Isbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions
aa r X i v : . [ m a t h . C T ] F e b Isbell adjunctions and Kan adjunctions via quantale-enrichedtwo-variable adjunctions
Lili Shen · Xiaoye TangAbstract
It is shown that every two-variable adjunction in categories enriched in a com-mutative quantale serves as a base for constructing Isbell adjunctions between functor cate-gories, and Kan adjunctions are precisely Isbell adjunctions constructed from suitable asso-ciated two-variable adjunctions. Representation theorems are established for fixed points ofthese adjunctions.
Keywords
Quantale · Quantale-enriched category · Two-variable adjunction · Isbelladjunction · Kan adjunction
Mathematics Subject Classification (2010) · · A distributor [2] ϕ : X / / ◦ Y between categories enriched in a commutative quantale [11,6,16,5] V = ( V , ⊗ , k ) may be described as a V -bifunctor ϕ : X op ⊗ Y / / V , and it induces three pairs of adjoint V -functors between the (co)presheaf V -categories of X and Y :(1) the Isbell adjunction [19] ϕ ↑ ⊣ ϕ ↓ : ( V Y ) op / / V X op ,(2) the Kan adjunctions [19,18] ϕ ∗ ⊣ ϕ ∗ : V X op / / V Y op and ϕ † ⊣ ϕ † : ( V X ) op / / ( V Y ) op .In this paper we demonstrate that every two-variable adjunction [4,20] between V -categories gives rise to (generalized) Isbell adjunctions and Kan adjunctions, and in factprovides a unified framework for these notions. Explicitly, for every two-variable adjunc-tion ( X , Y , Z , ⊛ , ւ , ց ) between V -categories (Definition 3.3), a V -bifunctor ϕ : A op ⊗ B / / Z This work was supported by National Natural Science Foundation of China (No. 12071319).L. Shen (cid:0)
School of Mathematics, Sichuan University, Chengdu 610064, ChinaE-mail: [email protected]. TangSchool of Mathematics, Sichuan University, Chengdu 610064, ChinaE-mail: [email protected] Lili Shen, Xiaoye Tang induces an Isbell adjunction ϕ ↑ ⊣ ϕ ↓ : ( X B ) op / / Y A op between the V -categories of V -functors (Proposition 5.1). As every two-variable adjunctionis associated with several others (Lemma 5.2), Isbell adjunctions constructed upon suitabletwo-variable adjunctions are precisely Kan adjunctions ψ ∗ ⊣ ψ ∗ : Z A op / / X B op and ζ † ⊣ ζ † : ( Y A ) op / / ( Z B ) op induced by V -bifunctors ψ : A op ⊗ B / / Y and ζ : A op ⊗ B / / X , respectively (Proposition 5.3). As revealed in Subsection 7.1, all these adjunctions are re-duced to the ones described in [19,18] when X = Y = Z = V , which are now unified in thegeneral framework of two-variable adjunction.The main results of this paper are concerned about the representation theorems for fixedpoints of Isbell adjunctions and Kan adjunctions. Explicitly, if ( X , Y , Z , ⊛ , ւ , ց ) is a two-variable adjunction between complete V -categories, then fixed points of ϕ ↑ ⊣ ϕ ↓ , ψ ∗ ⊣ ψ ∗ and ζ † ⊣ ζ † constitute complete V -categories, which are denoted by M ϕ , K ψ and K † ζ , respectively. By the aid of dense and codense V -functors, Theorem 6.2, Corollaries 6.6 and6.7 precisely characterize complete V -categories that are equivalent to M ϕ , K ψ and K † ζ ,respectively. It is worth pointing out that Theorem 6.2 is not a straightforward generalizationof [19, Theorem 4.16] or [8, Corollary 7.9] for the case of X = Y = Z = V , since the absenceof the Yoneda embedding in general V -categories of V -functors forces us to develop newtechniques to complete the proof, among which the key tools are the V -bifunctors ι : A ⊗ X / / X A op and ι † : A ⊗ X op / / ( X A ) op constructed for any V -category A and complete V -category X , which extend the notion of formal ball in V -categories (Remark 4.4) and satisfy a generalized version of the Yonedalemma (Propositions 4.7 and 4.8). V -categories Throughout, let V = ( V , ⊗ , k ) denote a commutative quantale [16]; that is, ( V , ⊗ , k ) is a commutative monoid with theunderlying set V being a complete lattice, such that x ⊗ (cid:16) _ i ∈ I y i (cid:17) = _ i ∈ I x ⊗ y i for all x , y i ∈ V ( i ∈ I ) . The right adjoint induced by the multiplication, denoted by → , ( x ⊗ − ) ⊣ ( x → − ) : V / / V , sbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions 3 satisfies x ⊗ y z ⇐⇒ x y → z for all x , y , z ∈ V .A V -category [11,6] consists of a set X and a map a : X × X / / V satisfying k a ( x , x ) and a ( x , y ) ⊗ a ( y , z ) a ( x , z ) for all x , y , z ∈ X . For simplicity, we abbreviate the pair ( X , a ) to X and write X ( x , y ) insteadof a ( x , y ) if no confusion arises.Every V -category X is equipped with an underlying (pre)order given by x y ⇐⇒ k X ( x , y ) for all x , y ∈ X . We write x ∼ = y iff x y and y x . A V -category X is separated if x = y whenever x ∼ = y in X .A map f : X / / Y between V -categories is a V -functor if X ( x , y ) Y ( f x , f y ) for all x , y ∈ X . With the order of V -functors given by f g : X / / Y ⇐⇒ ∀ x ∈ X : f x gx ⇐⇒ ∀ x ∈ X : k Y ( f x , gx ) , we obtain a 2-category V - Cat of V -categories and V -functors.Given a V -functor f : X / / Y , we say that: – f is fully faithful , if X ( x , y ) = Y ( f x , f y ) for all x , y ∈ X . – f is essentially surjective , if there exists x ∈ X such that y ∼ = f x for all y ∈ Y . – f is an equivalence (resp. isomorphism ) of V -categories, if there exists a V -functor g : Y / / X with g f ∼ = X (resp. g f = X ) and f g ∼ = Y (resp. f g = Y ), where 1 X and 1 Y refer to the identity maps on X and Y , respectively. In this case, X and Y are equivalent (resp. isomorphic ) V -categories, and denoted by X ≃ Y (resp. X ∼ = Y ).The following characterization of equivalences of V -categories is well known: Proposition 2.1. [6] A V -functor f : X / / Y is an equivalence of V -categories if, and onlyif, f is fully faithful and essentially surjective. Example 2.2.
Some basic examples of V -categories are listed below:(1) V itself is a separated V -category with V ( x , y ) = x → y for all x , y ∈ V .(2) Let [ , ∞ ] = ([ , ∞ ] , > , + , ) denote Lawvere’s quantale. Then [ , ∞ ] -categories are (gen-eralized) metric spaces [11].(3) Every V -category X has a dual X op , which has the same underlying set as X , and X op ( x , y ) = X ( y , x ) for all x , y ∈ X . Lili Shen, Xiaoye Tang (4) The product of V -categories X , Y , denoted by X × Y , has the cartesian product of theirunderlying sets as its set of objects, and ( X × Y )(( x , y ) , ( x ′ , y ′ )) = X ( x , x ′ ) ∧ Y ( y , y ′ ) for all x , x ′ ∈ X , y , y ′ ∈ Y .(5) The tensor product of V -categories X , Y , denoted by X ⊗ Y , has the cartesian productof their underlying sets as its set of objects, and ( X ⊗ Y )(( x , y ) , ( x ′ , y ′ )) = X ( x , x ′ ) ⊗ Y ( y , y ′ ) for all x , x ′ ∈ X , y , y ′ ∈ Y .(6) Given V -categories X , Y , we denote by Y X the V -category of V -functors from X to Y ,with Y X ( f , g ) = ^ x ∈ X Y ( f x , gx ) for all f , g ∈ Y X . In particular, V X op and ( V X ) op are called the presheaf and copresheaf V -categories of X , respectively, with V X op ( µ , µ ′ ) = ^ x ∈ X µ x → µ ′ x and ( V X ) op ( λ , λ ′ ) = ^ x ∈ X λ ′ x → λ x for all µ , µ ′ ∈ V X op , λ , λ ′ ∈ V X . Remark 2.3.
The dual of a V -category given in Example 2.2(3) may be expanded to anisomorphism ( − ) op : V - Cat / / ( V - Cat ) co of 2-categories, where “co” refers to the dualization of 2-cells. Explicitly, the dual of a V -functor f : X / / Y , denoted by f op : X op / / Y op , is the same map as f on objects, but ( f ′ ) op f op whenever f f ′ : X / / Y . Moreover, it is easy to see that ( Y X ) op ∼ = ( Y op ) X op and ( X ⊗ Y ) op = X op ⊗ Y op (2.i)for all V -categories X , Y . V -categories Recall that a pair of V -functors, f : X / / Y and g : Y / / X , form an adjunction in V - Cat ,denoted by f ⊣ g , if 1 X g f and f g Y ;or equivalently, if Y ( f x , y ) = X ( x , gy ) (3.i)for all x ∈ X , y ∈ Y . It is well known that the V -functoriality of f , g is implied by (3.i); thatis, maps f , g satisfying (3.i) are necessarily V -functors (see, e.g., [10, Theorem 2.10]). Inthis case, f is called a left adjoint of g , and g a right adjoint of f .The aim of this section is to introduce the notion of two-variable adjunction in thecontext of V -categories, which appears in [4,15] for ordinary categories and in [20] forenriched categories. First, it is straightforward to check the following lemma: sbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions 5 Lemma 3.1.
For V -categories X, Y , Z, a map ϕ : X ⊗ Y / / Z is a V -bifunctor if, and onlyif, (1) for any x ∈ X, ϕ ( x , − ) : Y / / Z is a V -functor, and (2) for any y ∈ Y , ϕ ( − , y ) : X / / Z is a V -functor. Suppose that X , Y , Z are V -categories and ⊛ : X ⊗ Y / / Z is a V -bifunctor. If the V -functors − ⊛ y : X / / Z and x ⊛ − : Y / / Z admit right adjoints in V - Cat for any y ∈ Y , x ∈ X , say, ( − ⊛ y ) ⊣ ( − ւ y ) : Z / / X and ( x ⊛ − ) ⊣ ( x ց − ) : Z / / Y , (3.ii)then Z ( x ⊛ y , z ) = X ( x , z ւ y ) = Y ( y , x ց z ) (3.iii)for all x ∈ X , y ∈ Y , z ∈ Z . Lemma 3.2. ւ : Z ⊗ Y op / / X and ց : X op ⊗ Z / / Y defined by (3.ii) (or equivalently,by (3.iii) ) are V -bifunctors.Proof. We check the V -bifunctoriality of ւ as an example. By Lemma 3.1, it suffices toshow that for each z ∈ Z , z ւ − : Y op / / X is a V -functor, which follows from Y ( y , y ′ ) Z (( z ւ y ′ ) ⊛ y , ( z ւ y ′ ) ⊛ y ′ ) Z (( z ւ y ′ ) ⊛ y , ( z ւ y ′ ) ⊛ y ′ ) ⊗ X ( z ւ y ′ , z ւ y ′ )= Z (( z ւ y ′ ) ⊛ y , ( z ւ y ′ ) ⊛ y ′ ) ⊗ Z (( z ւ y ′ ) ⊛ y ′ , z ) Z (( z ւ y ′ ) ⊛ y , z )= X ( z ւ y ′ , z ւ y ) , for all y , y ′ ∈ Y . Definition 3.3. [20] Let X , Y , Z be V -categories. A two-variable adjunction in V - Cat con-sists of V -bifunctors ⊛ : X ⊗ Y / / Z , ւ : Z ⊗ Y op / / X , ց : X op ⊗ Z / / Y such that Z ( x ⊛ y , z ) = X ( x , z ւ y ) = Y ( y , x ց z ) for all x ∈ X , y ∈ Y , z ∈ Z .For each V -category X , the Yoneda embedding (resp. co-Yoneda embedding ) refers tothe V -functor y : X / / V X op , x X ( − , x ) ( resp. y † : X / / ( V X ) op , x X ( x , − )) . The following
Yoneda lemma is well known:
Lemma 3.4 (Yoneda) . [6] For any x ∈ X, µ ∈ V X op , λ ∈ V X , it holds that V X op ( y x , µ ) = µ x and ( V X ) op ( λ , y † x ) = λ x . (3.iv) Lili Shen, Xiaoye Tang
The supremum (resp. infimum ) of µ ∈ V X op (resp. λ ∈ V X ), when it exists, is an objectsup µ ∈ X (resp. inf λ ∈ X ) satisfying X ( sup µ , x ) = V X op ( µ , y x ) = ^ y ∈ X µ y → X ( y , x ) (3.v) (cid:16) resp. X ( x , inf λ ) = ( V X ) op ( y † x , λ ) = ^ y ∈ X λ y → X ( x , y ) (cid:17) . (3.vi)for all x ∈ X . A V -category X is complete if every µ ∈ V X op admits a supremum, or equiv-alently, if y : X / / V X op has a left adjoint, given by sup : V X op / / X . It is well known that X is complete if and only if X op is complete [21], where the completeness of X op may betranslated as y † : X / / ( V X ) op admitting a right adjoint inf : ( V X ) op / / X .Every V -functor f : X / / Y induces a V -functor f → : V X / / V Y given by ( f → µ ) y = _ x ∈ X Y ( f x , y ) ⊗ µ x , (3.vii)and thus gives rise to V -functors ( f op ) → : V X op / / V Y op and ( f → ) op : ( V X ) op / / ( V Y ) op between (co)presheaf V -categories of X , Y . Proposition 3.5. [21] Let f : X / / Y be a V -functor between complete V -categories. Thenf is a left (resp. right) adjoint in V - Cat if, and only if, f preserves suprema (resp. infima) inthe sense that f sup X = sup Y ( f op ) → ( resp. f inf X = inf Y ( f → ) op ) . Therefore, if X , Y , Z are complete V -categories, then a two-variable adjunction ( X , Y , Z , ⊛ , ւ , ց ) in V - Cat is completely determined by a V -bifunctor ⊛ : X ⊗ Y / / Z that preserves suprema on both sides. V -categories In a V -category X , the tensor (resp. cotensor ) [6] of v ∈ V and x ∈ X , when it exists, is anobject v ⋆ x ∈ X (resp. v x ∈ X ) such that X ( v ⋆ x , y ) = v → X ( x , y ) ( resp. X ( y , v x ) = v → X ( y , x )) (4.i)for all y ∈ X . X is tensored (resp. cotensored ) if v ⋆ x (resp. v x ) exists for all v ∈ V , x ∈ X .A V -category X is order-complete if the underlying ordered set of X is complete. Proposition 4.1. [22] A V -category X is complete if, and only if, X is tensored, cotensoredand order-complete. Proposition 4.2. [22] Let f : X / / Y be a V -functor between complete V -categories. Thenf is a left (resp. right) adjoint in V - Cat if, and only if, f is a left (resp. right) adjoint betweenthe underlying ordered sets of X, Y , and preserves tensors (resp. cotensors) in the sense thatf ( v ⋆ x ) = v ⋆ f x (resp. f ( v x ) = v f x) for all v ∈ V , x ∈ X. sbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions 7 In fact, in a complete V -category X , (4.i) indicates that there are adjoint V -functors ( − ⋆ x ) ⊣ X ( x , − ) : X / / V and X ( − , x ) ⊣ ( − x ) : V op / / X (4.ii)for all x ∈ X . Then it follows immediately from Proposition 4.2 that X (cid:16) x , ^ i ∈ I y i (cid:17) = ^ i ∈ I X ( x , y i ) and X (cid:16) _ i ∈ I y i , x (cid:17) = ^ i ∈ I X ( y i , x ) (4.iii)for all x , y i ∈ X ( i ∈ I ), where ^ i ∈ I y i and _ i ∈ I y i are calculated in the underlying order of X .Let X be a complete V -category. For each V -category A , we define ι ( a , x ) : A op / / X , ι ( a , x ) b = A ( b , a ) ⋆ x , (4.iv) ι † ( a , x ) : A / / X , ι † ( a , x ) b = A ( a , b ) ⋆ x (4.v)for all a ∈ A , x ∈ X . Lemma 4.3. ι ( a , x ) : A op / / X and ι † ( a , x ) : A / / X are both V -functors.Proof. The V -functoriality of ι ( a , x ) follows from A ( b , c ) A ( c , a ) → A ( b , a ) ^ y ∈ Y ( A ( b , a ) → X ( x , y )) → ( A ( c , a ) → X ( x , y ))= ^ y ∈ Y X ( A ( b , a ) ⋆ x , y ) → X ( A ( c , a ) ⋆ x , y )= X ( A ( c , a ) ⋆ x , A ( b , a ) ⋆ x )= X ( ι ( a , x ) c , ι ( a , x ) b ) for all b , c ∈ A , and the V -functoriality of ι † ( a , x ) is obtained dually. Remark 4.4.
The V -functor ι ( a , x ) : A op / / X extends the notion of formal ball in V -categories [7], which originates from (generalized) metric spaces (i.e., when V = [ , ∞ ] ) [3,17]. In fact, if X = V , then tensors in V are given by v ⋆ r = v ⊗ r for all v , r ∈ V . Consequently, for any a ∈ A and r ∈ V , ι ( a , r ) : A op / / V is precisely a formal ball of center a and radius r in the sense of [7, Definition 5.1].A V -functor f : X / / Y is dense (resp. codense ) if, for every y ∈ Y , there exists µ ∈ V X op (resp. λ ∈ V X ) such that y = sup ( f op ) → µ ( resp. y = inf ( f → ) op λ ) . The following properties of (co)dense V -functors are useful later: Lemma 4.5. (cf. [6, Theorem 5.1] and [8, Proposition 4.12]) A V -functor f : X / / Y isdense (resp. codense) if, and only if,Y ( y , y ′ ) = ^ x ∈ X Y ( f x , y ) → Y ( f x , y ′ ) (cid:16) resp. Y ( y , y ′ ) = ^ x ∈ X Y ( y ′ , f x ) → Y ( y , f x ) (cid:17) for all y , y ′ ∈ X. Lili Shen, Xiaoye Tang
Lemma 4.6. (cf. [6, Proposition 5.9] and [8, Corollary 4.13]) (1)
If a V -functor f : X / / Y is essentially surjective, then f is both dense and codense. (2) If V -functors f : X / / Y and g : Y / / Z are both dense (resp. codense), and g is a left(resp. right) adjoint in V - Cat , then g f : X / / Z is dense (resp. codense).
Given a V -category A and a complete V -category X , Lemma 4.3 actually gives rise towell-defined maps ι : A ⊗ X / / X A op and ι † : A ⊗ X op / / ( X A ) op , and moreover: Proposition 4.7. ι : A ⊗ X / / X A op is a dense V -bifunctor, and ι † : A ⊗ X op / / ( X A ) op isa codense V -bifunctor.Proof. We only prove the claim for ι , as the claim for ι † can be obtained dually.First, ι is a V -bifunctor. This is because ( A ⊗ X )(( a , x ) , ( a ′ , x ′ )) = A ( a , a ′ ) ⊗ X ( x , x ′ ) ( A ( b , a ) → A ( b , a ′ )) ⊗ X ( x , x ′ ) ^ b ∈ A ^ y ∈ Y ( A ( b , a ′ ) → X ( x ′ , y )) → ( A ( b , a ) → X ( x , y ))= ^ b ∈ A ^ y ∈ Y X ( A ( b , a ′ ) ⋆ x ′ , y ) → X ( A ( b , a ) ⋆ x , y )= ^ b ∈ A X ( A ( b , a ) ⋆ x , A ( b , a ′ ) ⋆ x ′ )= X A op ( ι ( a , x ) , ι ( a ′ , x ′ )) for all a , a ′ ∈ A , x , x ′ ∈ X .Second, ι is dense. By Lemma 4.5 it suffices to show that X A op ( µ , µ ′ ) = ^ a ∈ A ^ x ∈ X X A op ( ι ( a , x ) , µ ) → X A op ( ι ( a , x ) , µ ′ ) for all µ , µ ′ ∈ X A op . To this end, note that for any c ∈ A , the V -functoriality of µ : A op / / X implies that A ( b , c ) X ( µ c , µ b ) for all b ∈ A , and consequently k ^ b ∈ A A ( b , c ) → X ( µ c , µ b ) = ^ b ∈ A X ( A ( b , c ) ⋆ µ c , µ b ) = X A op ( ι ( c , µ c ) , µ ) . It follows that ^ a ∈ A ^ x ∈ X X A op ( ι ( a , x ) , µ ) → X A op ( ι ( a , x ) , µ ′ ) X A op ( ι ( c , µ c ) , µ ) → X A op ( ι ( c , µ c ) , µ ′ ) X A op ( ι ( c , µ c ) , µ ′ ) X ( ι ( c , µ c ) c , µ ′ c ) = X ( A ( c , c ) ⋆ µ c , µ ′ c ) = A ( c , c ) → X ( µ c , µ ′ c ) X ( µ c , µ ′ c ) . Hence ^ a ∈ A ^ x ∈ X X A op ( ι ( a , x ) , µ ) → X A op ( ι ( a , x ) , µ ′ ) ^ c ∈ A X ( µ c , µ ′ c ) = X A op ( µ , µ ′ ) , which is in fact an equation since the reverse inequality is trivial. sbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions 9 Proposition 4.8.
Given a V -category A and a complete V -category X, it holds thatX A op ( ι ( a , x ) , µ ) = X ( x , µ a ) and ( X A ) op ( λ , ι † ( a , x )) = X ( x , λ a ) (4.vi) for all a ∈ A, x ∈ X, µ ∈ X A op , λ ∈ X A .Proof. The identity for ι follows from X A op ( ι ( a , x ) , µ ) = ^ b ∈ A X ( A ( b , a ) ⋆ x , µ b ) = ^ b ∈ A A ( b , a ) → X ( x , µ b ) = X ( x , µ a ) for all a ∈ A , x ∈ X , µ ∈ X A op , and the identity for ι † follows dually. Remark 4.9.
Proposition 4.8 may be viewed as a generalized version of the Yoneda lemma.In fact, if X = V , then for each V -category A , by setting x = k in (4.vi) we obtain the Yoneda embedding ι ( a , k ) = y A and the co-Yoneda embedding ι † ( a , k ) = y † A , and (4.vi) becomes exactly (3.iv) in Lemma 3.4. Let ( X , Y , Z , ⊛ , ւ , ց ) be a two-variable adjunction in V - Cat . For each V -bifunctor ϕ : A op ⊗ B / / Z , we define ϕ ↑ : Y A op / / ( X B ) op , ( ϕ ↑ µ ) b = ^ a ∈ A ϕ ( a , b ) ւ µ a , (5.i) ϕ ↓ : ( X B ) op / / Y A op , ( ϕ ↓ λ ) a = ^ b ∈ B λ b ց ϕ ( a , b ) . (5.ii) Proposition 5.1. ϕ ↑ ⊣ ϕ ↓ : ( X B ) op / / Y A op in V - Cat .Proof.
First, ϕ ↑ and ϕ ↓ are well-defined maps. Given µ ∈ Y A op , the V -functoriality of ϕ ↑ µ : B / / X follows from B ( b , b ′ ) ^ a ∈ A Z ( ϕ ( a , b ) , ϕ ( a , b ′ )) ( Lemma 3.1 ) ^ a ∈ A ^ x ∈ X Z ( x ⊛ µ a , ϕ ( a , b )) → Z ( x ⊛ µ a , ϕ ( a , b ′ ))= ^ a ∈ A ^ x ∈ X X ( x , ϕ ( a , b ) ւ µ a ) → X ( x , ϕ ( a , b ′ ) ւ µ a ) ( Definition 3.3 )= ^ a ∈ A X ( ϕ ( a , b ) ւ µ a , ϕ ( a , b ′ ) ւ µ a ) ^ a ∈ A X (cid:16) ^ a ′ ∈ A ϕ ( a ′ , b ) ւ µ a ′ , ϕ ( a , b ′ ) ւ µ a (cid:17) ( Equations (4.iii) ) = X (cid:16) ^ a ′ ∈ A ϕ ( a ′ , b ) ւ µ a ′ , ^ a ∈ A ϕ ( a , b ′ ) ւ µ a (cid:17) ( Equations (4.iii) )= X (( ϕ ↑ µ ) b , ( ϕ ↑ µ ) b ′ ) ( Equation (5.i) ) for all b , b ′ ∈ B ; and similarly, ϕ ↓ λ : A op / / Y is a V -functor whenever λ ∈ X B .Second, ϕ ↑ and ϕ ↓ are V -functors and ϕ ↑ ⊣ ϕ ↓ in V - Cat . Indeed, ( X B ) op ( ϕ ↑ µ , λ ) = ^ b ∈ B X ( λ b , ( ϕ ↑ µ ) b )= ^ b ∈ B X (cid:16) λ b , ^ a ∈ A ϕ ( a , b ) ւ µ a (cid:17) ( Equation (5.i) )= ^ b ∈ B ^ a ∈ A X ( λ b , ϕ ( a , b ) ւ µ a ) ( Equations (4.iii) )= ^ a ∈ A ^ b ∈ B Y ( µ a , λ b ց ϕ ( a , b )) ( Definition 3.3 )= ^ a ∈ A Y (cid:16) µ a , ^ b ∈ B λ b ց ϕ ( a , b ) (cid:17) ( Equations (4.iii) )= ^ a ∈ A Y ( µ a , ( ϕ ↓ λ ) a ) ( Equation (5.ii) )= Y A op ( µ , ϕ ↓ λ ) for all µ ∈ Y A op , λ ∈ X B , which completes the proof.For each V -bifunctor ϕ : X ⊗ Y / / Z , let ϕ ∂ : Y ⊗ X / / Z denote the V -bifunctor givenby ϕ ∂ ( y , x ) = ϕ ( x , y ) for all x ∈ X , y ∈ Y . Lemma 5.2.
For V -bifunctors ⊛ : X ⊗ Y / / Z, ւ : Z ⊗ Y op / / X, ց : X op ⊗ Z / / Y , thefollowing statements are equivalent: (i) ( X , Y , Z , ⊛ , ւ , ց ) is a two-variable adjunction in V - Cat . (ii) ( X , Z op , Y op , ց , ւ ∂ , ⊛ ) is a two-variable adjunction in V - Cat . (iii) ( Z op , Y , X op , ւ , ⊛ , ց ∂ ) is a two-variable adjunction in V - Cat .Proof.
This is an immediate consequence of Z ( x ⊛ y , z ) = X ( x , z ւ y ) = Y ( y , x ց z ) ⇐⇒ Y op ( x ց z , y ) = X ( x , y ւ ∂ z ) = Z op ( z , x ⊛ y ) ⇐⇒ X op ( z ւ y , x ) = Z op ( z , x ⊛ y ) = Y ( y , z ց ∂ x ) for all x ∈ X , y ∈ Y , z ∈ Z .Note that the duality of V -categories (cf. Remark 2.3) allows us to reformulate Proposi-tion 5.1 as follows: the dual of each V -bifunctor ϕ : A op ⊗ B / / Z op , i.e. , ϕ op : ( A op ) op ⊗ B op / / Z , sbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions 11 induces an adjunction ( ϕ op ) ↑ ⊣ ( ϕ op ) ↓ : ( X B op ) op / / Y ( A op ) op in V - Cat , given by (5.i) and (5.ii), which by duality corresponds to (( ϕ op ) ↓ ) op ⊣ (( ϕ op ) ↑ ) op : ( Y A ) op / / X B op . (5.iii)Applying (5.iii) to the two-variable adjunctions of Lemma 5.2(ii)(iii), we obtain that any V -bifunctors ψ : A op ⊗ B / / Y and ζ : A op ⊗ B / / X give rise to adjunctions given in the following Proposition 5.3, where ψ ∗ : X B op / / Z A op , ( ψ ∗ λ ) a = _ b ∈ B λ b ⊛ ψ ( a , b ) , (5.iv) ψ ∗ : Z A op / / X B op , ( ψ ∗ µ ) b = ^ a ∈ A µ a ւ ψ ( a , b ) (5.v)and ζ † : ( Z B ) op / / ( Y A ) op , ( ζ † λ ) a = ^ b ∈ B ζ ( a , b ) ց λ b , (5.vi) ζ † : ( Y A ) op / / ( Z B ) op , ( ζ † µ ) b = _ a ∈ A ζ ( a , b ) ⊛ µ a . (5.vii) Proposition 5.3. ψ ∗ ⊣ ψ ∗ : Z A op / / X B op and ζ † ⊣ ζ † : ( Y A ) op / / ( Z B ) op in V - Cat . As we will see in Subsection 7.1, adjunctions given by Propositions 5.1 and 5.3 general-ize Isbell adjunctions and Kan adjunctions induced by V -distributors between V -categories,respectively. Hence, following the terminologies of [19,18], adjunctions of the forms ϕ ↑ ⊣ ϕ ↓ and ψ ∗ ⊣ ψ ∗ , ζ † ⊣ ζ † are called Isbell adjunctions and
Kan adjunctions , respectively.
In this section, we assume that X , Y , Z are complete V -categories, and ( X , Y , Z , ⊛ , ւ , ց ) is a two-variable adjunction in V - Cat .Given a V -category A , a V -functor h : A / / A is a V -closure operator if1 A h and hh ∼ = h . In particular, each adjunction f ⊣ g : B / / A in V - Cat induces a V -closure operator g f : A / / A .We denote by Fix ( h ) : = { a ∈ A | ha ∼ = a } the V -subcategory of A consisting of fixed points of h . Proposition 6.1. [19]
Let h : A / / A be a V -closure operator. Then (1) the inclusion V -functor Fix ( h ) (cid:31) (cid:127) / / A is the right adjoint of the codomain restrictionh : A / / Fix ( h ) ; (2) Fix ( h ) is a complete V -category provided so is A. If X is a complete V -category, then for any V -category A , it is easy to see that X A isequipped with the pointwise tensors, cotensors and underlying joins inherited from X , andthus X A is also a complete V -category by Proposition 4.1. Therefore, by Proposition 5.1,every V -bifunctor ϕ : A op ⊗ B / / Z gives rise to a complete V -category M ϕ : = Fix ( ϕ ↓ ϕ ↑ ) = { µ ∈ Y A op | ϕ ↓ ϕ ↑ µ ∼ = µ } of the fixed points of the V -closure operator ϕ ↓ ϕ ↑ : Y A op / / Y A op induced by the Isbelladjunction ϕ ↑ ⊣ ϕ ↓ .The following theorem is the main result of this paper, which characterizes those com-plete V -categories that are equivalent to M ϕ : Theorem 6.2.
Let ϕ : A op ⊗ B / / Z be a V -bifunctor. A complete V -category C is equivalentto M ϕ if, and only if, there exist a dense V -bifunctor α : A ⊗ Y / / C and a codense V -bifunctor β : B ⊗ X op / / C such thatC ( α ( a , y ) , β ( b , x )) = Z ( x ⊛ y , ϕ ( a , b )) (6.i) for all a ∈ A, b ∈ B, x ∈ X, y ∈ Y .
Before proving this theorem, we present the following lemma as a preparation:
Lemma 6.3.
For any a ∈ A, b ∈ B, x ∈ X, y ∈ Y , it holds that ( ϕ ↑ ι A , Y ( a , y )) b ∼ = ϕ ( a , b ) ւ y and ( ϕ ↓ ι † B , X ( b , x )) a ∼ = x ց ϕ ( a , b ) . Proof.
The first isomorphism follows from X ( x ′ , ( ϕ ↑ ι A , Y ( a , y )) b ) = X (cid:16) x ′ , ^ a ′ ∈ A ϕ ( a ′ , b ) ւ ( A ( a ′ , a ) ⋆ y ) (cid:17) ( Equations (4.iv) and (5.i) )= ^ a ′ ∈ A X ( x ′ , ϕ ( a ′ , b ) ւ ( A ( a ′ , a ) ⋆ y )) ( Equations (4.iii) )= ^ a ′ ∈ A Y ( A ( a ′ , a ) ⋆ y , x ′ ց ϕ ( a ′ , b )) ( Definition 3.3 )= ^ a ′ ∈ A A ( a ′ , a ) → Y ( y , x ′ ց ϕ ( a ′ , b )) ( Equations (4.i) )= Y ( y , x ′ ց ϕ ( a , b ))= X ( x ′ , ϕ ( a , b ) ւ y ) ( Definition 3.3 ) for all x ′ ∈ X , where the penultimate equality holds since A ( a ′ , a ) Z ( ϕ ( a , b ) , ϕ ( a ′ , b )) Y ( x ′ ց ϕ ( a , b ) , x ′ ց ϕ ( a ′ , b )) by applying Lemmas 3.1 and 3.2 to ϕ : A op ⊗ B / / Z and ց : X op ⊗ Z / / Y . The secondisomorphism is obtained dually. sbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions 13 Proof of Theorem 6.2.
Necessity.
It suffices to prove the case C = M ϕ . We show that α : A ⊗ Y / / M ϕ and β : B ⊗ X op / / M ϕ given by α ( a , y ) = ϕ ↓ ϕ ↑ ι A , Y ( a , y ) and β ( b , x ) = ϕ ↓ ι † B , X ( b , x ) are dense and codense, respectively, and satisfies (6.i).First, the density of α and the codensity of β both follow from Lemma 4.6. Indeed, α is the composition of the dense V -bifunctor ι A , Y : A ⊗ Y / / Y A op (Proposition 4.7) and thecodomain restriction of the V -closure operator ϕ ↓ ϕ ↑ : Y A op / / Y A op (cf. Proposition 6.1),while β is the composition of the codense V -bifunctor ι † B , X : B ⊗ X op / / ( X B ) op (Proposi-tion 4.7) and the codomain restriction of the right adjoint V -functor ϕ ↓ : ( X B ) op / / Y A op .Second, Equation (6.i) follows from M ϕ ( α ( a , y ) , β ( b , x )) = Y A op ( ϕ ↓ ϕ ↑ ι A , Y ( a , y ) , ϕ ↓ ι † B , X ( b , x ))= ( X B ) op ( ϕ ↑ ϕ ↓ ϕ ↑ ι A , Y ( a , y ) , ι † B , X ( b , x )) ( ϕ ↑ ⊣ ϕ ↓ )= ( X B ) op ( ϕ ↑ ι A , Y ( a , y ) , ι † B , X ( b , x )) ( ϕ ↑ ⊣ ϕ ↓ )= X ( x , ( ϕ ↑ ι A , Y ( a , y )) b ) ( Proposition 4.8 )= X ( x , ϕ ( a , b ) ւ y ) ( Lemma 6.3 )= Z ( x ⊛ y , ϕ ( a , b )) ( Definition 3.3 ) for all a ∈ A , b ∈ B , x ∈ X , y ∈ Y . Sufficiency.
We show that h : M ϕ / / C , h µ = _ a ∈ A α ( a , µ a ) is a fully faithful and essentially surjective V -functor, and thus an equivalence of V -categoriesby Proposition 2.1.First, h is a fully faithful V -functor. Note that C ( h µ , β ( b , x )) = X ( x , ( ϕ ↑ µ ) b ) (6.ii)for all µ ∈ M ϕ , b ∈ B , x ∈ X , because C ( h µ , β ( b , x )) = C (cid:16) _ a ∈ A α ( a , µ a ) , β ( b , x ) (cid:17) = ^ a ∈ A C ( α ( a , µ a ) , β ( b , x )) ( Equations (4.iii) )= ^ a ∈ A Z ( x ⊛ µ a , ϕ ( a , b )) ( Equation (6.i) )= ^ a ∈ A X ( x , ϕ ( a , b ) ւ µ a ) ( Definition 3.3 )= X (cid:16) x , ^ a ∈ A ϕ ( a , b ) ւ µ a (cid:17) ( Equations (4.iii) )= X ( x , ( ϕ ↑ µ ) b ) . ( Equation (5.i) ) It follows that M ϕ ( µ , µ ′ ) = Y A op ( µ , ϕ ↓ ϕ ↑ µ ′ ) ( µ ′ ∈ M ϕ )= ( X B ) op ( ϕ ↑ µ , ϕ ↑ µ ′ ) ( ϕ ↑ ⊣ ϕ ↓ )= ^ b ∈ B X (( ϕ ↑ µ ′ ) b , ( ϕ ↑ µ ) b )= ^ b ∈ B ^ x ∈ X X ( x , ( ϕ ↑ µ ′ ) b ) → X ( x , ( ϕ ↑ µ ) b )= ^ b ∈ B ^ x ∈ X C ( h µ ′ , β ( b , x )) → C ( h µ , β ( b , x )) ( Equation (6.ii) )= C ( h µ , h µ ′ ) ( Lemma 4.5 ) for all µ , µ ′ ∈ M ϕ , where Lemma 4.5 is applied to the codense V -bifunctor β : B ⊗ X op / / C in the last equality.Second, h is essentially surjective. For any c ∈ C , note that C ( α ( a , − ) , c ) ∈ V Y op for all a ∈ A . Thus it makes sense to define µ : A op / / Y , µ a = sup Y C ( α ( a , − ) , c ) , which is a V -functor because A ( a , a ′ ) ^ y ∈ Y C ( α ( a , y ) , α ( a ′ , y )) ( V -functoriality of α ) ^ y ∈ Y C ( α ( a ′ , y ) , c ) → C ( α ( a , y ) , c )= V Y op ( C ( α ( a ′ , − ) , c ) , C ( α ( a , − ) , c )) Y ( µ a ′ , µ a ) ( V -functoriality of sup Y ) for all a , a ′ ∈ A . We claim that c ∼ = h ϕ ↓ ϕ ↑ µ . Indeed, C ( c , β ( b , x )) = C ( h ϕ ↓ ϕ ↑ µ , β ( b , x )) (6.iii)for all b ∈ B , x ∈ X , because the density of α : A ⊗ Y / / C guarantees that C ( c , β ( b , x )) = ^ a ∈ A ^ y ∈ Y C ( α ( a , y ) , c ) → C ( α ( a , y ) , β ( b , x )) ( Lemma 4.5 )= ^ a ∈ A ^ y ∈ Y C ( α ( a , y ) , c ) → Z ( x ⊛ y , ϕ ( a , b )) ( Equation (6.i) )= ^ a ∈ A ^ y ∈ Y C ( α ( a , y ) , c ) → Y ( y , x ց ϕ ( a , b )) ( Definition 3.3 )= ^ a ∈ A Y ( µ a , x ց ϕ ( a , b )) ( Equation (3.v) )= ^ a ∈ A X ( x , ϕ ( a , b ) ւ µ a ) ( Definition 3.3 )= X (cid:16) x , ^ a ∈ A ϕ ( a , b ) ւ µ a (cid:17) ( Equations (4.iii) ) sbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions 15 = X ( x , ( ϕ ↑ µ ) b ) ( Equation (5.i) )= X ( x , ( ϕ ↑ ϕ ↓ ϕ ↑ µ ) b ) ( ϕ ↑ ⊣ ϕ ↓ )= C ( h ϕ ↓ ϕ ↑ µ , β ( b , x )) . ( Equation (6.ii) ) Hence, by the codensity of β : B ⊗ X op / / C , C ( c , c ′ ) = ^ b ∈ B ^ x ∈ X C ( c ′ , β ( b , x )) → C ( c , β ( b , x )) ( Lemma 4.5 )= ^ b ∈ B ^ x ∈ X C ( c ′ , β ( b , x )) → C ( h ϕ ↓ ϕ ↑ µ , β ( b , x )) ( Equation (6.iii) )= C ( h ϕ ↓ ϕ ↑ µ , c ′ ) ( Lemma 4.5 ) for all c ′ ∈ C , which completes the proof. Remark 6.4.
The above proof for Theorem 6.2 is a direct one. Besides, an easier but indirectapproach to the sufficiency of Theorem 6.2 may be formulated by applying the results of [8].Explicitly, since we have dense V -functors α , ι A , Y and codense V -functors β , ι † B , X satisfying ( X B ) op ( ϕ ↑ ι A , Y ( a , y ) , ι † B , X ( b , x )) = Z ( x ⊛ y , ϕ ( a , b )) = C ( α ( a , y ) , β ( b , x )) A ⊗ Y Y A op ι A , Y ♦♦♦♦♦♦♦♦ B ⊗ X op ( X B ) op ι † B , X g g ❖❖❖❖❖❖❖ A ⊗ Y C α + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ B ⊗ X op C β s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ Y A op ( X B ) op ϕ ↑ / / ( X B ) op Y A op ϕ ↓ o o ⊥ for all a ∈ A , b ∈ B , x ∈ X , y ∈ Y , it follows from [8, Theorem 5.1] that C is equivalent to M ϕ .Now, in order to derive representation theorems for the complete V -categories of fixedpoints of Kan adjunctions given by Proposition 5.3, i.e., K ψ : = Fix ( ψ ∗ ψ ∗ ) = { λ ∈ X B op | ψ ∗ ψ ∗ λ ∼ = λ } , K † ζ : = Fix ( ζ † ζ † ) = { λ ∈ Z B | ζ † ζ † λ ∼ = λ } , let us apply Theorem 6.2 to the adjunction (5.iii). Indeed, for any V -bifunctor ϕ : A op ⊗ B / / Z op , the V -category of fixed points of (5.iii) is precisely ( M ϕ op ) op . By Theorem 6.2,the dual of a complete V -category C is equivalent to M ϕ op if, and only if, there exist a dense V -bifunctor α : A op ⊗ Y / / C op and a codense V -bifunctor β : B op ⊗ X op / / C op with C ( β ( b , x ) , α ( a , y )) = C op ( α ( a , y ) , β ( b , x )) = Z ( x ⊛ y , ϕ ( a , b )) for all a ∈ A , b ∈ B , x ∈ X , y ∈ Y . Therefore, by the duality of V -categories we have: Corollary 6.5.
Let ϕ : A op ⊗ B / / Z op be a V -bifunctor. A complete V -category C is equiv-alent to ( M ϕ op ) op if, and only if, there exist a dense V -bifunctor β : B ⊗ X / / C and acodense V -bifunctor α : A ⊗ Y op / / C such thatC ( β ( b , x ) , α ( a , y )) = Z ( x ⊛ y , ϕ ( a , b )) for all a ∈ A, b ∈ B, x ∈ X, y ∈ Y .
The representation theorems for K ψ and K † ζ are then obtained by applying Corollary6.5 to Proposition 5.3: Corollary 6.6.
Let ψ : A op ⊗ B / / Y be a V -bifunctor. A complete V -category C is equiv-alent to K ψ if, and only if, there exist a dense V -bifunctor β : B ⊗ X / / C and a codense V -bifunctor α : A ⊗ Z / / C such thatC ( β ( b , x ) , α ( a , z )) = Y ( ψ ( a , b ) , x ց z ) for all a ∈ A, b ∈ B, x ∈ X, z ∈ Z. Corollary 6.7.
Let ζ : A op ⊗ B / / X be a V -bifunctor. A complete V -category C is equiva-lent to K † ζ if, and only if, there exist a dense V -bifunctor β : B ⊗ Z op / / C and a codense V -bifunctor α : A ⊗ Y op / / C such thatC ( β ( b , z ) , α ( a , y )) = X ( ζ ( a , b ) , z ւ y ) for all a ∈ A, b ∈ B, y ∈ Y , z ∈ Z. X = Y = Z = V When X = Y = Z = V , the multiplication ⊗ of the quantale V obviously induces a two-variable adjunction on V , since V ( x ⊗ y , z ) = V ( x , y → z ) = V ( y , x → z ) for all x , y , z ∈ V . A V -bifunctor ϕ : A op ⊗ B / / V is precisely a V -distributor ϕ : A / / ◦ B ;that is, a map ϕ : A × B / / V satisfying B ( b , b ′ ) ⊗ ϕ ( a , b ) ⊗ A ( a ′ , a ) ϕ ( a ′ , b ′ ) for all a , a ′ ∈ A , b , b ′ ∈ B . Therefore, the Isbell adjunction ϕ ↑ ⊣ ϕ ↓ and Kan adjunctions ϕ ∗ ⊣ ϕ ∗ , ϕ † ⊣ ϕ † induced by ϕ are exactly the ones given in [19, Proposition 4.1] and [18,Proposition 6.2.1], respectively, when Q = V is a commutative quantale. In this case, M ϕ , K ϕ and K † ϕ are separated and complete V -categories since so is V , and the correspondingrepresentation theorems are formulated as follows: Corollary 7.1.
Let ϕ : A / / ◦ B be a V -distributor, and let C be a separated and complete V -category. (1) C is isomorphic to M ϕ if, and only if, there exist a dense V -bifunctor α : A ⊗ V / / Cand a codense V -bifunctor β : B ⊗ V op / / C and such thatC ( α ( a , y ) , β ( b , x )) = ( x ⊗ y ) → ϕ ( a , b ) for all a ∈ A, b ∈ B, x , y ∈ V . sbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions 17 (2) C is isomorphic to K ϕ if, and only if, there exist a dense V -bifunctor β : B ⊗ V / / Cand a codense V -bifunctor α : A ⊗ V / / C such thatC ( β ( b , x ) , α ( a , z )) = ϕ ( a , b ) → ( x → z ) for all a ∈ A, b ∈ B, x , z ∈ V . (3) C is isomorphic to K † ϕ if, and only if, there exist a dense V -bifunctor β : B ⊗ V op / / Cand a codense V -bifunctor α : A ⊗ V op / / C such thatC ( β ( b , z ) , α ( a , y )) = ϕ ( a , b ) → ( y → z ) for all a ∈ A, b ∈ B, y , z ∈ V . Remark 7.2.
As elaborated in [8, Section 7], (1) and (2) of Corollary 7.1 are actually thecommutative case of [8, Corollary 7.9], whose prototypes come from [1, Theorem 14(2)]and [14, Proposition 7.3], respectively.7.2 Tensors and cotensorsFor every complete V -category X , from (4.ii) we see that its tensors and cotensors gives riseto a two-variable adjunction ( V , X , X , ⋆, X op ( − , − ) , ) ; explicitly, X ( v ⋆ x , y ) = V ( v , X op ( y , x )) = X ( x , v y ) for all v ∈ V , x , y ∈ X . Let denote the singleton V -category. Considering a V -functor λ : A op / / X as a V -bifunctor λ : A op ⊗ / / X :(1) The induced Isbell adjunction is given by λ ↑ : X A op / / V op , λ ↑ µ = ^ a ∈ A X ( µ a , λ a ) = X A op ( µ , λ ) , λ ↓ : V op / / X A op , ( λ ↓ v ) a = v λ a , which satisfies X A op ( µ , λ ↓ v ) = V op ( λ ↑ µ , v ) = v → X A op ( µ , λ ) for all µ ∈ X A op , v ∈ V ; that is, λ ↓ v is the cotensor of v and λ in X A op (cf. (4.i)), givenby the pointwise cotensors inherited from X .(2) The induced Kan adjunction is given by λ ∗ : V / / X A op , ( λ ∗ v ) a = v ⋆ λ a , λ ∗ : X A op / / V , λ ∗ µ = ^ a ∈ A X ( λ a , µ a ) = X A op ( λ , µ ) , which satisfies X A op ( λ ∗ v , µ ) = V ( v , ϕ ∗ µ ) = v → X A op ( λ , µ ) for all µ ∈ X A op , v ∈ V ; that is, λ ∗ v is the tensor of v and λ in X A op (cf. (4.i)), given bythe pointwise tensors inherited from X . V = , the two-element Boolean algebra, a two-variable adjunctionwith respect to ordered sets X , Y , Z consists of maps ⊛ : X × Y / / Z , ւ : Z × Y / / X , ց : X × Z / / Y such that x ⊛ y z ⇐⇒ x z ւ y ⇐⇒ y x ց z for all x ∈ X , y ∈ Y , z ∈ Z ; that is, ( X , Y , Z , ⊛ , ւ , ց ) is an adjoint triple in the sense of [13,12]. For any sets A , B and maps ϕ : A × B / / Z , ψ : A × B / / Y and ζ : A × B / / X , – M ϕ is the multi-adjoint concept lattice of the context ( A , B , ϕ ) [13,9], – K ψ is the multi-adjoint property-oriented concept lattice of the context ( A , B , ψ ) [12,9], – K † ζ is the multi-adjoint object-oriented concept lattice of the context ( A , B , ζ ) [12,9].In particular, our Theorem 6.2 formalizes the representation theorem of the multi-adjointconcept lattice (see [13, Theorem 20]) when V = . Acknowledgements
The authors would like to thank Professor Hongliang Lai and Professor Dexue Zhangfor helpful discussions.
Conflict of interest
The authors declare that they have no conflict of interest.
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