Injective Objects and Fibered Codensity Liftings
aa r X i v : . [ c s . L O ] F e b Injective Objects and Fibered CodensityLiftings ⋆ Yuichi Komorida , The Graduate University for Advanced Studies, SOKENDAI, Tokyo, Japan National Institute of Informatics, Tokyo, Japan, [email protected]
Abstract.
Functor lifting along a fibration is used for several differentpurposes in computer science. In the theory of coalgebras, it is used todefine coinductive predicates, such as simulation preorder and bisimi-larity. Codensity lifting is a scheme to obtain a functor lifting along afibration. It generalizes a few previous lifting schemes including the Kan-torovich lifting. In this paper, we seek a property of functor lifting calledfiberedness. Hinted by a known result for Kantorovich lifting, we identifya sufficient condition for a codensity lifting to be fibered. We see thatthis condition applies to many examples that have been studied. As anapplication, we derive some results on bisimilarity-like notions.
In this paper, we focus on a category-theoretical gadget, called functor lifting ,and seek a property thereof, called fiberedness . As is often the case with suchmathematical topics, functor lifting comes up in several different places in com-puter science under various disguises (as mentioned in Section 1.6). Here we seeone of such places, bisimilarity and its generalizations on coalgebras , before weformally introduce functor lifting.
Computer programs work as we write them, not necessarily as we expect. Oneapproach to overcome this gap is to verify the systems so that we can makesure that they meet our requirements. Abstract mathematical methods are oftenuseful for the purpose, but before that, we have to model the target system bysome mathematical structure.
Coalgebra [28] is one of such mathematical structure with a broad scopeof application. It is defined in terms of the theory of categories and functors .Given a category C and an endofunctor F : C → C , an F -coalgebra is defined ⋆ The author was supported by ERATO HASUO Metamathematics forSystems Design Project (No. JPMJER1603), JST. This is the acceptedmanuscript of [19]. The final publication is available at Springer viahttp://dx.doi.org/10.1007/978-3-030-57201-3 7. Y. Komorida as an arrow c : X → F X . This simple definition includes many kinds of state-transition systems as special cases, e.g., Kripke frame (and model), Markov chain(and process), and (deterministic and non-deterministic) automata.Having modeled a system as a coalgebra, we can ask a fundamental question:which states behave the same?
Bisimilarity [26,27] is one of the notions to definesuch equivalence. (For an introduction, see, e.g., [29].) We sketch the idea in thecase where C = Set and F = Σ × ( − ). In this case, F -coalgebras are deterministicLTSs. Consider a coalgebra c : X → Σ × X and define l : X → Σ and n : X → X by ( l ( x ) , n ( x )) = c ( x ). The point here is the following observation: if x, y ∈ X behave the same, then l ( x ) = l ( y ) must hold, and n ( x ) and n ( y ) must behavethe same. This is almost the definition of bisimilarity: the bisimilarity relationis the greatest binary relation ∼⊆ X × X that satisfies x ∼ y = ⇒ l ( x ) = l ( y ) ∧ n ( x ) ∼ n ( y ) . For other functors F , the idea is roughly the same: in a coalgebra c : X → F X , for x, y ∈ X to behave the same, c ( x ) and c ( y ) must behave the same. Todefine bisimilarity precisely, however, we have to turn a relation R ⊆ X × X into R ′ ⊆ F X × F X . An elegant way to formulate this is the following: bundle binary relations on allsets into one fibration and use functor lifting as in [11]. We give ideas on themhere. The precise definitions are in Section 2.First, we gather all pairs (
X, R ) of a set X and a binary relation R ⊆ X × X into one category ERel (Example 8). It comes with a forgetful func-tor U : ERel → Set . (This is a fibration .) Any binary relation R on X is sent to X by U ; placing the things vertically, R is “above” X . Now let us assume thatthere exists a functor ˙ F : ERel → ERel satisfying U ◦ ˙ F = F ◦ U . This meansthat any binary relation R on X is sent to one on F X : ERel ˙ F / / U (cid:15) (cid:15) ERel U (cid:15) (cid:15) R ✤ / / ˙ F R
Set F / / Set X ✤ / / F X (This means that the functor ˙ F is a lifting of F along U .) The functor U : ERel → Set has an important structure: for any f : Y → X and a relation R on X , wecan obtain a relation f ∗ R on Y in a canonical way: f ∗ R = { ( y, y ′ ) ∈ Y × Y | ( f ( y ) , f ( y ′ )) ∈ R } . (This is called reindexing or pullback .) By using these, we can define the bisim-ulation relation on c : X → F X as the greatest fixed point of f ∗ ◦ ˙ F .An advantage of this approach is that we can readily generalize this to other“bisimilarity-like” notions. For example, by changing the fibration to PMet ⊤ → Set (Example 7), one can define a behavioral (pseudo)metric [2]. njective Objects and Fibered Codensity Liftings 3
Now we know that a functor lifting induces a bisimilarity-like notion. Then, howcan we obtain a functor lifting?
Codensity Lifting is a scheme to obtain suchliftings. It is first introduced by Katsumata and Sato [16] for monads using co-density monad construction [23]. It is later extended to general endofunctorsby Sprunger et al. [32]. It is parametrized in a set of data called a lifting pa-rameter . By changing lifting parameters, a broad class of functor liftings can berepresented as codensity liftings, as is shown, e.g., in Komorida et al. [18].As mentioned in the last section, we can define a bisimilarity-like notion usingcodensity lifting. It is called codensity bisimilarity in [18, Sections III and VI].
In some situations, we have to assume that ˙ F : E → E interacts well with thepullback operation between the fibers. In such a situation, ˙ F is required to be fibered (Definition 11). It means that pullbacks and ˙ F are “commutative,” in thesense that they satisfy ˙ F ( f ∗ P ) = ( F f ) ∗ ( ˙ F P ).Some of the existing works indeed require fiberedness. For example, Hasuoet al. [9, Definition 2.2] include fiberedness in their definition of predicate lifting.Fiberedness also plays a notable role in [3], where it is rephrased to isometry-preservation. However, there has been no systematic result on fiberedness ofcodensity lifting.
In the current paper, hinted by a result of Baldan et al. [3], we show a suffi-cient condition on the lifting parameter guaranteeing the resulting functor tobe fibered (Theorem 20). The scope of our fiberedness result is so broad that itcovers, e.g., most of the examples presented in [18] (Section 5).The condition involves a variation of the notion of injective object, which wecall c-injective object (Definition 15). To our knowledge, such a notion connectinginjective objects and fibrations is new. We study some basic properties of them.Using the fiberedness result, we show a property of codensity bisimilaritywhich we call stability under coalgebra morphisms (Proposition 49). As a corol-lary, we see that, when there is a final coalgebra, the codensity bisimilarity onany coalgebra is determined by that on the final coalgebra. Note that this kindof property is well-known for a conventional bisimilarity relation (Corollary 50).To summarize, our technical contributions are as follows: – We define c-injective objects for fibrations (Definition 15) and show someproperties of them. – We show a sufficient condition on the lifting parameter to guarantee fibered-ness of codensity lifting (Theorem 20 and Corollary 24). – We show a number of examples (Section 5) to which the condition above isapplicable. – As an application, we show that codensity bisimilarity is stable under coalge-bra morphisms (Proposition 49) in many cases, including a new one (Example 51).
Y. Komorida
Even though we focused on bisimilarity and coalgebra above, functor liftingcomes up in computer science here and there. To name a few, it has applica-tions in logical predicates [11,15], quantitative bisimulation [3], and differentialprivacy [30].As mentioned above, there have been many methods to obtain liftings offunctors.
Kantorovich lifting [2,20] and generalized Kantorovich metric [5] areboth special cases of the version of codensity lifting considered here.
Categorical ⊤⊤ -lifting [15] is the precursor of the original version of codensity lifting, but it isnot a special case of codensity lifting. For categorical ⊤⊤ -lifting, one uses internalHom-objects rather than Hom-sets like codensity lifting. Obtaining a sufficientcondition for fiberedness of categorical ⊤⊤ -lifting is future work. Wassersteinlifting [2] is another method that is somehow dual to Kantorovich lifting. Theyhave shown that any lifting obtained by this scheme is fibered. Klin [17] goes adifferent way: rather than showing fiberedness, they incorporate fiberedness inthe definition. They study the least fibered lifting along
EqRel → Set and showthat, in good situations, it coincides with the canonical relation lifting .The notion of injective object is first introduced in homological algebra as injective modules [1]. There are also some works about injective objects outsidehomological algebra: Scott [31] and Banaschewski and Bruns [4] have identifiedthe injective objects in
Top and Pos , respectively. We use their results in Sec-tion 4 (where the categories mentioned are defined). Injective objects w.r.t. iso-metric embeddings in the category of metric spaces are also well-studied andcalled hyperconvex spaces [7]. Finding a precise connection between them andc-injective objects in
PMet ⊤ → Set (Example 7) is future work. Recently, inhis preprint [8], Fujii has extended the above result in
Pos and characterizedinjective objects in the category of Q -categories with respect to the class of fullyfaithful Q -functors, for any quantale Q . In Section 2, we review
CLat ⊓ -fibrations and functor liftings . In Section 3, wereview the definition of codensity lifting and introduce the notion of c-injectiveobjects . We show a sufficient condition for a codensity lifting to be fibered. InSection 4, we show some general results on c-injective objects. In Section 5, welist several examples of fibered codensity liftings using the results in Section 4. InSection 6, we apply the fiberedness result to codensity bisimilarity . In Section 7,we conclude with some remarks and future work. We assume some knowledge of category theory , but the full content of the stan-dard reference [25] is not needed. The basic definitions and theorems, e.g., thosein Leinster [24], is enough. Even though we have explained our motivation njective Objects and Fibered Codensity Liftings 5 through coalgebra, no knowledge of coalgebra is needed for the main result inSection 3.In the following,
Set means the category of sets and (set-theoretic) functions. ⊓ -Fibrations Here we introduce
CLat ⊓ -fibrations , as defined in [18]. We use them to modelvarious “notions of indistinguishability” like preorder, equivalence relation, andpseudometric. Assuming full knowledge of the theory of fibrations, we coulddefine them as poset fibrations with fibered small meets. Instead, we give anexplicit definition below. This is mainly because we need the notion of Cartesianarrow . For a comprehensive account of the theory of fibrations, the reader canconsult, e.g., a book by Jacobs [13] or Hermida’s thesis [10], but in the following,we do not assume any knowledge of fibrations.We first define a fiber of a functor over an object. Basically, this is onlyconsidered in the case where the functor is a fibration.
Definition 1 (fiber).
Let p : E → C be a functor and X ∈ C be an object. The fiber over X is the subcategory of E – whose objects are P ∈ E such that pP = X and – whose arrows are f : P → Q such that pf = id X .We denote it by E X .Note that, if p is faithful, then each fiber is a thin category, i.e., a preorder.The following definition of poset fibration is a special case of that in [13]. Definition 2 (cartesian arrow and poset fibration).
Let p : E → C be afaithful functor.An arrow f : P → Q in E is Cartesian if the following condition is satisfied: – For each R ∈ E and g : R → Q , there exists h : R → P such that g = f ◦ h ifand only if there exists h ′ : pR → pP such that pg = pf ◦ h ′ .The functor p is called a poset fibration if the following are satisfied: – For each X ∈ C , the fiber E X is a poset. The order is denoted by ⊑ . Wedefine the direction so that P ⊑ Q holds if and only if there is an arrow P → Q in E X . – For each Q ∈ E and f : X → pQ , there exists an object f ∗ Q ∈ E X anda Cartesian arrow ˙ f : f ∗ Q → Q such that p ˙ f = f . (Such f ∗ Q and ˙ f arenecessarily unique.)The map Q f ∗ Q turns out to be a monotone map from E Y to E X . We call itthe pullback functor and denote it by f ∗ : E Y → E X .Intuitively, pullback functors model substitutions. Indeed, in many examples,they are just “assigning f ( x ) to y ”, as can be seen below. Y. Komorida
Example 3 (pseudometric).
Let ⊤ be a positive real number or + ∞ . Define acategory PMet ⊤ as follows: – Each object is a pair (
X, d ) of a set X and a [0 , ⊤ ]-valued pseudometric d : X × X → [0 , ⊤ ]. (A pseudometric is a metric without the condition d ( x, y ) = 0 = ⇒ x = y .) – Each arrow from (
X, d X ) to ( Y, d Y ) is a nonexpansive map f : X → Y . ( f isnonexpansive if, for all x and x ′ ∈ X , d X ( x, x ′ ) ≥ d Y ( f ( x ) , f ( x ′ )).)The obvious forgetful functor PMet ⊤ → Set is a poset fibration. For each X ∈ Set , the objects of the fiber (
PMet ⊤ ) X are the pseudometrics on X .However, the order is reversed: in our notation, the order is defined by( X, d ) ⊑ ( X, d ) ⇔ ∀ x, x ′ ∈ X, d ( x, x ′ ) ≥ d ( x, x ′ ) . An arrow f : ( X, d X ) → ( Y, d Y ) is Cartesian if and only if it is an isometry,i.e., d X ( x, x ′ ) = d Y ( f ( x ) , f ( x ′ )) holds for all x, x ′ . For ( Y, d Y ) ∈ PMet ⊤ and f : X → Y , the pullback f ∗ ( Y, d Y ) is the set X with the pseudometric ( x, x ′ ) d Y ( f ( x ) , f ( x ′ )).We list a few properties of pullback functors that we use: Proposition 4.
Let p : E → C be a poset fibration, f : X → Y be an arrow in C and P ∈ E X and Q ∈ E Y be objects in E . There exists an arrow g : P → Q such that pg = f if and only if P ⊑ f ∗ Q . Moreover, such g is Cartesian if andonly if P = f ∗ Q . Proposition 5.
Let p : E → C be a poset fibration. – For each X ∈ C , (id X ) ∗ : E X → E X is the identity functor. – For each composable pair of arrows X f −→ Y g −→ Z in C , ( g ◦ f ) ∗ = f ∗ ◦ g ∗ holds.Now we define the class that we are concerned about, CLat ⊓ -fibrations. Definition 6 (CLat ⊓ -fibration). A poset fibration p : E → C is a CLat ⊓ -fibration if the following conditions are satisfied: – Each fiber E X is small and has small meets, which we denote by d . – Each pullback functor f ∗ preserves small meets.Note that, in the situation above, each fiber E X is a complete lattice: thesmall joins can be constructed using small meets. Example 7 (pseudometric).
The poset fibration
PMet ⊤ → Set in Example 3 isa
CLat ⊓ -fibration. Indeed, meets can be defined by sups of pseudometrics: if welet ( X, d ) = d a ∈ A ( X, d a ), then d ( x, x ′ ) = sup a ∈ A d a ( x, x ′ )holds. njective Objects and Fibered Codensity Liftings 7 Example 8 (binary relations).
Define a category
ERel of sets with an endorela-tion as follows: – Each object is a pair (
X, R ) of a set X and a binary relation R ⊆ X × X . – Each arrow from (
X, R X ) to ( Y, R Y ) is a map f : X → Y preserving therelations; that is, we require f to satisfy ( x, x ′ ) ∈ R X = ⇒ ( f ( x ) , f ( x ′ )) ∈ R Y .The obvious forgetful functor ERel → Set is a
CLat ⊓ -fibration. For each X ∈ Set , the fiber
ERel X is the complete lattice of subsets of X × X .An arrow f : ( X, R X ) → ( Y, R Y ) is Cartesian if and only if it reflects the re-lations, i.e., ( x, x ′ ) ∈ R X ⇔ ( f ( x ) , f ( x ′ )) ∈ R Y holds for all x, x ′ . For ( Y, R Y ) ∈ ERel and f : X → Y , the pullback f ∗ ( Y, R Y ) is the set X with the relation { ( x, x ′ ) ∈ X × X | ( f ( x ) , f ( x ′ )) ∈ R Y } .Define the following full subcategories of ERel : – The category
Pre of preordered sets and monotone maps. – The category
EqRel of sets with an equivalence relation and maps preserv-ing them.The forgetful functors
Pre → Set and
EqRel → Set are also
CLat ⊓ -fibrations. CLat ⊓ -fibrations are not necessarily “relation-like”. There also is an examplewith a much more “space-like” flavor. Example 9.
The forgetful functor
Top → Set from the category
Top of topo-logical spaces and continuous maps is a
CLat ⊓ -fibration. Another pivotal notion in the current paper is functor lifting . In Section 1.2 wehave seen that it is used to define bisimilarity, or more generally bisimilarity-likenotions, as a way to turn a relation (or pseudometric, etc.) on X into one on F X . Here we review the formal definition in a restricted form that only considers
CLat ⊓ -fibration. (Note that, usually it is defined more generally, and there areindeed applications of such general definition.) Definition 10 (lifting of endofunctor).
Let p : E → C be a CLat ⊓ -fibrationand F : C → C be a functor. A lifting of F along p is a functor ˙ F : E → E suchthat p ◦ ˙ F = F ◦ p holds: E ˙ F / / p (cid:15) (cid:15) E p (cid:15) (cid:15) C F / / C . We then define fiberedness of a lifting. This means that the lifting interactswell with the pullback structure of the fibration, but we first give a definitionfocusing on Cartesian arrows. Here we define it in a slightly more general wayso that we can use them later (Section 4).
Y. Komorida
Definition 11 (fibered functor [13, Definition 1.7.1]).
Let p : E → C and q : F → D be CLat ⊓ -fibrations. A fibered functor from p to q is a functor ˙ F : E → F such that there is another functor F : C → D satisfying q ◦ ˙ F = F ◦ p and ˙ F sends each Cartesian arrow to a Cartesian arrow.Note that, in the situation above, such F is uniquely determined by p , q , and˙ F . Now we see a characterization of fiberedness by means of pullback. Proposition 12.
Let p : E → C and q : F → D be CLat ⊓ -fibrations and ˙ F : E → F and F : C → D be functors satisfying q ◦ ˙ F = F ◦ p . ˙ F is a fibered functor if andonly if, for any f : X → Y in C and P ∈ E Y , ˙ F ( f ∗ P ) = ( F f ) ∗ ( ˙ F P ) holds.We use this in the proof of the main result.
Before we formulate our main result, we introduce codensity lifting of endofunc-tors [16,32]. Here we use an explicit definition for a narrower situation than theoriginal one.
Definition 13 (codensity lifting (as in [18])).
Let – p : E → C be a CLat ⊓ -fibration, – F : C → C be a functor, – Ω ∈ E be an object above Ω ∈ C , and – τ : F Ω → Ω be an F -algebra.Define a functor F Ω ,τ : E → E , which is a lifting of F along p , by F Ω ,τ P = l f ∈ E ( P, Ω ) ( F ( pf )) ∗ τ ∗ Ω for each P ∈ E . The functor F Ω ,τ is called a codensity lifting of F . Note that,for each P ∈ E and f : P → Ω , the situation is as follows: Ω F pP F ( pf ) / / F Ω τ / / Ω and we can indeed obtain the pullback ( F ( pf )) ∗ τ ∗ Ω .We have given only the object part of F Ω ,τ above, but the arrow part, ifit is well-defined, should be determined uniquely since p is faithful. We give aproof that it is indeed well-defined. For each f : P → Q , we need another arrow g : F Ω ,τ P → F Ω ,τ Q such that pg = F ( pf ). By Proposition 4, it suffices to showthe following proposition: njective Objects and Fibered Codensity Liftings 9 Proposition 14.
For any f : P → Q , F Ω ,τ P ⊑ ( F ( pf )) ∗ (cid:0) F Ω ,τ Q (cid:1) holds. Proof.
By definition, the l.h.s. satisfies F Ω ,τ P = l g ∈ E ( P, Ω ) ( F ( pg )) ∗ τ ∗ Ω . On the other hand, the r.h.s. satisfies( F ( pf )) ∗ (cid:0) F Ω ,τ Q (cid:1) = ( F ( pf )) ∗ l h ∈ E ( Q, Ω ) ( F ( ph )) ∗ τ ∗ Ω = l h ∈ E ( Q, Ω ) ( F ( pf )) ∗ ( F ( ph )) ∗ τ ∗ Ω = l h ∈ E ( Q, Ω ) ( F ( p ( h ◦ f ))) ∗ τ ∗ Ω . Here, since { g ∈ E ( P, Ω ) } ⊇ { h ◦ f | h ∈ E ( Q, Ω ) } holds, we have l g ∈ E ( P, Ω ) ( F ( pg )) ∗ τ ∗ Ω ⊑ l h ∈ E ( Q, Ω ) ( F ( p ( h ◦ f ))) ∗ τ ∗ Ω . This means F Ω ,τ P ⊑ ( F ( pf )) ∗ (cid:0) F Ω ,τ Q (cid:1) . In the proof of the functoriality of F Ω ,τ , ultimately we use the fact that, for any f : P → Q , any “test” k : Q → Ω can be turned into another “test” k ◦ f : P → Ω .On the other hand, when we try to prove fiberedness of F Ω ,τ , we have to somehowlift a “test” g : P → Ω along a Cartesian arrow f : P → Q and obtain another“test” h : Q → Ω . This observation leads us to the following definition of c-injective object . (The letter c here comes from Cartesian .) Definition 15 (c-injective object).
Let p : E → C be a fibration. An object Ω ∈ E is a c-injective object if the functor E ( − , Ω ) : E op → Set sends everyCartesian arrow to a surjective map.Equivalently, Ω ∈ E is a c-injective object if, for any Cartesian arrow f : P → Q in E and any (not necessarily Cartesian) arrow g : P → Ω , there is a (notnecessarily Cartesian) arrow h : Q → Ω satisfying g = h ◦ f .Some basic objects can be shown to be c-injective objects. Example 16 (the two-point set).
In the fibration
EqRel → Set , (2 , =) is a c-injective object. Here, 2 = {⊥ , ⊤} is the two-point set and = means the equalityrelation. Indeed, for any Cartesian f : ( X, R X ) → ( Y, R Y ) and any g : ( X, R X ) → (2 , =), if we define h : ( Y, R Y ) → (2 , =) by h ( y ) = ( g ( x ) if ( y, f ( x )) ∈ R Y ⊤ otherwise,then this turns out to be well-defined and satisfies h ◦ f = g . Example 17 (the two-point poset of truth values).
In the fibration
Pre → Set ,(2 , ≤ ) is a c-injective object. Here, ≤ is the unique partial order satisfying ⊥ ≤ ⊤ and ⊤ (cid:2) ⊥ . Indeed, for any Cartesian arrow f : ( X, R X ) → ( Y, R Y ) and any g : ( X, R X ) → (2 , ≤ ), if we define h : : ( Y, R Y ) → (2 , ≤ ) by h ( y ) = ( ⊥ if ( y, f ( x )) ∈ R Y for some x such that g ( x ) = ⊥⊤ otherwise,then this turns out to be well-defined and satisfies h ◦ f = g . Example 18 (the unit interval as a pseudometric space [3, Theorem 5.8]).
In thefibration
PMet ⊤ → Set , [0 , ⊤ ] is a c-injective object. Indeed, for any arrow g : ( X, d X ) → ([0 , ⊤ ] , d e ) and any Cartesian arrow f : ( X, d X ) → ( Y, d Y ), we canshow that the map h : Y → [0 , ⊤ ] defined by h ( y ) = inf x ∈ X ( g ( x ) + d Y ( f ( x ) , y ))is nonexpansive from ( Y, d Y ) to ([0 , ⊤ ] , d e ).The following non-example shows that c-injectivity crucially depends on thefibration we consider. Example 19 (non-example).
In contrast to Example 17, in the fibration
ERel → Set , (2 , ≤ ) is not c-injective, where 2 = {⊥ , ⊤} is the two-point set and ≤ is theunique partial order satisfying ⊥ ≤ ⊤ and ⊤ (cid:2) ⊥ .This can be seen as follows. Let X = { a, b } , Y = { x, y, z } , R X = ∅ , and R Y = { ( x, z ) , ( z, y ) } . Then ( X, R X ) and ( Y, R Y ) are objects of ERel . Considerthe maps f : ( X, R X ) → ( Y, R Y ) and g : ( X, R X ) → (2 , ≤ ) defined by f ( a ) = x , f ( b ) = y , g ( a ) = ⊤ , and g ( b ) = ⊥ . Note that f is Cartesian. However, there isno h : ( Y, R Y ) → (2 , ≤ ) such that h ◦ f = g : such h would satisfy ⊤ = h ( f ( a )) = h ( x ) ≤ h ( z ) ≤ h ( y ) = h ( f ( b )) = ⊥ , which contradict ⊤ (cid:2) ⊥ .The same example can also be used to show that, in contrast to Example 16,(2 , =) is not c-injective, where = means the equality relation. Now we are prepared to state the following main theorem of the current paper.The strategy of the proof is roughly as mentioned earlier.
Theorem 20 (fiberedness from injective object).
In the setting of Definition 13,if Ω is a c-injective object, then F Ω ,τ is fibered. Proof.
Let f : P → Q be any Cartesian arrow. By Proposition 12, it sufficesto show F Ω ,τ P = ( F ( pf )) ∗ (cid:0) F Ω ,τ Q (cid:1) . Here, F Ω ,τ P ⊑ ( F ( pf )) ∗ (cid:0) F Ω ,τ Q (cid:1) has al-ready been proven. Thus, our goal is the inequality F Ω ,τ P ⊒ ( F ( pf )) ∗ (cid:0) F Ω ,τ Q (cid:1) .Here, since Ω is c-injective and f is Cartesian, the following inclusion holds: { g ∈ E ( P, Ω ) } ⊆ { h ◦ f | h ∈ E ( Q, Ω ) } . By the definition of the meet, we have l g ∈ E ( P, Ω ) ( F ( pg )) ∗ τ ∗ Ω ⊒ l h ∈ E ( Q, Ω ) ( F ( p ( h ◦ f ))) ∗ τ ∗ Ω . njective Objects and Fibered Codensity Liftings 11 By the calculation in the proof of Proposition 14, this implies F Ω ,τ P ⊒ ( F ( pf )) ∗ (cid:0) F Ω ,τ Q (cid:1) . Remark 21.
A refinement of Theorem 20 to an if-and-only-if result seems hard.At least there is a simple counterexample to the most naive version of it: Considera
CLat ⊓ -fibration Id : C → C , an endofunctor Id : C → C , an object C ∈ C , andan arrow τ : C → C . The codensity lifting Id C,τ is always equal to Id, which isfibered. However, since any arrow in C is a Cartesian arrow w.r.t. Id, it is nothard to find an example of C and C such that C is not c-injective w.r.t. Id. Example 22 (Kantorovich lifting).
Baldan et al. [3, Theorem 5.8] have shownthat any Kantorovich lifting preserves isometries. In terms of fibrations, thismeans that such functor is a fibered endofunctor on the fibration
PMet ⊤ → Set .Since Kantorovich lifting is a special case of codensity lifting where Ω =([0 , ⊤ ] , d R ), Theorem 20 and Example 18 recover the same result. Actually, thishas inspired Theorem 20 as a prototype.The argument above also applies to situations with multiple parameters. Definition 23 (codensity lifting with multiple parameters (as in [18])).
Let E , C , p , and F be as in Definition 13. Let A be a set. Assume that, for each a ∈ A , we are given Ω a ∈ E above Ω a ∈ C and τ a : F Ω a → Ω a . Define a functor F Ω ,τ : E → E by F Ω ,τ P = l a ∈ A F Ω a ,τ a P for each P ∈ E . Corollary 24.
In the setting of Definition 23, if, for each a ∈ A , Ω a is a c-injective object, then F Ω ,τ is fibered. Proof.
For any P ∈ E above X ∈ C and f : Y → X in C , using Theorem 20, wecan see( F f ) ∗ F Ω ,τ P = ( F f ) ∗ l a ∈ A F Ω a ,τ a P = l a ∈ A ( F f ) ∗ F Ω a ,τ a P = l a ∈ A F Ω a ,τ a f ∗ P = F Ω ,τ f ∗ P. Example 25 (Kantorovich lifting with multiple parameters).
In [20], K¨onig andMika-Michalski introduced a generalized version of Kantorovich lifting.Since it is a special case of Definition 23 where p is the fibration PMet ⊤ → Set and Ω = ([0 , ⊤ ] , d R ), Corollary 24 and Example 18 imply that such liftingalways preserves isometries. Here we seek properties of c-injective objects, mainly to obtain more examplesof them. We also see that, in a few fibrations, c-injective objects have beenessentially identified by previous works. M -injective Objects To connect c-injectivity with existing works, we consider a more general notionof M -injective object. The following definition is found e.g. in [14, Section 9.5]. Definition 26.
Let C be a category and M be a class of arrows in C . An object X ∈ C is an M -injective object if the functor C ( − , X ) : C op → Set sends everyarrow in M to a surjective map.The definition of c-injective objects is a special case of the definition abovewhere M is the class of all Cartesian arrows.The following is a folklore result. The dual is found e.g. in [12, Proposition10.2]. Proposition 27.
Let C , D be categories, M C , M D be classes of arrows, and L ⊣ R : C → D be a pair of adjoint functors. Assume that L sends any arrow in M D to one in M C . For any M C -injective C ∈ C , RC ∈ D is M D -injective. Proof.
It suffices to show that D ( − , RC ) : D op → Set sends each arrow in M D toa surjective map. By the assumption, the functor above factorizes to L : D → C and C ( − , C ) : D op → Set . The former sends each arrow in M D to one in M C and the latter sends one in M C to a surjective map. Thus, the composition ofthese sends each arrow in M D to a surjective map.For epireflective subcategories, we have a sharper result: Proposition 28.
In the setting of Proposition 27, assume, in addition, – R is fully faithful, – R sends each arrow in M C to one in M D , and – each component of the unit η : Id → RL is an epimorphism in M D .Then, D ∈ D is M D -injective if and only if it is isomorphic to RC for some M C -injective C ∈ C . Proof.
The “if” part is Proposition 27. We show the “only if” part.Let D ∈ D be any M D -injective object. Since η D : D → RLD is in M D , wecan use the M D -injectiveness of D to obtain f : RLD → D such that f ◦ η D =id D . Here, η D ◦ f ◦ η D = η D and, by epi-ness of η D , η D ◦ f = id RLD . Thus, η D is an isomorphism.Now we show that LD is M C -injective. Let f : C → LD and g : C → C ′ beany arrow in C and assume that g is in M C . Send these by R to D and consider Rf and Rg . By the assumption, Rg is in M D . Since RLD is isomorphic to D ,it is also M D -injective. Using these, we can obtain h ′ : RC ′ → RLD such that h ′ ◦ Rg = Rf . Since R is full, there is h : C ′ → LD such that Rh = h ′ . Thefaithfulness of R implies h ◦ g = f . Thus LD is M C -injective.Using this result, we can identify c-injective objects in a few situations. njective Objects and Fibered Codensity Liftings 13 Example 29 (continuous lattices in
Top → Set [31]).
In the setting of Proposition 28,consider the case where D = Top , C = Top . Here Top is the full subcategoryof Top of T spaces. Let R be the inclusion. It has a left adjoint L , taking eachspace to its Kolmogorov quotient. Let M C be the class of topological embeddings(i.e. homeomorphisms to their images) and M D be the class of Cartesian arrows(w.r.t. the fibration Top → Set ). Then the assumptions in Proposition 28 aresatisfied and we can conclude that c-injective objects in
Top are precisely injec-tive objects in
Top w.r.t. embeddings.The latter has been identified by Scott [31]. According to his result, suchobjects are precisely continuous lattices with the Scott topology. Thus, we cansee that c-injective objects in Top are precisely such spaces.
Example 30 (complete lattices in
Pre → Set [4]).
In the setting of Proposition 28,consider the case where D = Pre , C = Pos . Here
Pos is the full subcategoryof
Pre of posets. Let R be the inclusion. It has a left adjoint L , taking eachpreordered set to its poset reflection. Let M C be the class of embeddings and M D be the class of Cartesian arrows (w.r.t. the fibration Pre → Set ). Then theassumptions in Proposition 28 are satisfied and we can conclude that c-injectiveobjects in
Pre are precisely injective objects in
Pos w.r.t. embeddings.The latter has been identified by Banaschewski and Bruns [4] . According totheir result, such objects are precisely complete lattices. Thus, we can see thatc-injective objects in
Pre are precisely complete lattices.
To develop the theory of c-injective objects further, we establish some preserva-tion results for c-injectivity. Based on the two propositions of the last section,we show two propositions specific to fibrations and c-injective objects.From Proposition 27, we can derive the following:
Proposition 31.
Let p : E → C , q : F → D be CLat ⊓ -fibrations and L ⊣ R : E → F be a pair of adjoint functors. If L is fibered (from q to p ), then RE ∈ F is c-injective (in q ) for each c-injective E ∈ E . Proof.
Let M E be the class of all arrows Cartesian w.r.t. p and M F be the classof all arrows Cartesian w.r.t. q . Then, use Proposition 27 to the pair L ⊣ R ofadjoint functors.From Proposition 28, we can derive the following: Proposition 32.
In the setting of Proposition 31, assume in addition that both L and R are fibered and that η : Id → RL is componentwise epi. Then, F ∈ F isc-injective if and only if it is isomorphic to RE for some c-injective E ∈ E . Proof.
Use Proposition 28 in the same setting as the proof of Proposition 31.
We list several examples of Theorem 20. Indeed, most of the examples listedin [18, Table VI] turn out to be fibered by Theorem 20. Since the conditions inTheorem 20 only refer to p : E → C and Ω , we sort the examples by these data.We here recall some basic functors considered: Definition 33.
Let P : Set → Set be the covariant powerset functor and D ≤ : Set → Set be the subdistribution functor. Here, a subdistribution p ∈ D ≤ X is a mea-sure on the σ -algebra of all subsets of X with total mass ≤
1. We abbreviate p ( { x } ) to p ( x ). In Example 18 we have seen that, in the fibration
PMet ⊤ → Set , the object([0 , ⊤ ] , d R ) is c-injective. We gather examples of this case here. As mentionedin Example 22 and Example 25, this class of examples has been already studiedand shown to be fibered in [3,20]. Example 34 (Hausdorff pseudometric).
Let inf : P [0 , ⊤ ] → [0 , ⊤ ] be the map tak-ing any set to its infimum. Then, the codensity lifting P ([0 , ⊤ ] ,d R ) , inf : PMet ⊤ → PMet ⊤ turns out to induce the Hausdorff distance : for any (
X, d X ) ∈ PMet ⊤ ,if we let ( P X, d P X ) = P ([0 , ⊤ ] ,d R ) , inf ( X, d X ), then d P X ( S, T ) = max (cid:18) sup x ∈ S inf y ∈ T d X ( x, y ) , sup y ∈ T inf x ∈ S d X ( x, y ) (cid:19) holds for any S, T ∈ P X . By Theorem 20, this functor is fibered. Example 35 (Kantorovich pseudometric).
Let e : D ≤ [0 , ⊤ ] → [0 , ⊤ ] be the maptaking any distribution to its expected value. Then, the codensity lifting D ≤ , ⊤ ] ,d R ) ,e : PMet ⊤ → PMet ⊤ turns out to induce the Kantorovich distance : for any (
X, d X ) ∈ PMet ⊤ , if welet ( D ≤ X, d D ≤ X ) = D ≤ , ⊤ ] ,d R ) ,e ( X, d X ), then d D ≤ X ( p, q ) = sup f : ( X,d X ) → ([0 , ⊤ ] ,d R ) nonexpansive (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ X f ( x ) p ( x ) − X x ∈ X f ( x ) q ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) holds for any p, q ∈ D ≤ X . By Theorem 20, this functor is fibered. In Example 30, we have identified complete lattices as c-injective objects in thefibration
Pre → Set . In particular, the two-point set (2 , ≤ ) is a c-injective object(Example 17).Katsumata and Sato [16, Section 3.1] used codensity lifting to recover the lower , upper , and convex preorders on powersets. Here we see that our resultapplies to them: all of the following liftings are fibered. njective Objects and Fibered Codensity Liftings 15 Example 36 (lower preorder).
Define ♦ : P → ♦ S = ⊤ if and only if ⊤ ∈ S . Then, the codensity lifting P (2 , ≤ ) , ♦ : Pre → Pre turns out to induce the lower preorder : if we let ( P X, ≤ ♦ P X ) = P (2 , ≤ ) , ♦ ( X, ≤ X ), then, for any S, T ∈P X , S ≤ ♦ P X T ⇔ ∀ x ∈ S, ∃ y ∈ T, x ≤ X y. Example 37 (upper preorder).
Define (cid:3) : P → (cid:3) S = ⊤ if and only if ⊥ / ∈ S . Then, the codensity lifting P (2 , ≤ ) , (cid:3) : Pre → Pre turns out to induce the upper preorder : if we let ( P X, ≤ (cid:3) P X ) = P (2 , ≤ ) , (cid:3) ( X, ≤ X ), then, for any S, T ∈P X , S ≤ (cid:3) P X T ⇔ ∀ y ∈ T, ∃ x ∈ S, x ≤ X y. Example 38 (convex preorder).
Denote the family of the two lifting parametersabove by ((2 , ≤ ) , { ♦ , (cid:3) } ). Then, the codensity lifting (with multiple parame-ters, Definition 23) P (2 , ≤ ) , { ♦ , (cid:3) } : Pre → Pre is simply the meet of P (2 , ≤ ) , ♦ and P (2 , ≤ ) , (cid:3) . This is what is called the convex preorder . Remark 39.
The original formulation [16, Section 3.1] is based on codensity lift-ing of monads, so apparently different to ours. In our terms, they used themultiplication µ : PP → P P
1. Using twodifferent bijections between P In Example 16 we have seen that, in the fibration
EqRel → Set , the object(2 , =) is c-injective. We gather examples of this case here. All of the followingliftings are fibered. Details on the following examples can be found in [18]. Example 40 (lifting for bisimilarity on Kripke frames).
Consider the codensitylifting P (2 , =) , ♦ : EqRel → EqRel , where ♦ is as defined in Example 36. Thisturns out to satisfy the following: if we let ( P X, ∼ P X ) = P (2 , =) , ♦ ( X, ∼ X ), then S ∼ P X T ⇔ ( ∀ x ∈ S, ∃ y ∈ T, x ∼ X y ) ∧ ( ∀ y ∈ T, ∃ x ∈ S, x ∼ X y )holds for any S, T ∈ P X . This can be used to define (the conventional notionof) bisimilarity on Kripke frames ( P -coalgebras). Example 41 (lifting for bisimilarity on Markov chains).
For each r ∈ [0 , r : D ≤ → r ( p ) = ⊤ if and only if p ( ⊤ ) ≥ r . These definea [0 , , =) , thr r ) r ∈ [0 , . The codensitylifting D ≤ , =) , thr defined by this family can be used to define probabilisticbisimilarity on Markov chains ( D ≤ -coalgebras). In Example 29, we have identified c-injective objects in the fibration
Top → Set .In particular, the
Sierpinski space , defined as follows, is a c-injective object:
Definition 42 (Sierpinski space).
The
Sierpinski space is a topological space(2 , O O ) where 2 = {⊥ , ⊤} and the family O O of open sets is {∅ , {⊤} , } . Wedenote this space by O .The following liftings of P have appeared in [16, Section 3.2]. All of them arefibered: in other words, they send embeddings to embeddings. Example 43 (lower Vietoris lifting).
Consider the codensity lifting P O , ♦ : Top → Top , where ♦ is as defined in Example 36. For each ( X, O X ) ∈ Top , if we let( P X, O ♦ P X ) = P O , ♦ ( X, O X ), then the topology O ♦ P X is the coarsest one suchthat, for each U ∈ O X , the set { V ⊆ X | V ∩ U = ∅} is open. This is called lower Vietoris lifting in [16]. Example 44 (upper Vietoris lifting).
Consider the codensity lifting P O , (cid:3) : Top → Top , where (cid:3) is as defined in Example 37. For each ( X, O X ) ∈ Top , if we let( P X, O (cid:3) P X ) = P O , (cid:3) ( X, O X ), then the topology O (cid:3) P X is the coarsest one suchthat, for each U ∈ O X , the set { V ⊆ X | V ⊆ U } is open. This is called upperVietoris lifting in [16]. Example 45 (Vietoris lifting).
Define the codensity lifting P O , { ♦ , (cid:3) } : Top → Top like one in Example 38. We call this
Vietoris lifting .This turns out to be connected to
Vietoris topology [21] as follows. For each( X, O X ) ∈ Top , let ( P X, O ♦ , (cid:3) P X ) = P O , { ♦ , (cid:3) } ( X, O X ). The set K ( X, O X ) ofclosed subsets of ( X, O X ) is a subset of P X . Here, the topology on K ( X, O X )induced from O ♦ , (cid:3) P X is the same as the Vietoris topology.This coincidence and the fiberedness of P O , { ♦ , (cid:3) } implies that the Vietorisfunctor V : Stone → Stone , defined in [21], sends embeddings to embeddings.In [18], we considered another lifting:
Example 46 (lifting for bisimulation topology).
Fix any set Σ . Let A Σ : Set → Set be the functor defined by A Σ X = 2 × X Σ . Define acc : A Σ → t, ρ ) = t . For each a ∈ Σ , define h a i : A Σ → h a i ( t, ρ ) = ρ ( a ). Here,( O , acc) and ( O , h a i ) for each a ∈ Σ consist of a family of lifting parameters. Thecodensity lifting (with multiple parameters, Definition 23) A Σ O , { acc }∪{h a i| a ∈ Σ } : Top → Top was used to define bisimulation topology for deterministic automata ( A Σ -coalgebras). This is fibered. This fact is used in Example 51, where we will lookat bisimulation topology again. Now we present an application of our main result. Based on codensity lifting,we defined codensity bisimilarity in [18]. It subsumes bisimilarity, simulation njective Objects and Fibered Codensity Liftings 17 preorder, and behavioral metric as special cases. Here we see that, in the casesto which our fiberedness result applies, codensity bisimilarity interacts well withcoalgebra morphisms. In particular, the codensity bisimilarity on any coalgebrais determined by that on the final coalgebra, if it exists.Recall the definition of coalgebra:
Definition 47 (coalgebra of an endofunctor).
Let F : C → C be an endo-functor on a category C . An F -coalgebra is a pair of an object X ∈ C and anarrow c : X → F X .Let c : X → F X and d : Y → F Y be F -coalgebras. A morphism of coalgebras from ( X, c ) to (
Y, d ) is an arrow f : X → Y in C such that d ◦ f = F f ◦ c holds.As sketched in Section 1, functor lifting can be used to define a “bisimilarity-like notion”. If we use codensity lifting in this construction, we obtain the fol-lowing definition: Definition 48 (codensity bisimilarity [18, Definitions III.6 and III.8]).
Assume the setting of Definition 23. Let c : X → F X be any F -coalgebra. Define Φ Ω ,τc : E X → E X by Φ Ω ,τc P = c ∗ (cid:0) F Ω ,τ P (cid:1) .The ( ( Ω , τ ) -)codensity bisimilarity is the greatest fixed point (w.r.t. ⊑ ) of Φ Ω ,τc . We denote this by νΦ Ω ,τc .Note that the greatest fixed point of Φ Ω ,τc always exists. This can be seen,for example, by the Tarski fixed point theorem. Another option is to use theconstructive fixed point theorem by Cousot and Cousot [6]. We use their char-acterization of the greatest fixed point to prove the following proposition: Proposition 49 (stability of codensity bisimilarity).
Assume the settingof Definition 23 (codensity lifting with multiple parameters). Assume also thateach Ω a is a c-injective object. Then, codensity bisimilarity is stable under coal-gebra morphisms: for any morphism of coalgebras f from ( X, c ) to (
Y, d ), wehave νΦ Ω ,τc = f ∗ (cid:16) νΦ Ω ,τd (cid:17) . Proof.
Define a transfinite sequence (cid:0) ν α Φ Ω ,τc (cid:1) α is an ordinal of elements of E X bythe following: ν α Φ Ω ,τc = l β<α Φ Ω ,τc (cid:0) ν β Φ Ω ,τc (cid:1) . Define another transfinite sequence (cid:16) ν α Φ Ω ,τd (cid:17) α is an ordinal by a similar manner.By the result in [6], there is an ordinal γ such that ν γ Φ Ω ,τc = νΦ Ω ,τc and ν γ Φ Ω ,τd = νΦ Ω ,τd . Thus, it suffices to show the following claim:
Claim.
For any ordinal α , we have ν α Φ Ω ,τc = f ∗ (cid:16) ν α Φ Ω ,τd (cid:17) . This formulation differs slightly from the conventional one where successor and limitordinals are distinguished, but the result also holds under this definition.8 Y. Komorida
We show this by transfinite induction on α . Assume the claim holds for all β < α .Using the assumption that f is a morphism of coalgebras, the fiberedness of F Ω ,τ (Corollary 24), and the functoriality of pullback (Proposition 5), we have f ∗ ◦ Φ Ω ,τd = Φ Ω ,τc ◦ f ∗ . It implies the claim for αf ∗ (cid:16) ν α Φ Ω ,τd (cid:17) = f ∗ l β<α Φ Ω ,τd (cid:16) ν β Φ Ω ,τd (cid:17) = l β<α f ∗ (cid:16) Φ Ω ,τd (cid:16) ν β Φ Ω ,τd (cid:17)(cid:17) = l β<α Φ Ω ,τc (cid:16) f ∗ ν β Φ Ω ,τd (cid:17) = l β<α Φ Ω ,τc (cid:0) ν β Φ Ω ,τc (cid:1) = ν α Φ Ω ,τc . In particular, the codensity bisimilarity is determined by that on the finalcoalgebra:
Corollary 50.
Assume the setting of Proposition 49. Assume also that thereexists a final F -coalgebra z : Z → F Z . Then, for any F -coalgebra c : X → F X ,the unique coalgebra morphism ! X : X → Z satisfies νΦ Ω ,τc = (! X ) ∗ (cid:0) νΦ Ω ,τz (cid:1) . Example 51 (bisimulation topology for deterministic automata).
Recall Example 46.For any A Σ -coalgebra c : X → A Σ X , we defined the codensity bisimilarity on X by νΦ O , { acc }∪{h a i| a ∈ Σ } c ∈ Top X [18].The functor A Σ has a final coalgebra: the set 2 Σ ∗ of all languages on thealphabet Σ can be given an A Σ -coalgebra structure and it is final. For an A Σ -coalgebra c : X → A Σ X , the unique coalgebra morphism l : X → Σ ∗ assigns toeach state the recognized language when started from it.Corollary 50 implies that this map l determines the bisimulation topology on X . We believe that this fact is new, and it supports our use of the term languagetopology in [18, § VIII-C].
Inspired by the proof of fiberedness of Kantorovich lifting [3], we showed a suffi-cient condition for codensity lifting to be fibered. We listed a number of examplesthat satisfy the sufficient condition. In addition, we apply the fiberedness to showa result on codensity bisimilarity.One possible direction of research is to investigate the notion of c-injectivenessin more depth. The existing work on injective objects in homological algebra andtopos theory can be a clue for that. In particular, we have not studied which cat-egory has enough c-injectives . This may be connected with some deep fibrationalproperty.Another possible direction is to generalize the main result. In [16], codensitylifting of a monad was introduced for a general fibration in terms of right Kan njective Objects and Fibered Codensity Liftings 19 extension. This definition can readily be adapted to endofunctors, but in thecurrent paper, we considered only
CLat ⊓ -fibrations. Extending the main resultto this general situation, in particular, to non-poset fibrations, may broaden thescope of application. It can also be fruitful to extend the definition of codensitylifting itself: for example, in Definition 13, we could substitute τ ∗ Ω with otherobjects above F Ω . Seeking consequences and examples of this version of thedefinition is future work. Another related research direction is to obtain a similarsufficient condition for fiberedness of categorical ⊤⊤ -lifting [15].Last but not least, we have to seek other applications. As mentioned in Sec-tion 1, functor lifting is used in many situations. Using codensity lifting thereand seeing what can be implied by the current result seems to be a promisingresearch direction. In particular, codensity lifting seems to be intimately con-nected to coalgebraic modal logic , where τ : F Ω → Ω is regarded as a modality .Recently, Kupke and Rot [22] have identified a sufficient condition for a logicto expressive w.r.t. a coinductive predicate (like bisimilarity, behavioral metricetc.). They used fiberedness of lifting in a crucial way (they use the term fi-bration map ), which suggests that the current work can play a pivotal role ininvestigating modal logics. Acknowledgments
The author is grateful to Ichiro Hasuo and Shin-ya Kat-sumata for fruitful discussions on technical and structural points. The author isalso indebted to anonymous reviewers for clarifying things and pointing out pos-sible future directions, including a topos-theoretic viewpoint and an alternativedefinition of codensity lifting.
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