x , from k ≤ V x ∈ X b ( f ( x ) , g ( x )) we get k ≤ b ( f ( x ) , g ( x )) . By applying composition we get: b ( f ( x ) , y ) ≤ b ( f ( x ) , g ( x )) ⊗ b ( g ( x ) , y ) ≤ b ( f ( x ) , y ) . ⇐ ) . For every x we get: k ≤ b ( g ( x ) , g ( x )) ≤ b ( f ( x ) , g ( x )) ,hence: k ≤ ^ x ∈ X b ( f ( x ) , g ( x )) .The second point follows from the previous lemma.As a corollary we get: Corollary 2.16.
A function f : ( X , a ) → ( Y , b ) between two V -categories is a V -functor iff f ∗ : X −7−→ Y is aV -distributor; equivalently iff f ∗ : Y −7−→ X is a V -distributor.
This theorem allows us to "mirror" all the good features V - Dist has in V - Cat , giving us an "higher"perspective on notions like: weighted (co)-limits, (co)-ends, Kan extensions, and adjoint V -functors.The key fact is that V - Dist is "closed", meaning that composition as adjoint in each variable, more precisely,given: α : Z −7−→ X , β : X −7−→ Y ,one has: • ( − ) · α ⊣ ( − )  α ; • β · (=) ⊣ β à (=) .Where: ( − ) · α : V - Dist ( X , Y ) → V - Dist ( Z , Y ) β β · α . β · (=) : V - Dist ( Z , X ) → V - Dist ( Z , Y ) α β · α and: ( γ  α )( x , y ) : = ^ z ∈ Z [ α ( z , x ) , γ ( z , y )] , ( β à γ )( z , x ) : = ^ y ∈ Y [ β ( x , y ) , γ ( z , y )] .7 efinition 2.17. Let f : ( X , a ) → ( Y , b ) be a V -functor, ( Z , c ) a V - Cat , and j : X −7−→ Z be a distributor. Wesay that a V -functor g : ( Z , c ) → ( Y , b ) is the colimt of f weighted by j , and denote it by: colim ( j , f ) , if g ∗ ≃ f ∗  j , that is to say: g ∗ ( z , y ) : = b ( g ( z ) , y ) = ^ x ∈ X [ j ( x , z ) , f ∗ ( x , y )] , ( ) .Dually, if we have a distributor l : Z −7−→ X , we say that a V -functor h : ( X , a ) → ( Y , b ) is the limit of fweighted by l , and denote it by: lim ( l , f ) , if h ∗ ≃ l à f ∗ , that is to say: h ∗ ( y , z ) : = b ( y , h ( z )) = ^ x ∈ X [ l ( z , x ) , f ∗ ( y , z )] , ( ) .A V -category ( Y , b ) is called (co)-complete , if it has all weighted (co)-limits; meaning that, for all V -functors f : ( X , a ) → ( Y , b ) and for all distributors ( j : X −7−→ Z ) h : Z −7−→ X , the (co)-limits of f weighted by ( j ) - h exists. Remark 2.18.
Using the fact that: D ( X )( j , l ) : = ^ x ∈ X [ j ( x ) , l ( x )] ,(and similarly for U ( X ) ), we can re-write ( ) and ( ) as: b ( colim ( j , f )( z ) , y ) ≃ D ( X )( j ( − , z ) , b ( f ( − ) , y )) , b ( y , lim ( l , f )( z )) ≃ U ( X )( b ( y , f (=)) , l ( z , =)) . Examples 2.19.
Let ( X , a ) be a V -category and x : K → X be a point (seen as a V -functor from the onepoint V -category ( k ) ). Let u ∈ V and view it as a distributor ˜ u : K −7−→ K by defining u (
1, 1 ) : = u .1. The colimit of x , weighted by ˜ u , usually denoted as x ⊙ u , is called co-power . Unravelling the definitionof colimits, we get (for all y ∈ Y ): a ( x ⊙ u , y ) = [ u , a ( x , y )] .A category in which all the co-powers exist is called co-powered.2. Dually, due to the form of ˜ u , we can consider also the limit of x weighted by ˜ u . This is usually denotedas x ⋔ u and it is called power .A category in which all the powers exist is called powered.Let f : I → X , i x i be a family of elements of X :1. Let α : I −7−→ K be the constant distributor α ( i ) : = k . The colimit of f weighted by α is an element ˜ x ,such that (for all y ∈ Y ): a ( ˜ x , y ) = ^ i a ( x i , y ) .Such colimit is called conical supremum , and due to its universal property (which coincides with theusual suprema’s one in the case in which V =
2) is usually denoted as ˜ x : = W i x i .2. Dually, let β : K −7−→ X be the same distributor as above, but seen as an arrow from K to X . The limitof f weighted by β is an element ˜ x , such that (for all y ∈ Y ): a ( y , ˜ x ) = ^ i a ( y , x i ) .Such limit is called conical infumum ; as before, in the case in which V =
2, it coincides with theinfimum of the family { x i } i , and for this reason is usally denoted as ˜ x : = V i x i .8ll the (co)-limits we described above play a very important role, as they are basic building blocks: every(co)-limits can be written as a suitable composite of them.Let f : ( X , a ) → ( Y , b ) be a V -functor and j : X −7−→ Z be a distributor. If ( Y , b ) is co-powered and colim ( j , f ) exists, then we have: colim ( j , f )( z ) ≃ _ x ∈ X j ( x , z ) ⊙ f ( x ) .Dually, if ( Y , b ) is powered and lim ( l , f ) exists (for a distributor l : Z −7−→ X ), one has: lim ( l , f )( z ) ≃ ^ x ∈ X l ( z , x ) ⋔ f ( x ) . Remark 2.20.
Just as the set of downsets and the set of upsets of an order set X have all suprema andinfima, for a V -category ( X , a ) , D ( X ) and D ( X ) are both complete and (co)-complete V -categories.Another important concept we are going to use is the concept of adjunction between a pair of V -functors ( X , a ) ( Y , b ) . fg ⊣ This concept generalize the classical notion of Galois connection from order theory.
Definition 2.21.
Given two V -functors f : ( X , a ) → ( Y , b ) , g : ( Y , b ) → ( X , a ) ; f is left adjoint to g (viceversa: g is right adjoint to f ) and denoted: f ⊣ g , if f ∗ ≃ g ∗ . Pointwise this means that, for all x , y ∈ X , Y : b ( f ( x ) , y ) = a ( x , g ( y )) . Examples 2.22.
Let ( X , a ) be a V -category.1. If x ⊙ u exists for all u ∈ V , then the universal property of the colimits: a ( x ⊙ u , y ) = [ u , a ( x , y )] ,can be rephrased by saying that x ⊙ u ⊣ a ( x , =) .Thus, co-powered categories can be equivalently described as those for which a ( x , =) admits a leftadjoint for all x ∈ X .2. If x ⋔ u exists for all u ∈ V , then the universal property of the limits: [ u , a ( y , x )] = a ( y , x ⋔ u ) ,can be rephrased by saying that − ⋔ u ⊣ a ( − , x ) .Thus, powered categories can be equivalently described as those for which a ( − , x ) admits a rightadjoint for all x ∈ X .Let f : ( X , a ) → ( Y , b ) and g : ( Y , b ) → ( Z , c ) two V -functors and j : X −7−→ D be a distributor. From thedefinition of weighted colimit it follows that colim ( j , g f ) ≤ g ( colim ( j , f )) .Vice versa, if we consider a distributor l : D −7−→ X , from the definition of weighted limit it follows that g ( lim ( l , f )) ≤ lim ( h , g f ) .If the equality holds, we say that g preserves the (co)-limit of f weighted by ( j ) - h . If g preserves all(co)-limits, we say that is (co)-continuous . Examples 2.23. a ( x , =) : X → V , is continuous.2. a ( − , x ) : X op → V sends weighted colimits to weighted limits. Meaning that a ( colim ( j , f ) , x ) = lim ( j ◦ , a ( − , x )) , for suitable j and f ; where j ◦ is the distributor obtained from j by applying theinvolution ( − ) ◦ . 9. Left adjoints V -functors are continuous; vice versa, right adjoints are (co)-continuous.Adjunctions allow to give a nice characterization of (co)-complete V -categories: Theorem 2.24.
Let ( X , a ) be a V -category. Then ( X , a ) is complete iff the (co)-Yoneda has a right adjoint;dually, it is cocomplete if the Yoneda has a left adjoint. Moreover, in case they exist, they have the followingform: lim : U ( X ) → X , l lim ( l , I d ) , colim : D ( X ) → X , j colim ( j , I d ) . Remark 2.25.
Since both U ( − ) and D ( − ) define Kock–Zöberlein monads (see [ Koc95 ] for a far reachingtreatment on them) on V - Cat , in order to prove that the (co)-Yoneda has a (right)-left adjoint is sufficientto provide a left inverse of it.
Recall that the list monad is defined as: L : Sets → Sets , f : X → Y L f : ∐ n ≥ X n → ∐ m ≥ Y m , x : = ( x , ..., x n ) ( f ( x ) , ..., f ( x n )) ,Whit unity and multiplication given by: • e X : X → L ( X ) , x → ( x ) ; • m X : L ( X ) → L ( X ) , ( x , ..., x n ) ( x , ..., x k , ..., x n , ..., x ln ) . Remark 2.26.
Let x , w be two lists; in order to avoid possible confusion with the list of list y = ( x , w ) , wewill denote the list obtained by concatenating x and w as ( x ; w ) . Moreover, in the case in which one of ofthe two is the single element list, we will use the shortcut ( x ; w ) , instead of ( x ; ( w )) .We can extend (in a functorial way) the list monad L to V -relation by defining, for r : X −7−→ Y :˜ Lr : L ( X ) −7−→ L ( Y ) , ( x , y ) ¨ ⊥ if the two lists haven’t the same lenght r ( x , y n ) ⊗ ... ⊗ r ( x n , y n ) . otherwiseIt can be easely proven that this particular extension defines a monad on ( L , V ) - Rel that preserves theinvolution.
Remark 2.27.
From now on, we will use L for both (the ordinary list monad and its extension).This allows us to define an order-enriched category ( L , V ) - Rel in which a morphism r : X − * ◦ Y is a V -relation of the form: r : L ( X ) −7−→ Y ,and in which composition is given by: s ◦ r : = s · Lr · m ◦ X ,and in which e ◦ X : X − * ◦ X is the identity. Remark 2.28.
Note that, due to the Kleisli-style composition we defined, − ◦ r preserves suprema, but (ingeneral) s ◦ (=) doesn’t. That’s the reason why ( L , V ) - Rel is not a quantaloid, but just an order-enrichedcategory.
Definition 2.29. An ( L , V ) -category is a pair ( X , a ) , where X is a set and a : X − * ◦ X is a ( L , V ) - Rel satisfying: • e ◦ X ≤ a ; 10 a ◦ a ≤ a . Definition 2.30.
Given two ( L , V ) -categories ( X , a ) , ( Y , b ) , an ( L , V ) -functor f : ( X , a ) → ( Y , b ) , is a func-tion between the two underlying sets such that: a ≤ f ◦ · b · L f ,which pointwise means, for all x ∈ LX , y ∈ X : a ( x , y ) ≤ b ( L f ( x ) , f ( y )) .If the equality holds, we call it fully-faithful.In this way we can define ( L , V ) - Cat as the category whose object are ( L , V ) -categories and whosearrows are ( L , V ) -functors among them; moreover, ( L , V ) - Cat becomes an order-enriched category, bydefining, for two ( L , V ) -functors f , g : ( X , a ) → ( Y , b ) : f ≤ g iff k ≤ ^ x ∈ LX b ( L f ( x ) , g ( x )) . Examples 2.31.
1. Every set X defines an ( L , V ) -category by taking e ◦ X as ( T , V ) -structure. In particularwe can define the one-point ( L , V ) -category E : = ( e ◦ ) .2. For two ( L , V ) -categories ( X , a ) , ( Y , b ) , we can form their tensor product X ⊠ Y : = ( X × Y , a ⊠ b ) ,where: a ⊠ b ( γ , ( x , y )) : = a ( L π ( γ ) , x ) ⊗ b ( L π ( γ ) , y ) ;where γ ∈ L ( X × Y ) and π , π are the obvious projections. Unluckily, in general it is not true that X ⊠ E ≃ E .3. V itself defines an ( L , V ) -category with: [ v , w ] : = [ v ⊗ ... ⊗ v n , w ] .4. In general any monoidal V -category ( X , a , ∗ , u X ) defines an ( L , V ) -category in a similar way by defin-ing a ( x , y ) : = a ( x ∗ ... ∗ x n , y ) . ( L , V ) -categories defined in this way are called representables andtheir ( L , V ) -structure will be denoted by ˆ a : = a · α , where: α : L ( X ) → X , x x ∗ ... ∗ x n .In this way we can define a 2-functor K : V - Cat L → ( L , V ) - Cat , which has a left adjoint M : ( L , V ) - Cat → V - Cat that sends an ( L , V ) -category ( X , a ) to ( LX , La · m ◦ X , m X ) and an ( L , V ) -functor f to T f . M is also a 2-functor.Using the aforementioned adjuction, is it possible to extend the monad L to a monad on ( L , V ) - Cat ,denoted L as well. Moreover one can prove (see [ CCH15 ] ) that there is an equivalence: V - Cat L ≃ ( L , V ) - Cat L .5. A priori, due to the non-symmetric form arrows in ( L , V ) - Rel have, it is not clear how to definedefine an ( L , V ) -category that seems to play the role of a dual. Luckily, one can use the adjunction K ⊣ M and the involution in V - Rel to define, for an ( L , V ) -category ( X , a ) , its opposite categoryas: X op : = ( LX , m X · La ◦ · m X ) . At first this might seen as an ad hoc definition, but if we apply thisconstruction to a V -category ( X , a ) , seen as an ( L , V ) -category ( LX , e ◦ X · a ) , we get: X op : = K ( LX , La ◦ ) ,where ( LX , La ◦ ) is the dual, as a V -category, of ( LX , La ) . In particular a (( − ) , y ) : = a ( u X , y ) . When it is clear from the context that we’re dealing with representables ( L , V ) -categories we will indiscriminately use a to denoteboth.
11. For any ( L , V ) -category ( X , a ) , we can form the ( L , V ) -category, denoted ( D L ( X ) , D L ( X )[ − , =]) whoseunderlying set consists of all ( L , V ) -functors of the form: f : X op ⊠ E → V and whose ( L , V ) -structureis given by: D L ( X )[ f , g ] : = ^ x ∈ LX [( f ( x ) , ..., f n ( x )) , g ( x ))] .We have a fully-faithful functor, called the Yoneda embedding, y X : X D L ( X ) , defined as: x a ( − , x ) .Moreover, it can be proved that: D L ( X )[ L y X ( x ) , g ] = g ( x ) .The last result is known as the Yoneda Lemma and justifies the term "embedding". ( L , V ) - Cat
The relational point of view used for introducing V -categories allows us introduce the corresponding no-tion of distributor for ( L , V ) -categories by considering the composition ◦ defined in the previous section.Unfortunately, due to the non-symmetric form ◦ has (and due to other technical details) we won’t be ableto provide a theory of weighted limits, but only one for weighted colimits. Definition 2.32. [ CH09a ] Given two ( L , V ) -categories ( X , a ) , ( Y , b ) a ( L , V ) -distributor j : ( X , a ) − * ◦ ( Y , b ) is a ( L , V ) -relation between them such that: • j ◦ a ≤ j ; • b ◦ j ≤ j .Just as in the V -case, we can define an order-enriched category ( L , V ) - Dist , where the composition isthe one defined in ( L , V ) - Rel . Remark 2.33.
As in the V -case, one can prove the equivalence between: ( L , V ) - Dist ( X , Y ) ≃ ( L , V ) - Cat ( X op ⊠ Y , V ) ≃ ( L , V ) - Cat ( Y , D L ( X )) .In particular, to every ( L , V ) -distributor j : X − * ◦ Y we can associate its mate : ð j ñ : Y → D ( X ) , y j ( − , y ) .Just as in the V -case, we have the equivalent of Theorem 2.15. Where now, for an ( L , V ) -functor f : ( X , a ) → ( Y , b ) , the two (adjoint) associated ( L , V ) -distributors are:1. f ⊛ : X − * ◦ Y , f ⊛ ( x , y ) : = b ( L f ( x ) , y ) .2. f ⊛ : Y − * ◦ X , f ⊛ ( y , x ) : = b ( y , f ( x )) .As already noticed in Remark 2.28, due to the intrinsic asymmetry Kleisli composition has, contrary to the V -case, we can only prove that: ( − ) ◦ α : ( L , V ) - Dist ( X , Y ) → ( L , V ) - Dist ( Z , Y ) β β ◦ α .Has a right adjoint: ( − ) ◮ α : ( L , V ) - Dist ( Z , Y ) → ( L , V ) - Dist ( X , Y ) β β ◮ α : = β Â ˆ α ,where ˆ α : = L α · m ◦ X . 12 efinition 2.34. Let f : ( X , a ) → ( Y , b ) be a ( L , V ) -functor and j : X − * ◦ Z be an ( L , V ) -distributor. We saythat a ( L , V ) -functor g : ( Z , c ) → ( Y , b ) is the colimt of f weighted by j , and denote it by: L - colim ( j , f ) , if g ⊛ ≃ f ⊛ ◮ j , that is to say: g ⊛ ( z , y ) : = b ( L g ( z ) , y ) = ^ x ∈ LX [ ˆ j ( x , z ) , f ⊛ ( x , y )] , ( ) .An ( L , V ) -category ( Y , b ) is called cocomplete , if it has all weighted colimits; meaning that, for all ( L , V ) -functors f : ( X , a ) → ( Y , b ) and for all ( L , V ) -distributors h : Z − * ◦ X , the colimits of f weighted by h exists.Contrary to what happens in V - Cat , in ( L , V ) - Cat cocompleteness cannot be reduced to distributors into K (a.k.a. presheaves), nevertheless the corresponding of Theorem 2.24 still holds also in the ( L , V ) -case. Examples 2.35.
1. Using the Yoneda Lemma, one can prove that every element of D L ( X ) can be writtenhas the colimit of y X weighted over itself.2. Consider the V - Rel : u : E − * ◦ E , where u ∈ V . Given an ( L , V ) -category ( X , a ) and a point x ∈ ( X , a ) , L - colim ( u · a ( − , x ) , I d )( ) (if exists) coincides with the co-power x ⊙ u of the underling V -category ( X , e X · a ) .In the case the target category is representable, we have the following: Proposition 2.36.
Let f : ( X , a ) → ( Y , b · α ) be an ( L , V ) -functor, with ( Y , b · α ) be a representable ( L , V ) -category, and j : X − * ◦ E be an ( L , V ) -distributor. If L- colim ( j , f ) exists, then:L- colim ( j , f )( ) ≃ colim ( e ◦ · ˆ j , α L f ) . Where α L f : K ( X ) → ( Y , b ) .Before the proof we state a useful Lemma. Lemma 2.37.
In the hypotesis above, α L f : K ( X ) → ( Y , b ) . is a V -functor and e ◦ · ˆ j is a V -distributor.Proof. Sine f is an ( L , V ) -functor, then f ⊛ : = b · α L f = : ( α L f ) ∗ is an ( L , V ) -distributor, hence (in particular): f ⊛ · La · m ◦ X ≤ f ⊛ ;moreover, because b · b ≤ b , then we have also: b · b · α L f ≤ b · α L f ,from which follows that ( α L f ) ∗ is a V -distributor, hence that α L f is a V -functor.A quick calculation shows that e ◦ · ˆ j = j ; moreover, since j is an ( L , V ) -distributor we have: e ◦ · ˆ j · La · m ◦ X = j · La · m ◦ X = j ◦ a ≤ j .Which shows that e ◦ · ˆ j : LX − * ◦ E is a V -distributor. Hence by Corollary 2.16 the result follows. Proof. (Of the proposition) We have:˜ b ( e Y ( L - colim ( j , f )( )) , c ) = b ( L - colim ( j , f )( ) , c )= ^ x ∈ LX [ ˆ j ( x , e ( )) , b ( α L f ( x ) , c )] , = ^ x ∈ LX [ j ( x ) , b ( α L f ( x ) , c )] , = b ( colim ( e ◦ · ˆ j , α L f ) , c ) .From which the result follows. 13 orollary 2.38. Assuming the same hypotesis of the previous proposition, ˜ b ( e Y ( L- colim ( j , f )( )) , c ) = lim ( e ◦ · ˆ j , b ( α L f ( − ) , c )) . Proof.
It follows from Example 2 of 2.23.
Corollary 2.39.
Assuming the same hypotesis of the previous proposition, L- colim ( j , f )( ) ≃ W x ∈ LX j ( x ) ⊙ α L f ( x ) . Where the conical colimit and the co-power are taken in the underlying V -category ( Y , b ) .Proof. Follows from the discussion on weighted colimits done in 2.19.
Definition 2.40.
Given two ( L , V ) -functors f : ( X , a ) → ( Y , b ) , g : ( Y , b ) → ( X , a ) ; f is left adjoint to g (vice versa: g is right adjoint to f ) and denoted: f ⊣ g , if f ⊛ ≃ g ⊛ . Pointwise this means that, for all x , y ∈ LX , Y : b ( L f ( x ) , y ) = a ( x , g ( y )) . Examples 2.41.
1. Any ( L , V ) -functor f : ( X , a ) → ( Y , b ) defines an adjoint pair of ( L , V ) -functors: D L ( X ) D L ( Y ) D ( f ) : = −◦ f ⊛ −◦ f ⊛ ⊣
2. Given an ( L , V ) -functor f : ( X , a ) → ( Y , b ) , with ( Y , b ) cocomplete, we can consider the followingfunctor: Lan y ( f ) : = L - colim ( y ⊛ , f ) : D L ( X ) → Y ,which is called the left Kan extension of f along y X ; it can be shown that Lan y ( f ) has a right adjoint R given by: R ( y ) : = b ( L f ( − ) , y ) : Y → D L ( X ) .Let f : ( X , a ) → ( Y , b ) and g : ( Y , b ) → ( Z , c ) two ( L , V ) -functors and j : X −7−→ D be a distributor. Fromthe definition of weighted colimit it follows that L - colim ( j , g f ) ≤ g ( T - colim ( j , f )) .If the equality holds, we say that g preserves the colimit of f weighted by h . If g preserves all colimits, wesay that is co-continuous. As in the V -case, we have that left adjoint ( L , V ) -functors are co-continuous. In this subsection we are going to define the V -version of quantales, state some of their properties andintroduce lax monoidal monads along with their properties.It is well known that quantales can be described as monoids in the monoidal category of sup-lattices, wherethe monoidal structure on Sup could be described as the one induced by the strong-commutative monad : P : Sets → Sets , P ( X ) : = X .(see [ JT84 ] for details).Since Sup is equivalent to the category of cocomplete 2-categories (with co-continuous 2-functors as ar-rows), it is natural to define an enriched quantale as a monoid in the category of cocomplete V -category,where the monoidal structure is the one induced by the strong commutative monad: P : Sets → Sets , P ( X ) : = V X .This motivates the following definition: 14 efinition 2.42. An enriched quantale (for simplicity, from now on we will call it: a "quantale") ( Q , ∗ , k ) is an object of Mon ( Sets P V , ⊗ V , V ) . Remark 2.43.
From the definition of quantale, it follows that − ∗ x : Q → Q and x ∗ = : Q → Q are twoco-continuous V -functors. Since Q is cocomplete, it follows that both have right adjoints, given by: y  (=) : = colim ( Q ( − ∗ y , =) , I d ) , x à (=) : = colim ( Q ( x ∗ = , =) ; I d ) ,which can be re-written pointwise as: y  z : = _ x ∈ X Q ( x ∗ y , z ) ⊙ x , x à z : = _ y ∈ Y Q ( x ∗ y , z ) ⊙ y .Note that in the ordered case, the last one correspond to the familiar notion of left and right implication respectively.Recall that a V -functor f : ( X , a , ∗ X , k X ) → ( Y , b , ∗ Y , k Y ) between two monoidal V -categories is called lax-monoidal if: k Y ≤ f ( k X ) , f ( x ) ∗ Y f ( y ) ≤ f ( x ∗ X y ) .Let Mon (( V - Cat )) lax , be the category formed by monoidal V -category with lax-monoiadal functors betweenthem. Proposition 2.44. K : Mon (( V -
Cat )) lax → ( L , V ) - Cat is fully-faithful. Here K is the functor defined inExample 4Proof.
The fact that K is faithful is straightforward.Let f : K ( X ) → K ( N ) , be a ( L , V ) -functor with ˜ a , ˜ b being their corresponding ( L , V ) -structure. By definitionwe have that: ˜ a ( x , y ) ≤ ˜ b ( L f ( x ) , f ( y )) , where L f ( x ) : = ( f ( x ) , ..., f ( x n )) ,which by definitions implies: a ( x ∗ X ... ∗ X x n , y ) ≤ b ( f ( x ) ∗ Y ... ∗ Y f ( x n ) , f ( y )) .By taking y = x ∗ X ... ∗ X x , from k ≤ a ( x , x ) , it follows that: f ( x ) ∗ Y f ( y ) ≤ f ( x ∗ X y ) .While, by taking as first argument the empty list: k Y ≤ f ( k X ) .By relaxing the definition of quantale we can define another useful category to which ideas we are goingto develop in the following sections apply. Definition 2.45.
A monoidal V -category ( X , a , ∗ X , k X ) is called residuated if for every x ∈ X x ∗ X (=) , ( − ) ∗ X x have right adjoints: x  (=) , x à (=) . The category formed by residuated monoidal V -categories with laxmonoidal functor between them is denoted by: ResMon (( V - Cat )) lax Remark 2.46.
From the definitinon is clear that quantales form a full-subcategory of both
ResMon (( V - Cat )) lax and Mon (( V - Cat )) lax . 15 emark 2.47. Notice that a morphism f in ResMon (( V - Cat )) lax satisfies: f ( y  z ) ≤ f ( y )  f ( z ) , f ( y à z ) ≤ f ( y ) à f ( z ) ;Indeed, from: f ( y  z ) ∗ f ( y ) ≤ f (( y  z ) ∗ y ) , and ( y  z ) ∗ y ≤ z ,it follows: f ( y  z ) ≤ f ( y )  f ( z ) .In the same way, but starting from: f ( y ) ∗ f ( y à z ) ≤ f ( y ∗ ( y à z )) ,we get: f ( y à z ) ≤ f ( y ) à f ( z ) . Definition 2.48.
Given a quantale Q , a lax-monoidal monad T : Q → Q is a lax-monoidal functor such that: • k Q ≤ T ( k Q ) ; • T = T ; Remark 2.49.
Lax-monoidal monads are generalization, for a general quantale, of the notion of quanticnuclei.
Theorem 2.50.
Let Q be a quantale and t : Q → Q be lax-monoidal monad, then:Q T : = { x ∈ Q such that T ( x ) = x } , is a quantale; moreover the inclusion Q T Q is a V -functor that has a left adjoint π T which is a morphism ofquantale.Proof. The proof is a straightforward generalization of the corresponding proof for quantic nuclei. The V -structure on Q T is the one induced by Q while the right adjoint to the inclusion functor is: π T ( x ) : = T ( x ) . In this section we recall how injectives and injective hulls are built in the category of ordered sets andmonotone maps between them. We show how the same ideas apply to the enriched case, for a generalcommutative quantale V .Recall that in a category C , an object X , is called injective (w.r.t a class of moprhims J ), if for everydiagram of the form: Y XZ where Y → X is in J , there exists an extension: Y XZ making the diagram commutes.Given an object Y , E is called an injective hull if there exists a monomorphism e : A → E such that:16 E is J -injective; • For every f : E → X , f e ∈ J implies f ∈ J .Let ( X , ≤ ) be an ordered set. Recall that we have two embedding: ↓ : X D ( X ) , x x : = { w ∈ X , w ≤ x } , ↑ : X U ( X ) , x x : = { w ∈ X , x ≤ w } .It is well known that there exists an adjunction: D ( X ) U ( X ) LR ⊣ Where, for j ∈ D ( X ) , l ∈ U ( X ) , one has: L ( j ) : = { x ∈ X : ∀ w ∈ X w ∈ j ⇒ w ≤ x } , R ( l ) : = { x ∈ X : ∀ w ∈ X w ∈ l ⇒ x ≤ w } .This adjunction induces a monad (closure operator) T on D ( X ) defined, for j ∈ D ( X ) , as follow: T ( j ) : = { x ∈ X : ∀ y ∈ X y ∈ L ( j ) ⇒ x ≤ y } , = { x ∈ X : ∀ y ∈ X y ( ∀ w ∈ X w ∈ j ⇒ w ≤ y ) ⇒ x ≤ y } , = ^ y ∈ X { y : ∀ w ∈ X w ∈ j ⇒ w ≤ y } , = ^ y ∈ X { y : j ≤↓ y } .In particular is immediate from the definition that, for all x ∈ X : T ( ↓ x ) : = ^ y ∈ X { y : ↓ x ≤↓ y } = ↓ x .Thus we have the following commutative diagram: X D ( X ) T D ( X ) ↓ ( − ) ˜ ↓ ( − ) T The fixed point of this monad for an ordered set D ( X ) T called the Dedekind-MacNeille completion of X . Theterm "completion" refers to the property D ( X ) T has: every element of D ( X ) T is both a supremum of ele-ments of X and an infimum of elements of X , and D ( X ) T is the smallest complete ordered set that has thisproperty. Moreover, in [ Mac37 ] , MacNeille proved that w.r.t monotone embedding D ( X ) T is the injectivehull of X .This construction admits an "easy" generalization to the enriched case. First of all, we recall that injec-tives in V - Cat w.r.t fully-faithful functors are cocomplete V -categories; this can be easily showed using thecharacterization of cocompleteness we mentioned in 2.24. Remark 3.1.
Injectives in V - Cat (w.r.t fully-faithful functors) can be equivalently characterized as algebrasfor the Koch-Zöberlein monad D ( − ) , as explained in [ Esc97, Esc98, EF99 ] in a more general context.17et ( X , a ) be a V -category. As in the ordered case, we have an adjunction, called the Isbell duality : D ( X ) U ( X ) LR ⊣ Where, for j ∈ D ( X ) , l ∈ U ( X ) , one has: L ( j ) : = ^ x ∈ X [ j ( x ) , a ( x , =)] ≃ ( y X ) ∗ · j ∗ , R ( l ) : = ^ x ∈ X [ j ( x ) , a ( − , x )] ≃ l ∗ · ( λ X ) ∗ .The monad induced by this adjunction, T , can be calculated as before: T ( j ) : = ^ x ∈ X [ L ( j )( x ) , a ( − , x )] , = ^ x ∈ X L ( j )( x ) ⋔ y x ( x ) , = lim ( L ( j ) , y X ) , = lim (( y X ) ∗ · j ∗ , y X ) .In particular is immediate from the definition that, for all x ∈ X : T ( y X ( x )) : = lim (( y X ) ∗ · j ∗ , y X ) ≃ y X ( x ) .Thus as before, the Yoneda factorizes: X D ( X ) T D ( X ) y X ˜ y X T The fixed point of this monad form a V -category D ( X ) T in which every element j can be written as: j ≃ colim ( j ∗ · ( y X ) ∗ , y X ) , j ≃ lim (( y X ) ∗ · j ∗ , y X ) ,where the first isomorphism is the well know fact that every presheaves is the weighted colimit over itselfof the Yoneda ( [ Kel82 ] ), and the second one follows from being j a fixed point of T .This motivates the following definition: Definition 3.2.
A fully-faithful V -functor i : ( X , a ) ( Y , b ) is called dense , if every element y ∈ ( Y , b ) is ofthe form: y ≃ colim ( y ∗ · i ∗ , i ) , y ≃ lim ( i ∗ · y ∗ , i ) ,In this way we have a nice characterization of essential morphisms in V - Cat as contained in the follow-ing:
Theorem 3.3.
A fully-faithful V functor i : ( X , a ) ( Y , b ) is dense iff is essential. Before proving it, we state a straightforward lemma:
Lemma 3.4.
Suppose we have a fully faithful V -functor f : ( Y , b ) → ( Z , c ) and a functor g : ( X , a ) → ( Y , b ) .If f g is dense, then g is dense too.Proof. (Of the theorem) ⇒ ) . Let f : ( Y , b ) → ( Z , c ) be a V -functor such that f (cid:12)(cid:12) X is fully-faithful. We haveto show that: c ( f ( x ) , f ( y )) ≤ b ( x , y ) , for all x,y ∈ Y .18ince i is dense, we can write: x ≃ colim ( x ∗ · i ∗ , i ( x )) , y ≃ lim ( i ∗ · y ∗ , i ( y )) .By playing with the properties of (co)-limits, we have: b ( x , y ) = b ( colim ( x ∗ · i ∗ , i ( w )) , lim ( i ∗ · y ∗ , i ( q ))) , = lim ( x ∗ · i ∗ , lim , ( i ∗ · y ∗ ) , b ( i ( w ) , i ( q )))) ,(since i is fully-faithful), = lim ( x ∗ · i ∗ , lim , ( i ∗ · y ∗ ) , a ( w , q ))) (since f (cid:12)(cid:12) X is fully-faithful), = lim ( x ∗ · i ∗ , lim , ( i ∗ · y ∗ ) , c ( f i ( w ) , f i ( q )))) , = c ( colim ( x ∗ · i ∗ , f i ( w )) , lim ( i ∗ · y ∗ , f i ( q ))) , ≥ c ( f ( colim ( x ∗ · i ∗ , i ( w )) , f ( lim ( i ∗ · y ∗ , i ( q ))) , ≥ c ( f ( x ) , f ( y )) .Which shows that i is essential. ⇐ ) . Because D ( X ) T is injective there exists a V -functor Y → D ( X ) T that makes the following commutative: Y X D ( X ) T And because i is essential, then we have that the dashed arrow is an embedding. The result follows fromthe previous lemma.This theorem ends our intermezzo section. In the next section we will apply similar ideas in order toconstruct the injective hulls in ( L , V ) - Cat . Unfortunately the procedure will not be so smooth as for V -categories.The first inconvenience is due to the fact that (as for now), since we don’t have an equivalent of U ( X ) for an ( L , V ) -category ( X , a ) , we don’t have an analogue of the Isbell duality and thus we will have to de-fine the monad T "manually"; Moreover, since we need the category of algebras for this monad to be an ( L , V ) -category we need T to be a lax-monoidal monad.The second inconvenience follows directly from the first one. In order to prove that T is lax-monoidalwe will need to reduce our attention to a particular sub-category of ( L , V ) - Cat : the category of
QuantumB-algebras . For this category we will be able to mimic all the construction done in this section and buildinjective hulls as algebras for a lax-monoidal monad which resembles the one we introduced before. With aslight modification of definition 3.2, where we will substitute the V -colimit part with an ( L , V ) -colimit, wewill be able to proof the equivalent of 3.3. Luckily the restriction to quantum B-algebras will not preventus to construct the injective hull for all ( L , V ) -categories, by embedding every ( L , V ) -category in a quantumB-algebras we will provide an injective hull for all ( L , V ) -categories. Promonoidal categories were introduced by Day in his thesis [ Day71 ] . They originated from the observa-tion that, in order to define a monoidal structure on [ X , V ] , what you need is only a promonoidal structureon X . As we are going to describe, a promonoidal category is a monoid in V - Dist , with the monoidal19tructure inherited from the on in V - Cat we described briefly in the previous section.Quantum B-Algebras were introduced in [ Rum13 ] . The motivation comes from non-commutative alge-braic logic, for which they provide an unified semantic as better explained in [ Rum13, RY14 ] . They weredefined as orderded sets equipped with two implications that mimic the residuals in an quantale.Since ordered monoids (and more general their enriched counterpart) are promonoidal categories forwhich (part of) the promonoidal structure is representable, is natural to ask if the same holds for quantumB-algebras. Unsurprisingly the answer, as briefly shown in the last section of [ Rum16 ] , is positive: quantumB-algebras can be seen as representables promonoidal category.In the following section we’ll take this point of view in order to present the enriched version of quantumB-algebras, and we’ll show how ( L , V ) -categories provide a common roof for both of them by showinghow some categorical constructions become natural when one consider both as being fully-faithful sub-categories of ( L , V ) - Cat .The section is structured as follows: • In the first subsection are devoted to introduce promonoidal categories and quantum B-algebras alongwith some of their basics properties. • In the second subsection we show how injectives in both categories can be easily characterize fromthe characterization of injectives in ( L , V ) - Cat , and we generalize some constructions of [ Rum16 ] tothe enriched case, by showing that every (enriched) quantum B-algebra admits an injective hulls.We would like to stress the fact that results of 4.2 are not a mere exercise in fuzzyfication , that is to say, arephrase of the ones in [ Rum16 ] for a general V . In order to be able to generalize them, on needs a moregeneral theory of colimits, which is provided by the notion of ( L , V ) -colimits introduced in the previoussection. This shred light on some universal constructions that in the ordered case are somehow "hidden". V -Categories and Quantum B-Algebras In the previous section we briefly explained how V - Cat can be equipped with a monoidal structure, bydefining, for two V -categories ( X , a ) , ( Y , b ) , their tensor product as: X ⊠ Y = ( X ⊠ Y , a ⊗ b ) .This monoidal structure induces a monoidal structure on V - Dist , where for j : X −7−→ Y , l : Z −7−→ W : j ⊠ l : X ⊠ Z −7−→ Y ⊠ W , j ⊠ l (( x , z ) , ( y , w )) : = j ( x , y ) ⊗ l ( y , w ) .With this in mind, is possible to talk about monoids in V - Dist . Definition 4.1. A promonoidal V -category is a monoid in V - Dist . This means it is a V -category ( X , a ) together with two distributors: • P : X ⊠ X −7−→ X ; • J : 1 −7−→ X .Such that: • P · ( P ⊠ I d ) = P · ( I d ⊠ P ) ; • P · ( J ⊠ I d ) =
I d , P · ( I d ⊠ J ) = Id.
Remark 4.2.
The last two conditions pointwise mean:20 W x ∈ X P ( a , x , d ) ⊗ P ( b , c , x ) = W x ∈ X P ( x , a , d ) ⊗ P ( a , b , x ) . • W z ∈ X J ( z ) ⊗ P ( x , z , w ) = a ( x , w ) , W z ∈ X J ( z ) ⊗ P ( z , x , w ) = a ( x , w ) . Definition 4.3.
Following Day [ DS95 ] , given two promonoidal categories ( X , a , P , J ) , ( Y , b , R , U ) , a promonoidalfunctor between them is a V -functor f : ( X , a ) → ( Y , b ) , such that, for all x , y , z ∈ X : P ( x , y , z ) ≤ R ( f ( x ) , f ( y ) , f ( z )) , J ( x ) ≤ U ( f ( x )) .If the reverse inequalities hold, we call it strong . Examples 4.4.
1. Any monoidal V -category ( X , a , ∗ X , u X ) defines a promonoidal one by taking: • P ( x , y , z ) : = a ( x ∗ X y , z ) . • J ( x ) : = a ( u x , x ) .2. Any quantales ( Q , ∗ , u Q ) defines a promonoidal catgory in two (equivalents) ways: the first one is theone described before, while the second one is as follows: • P ( x , y , z ) : = Q ( x , y  z ) = Q ( y , x à z ) . • J ( x ) : = Q ( u Q , x ) .The equivalence of the two formulations follows by adjunction.The last example will be explored further when we will introduce Quantum B-Algebras . Remark 4.5.
As mentioned in the introduction of the section, a promonoidal structure induces a monoidalstructure on the corresponding category of presheaves. More precisely, given a promonoidal category ( X , a , P , J ) we can define a monoidal structure on ( D ( X ) , ∗ D , J ◦ ) , as follows: j ∗ D l ( x ) : = _ w , z P ◦ ( w , z , x ) ⊗ j ( w ) ⊗ l ( w ) .In particular, when the promonoidal structure comes from a monoidal one, the previous formula reads as: j ∗ D l ( x ) : = _ w , z a ( x , w ∗ z ) ⊗ j ( w ) ⊗ l ( w ) .In this case ( D ( X ) , ∗ D , J ◦ ) becomes a quantale, where as an example, the right implication is: l  h ( x ) : = ^ y [ l ( y ) , h ( x ∗ y )] . Proposition 4.6.
V -
Pro is a fully-faithful subcategory of ( L , V ) - Cat .Proof.
Given a promonoidal V -category ( X , a , P , J ) defines an ( L , V ) -category ( X , ˜ a : = ∐ n ≥ a n ) in the fol-lowing inductive way: • a : = J ; • a : = a ; • a : = P ; • a n : = P · ( I d ⊠ a n − ) . 21ndeed, e ◦ X ≤ ˜ a follows directly from the definition, while ˜ a ◦ ˜ a ≤ ˜ a follows from the inductive definitionof ˜ a , from the properties of P and J , and from the fact that a is a V -structure, in the following way: Let x : = ( x , ..., x n ) ∈ L LX , y : = ( y , ..., y n ) ∈ LX , and z ∈ X . We want to show: L ˜ a ( x , y ) ⊗ ˜ a ( y , z ) ≤ ˜ a ( m X ( x ) , z ) .Unravelling the definition this means: a l ( x ) ( x , y ) ⊗ ... ⊗ a l ( x n ) ( x , y ) ⊗ a n ( y , z ) ≤ a l ( x )+ ... + l ( x n ) ( m X ( x ) , z ) , ( ) where l ( − ) denotes the length of a list.The prove of ( ) is a long and frustrating matter of bookkeeping relying on the properties P and a have.Instead of proving ( ) , we prefer to illustrate the properties used in a simple case which is sufficient generalto encompass ( ) .Consider x : = (( x , x ) , x ) and y = ( y , y ) . Then ( ) becomes: P ( x , x , y ) ⊗ a ( x , y ) ⊗ P ( y , y , z ) = _ d P ( x , x , y ) ⊗ a ( y , d ) ⊗ a ( x , y ) ⊗ P ( d , y , z ) , ( from a · P ≤ P ) ≤ _ d P ( x , x , d ) ⊗ a ( x , y ) ⊗ P ( d , y , z ) , ( from P · ( I d ⊠ a ) ≤ P · ( a ⊠ a ) ≤ P ) ≤ _ d P ( x , x , d ) ⊗ P ( d , x , z ) , = ˜ a (( x , x , x ) , z ) Suppose that f : ( X , a , P , J ) → ( Y , b , R , U ) is a promonoidal functor. Then, for all x ∈ LX , y ∈ X (with l ( x ) = n ):˜ a ( x , y ) : = a n ( x , y ) : = _ c a n − (( x , ..., x n − ) , c ) ⊗ P ( c , x n + , y ) , ( by induction on a n , using the fact that f is a promonoidal functor ) ≤ _ c b n − (( f ( x ) , ..., f ( x n − )) , f ( c )) ⊗ R ( f ( c ) , f ( x n + ) , f ( y )) ,: = b n ( L f ( x ) , f ( y )) = b ( L f ( x ) , f ( y )) .This shows that f is an ( L , V ) -functor.If: ˜ a ( x , y ) ≤ b ( L f ( x ) , f ( y )) , take: x = ( x , x ) ⇒ P ( x , x , y ) ≤ R ( f ( x ) , f ( x ) , f ( y )) , x = ( x ) ⇒ a ( x , y ) ≤ b ( f ( x ) , f ( y )) , x = ( − ) ⇒ J ( x ) ≤ U ( f ( x )) .This shows that V - Pro is a fully-faithful subcategory of ( L , V ) - Cat . Definition 4.7. A quantum B-algebra ( X B , a ) is a representable promonoidal V -category. This means it is apromonoidal V -category ( X , a , P , J ) equipped with two binary operations:  , à : X × X → X and an element u X ∈ X , such that: • P ( x , − , y ) ≃ a ( − , x à y ) ; 22 P ( − , x , y ) ≃ a ( − , x  y ) ; • J ( x ) = a ( u , x ) . Remark 4.8.
From the representability condition it directly follows that  , à are two bi-functors of theform:  , à : X op ⊠ X → X .Moreover, from representability, it follows that: • a ( x , y  z ) = a ( y , x à z ) ; • x  ( y à z ) = y à ( x  z ) ; • u X  (=) ( u X à (=) ) is the identity.In this way we can recover the definition of (unital) quantum B-algebras given in [ Rum13 ] .By expressing what it means to be promonoidal functor for quantum B-algebras, we have: Definition 4.9.
A morphism between two quantum B-algebras f : ( X B , a ) → ( Y B , b ) is V -functor satisfying: f ( x  y ) ≤ f ( x )  f ( y ) , f ( x à y ) ≤ f ( x ) à f ( y ) , u Y ≤ f ( u X ) ( ) If also: f ( x )  f ( y ) ≤ f ( x  y ) , f ( x ) à f ( y ) ≤ f ( x à y )) , ( ) then f is called strict. Remark 4.10.
It is relative easy to derive these conditions from the promonoidal functor’s one using rep-resentability. As an example we show how to derive the condition on the unities: u Y ≤ f ( u X ) .(from f ∗ ⊣ f ∗ ) J ( − ) ≤ U ( f ( − ))( u x ) ∗ ≤ f ∗ · ( u Y ) ∗ f ∗ · ( u X ) ∗ ≤ ( u Y ) ∗ b ( f ( u X ) , =) ≤ b ( u Y , =) u Y ≤ f ( u X ) From 4.6, it follows:
Proposition 4.11.
The category of quantum B-algebras is a fully-faithful subcategory of ( L , V ) - Cat . Remark 4.12.
Let ( X B , a ) be a quantum B-algebra, by unravelling the definition contained in 4.6, thecorresponding ( L , V ) -structure is given as follows. For a pair ( x , y ) defines:˜ a (( x , ..., x n ) , y ) : = ˜ a (( x , ..., x n − ) , x n  y ) : = ... : = a ( x , x  ... x n  y ) ,˜ a (( − ) , y ) : = a ( u X , y ) , where ( − ) is the empty list. Remark 4.13.
Let Q be a quantale. We can see it as a quantum B-algebra and as an ( L , V ) -category. It iseasy to see that the two structure (call them Q B and Q L ) coincide; indeed: Q B (( q , ..., q n ) , w ) : = Q ( q , q n  ...  w ) = Q ( q ∗ ... ∗ q n , w ) = : Q L (( q , ..., q n ) , w ) .23 .2 Injectives and Injective Hulls In the previous section we recalled the notion of injective object and of injective hull w.r.t a class of mor-phism J . As we did for V - Cat from now on, when we write "injective" we always mean injective w.r.tfully-faithful functors / embedding. Since ( L , V ) - Cat serves as a common roof under which all the con-structions are performed, when we are dealing with: promonoidal categories, quantum B-algebras, etc."injective" will always mean w.r.t to the corresponding notion in ( L , V ) - Cat .In the context of ( T , V ) -categories in [ Hof11 ] it was shown that: Theorem 4.14. An ( L , V ) -category ( X , a ) is injective iff is cocomplete. We can go further in our analysis and characterize them as quantales:
Theorem 4.15. An ( L , V ) -category ( X , a ) is injective w.r.t fully-faithful functors iff is a quantale. Before giving a direct proof, we state a useful lemma:
Lemma 4.16.
Let Q be a quantale and ( Y , b ) be an ( L , V ) -category. Suppose an ( L , V ) -functor f : Y → Q isgiven. Then there exists an ( L , V ) -functor g : D L ( Y ) → Q making the following commute:Y Q D L ( Y ) y Y fg Proof.
Let g : = Lan y Y ( f ) . Since Lan y Y ( f )( j ) : = L - colim ( y ⊛ , f )( ) , which by Proposition 2.36 is the sameas colim ( e ◦ · ˆ y ⊛ , α L f ) and Q is a quantale, g is well defined (as a function). Moreover it makes the diagramcommute.If we prove that there exists an ( L , V ) -functor R : Q → D L ( Y ) such that R ⊛ ≃ g ⊛ we can conclude that g isan ( L , V ) -functor too.By unravelling the definition of colimits, we have that: g ( j ) ≃ _ y ∈ LY D L ( Y )[ L y Y ( y ) , j ] ⊙ ( f ( y ) ∗ ... ∗ f ( y n )) (by Yoneda lemma) ≃ _ y ∈ LY j ( y ) ⊙ ( f ( y ) ∗ ... ∗ f ( y n )) Define R ( w ) : = Q ( L f ( − ) , w ) . Since R is the mate of f ⊛ , by Remark 2.33 it follows that R is an ( L , V ) -functor.In order to conclude, we have to show that: g ⊛ ( j , w ) = R ⊛ ( j , w ) .Without loos of generality we can suppose that j = ( j , j ) .24e can compute: g ⊛ ( j , w ) = Q ( g ( j ) ∗ g ( j ) , w )= Q ( _ y ∈ LY j ( y ) ⊙ ( f ( y ) ∗ ... ∗ f ( y n )) ∗ _ z ∈ LY j ( z ) ⊙ ( f ( z ) ∗ ... ∗ f ( z n )) , w ) (since the multiplication of Q preserves colimits being a quantale) = ^ y ∈ LY ^ z ∈ LY [ j ( y ) ⊗ j ( z ) , Q (( f ( y ) ∗ ... ∗ f ( y n )) ∗ ( f ( z ) ∗ ... ∗ f ( z n )) , w )]= ^ ( y ; z ) ∈ LY [ L j ( y ; z ) , R ( w )( y ; z )]= ^ x ∈ LY [ L j ( x ) , R ( w )( x )] (by the definition of the ( L , V ) -structure on D L ( Y ) , as explained in Example 6 of 2.31) = D L ( Y )[ j , R ( w )]= R ⊛ ( j , w ) Thus g is an ( L , V ) -functor as we wanted. Proof. (Of the theorem). ⇐ ) . Suppose Q is a quantale and Let g : ( X , a ) → Q be an ( L , V ) -functor, with ( X , a ) be an ( L , V ) -category.Suppose that f : ( X , a ) ( Y , b ) is a fully-faithful ( L , V ) -functor. Consider the Yoneda embedding y Y : Y D L ( Y ) , by the previous lemma there exists an extension such that the following diagram commutes: X QY D L ( Y ) gfy Y Hence Q is injective. ⇒ ) . Suppose that Q is injective. By 4.14, it follows that Q is a cococomplete ( L , V ) -category, hence that isrepresentable. Since the 2-functor: ( − ) : ( L , V ) - Cat → V - Cat ,sends injectives objects in ( L , V ) - Cat to injective objects in V - Cat and since injectives in V - Cat are cocom-plete V -categories (as explained in the previous section), it follows that Q is a cocomplete V -category. Itremains to show that the monoidal structure on Q preserves V -colimits (in each variable).Since Q is representable D ( Q ) becomes a quantale w.r.t Day’s convolution product (as explained in 4.5);moreover, the Yoneda embedding: y Q : Q D ( Q ) ,is a strong monoidal V -functor. Thus, since we have: Q ( x , w ) : = Q ( x ∗ ... ∗ x n , w )= D ( Q )( y Q ( x ∗ ... ∗ x n ) , y Q ( w ))= D ( Q )( y Q ( x ) ∗ ... ∗ y Q ( x n ) , y Q ( w )) : = D ( Q )( L y Q ( x ) , y Q ( w )) Q is also an embedding in ( L , V ) - Cat .Because Q is injective, there exists an extension: Q Q D ( Q ) y Q h With h · y Q = I d Q .Since the same holds in V - Cat , by Theorem 2.24, h must be of the form: h : D ( Q ) → Qj colim ( j , I d ) In order to show that Q is a quantale, by Theorem 2.24, it is sufficient to show that the monoidal structurepreserves colimits of the form colim ( j , I d ) , for j ∈ D ( Q ) . Let x ∈ Q and j ∈ D ( Q ) . We have: x ∗ h ( j ) : = h ( y Q ( x )) ∗ h ( j ) (since h is lax-monoidal) ≤ h ( y Q ( x ) ∗ j ) (since every presheaves is the colimit of the Yoneda) = h ( y Q ( x ) ∗ colim ( j , Y Q )) (since Day’s convolution preserves colimits) = h ( colim ( j , Y Q ( x ) ∗ Y Q (=))) (since the Yoneda is strong) = h ( colim ( j , Y Q ( x ∗ =))) (since h preserves colimits being left adj. to y Q in V - Cat ) ≃ colim ( j , h ( Y Q ( x ∗ =))) ≃ colim ( j , x ∗ (=)) Hence, because colim ( j , x ∗ (=)) ≤ x ∗ colim ( j , I d ) follows from the u.p of colimits, we have: x ∗ colim ( j , I d ) ≃ colim ( j , x ∗ (=)) .Thus Q is a quantale.With the aid of the previous theorem, we can characterize those full sub-categories of ( L , V ) - Cat whoseinjectives are quantales in the following and simple way:
Theorem 4.17.
Let C be a full sub-category of ( L , V ) - Cat which contains quantales. Then an object ( X , a ) isinjective iff is a quantale.Proof. ⇐ ) . Quantales are injectives in ( L , V ) - Cat and C is a fully-faithful subcategory of it. Thus, if X is aquantale, then it is injective in C too. ⇒ ) . If X is an injective object in C , then in ( L , V ) - Cat we have the following diagram:
X X D L ( X ) y X ∃ h h · y X = I d . This means that h : D L ( X ) → X B defines analgebra structure for the monad D L ( − ) , but because algebras for this monads are quantales (as shown in ),this implies that X is a quantale too.As corollaries of the previous theorem, we get a characterization of injectives as quantales in each ofthe categories displayed in the following diagram: ( L , V ) - Cat V - ProMon (( V - Cat )) lax QBalgResMon (( V - Cat )) lax In order to build injective hulls we mimic what is done for V - Cat and introduce the notion of dense mor-phism . The difference here is that, due to the nature colimits in our "roof" category, in our definition weconsider L - colim instead of colim . Definition 4.18.
Let ( X , a ) be in ( L , V ) - Cat . An embedding i : ( X , a ) Q , where Q is a quantale (seen asa ( L , V ) - Cat ) is called dense , if every q ∈ Q can be written as: q ≃ lim ( i ∗ · q ∗ , i ) , q ≃ L - colim ( i ⊛ ◦ q ⊛ , i )( ) .If only the latter holds, we call it pre-dense . Remark 4.19.
Notice that the limit in the previous definition is a V -enriched limit. We consider the 2-functor: ( − ) : ( L , V ) - Cat → V - Cat ,and take the limit accordingly.
Remark 4.20.
Using the decomposition of limits (see Example 2.19), we can write the first condition as: lim ( i ∗ · q ∗ , i ) ≃ ^ x ∈ X Q ( q , i ( x )) ⋔ i ( x ) ,which in the ordered case reduce to the more familiar: q = ^ { x : q ≤ i ( x ) } i ( x ) .While, unravelling the second condition and using Corollary 2.39, one has: L - colim ( i ⊛ ◦ q ⊛ , i )( ) ≃ _ x ∈ LX Q ( α Li ( x ) , q ) ⊙ α Li ( x ) , = _ x ∈ LX Q ( i ( x ) ∗ ... ∗ i ( x n ) , q ) ⊙ ( i ( x ) ∗ ... ∗ i ( x n )) ,which in the order case reduce to: q = _ { x : i ( x ) ∗ ... ∗ i ( x n ) ≤ q } i ( x ) ∗ ... ∗ i ( x n ) .The last condition shows how, in the ordered case, X generates the quantale Q . This also shows how, inthe enriched case, ( L , V ) -categories (more specifically the theory of ( L , V ) -colimits) are able to "capture",in an elegant and concise way, what we might call "monoidal colimits"; that is to say to provide a way tocapture the idea of "quantale generated by a V -category", an intuitive construction that would be difficultto formalize without the notion of ( L , V ) -categories.27 efinition 4.21. An embedding i : ( X B , a ) Q , where Q is a quantale, of a quantum B-algebra is dense, ifit is dense in ( L , V ) - Cat . Remark 4.22.
We stress the fact that we are considering embedding of ( L , V ) -categories into quantales,which are cocomplete ( L , V ) -categories and complete V -categories. Lemma 4.23.
Let i : ( X , a ) ( Y , b ) be a fully-faithful V -functor between two V -categories, with ( Y , b ) complete. Then: T : ( Y , b ) → ( Y , b ) , y lim ( i ∗ · y ⋆ , i ) , is a V -functor.Proof. Notice that T can be written as the composite of the following V -functors: Y λ Y / / U ( Y ) i − / / U ( X ) U ( i ) / / U ( Y ) lim / / Y .Where: • λ Y : y y ∗ : = b ( y , =) ; • i − : ψ i ∗ · ψ ; • U ( i ) : φ i ∗ · φ ; • lim : γ lim ( γ , I d ) .This is because: lim ( i ∗ · i ∗ · y ∗ , I d ) ≃ lim ( i ∗ · y ⋆ , i ) Lemma 4.24.
With the same notations as before we have: T · i ≃ i, meaning that, for all x ∈ ( X , a ) ,T ( i ( x )) = i ( x ) .Proof. The result follows by noticing that, since i is fully-faithful: i ∗ · i ( x ) ∗ ≃ x ∗ ,and thus that: x ∗ · i ( x ) ∗ = x ∗ · x ∗ · i ∗ ≤ i ∗ , in a universal way. This proves our desired result. Proposition 4.25.
Let i : ( X B , a ) Q, where Q is a quantale, be a pre-dense strict morphism in
QBalg .Then T : Q → Q defined before is a lax-monoidal monad.Proof.
We want to prove that, for all a ∈ Q : • a ≤ T ( a ) ; • k Q ≤ T ( k Q ) ; • T ( a ) = T ( a ) ; • T ( a ) ∗ T ( b ) ≤ T ( a ∗ b ) .We have the following chain of adjunctions: U ( X ) U ( Q ) Q U ( i ) i − ⊣ lim ( − ) ⋆ : = λ Q ⊣ i − : ψ i ∗ · ψ .Since T ( a ) ≃ lim ( i ∗ · a ⋆ , i ) , it follows that: k ≤ Q ( z , T ( a )) = U ( X B )( i ∗ · z ∗ , i ∗ · a ∗ ) iff for all x ∈ X Q ( a , i ( x )) ≤ Q ( z , i ( x )) .From this immediately follows: a ≤ T ( a ) , hence also k Q ≤ T ( k Q ) .From the latter, being T a V -functor, it follows that T ( a ) ≤ T ( a ) . For the other direction, we need to provethat: Q ( a , i ( x )) ≤ Q ( T ( a ) , i ( x )) .Being T a V -functor, from the previous lemma, it follows that: Q ( a , i ( x )) ≤ Q ( T ( a ) , T ( i ( x ))) = Q ( T ( a ) , i ( x )) ≤ Q ( T ( a ) , i ( x )) .Let’s prove that T ( a ) ∗ T ( b ) ≤ T ( a ∗ b ) . For x ∈ X , by using the pre-denseness of i , we have: Q ( a ∗ b , i ( x )) = Q ( a ∗ L - colim ( φ , I d ) , i ( x )) ,where, for the sake of notation, φ : = i ⊛ ◦ b ⊛ .By unravelling the definition of colimit, we get: Q ( a ∗ b , i ( x )) = Q ( a ∗ L - colim ( φ , i ( w ) , i ( x )) ,(from Proposition 2.36), = Q ( a ∗ colim ( e ◦ · φ , α Li ( w )) , i ( x )) , = lim ( e ◦ · φ , Q ( a ∗ α Li ( w ) , i ( x ))) , = lim ( e ◦ · φ , Q ( a ∗ ( i ( w ) ∗ ... ∗ i ( w n )) , i ( x ))) ,(by inductively applying adjunction) = lim ( e ◦ · φ , Q ( a , i ( w )  ( ...  i ( w n )  i ( x )))) , ( by inductively applying i ( x  y ) = i ( x )  i ( y ))= lim ( e ◦ · φ , Q ( a , i ( w  ...  ( w n  x )))) ,(being T a V -functor and from the previous lemma) ≤ lim ( e ◦ · φ , Q ( T ( a ) , i ( w  ...  ( w n  x )))) ,(by inductively applying adjunction, plus i ( x  y ) = i ( x )  i ( y ) backward) lim ( e ◦ · φ , Q ( T ( a ) ∗ ( i ( w ) ∗ ... ∗ i ( w n )) , i ( x ))) , = Q ( T ( a ) ∗ b , i ( x )) .Now, by repeating the same argument, but now expressing T ( a ) as a weighted colimit, and by using à instead of  , we get: Q ( a ∗ b , i ( x )) ≤ Q ( T ( a ) ∗ b , i ( x )) ≤ Q ( T ( a ) ∗ T ( b ) , i ( x )) . Corollary 4.26.
In the same hypotesis of the previous proposition, i ′ : X B → Q T is dense.Proof. From T being a left adjoint (as V -functor) and from begin a morphism of quantale, it follows that T is left adjoint also as ( L , V ) -functor, as: Q T ( T ( a ) ∗ ′ ... ∗ ′ T ( a n ) , b ) = Q T ( T ( a ∗ ... ∗ a n ) , b ) = Q ( a ∗ ... ∗ a n , g ( b )) .This implies that T preserves ( L , V ) -colimits as well, hence that i ′ is pre-dense.Denseness follows by noticing that, by construction, every element of Q T is such that T ( a ) ≃ a .29 roposition 4.27. Every quantum B-algebra X admits a strict embedding into a quantale.Proof.
Let ( X , a ) be the corresponding object in ( L , V ) - Cat . Consider the Yoneda embedding in ( L , V ) - Cat : y X : X D L ( X ) x a ( − , x ) .We have that: a (( x , w ) , z ) = D L ( X )[ L y X (( x ; w )) , y X ( z )] , = D L ( X )[( α L y X ( x )) ∗ y X ( w ) , y X ( z )] , = D L ( X )[ α L y X ( x )) , y X ( w )  y X ( z ))] , = D L ( X )[ L y X ( x )) , y X ( w )  y X ( z ))] .Since, by definition we have that: a (( x ; w ) , z ) = a ( x , w  z ) = D L ( X )[ L y X ( x ) , y X ( w  z )] ;from which it follows that, for all x ∈ LX : D L ( X )[ L y X ( x )) , y X ( w )  y X ( z ))] = D L ( X )[ L y X ( x ) , y X ( w  z )] ;hence, by representability: y X ( x )  y X ( x ) = y X ( x  x ) .By using the same argument, but starting from: a (( w ; x ) , z ) , we can also prove that: y X ( x ) à y X ( x ) = y X ( x à x ) . Remark 4.28.
Notice that D L ( X ) is an object of QBalg and when viewed as ( L , V ) - Cat its ( L , V ) -structure˜ Q D L ( X ) is the same as α · Q D L ( X ) by adjointness. Corollary 4.29. X → D L ( X ) is pre-dense strict morphism.Proof. Every element of D L ( X ) is a weighted colimit of y X in a canonical way, by Yoneda lemma it follows: φ ≃ L - colim ( y X , φ ) ≃ L - colim ( y X , ( y X ) ⊛ ◦ φ ⊛ ) . Corollary 4.30.
For every quantum B-algebra X B , there exists a quantale Q and a dense embedding i : X B Q.Proof.
Apply 4.25 to the Yoneda embedding: y X : X B D L ( X ) . Remark 4.31.
As already remarked before, our class J is the class of fully-faithful functors in ( L , V ) - Cat . Proposition 4.32.
Every dense morphism in ( L , V ) - Cat is essential.Proof.
Suppose that ( X , a ) → Q is dense and let g : Q → ( Z , b ) such that g (cid:12)(cid:12) X is fully-faithfulWe want to prove that: b ( L g ( a ) , g ( a )) ≤ Q ( a ∗ a , a ) , (where, for simplicity a = ( a , a )) .30ince the embedding is dense, we can write: a = L - colim ( i ⊛ ◦ a ⊛ , i )( ) , a = L - colim ( i ⊛ ◦ a ⊛ , i )( ) , a = lim ( i ∗ · a ∗ , i ) .In order to increase readability, we denote: j : = i ⊛ ◦ a ⊛ , j : = i ⊛ ◦ a ⊛ , j : = i ∗ · a ∗ .We have that: Q ( a ∗ a , a ) = Q ( colim ( e ◦ · ˆ j , α Li ( x )) ∗ colim ( e ◦ · ˆ j , α Li ( w )) , lim ( j , i ( x ))) (since ∗ preserves colimits, being Q a quantale), = lim ( e ◦ · ˆ j , lim ( e ◦ · ˆ j , lim ( j , Q ( α Li ( x ) ∗ α Li ( w ) , i ( x ))))) ,(being i fully faithful), = lim ( e ◦ · ˆ j , lim ( e ◦ · ˆ j , lim ( j , a (( x ; w ) , x )))) ,(since g (cid:12)(cid:12) X is fully-faithful), = lim ( e ◦ · ˆ j , lim ( e ◦ · ˆ j , lim ( j , b ( L g ( x ; w ) , g ( x ))))) .By using the Yoneda embedding y Z : ( Z , b ) → D L ( Z ) , we have that y Z g is an embedding (restricted to X ,of course). Moreover we have (denote y Z gi : = f ): y Z g ( lim ( j , i ( x ))) ≤ lim ( j , f ( x )) , ( ) colim ( e ◦ · ˆ j , α L f ( x )) ≤ y Z g ( colim ( e ◦ · ˆ j , α Li ( x ))) , colim ( e ◦ · ˆ j , α L f ( w )) ≤ y Z g ( colim ( e ◦ · ˆ j , α Li ( w ))) ,which combined together give: colim ( e ◦ · ˆ j , α L f ( x )) ∗ colim ( e ◦ · ˆ j , α L f ( w )) ≤ y Z g ( colim ( e ◦ · ˆ j , α Li ( x ))) ∗ y Z g ( colim ( e ◦ · ˆ j , α Li ( w ))) , ( ) .Thus: Q ( a ∗ a , a ) = lim ( e ◦ · ˆ j , lim ( e ◦ · ˆ j , lim ( j , b ( L g ( x ; w ) , g ( x k ))))) ,(using the Yoneda embedding), = lim ( e ◦ · ˆ j , lim ( e ◦ · ˆ j , lim ( j , D L ( Z )[( α L f ( x ; w ) , f ( x )]))) , = lim ( e ◦ · ˆ j , lim ( e ◦ · ˆ j , lim ( j , D L ( Z )[( α L f ( x )) ∗ ( α L f ( w )) , f ( x )]))) ,(since ∗ preserves colimits, being D L ( Z ) a quantale), = D L ( Z )[ colim ( e ◦ · ˆ j , α L f ( x )) ∗ colim ( e ◦ · ˆ j , α L f ( w )) , lim ( j , f ( x )]) ,(using (1) + (2)), ≥ D L ( Z )[ y Z g ( a ) ∗ y Z g ( a ) , y Z g ( a )] , = D L ( Z )[ L ( y Z g )( a ) , y Z g ( a )] ,(since the Yoneda is an embedding), = b ( L g ( a ) , g ( a )) .Which proves that: b ( L g ( a ) , g ( a )) ≤ Q ( a ∗ a , a ) . Remark 4.33.
Since the notion of denseness in
QBalg and ( L , V ) - Cat coincide, the same results holds fordense morphisms in
QBalg . 31 heorem 4.34. In QBalg i : ( X B , a ) Q is dense iff is esssential.Proof.
The first part is the previous proposition, while for the converse implication suppose that i : ( X B , a ) Q is essential. Consider the left Kan extension along the Yoneda embedding of i and take its image factor-ization, the result will be the quantale generated by X in Q (and in particular will be a strict embedding): X D L ( X ) Q Im ( Lan y ( i )) y X Lan y ( i ) Now, consider the following: X Im ( Lan y ( f )) Q ( Im ( Lan y ( f ))) Ty X Lan y ( f ) f where f is obtained by applying the injective property to j . In particular, being i essential it follows that f is fully-faithful, but since j is full and essentially on objects, it follows f is too, hence f is an equivalence.Hence Q ≃ W j which implies i is dense. In this section we apply the results proven in the last one to ( L , V ) - Cat (and more general to particularsub-categories of it). The idea is to use the Yoneda embedding to view any ( L , V ) -category as a subset ofthe quantale D L ( X ) and consider the quantum B-algebra generated by it.Due to the equational definition quantum B-algebras admit, if we have a subset X ( Q of a quantale Q containing the unit u Q , the quantum B-algebra generated by X (inside Q ) is obtained by inductively add allthe "missing" implications until the process reaches a "saturation". If the unit is not included in X we haveto add it in the first step of the construction; this might cause a problem, since we are "artificially" addingan element which is "alien" and can not be constructed inductively starting from elements of X .In order to overcome this problem, we introduce a slight variation of quantum B-algebras: pre-QuantumB-Algebras . A pre-quantum B-algebra is a promonoidal category ( X , P , J ) where only P is representable. Inthis way we obtain a new sub-category of ( L , V ) - Cat where all the constructions made in the last sectionstill work, since all it requires for them to work is the presence of an "implicational structure"; moreover,in the process of building the quantum B-algebra generated by a subset, we won’t have to "artificially" addthe unit of the quantale as we should have if we considered vanilla quantum B-algebras.
Definition 5.1. A pre-quantum B-algebra ( X B , a , J ) is a pre-representable promonoidal V -category. Thismeans it is a promonoidal V -category ( X , a , P , J ) equipped with two binary operations:  , à : X × X → X such that: • P ( x , − , y ) ≃ a ( − , x à y ) ; • P ( − , x , y ) ≃ a ( − , x  y ) ;By expressing what it means to be promonoidal functor for pre-quantum B-algebras, we have:32 efinition 5.2. A morphism between two pre-quantum B-algebras f : ( X B , a , J ) → ( Y B , b , U ) is V -functorsatisfying: f ( x  y ) ≤ f ( x )  f ( y ) , f ( x à y ) ≤ f ( x ) à f ( y ) , J ( x ) ≤ U ( f ( x )) ( ) If also: f ( x )  f ( y ) ≤ f ( x  y ) , f ( x ) à f ( y ) ≤ f ( x à y )) , ( ) then f is called strict.As for vanilla quantum B-algebras, we have: Proposition 5.3.
The category of pre-quantum B-algebras,
PQBalg , is a fully-faithful subcategory of ( L , V ) - Cat . Remark 5.4.
Notice that, since the strictness condition involves only the implications, the key results con-tained in Proposition 4.25 and Proposition 4.27 remain true when applied to pre-quantum B-algebras. Thuswe can construct the injective hull of a pre-quantum B-algebra ( X B , a , J ) as done in the previous section.The last step is to descrive what is the pre-quantum B-algebra generated by a subset of a quantale Q . Definition 5.5.
Let X ⊆ Q be a subset of a quantale ( Q , ∗ , u Q ) . The pre-quantum B-algebra generated by X , denoted as X B , is the smallest pre-quantum B-algebra containing X . Proposition 5.6. X B = ( S i X i , ˜ Q , J ) , where: • X : = X , X i + : = { a  b , c à d , a , b , c , d ∈ X i } ∪ X i . • ˜ Q is the restriction of the V -structure on Q to S i X i , . • J ( x ) : = Q ( u Q , x ) . Proof.
Is straightforward to verify that S i X i is a pre-quantum B-algebra; hence that X B ⊆ S i X i .For the other inclusion is sufficient to prove by induction that X i ⊆ X B . Remark 5.7.
Note that X B Q is a strict embedding.Let ( X , a ) be in ( L , V ) - Cat . Consider the Yoneda embedding: y X : ( X , a ) D L ( X ) . Define X B to be thepre-quantum B-algebra generated by y X ( X ) in D L ( X ) . By definition the Yoneda, factorizes as: X X B D L ( X ) .Let ˜ X B be its injective hull in PQBalg . We have that: X B ˜ X B is essential (and also dense) in ( L , V ) - Cat by 4.32. Consider the composite: ( X , a ) X B ˜ X B ,We have: Proposition 5.8. ( X , a ) X B ˜ X B is essential in ( L , V ) - Cat . Proof.
Let f : ˜ X → ( Z , b ) be such that f (cid:12)(cid:12) X is an embedding. By composing with y Z we can suppose that Z is a quantale, and thus that f is a morphism in ( V - Cat L ) lax .If we show that f (cid:12)(cid:12) X B is an embedding, being ˜ X the injective hull of X B , it will follow that f is an embeddingtoo, hence the result.First we are going to prove that: Z ( L f ( a ) , f ( w )) ≤ ˜ a X B (( a , ..., a n ) , w ) , with a , ..., a n ∈ X B and w = y X ( x ) , with x ∈ X .33ith a calculation similar to the one done in Proposition 4.32, by using the properties of f and the fact that f (cid:12)(cid:12) X is an embedding, we have (where we consider a = ( a , a ) for a matter of convenience):˜ a X B (( a , a ) , w ) : = a X B ( a , a  w ) ,(because X B is the pre-quantum B-algebra generated by Y X ( X ) in D L ( X ) ) = D L ( X )[ a , a  y X ( x )] , = D L ( X )[ a ∗ a , y X ( x )] , = lim ( e ◦ · j , lim ( e ◦ · j , a (( x ; x ) , x ))) , = lim ( e ◦ · j , lim ( e ◦ · j , Z ( L f ( x ; x )) , f ( w ))) , ≥ Z ( f ( a ) ∗ f ( a ) , f ( w )) .Hence: Z ( L f ( a ) , f ( w )) ≤ ˜ a X B ( a , w ) .If w / ∈ X , by induction, we can suppose is of the form w = y  z (or y à z ), for y , z ∈ y X ( X ) .From f ( w ) ∗ f ( y ) ≤ f ( w ∗ y ) it follows that f ( w ) ≤ f ( y )  f ( z ) ; hence that: Z ( L f ( a ) , f ( w )) ≤ Z ( L f ( a ) , f ( y )  f ( z )) ,: = Z ( f ( a ) ∗ f ( a ) , f ( y )  f ( z ) , = Z ( f ( a ) ∗ f ( a ) ∗ f ( y ) , f ( z )) , = : Z ( L f ( a ; y ) , f ( z )) .Since we have supposed z ∈ y X ( X ) , by the case we’ve already analyzed, it follows: Z ( L f ( a ; y ) , f ( z )) ≤ ˜ a X B ( a ; y , z ) : = ˜ a X B ( a , y  z ) ,hence that: Z ( L f ( a ) , f ( w )) ≤ ˜ a X B ( a , w ) ,for w ∈ X B . Remark 5.9.
Now it should be clear why we have introduced pre-quantum B-algebras. If we had considered vanilla quantum B-algebras, in the last proposition we would have faced a problem, since we would not beable to prove that: Z ( f ( a ) ∗ f ( a ) , f ( u )) ≤ ˜ a X B (( a , a ) , u ) ,where u is the unit of D L ( X ) . Remark 5.10.
One might ask why we did not consider pre-quantum B-algebras in the first place insteadof introducing vanilla quantum B-algebras. The reason for our choice is due to the fact that pre-quantumB-algebras doe not look "natural" as quantum B-algebras do; requiring the representability of only the"multiplicative" part of a promonoidal category is an artifice we employed in order to overcome the minorproblem one would have with the unit in the previous proposition.With the aid of this proposition we can prove our main theorem:
Theorem 5.11.
Let C be a full sub-category of ( L , V ) - Cat which contains quantales. Then every object ( X , a ) admits an injective hull.Proof. From Theorem 4.17 we get that the injectives in C are quantales. From the previous theorem, forevery object of C there exists an essential embedding of it into a quantale. Hence the result.34n particular this theorem applies to all the categories displayed in the following: ( L , V ) - Cat V - ProMon (( V - Cat )) lax QBalgResMon (( V - Cat )) lax Finally, we mention the case of topological spaces which, by [ Bar70 ] , can be also seen as examples of“generalised multicategories”. Injective topological spaces are characterised in [ Sco72 ] as precisely thecontinuous lattices, and [ Hof11 ] it is observed that injective topological spaces are those spaces where the(topological analogon of the) Yoneda embedding has a left adjoint in the ordered category of topologicalspaces and continuous maps. Here one considers the space2 ( UX ) op where 2 is the Sierpi´nski space and U X denotes the set of all ultrafilters on X which, with a certain topology,becomes the space ( U X ) op . The Yoneda map y : X → ( UX ) op sends a point x ∈ X to the set { x ∈ U X | x → x } of all ultrafilters convergent to x . In [ HT10, Example 4.10 ] it is observed that2 ( UX ) op −→ F X , A 7−→ \ A is am isomorphism; here F X = { filters of open subsets of X } with the sets U = { f ∈ F X | U ∈ f } , ( U open in X ) forming a basis for the topology of F X (for instance, see [ Esc97 ] ). On the other hand, the topologicalanalogous to the covariant presheaf category (see [ Hof14 ] ) is the lower Vietories space V X = { A ⊆ X | A closed } ,here the topology is generated by the sets B ◊ = { A ∈ V X | A ∩ B = ∅ } , ( B open in X ) .Injective hulls for topological spaces are described in [ Ban73 ] , we point out here that, similarly to thesituation for V -categories, this description is ultimately linked to the Isbell adjunction . For a topologicalspace X , the Isbell adjunction is given by the monotone maps F X V X ( − ) + ( − ) − ⊥ where A − = \ { x ∈ U X | x → x ∈ A } and f + = lim f ,for all closed subsets A ⊆ X and all filters of open subsets f . Note that ( − ) − : V X → F X is continuous but ( − ) + : F X → V X is in general not. We also recall from [ Hof13 ] the following definition.35 efinition 6.1. A topological space X is F -core-compact whenever, for all x ∈ X and all open neighbour-hoods U of x , there exists an open neighbourhood V of x so that V ≪ F U . Here V ≪ F U whenever, for allfilters f of opens with V ∈ f , lim f ∩ U = ∅ . Remark 6.2.
For open subsets U , V of X , V ≪ F U if and only if there exists some x ∈ X with V ⊆ ↓ x ⊆ U .We also note that F -core-compact spaces were introduced in [ Ern91 ] under the name C-space . Moreover,in [ Ban73, Proposition 3 ] it is shown that these are exactly the topological spaces which admit an injectivehull. Proposition 6.3.
A topological space X is F -core-compact if and only if the map ( − ) + : F X → V X is continuous.Proof.
Assume first that X is F -core-compact. Let B ⊆ X be open and f ∈ F X with lim f ∈ B ◊ , that islim f ∩ B = ∅ . Let x ∈ lim f ∩ B . By hypothesis, there is an open neighbourhood U of x with U ≪ F V . Then f ∈ U and, for every g ∈ U , lim g ∈ B ◊ .Assume now that ( − ) + : F X → V X is continuous. Let x ∈ X and let B be an open neighbourhood of x . Then, with f being the open neighbourhood filter of x , f → x ∈ B , hence lim f ∈ B ◊ . Since ( − ) + iscontinuous, there is some open neighbourhood U of x so that, for all g ∈ U , lim g ∈ B ◊ .With this notation, we can reformulate [ Ban73, Proposition 3 ] . Theorem 6.4.
A topological space X has an injective hull if and only if ( − ) + : F X → V X is continuous.
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E-mail adress : [email protected]@ua.pt