OON GENERALIZED EQUILOGICAL SPACES
WILLIAN RIBEIRO
Abstract.
In this paper we carry the construction of equilogical spaces into an arbitrary category X topological over Set , introducing the category X - Equ of equilogical objects. Similar to what isdone for the category
Top of topological spaces and continuous functions, we study some featuresof the new category as (co)completeness and regular (co-)well-poweredness, as well as the fact that,under some conditions, it is a quasitopos. We achieve these various properties of the category X - Equ by representing it as a category of partial equilogical objects, as a reflective subcategory of theexact completion X ex , and as the regular completion X reg . We finish with examples in the particularcases, amongst others, of ordered, metric, and approach spaces, which can all be described usingthe ( T , V )- Cat setting.
Introduction
As a solution to remedy the problem of non-existence of general exponentials in
Top , Scottpresents first in [Sco96], and later with his co-authors Bauer and Birkedal in [BBS04], the category
Equ of equilogical spaces. Formed by equipping topological T -spaces with arbitrary equivalencerelations, Equ contains
Top ( T -spaces and continuous functions) as a full subcategory and it iscartesian closed. This fact is directly proven by showing an equivalence with the category PEqu ofpartial equilogical spaces, which is formed by equipping algebraic lattices with partial (not neces-sarily reflexive) equivalence relations. Also in [BBS04], equilogical spaces are presented as modestsets of assemblies over algebraic lattices, offering a model for dependent type theory.Contributing to the study of
Equ , a more general categorical framework, explaining why such(sub)categories are (locally) cartesian closed, was presented in [BCRS98, CR00, Ros99]. It turnedout that
Equ is related to the free exact completion (
Top ) ex of Top [Car95, CM82, CV98]. By thesame token, suppressing the T -separation condition on the topological spaces, the category Equ is a full reflective subcategory of the exact completion
Top ex of Top . More, the reflector preservesproducts and special pullbacks, from where it is concluded in [BCRS98] that
Equ is locally cartesianclosed, since
Top ex is so [BCRS98, Theorem 4.1]. It is shown in [Ros98] that Equ can be presented asthe free regular completion of
Top [Car95, CV98], and [Men00] provides conditions for such regularcompletions to be quasitoposes.
Mathematics Subject Classification.
Key words and phrases. equilogical space, topological category, exact completion, regular completion, quasitopos,( T , V )-category, modest set.Research supported by Centro de Matemática da Universidade de Coimbra – UID/MAT/00324/2013 and by theFCT PhD Grant PD/BD/128059/2016, funded by the Portuguese Government through FCT/MCTES and co-fundedby the European Regional Development Fund through the Partnership Agreement PT2020. a r X i v : . [ m a t h . C T ] N ov WILLIAN RIBEIRO
In this paper we start with a category X and a topological functor | - | : X → Set , and, equippingeach object X of X with an equivalence relation on its underlying set | X | , we define the category X - Equ of equilogical objects and their morphisms. Recovering the results for the particular caseof
Top , X - Equ is (co)complete and regular (co-)well-powered. Under the hypothesis of pre-order-enrichment, we explore the concepts of separated and injective objects of X , leading us to thedefinition of a category X - PEqu of partial equilogical objects. In the presence of a separationcondition, we proceed presenting X - Equ as modest sets of assemblies over injective objects; fromthat, we verify its properties of cartesian closedness and regularity. This is the subject of our firstsection.In Section 2, analogously to the case of
Top , we get similar results when considering the exactcompletion X ex and the regular completion X reg of X , culminating in the fact that X - Equ is aquasitopos, by the results of [Men00]. To do so, we use a general approach to study weak cartesianclosedness of topological categories (see [CHR18]).We finish with Section 3, where we briefly recall the ( T , V )- Cat setting, which was introducedin [CT03] and further investigated in other papers [CH03, Hof07], and study the case when X =( T , V )- Cat , for a suitable monad T and quantale V , satisfying all conditions needed throughoutthe paper. Examples of such categories are Ord of preordered sets,
Met of Lawvere generalizedmetric spaces [Law02] and
App of Lowen approach spaces [Low97], amongst others. Based on fullembeddings among those categories, we place full embeddings among their categories of equilogicalobjects. 1.
The category of (partial) equilogical objects
Let X be a category and | - | : X → Set be a topological functor. In particular X is complete,cocomplete, and | - | preserves both limits and colimits. Definition 1.1.
The category X - Equ is defined as follows. • The objects are structures X = D X, ≡ | X | E , where X ∈ X and ≡ | X | is an equivalence relation onthe set | X | ; they are called equilogical objects of X . • A morphism from X = D X, ≡ | X | E to Y = D Y, ≡ | Y | E is the equivalence class of a morphism f : X → Y in X such that | f | is an equivariant map, i.e. x ≡ | X | x implies | f | ( x ) ≡ | Y | | f | ( x ), for all x, x ∈ | X | , with the equivalence relation on morphisms defined by f ≡ X→Y g ⇐⇒ ∀ x, x ∈ | X | , ( x ≡ | X | x = ⇒ | f | ( x ) ≡ | Y | | g | ( x )) . One can see that ≡ X→Y is indeed an equivalence relation; reflexivity follows from the fact thatthe underlying maps are equivariant, symmetry and transitivity follow from the same properties for ≡ | X | and ≡ | Y | .Identity of X is given by [1 X ] and composition of classes [ f ] : X → Y and [ g ] : Y → Z is given by[ g ] · [ f ] = [ g · f ], which is well-defined. N GENERALIZED EQUILOGICAL SPACES 3
Theorem 1.1. X - Equ is complete, cocomplete, regular well-powered and regular co-well-powered.
The proof of Theorem 1.1 goes along the general lines of the proof of [BBS04, Theorem 3.10].Limits and colimits are computed in X and their underlying sets are endowed with appropriateequivalence relations. The properties of regular well- and regular co-well-poweredness follow fromthe description of equalizers and coequalizers in X - Equ .In general X - Equ is neither well-powered nor co-well-powered, as observed in [BBS04] for topo-logical spaces. A morphism [ m ] : X → Y is a monomorphism in X - Equ if, and only if, x ≡ | X | x ⇐⇒ | m | ( x ) ≡ | Y | | m | ( x ) , ∀ x, x ∈ | X | . A morphism [ f ] : X → Y is an epimorphism in X - Equ if, and only if, y ≡ | Y | y ⇐⇒ ∃ x, x ∈ | X | ; x ≡ | X | x & y ≡ | Y | | f | ( x ) ≡ | Y | | f | ( x ) ≡ | Y | y . Having the embedding and the extension theorems configured for powersets [BBS04, Theorems 3.6,3.7], according to the authors, Scott has pointed out that those results in fact hold more generallyto continuous lattices. Powersets can be generalized to algebraic lattices, and it is explained that “The reason for considering algebraic lattices is that the lattice of continuous functions betweenpowerset spaces is not usually a powerset space, but it is an algebraic lattice. And this extends to allalgebraic lattices.” , culminating in the well known fact that the category
ALat of algebraic latticesand Scott-continuous functions is cartesian closed [GHK+80, Chapter II, Theorem 2.10].Algebraic lattices are in particular continuous lattices, therefore injective objects in
Top . Weshow below that these – injectivity and separation – are the crucial properties in order to extendthe arguments of [BBS04]. Next we assume that (a) X is a pre-order enriched category. Definition 1.2.
An object X of X is said to be separated if, for each morphisms f, g : Y → X in X ,whenever f ’ g ( f ≤ g and g ≤ f ), then f = g .Hence an object X is separated if, for each object Y , the pre-ordered set of morphisms X ( Y, X )is anti-symmetric. One can check that this is equivalent to the pre-ordered set X ( , X ) to beanti-symmetric, where = L {∗} , with L : Set → X the left adjoint of | - | .The full subcategory X sep of separated objects is replete and closed under mono-sources. Since(RegEpi , M ) is a factorization system for sources in the topological category X , where M stands forthe class of mono-sources [AHS90, Proposition 21.14], closure under mono-sources then implies that X sep is regular epi-reflective in X [HST14, II-Proposition 5.10.1].We will consider (pseudo-)injective objects of X with respect to | - | -initial morphisms. Hence,denoting by X inj the full subcategory on the injectives, Z ∈ X inj if, and only if, for each | - | -initialmorphism y : X → Y and morphism f : X → Z , there exists a morphism ˆ f : Y → Z such that WILLIAN RIBEIRO ˆ f · y ’ f ; X y (cid:47) (cid:47) f (cid:32) (cid:32) ’ Y ˆ f (cid:127) (cid:127) Z ˆ f is called an extension of f along y ; if Z is separated, then ˆ f · y = f . Moreover, assume that (b) for each X, Y ∈ X , x, x ∈ X ( , X ) and | - | -initial f ∈ X ( X, Y ) , f · x ’ f · x = ⇒ x ’ x . Thus, for X separated, if f is | - | -initial, then f is an embedding (regular monomorphism, whichin our case is equivalent to | - | -initial with | f | an injective map); hence restricting ourselves to theseparated objects, injectivity with respect to | - | -initial morphisms coincides with injectivity withrespect to embeddings. Definition 1.3.
The category X - PEqu of partial equilogical objects of X consists of: • objects are structures X = D X, ≡ | X | E , where X ∈ X inj and ≡ | X | is a partial (not necessarilyreflexive) equivalence relation on the set | X | ; • a morphism from D X, ≡ | X | E to D Y, ≡ | Y | E is the equivalence class of an X -morphism f : X → Y such that | f | is an equivariant map, with the equivalence relation on morphisms as in Definition1.1.In order to verify an equivalence between the categories of equilogical and partial equilogicalobjects, we will restrict ourselves to the separated objects, so we consider now that the objects inthe structures of Definitions 1.1 and 1.3 are all separated, and denote the resulting categories by X - Equ sep and X - PEqu sep , respectively. We also assume that (c) X has enough injectives, meaning that, for each X ∈ X , there exists an | - | -initial morphism y X : X → ˆ X , with ˆ X ∈ X inj , and, if X is separated, so is ˆ X . When X is separated, as we have seen before, y X is an embedding. Theorem 1.2. X - Equ sep and X - PEqu sep are equivalent.Proof.
As in the proof of [BBS04, Theorem 3.12], a functor R : X - PEqu sep → X - Equ sep is definedtaking each separated partial equilogical object X to R X = D RX, ≡ | RX | E , where | RX | = { x ∈| X | | x ≡ | X | x } and ≡ | RX | is the restriction of ≡ | X | . For a morphism [ f ] : X → Y , | f | ( | RX | ) ⊆ | RY | ,so we take the (co)restriction | f | : | RX | → | RY | , lift to an X -morphism f : RX → RY and set R [ f ] = [ f ].To prove that R is faithful one only needs to observe that, for elements x, x ∈ X , if x ≡ | X | x ,then x ≡ | X | x , and consequently x ≡ | X | x and x ≡ | X | x , whence x, x ∈ | RX | ; and to prove that R is full one uses the injectivity of Y , providing an extension ˆ f : X → Y of i RY · f along i RX . N GENERALIZED EQUILOGICAL SPACES 5
Finally, for essential surjectivity let X = D X, ≡ | X | E ∈ X - Equ sep and consider the embedding y X : X → ˆ X , ˆ X ∈ X sep , inj . Endow | ˆ X | with the following partial equivalence relation ϕ ≡ | ˆ X | ψ ⇐⇒ ∃ x, x ∈ | X | ; ϕ = | y X | ( x ) , ψ = | y X | ( x ) & x ≡ | X | x , that is, two elements of | ˆ X | are related if, and only if, they are the images by | y X | of elements thatare related in | X | . The sets | R ˆ X | and | X | are in bijection; using the | - | -initiality of y X and i R ˆ X ,this bijection proves to be an isomorphism in X , and consequently in X - Equ sep , by the definition of ≡ | ˆ X | , so R D ˆ X, ≡ | ˆ X | E ∼ = X . (cid:3) For our next result we must assume that (d) every injective object of X is exponentiable. Binary products and exponentials of injective objects are again injective, so X inj is a cartesian closedsubcategory of X . We also assume that (e) the reflector from X to X sep preserves finite products; whence the exponential of separated objects, when it exists, is again separated [Day72, Sch84]. Theorem 1.3. X - PEqu sep is cartesian closed.Proof.
Let X = D X, ≡ | X | E and Y = D Y, ≡ | Y | E be partial equilogical separated objects. We build theexponential Y X in X sep , inj and endow | Y X | with the partial equivalence relation: α ≡ | Y X | β if, andonly if, for all x, x ∈ X , x ≡ | X | x = ⇒ α ( x ) = | ev | ( α, x ) ≡ | Y | | ev | ( β, x ) = β ( x ) , for each α, β ∈ | Y X | , where ev : Y X × X → Y is the evaluation morphism in X . Then Y X = D Y X , ≡ | Y X | E ∈ X - PEqu sep and | ev | : | Y X | × | X | → | Y | is equivariant, so [ev] : Y X × X → Y isa valid morphism in X - PEqu sep . More, [ev] satisfies the universal property: for each morphism[ f ] : Z × X → Y , Z = D Z, ≡ | Z | E ∈ X - PEqu sep , there exists a unique [ f ] : Z → Y X commuting thediagram below. Y X × X [ev] (cid:47) (cid:47) YZ × X [ f ] (cid:57) (cid:57) [ f ] × X (cid:79) (cid:79) The morphism f : Z → Y X is the transpose of f : Z × X → Y , so thatev · ( f × X ) = f, WILLIAN RIBEIRO and [ f ] is indeed unique, for if [ f ] : Z → Y X is such that [ev · ( f × X )] = [ f ], then for each z ≡ | Z | z in Z and x ≡ | X | x in X , f ( z )( x ) = ev · ( f × X )( z, x ) = f ( z, x ) ≡ | Y | f ( z , x ) ≡ | Y | ev · ( f × X )( z , x ) = f ( z )( x ) , hence f ( z ) ≡ | Y X | f ( z ), i.e. [ f ] = [ f ]. (cid:3) Therefore, by Theorem 1.2, X - Equ sep is cartesian closed. We remark that the proof of Theorem1.3 also applies to X - PEqu without separation.To finish this section we discuss the presentation of equilogical objects as modest sets of assemblies,following what is done in [BBS04, Section 4].
Definition 1.4.
The category of assemblies
Assm ( X inj ) over injective objects of X consists of thefollowing data: objects are triples ( A, X, E A ), where A is a set, X ∈ X inj , and E A : A → P| X | isa function such that E A ( a ) = ∅ , for each a ∈ A , with P| X | the powerset of | X | . The elements of E A ( a ) are called realizers for a . A morphism between assemblies ( A, X, E A ) and ( B, Y, E B ) is a map f : A → B for which there exists a morphism g : X → Y in X such that | g | ( E A ( a )) ⊆ E B ( f ( a )); g issaid to be a realizer for f , and we say that | g | tracks f . Definition 1.5.
An object (
A, X, E A ) ∈ Assm ( X inj ) is called a modest set if, for all a, a ∈ A , a = a implies E A ( a ) ∩ E A ( a ) = ∅ . The full subcategory of the assemblies that are modest sets is denotedby Mdst ( X inj ).With these definitions, we get the same properties as those for the particular case of topologicalspaces, which we highlight in the following items, omitting some of the proofs that follow directlyfrom the ones in [BBS04].(1) Mdst ( X inj ) and Assm ( X inj ) have finite limits and the inclusion from modest sets to assembliespreserves them. (2) Mdst ( X inj ) and Assm ( X inj ) are cartesian closed and Mdst ( X inj ) → Assm ( X inj ) preserves expo-nentials. For (
A, X, E A ) and ( B, Y, E B ) in Assm ( X inj ), the exponential is ( C, Y X , E C ), where C = { f : A → B | ∃ g : X → Y ∈ X realizer for f } , and E C ( f ) = { α ∈ | Y X | | α tracks f } ;here, for simplicity, we denote also by α the map from | X | to | Y | , given by x
7→ | ev | ( α, x ), for each x ∈ | X | , where ev : Y X × X → Y is the evaluation map. If ( B, Y, E B ) is a modest set, then so is( C, Y X , E C ), for if f, f : A → B are tracked by α ∈ | Y X | , then for each a ∈ A , take x ∈ E A ( a ) = ∅ ,then α ( x ) ∈ E B ( f ( a )) ∩ E B ( f ( a )) = ∅ , whence f ( a ) = f ( a ) and then f = f .(3) Mdst ( X inj ) is a reflective subcategory of Assm ( X inj ) . (4) The regular subobjects of ( A, X, E A ) in Assm ( X inj ) , or in Mdst ( X inj ) , are in bijective correspon-dence with the powerset of A . N GENERALIZED EQUILOGICAL SPACES 7 (5)
Mdst ( X inj ) and Assm ( X inj ) are regular categories. Theorem 1.4. X - PEqu and
Mdst ( X inj ) are equivalent.Proof. Define the functor F : Mdst ( X inj ) → X - PEqu assigning to (
A, X, E A ) the object D X, ≡ | X | E ,where x ≡ | X | x ⇐⇒ ∃ a ∈ A ; x, x ∈ E A ( a ) , and on morphisms F assigns to each f : ( A, X, E A ) → ( B, Y, E B ) the class of a realizer g : X → Y for f ; ≡ | X | is indeed an equivalence relation and two realizers for f are in the same equivalenceclass, so F is well-defined.Faithfulness of F follows from the observation in item (2) above: two maps tracked by the samerealizer must be equal. To see that F is full, take a morphism [ g ] : F ( A, X, E A ) → F ( B, Y, E B ) in X - PEqu . For each a ∈ A , let x ∈ E A ( a ) = ∅ ; then x ≡ | X | x and so | g | ( x ) ≡ | Y | | g | ( x ), that is, thereexists b ∈ B such that | g | ( x ) ∈ E B ( b ), whence we set f ( a ) = b ; this b is uniquely determined since( B, Y, E B ) is a modest set, therefore we have a map f : A → B , which, by definition, has g as arealizer.Now let D X, ≡ | X | E be a partial equilogical object and define ( A, X, E A ) by A = { x ∈ | X | | x ≡ | X | x } / ≡ | X | and E A ([ x ]) = [ x ] ⊆ P| X | . Hence F ( A, X, E A ) = D X, ≡ | X | E and F is essentially surjective. (cid:3) The same argument can be repeated replacing X inj with X sep , inj , so we obtain Mdst ( X sep , inj ) ∼ = X - PEqu sep ∼ = X - Equ sep . Properties from items (1) to (5) remain valid, so they also hold for X - Equ sep .2.
Equilogical objects and exact completion
The category
Equ of equilogical spaces can also be obtained as a full reflective subcategory of theexact completion [BCRS98, Car95, CM82]
Top ex of the category of topological spaces [Ros99], andthis is a particular instance of a general process to obtain such categories [BCRS98].We can describe the exact completion X ex of X as: objects are pseudo-equivalence relations on X ,that is, parallel pairs of morphisms X r (cid:47) (cid:47) r (cid:47) (cid:47) X of X satisfying(i) reflexivity : there exists a morphism r : X → X such that r · r = 1 X = r · r ; X r (cid:126) (cid:126) r (cid:32) (cid:32) X X X (cid:111) (cid:111) X (cid:47) (cid:47) r (cid:79) (cid:79) X WILLIAN RIBEIRO (ii) symmetry : there exists a morphism s : X → X such that r · s = r and r · s = r ; X r (cid:126) (cid:126) r (cid:32) (cid:32) X X r (cid:111) (cid:111) r (cid:47) (cid:47) s (cid:79) (cid:79) X (iii) transitivity : for r , r : X → X a pullback of r , r , there exists a morphism t : X → X commuting the following diagram X r (cid:6) (cid:6) r (cid:24) (cid:24) X r (cid:126) (cid:126) r (cid:32) (cid:32) t (cid:79) (cid:79) (cid:74) X r (cid:126) (cid:126) r (cid:32) (cid:32) X r (cid:126) (cid:126) r (cid:33) (cid:33) X X X . A morphism from X r (cid:47) (cid:47) r (cid:47) (cid:47) X to Y s (cid:47) (cid:47) s (cid:47) (cid:47) Y is an equivalence classe [ f ] of an X -morphism f : X → Y such that there exists g : X → Y in X satisfying f · r i = s i · g , i = 1 , X r (cid:15) (cid:15) r (cid:15) (cid:15) g (cid:47) (cid:47) Y s (cid:15) (cid:15) s (cid:15) (cid:15) X f (cid:47) (cid:47) Y . Here two morphisms f , f : X → Y are related if, and only if, there exists a morphism h : X → Y such that f i = s i · h , i = 1 , X r (cid:15) (cid:15) r (cid:15) (cid:15) Y s (cid:15) (cid:15) s (cid:15) (cid:15) X f (cid:47) (cid:47) f (cid:47) (cid:47) h (cid:63) (cid:63) Y Since it is topological over
Set , X has a stable factorization system for morphisms given by(Epi , RegMono) [AHS90, Remark 15.2(3), Proposition 21.14] (see also [CHR18]). Let us denote byPER( X , RegMono) the full subcategory of X ex of the pseudo-equivalence relations X r (cid:47) (cid:47) r (cid:47) (cid:47) X such that h r , r i : X → X × X is a regular monomorphism. Lemma 2.1. X - Equ and
PER( X , RegMono) are equivalent.
N GENERALIZED EQUILOGICAL SPACES 9
Proof.
For each equilogical object D X, ≡ | X | E , consider E X = { ( x, x ) ∈ | X | × | X | | x ≡ | X | x } andthe source ( π X i : E X → | X | ) i =1 , of the projections from E X onto | X | . Take its | - | -initial lifting,which by abuse of notation we denote by ( π X i : E X → X ) i =1 , . Hence E X π X (cid:47) (cid:47) π X (cid:47) (cid:47) X belongs toPER( X , RegMono) and each morphism [ f ] : X → Y in X - Equ is a valid morphism[ f ] : ( E X , X, π X , π X ) → ( E Y , Y, π Y , π Y )in PER( X , RegMono).That correspondence defines a functor which is fully faithful and, for a pseudo-equivalence relation X r (cid:47) (cid:47) r (cid:47) (cid:47) X in PER( X , RegMono), define the equilogical object D X , ≡ | X | E by x ≡ | X | x ⇐⇒ ( ∃ (unique) x ∈ X ) | r | ( x ) = x & | r | ( x ) = x , for each x , x ∈ X . Then E X π X (cid:47) (cid:47) π X (cid:47) (cid:47) X is isomorphic to X r (cid:47) (cid:47) r (cid:47) (cid:47) X in PER( X , RegMono)and the functor is essentially surjective. (cid:3)
Hence [BCRS98, Theorem 4.3] states that
Theorem 2.1. X - Equ ∼ = PER( X , RegMono) is a full reflective subcategory of X ex ; the reflectorpreserves finite products and commutes with change of base in the codomain. Next we wish to prove that X - Equ is cartesian closed, so by Theorem 2.1 and [Sch84, Theorem1.2], it suffices to show that X ex is cartesian closed. To do so, we will apply the following resultderived from [Ros99, Theorem 1, Lemma 4] (see also [CHR18, Theorem 1.1]). Theorem 2.2.
Let X be a complete, infinitely extensive and well-powered category with factoriza-tions (RegEpi , Mono) such that f × is an epimorphism whenever f is a regular epimorphism. Then X ex is cartesian closed provided X is weakly cartesian closed. Since X is topological over Set , in order to use the latter theorem, we will assume that (f) X is infinitely extensive; more, assuming also conditions (a) to (e) from the previous section, following the same steps ofthe proofs of [CHR18, Theorems 5.3 and 5.5], we deduce the following result. Proposition 2.1. X is weakly cartesian closed. Furthermore, we can verify that X ex is actually locally cartesian closed. Consider the restrictionfunctor | - | : X inj → Pfn , where
Pfn is the category of sets and partial functions. The category F ( X inj , | - | ), or simply F ( X inj ), is described in [CR00] as follows: objects are triples ( X, A, σ : A → | X | ), where X is an injective object of X , A is a set and σ is a function; a morphism f : ( X, A, σ : A → | X | ) → ( Y, B, δ : B → | Y | ) is a map f : A → B such that there exists g : X → Y in X commuting thediagram A σ (cid:15) (cid:15) f (cid:47) (cid:47) B δ (cid:15) (cid:15) | X | | g | (cid:47) (cid:47) | Y | . Proposition 2.2.
The categories X and F ( X inj ) are equivalent.Proof. Define the functor G : X → F ( X inj ) by GX = ( ˆ X, | X | , σ X = | y X | : | X | → | ˆ X | ) , where y X is the | - | -initial morphism assured by condition (c) in the previous section; for eachmorphism f : X → Y , injectivity of ˆ Y implies the existence of a morphism g : ˆ X → ˆ Y extending y Y · f along y X X y X (cid:15) (cid:15) f (cid:47) (cid:47) Y y Y (cid:15) (cid:15) ˆ X g (cid:47) (cid:47) ˆ Y , hence we set Gf = | f | . G is faithful and to see it is full, let f : | X | → | Y | be a map commuting thediagram | X | | y X | (cid:15) (cid:15) f (cid:47) (cid:47) | Y | | y Y | (cid:15) (cid:15) | ˆ X | | g | (cid:47) (cid:47) | ˆ Y | , for some g : ˆ X → ˆ Y in X , then | - | -initiality of y Y implies the existence of a unique f : X → Y suchthat Gf = | f | = f .For essential surjectivity, let ( X, A, σ : A → | X | ) in F ( X inj ) and take the | - | -initial lifting of σ , thatwe denote by σ ini : A ini → X , so | A ini | = A and | σ ini | = σ . Hence GA ini = ( ˆ A ini , A, | y A ini | : A → | ˆ A ini | )and we verify that the identity map 1 A : A → A is a morphism from ( X, A, σ ) to ( ˆ A ini , A, | y A ini | ),and vice-versa. The latter fact comes readly from injectivity of ˆ A ini and | - | -initiality of σ ini : A σ (cid:15) (cid:15) A (cid:47) (cid:47) A | y A ini | (cid:15) (cid:15) | X | | g | (cid:47) (cid:47) | ˆ A ini | , N GENERALIZED EQUILOGICAL SPACES 11 for some morphism g : X → ˆ A ini , and by injectivity of X and | - | -initiality of y A ini : A | y A ini | (cid:15) (cid:15) A (cid:47) (cid:47) A σ (cid:15) (cid:15) | ˆ A ini | | g | (cid:47) (cid:47) | X | , for some morphism g : ˆ A ini → X . Therefore, GA ini ∼ = ( X, A, σ : A → | X | ) in F ( X inj ). (cid:3) Since X inj is (weakly) cartesian closed, as shown in [CR00], X ∼ = F ( X inj ) has all weak simpleproducts (in particular it is weakly cartesian closed), and more, X ∼ = F ( X inj ) is weakly locallycartesian closed, i.e. it has weak dependent products, whence by [CR00, Theorem 3.3], X ex ∼ = F ( X inj ) ex is locally cartesian closed.Therefore, by Theorem 2.1, we conclude that X - Equ is locally cartesian closed (see for instance[HST14, III-Corollary 4.6.2]), and, being complete and cocomplete, one may ask whether this cate-gory is actually a quasitopos.As discussed in [Ros98], “... the full subcategory of C ex consisting of those equivalence spanswhich are kernel pairs in C gives the free regular completion C reg of C .” , where in that context equivalence span means pseudo-equivalence relation. Hence, similar to what is observed in [Men00]for topological spaces, the category X - Equ , presented by PER( X , RegMono), is equivalent to theregular completion X reg of X .It is easy to depict the latter equivalence using the classical description of X reg [Car95]: objectsare X -morphisms f : X → Y , and a morphism from f : X → Y to g : Z → W is an equivalence class[ l ] of an X -morphism l : X → Z such that g · l · f = g · l · f , where f , f form the kernel pair of f .Ker( f ) f (cid:15) (cid:15) f (cid:47) (cid:47) (cid:74) X f (cid:15) (cid:15) X f (cid:47) (cid:47) Y Two such arrows l and m are equivalent if g · l = g · m . X f (cid:15) (cid:15) Z g (cid:15) (cid:15) [ l ] (cid:47) (cid:47) Y W
Lemma 2.2. X reg and PER( X , RegMono) are equivalent.
Proof.
Define F : X reg → PER( X , RegMono) as in the diagram below,( f : X → Y ) (cid:31) (cid:47) (cid:47) [ l ] (cid:15) (cid:15) (Ker( f ) , X, f , f ) [ l ] (cid:15) (cid:15) ( g : Z → W ) (cid:31) (cid:47) (cid:47) (Ker( g ) , Z, g , g )so it is a well-defined functor, since l : X → Z satisfies g · l · f = g · l · f if, and only if, there existsa unique l : Ker( f ) → Ker( g ) such that g · l = l · f and g · l = l · f .Ker( f ) l · f (cid:40) (cid:40) l (cid:36) (cid:36) l · f (cid:31) (cid:31) Ker( g ) g (cid:15) (cid:15) g (cid:47) (cid:47) (cid:74) Z g (cid:15) (cid:15) Z g (cid:47) (cid:47) W Then F is fully faithful, and it is essentially surjective because each pseudo-equivalence relation X r (cid:47) (cid:47) r (cid:47) (cid:47) X with h r , r i : X → X × X a regular monomorphism is seen to form the kernelpair of the | - | -final lifting p : X → ˜ X of the projection map p : | X | → | X | / ∼ , where ∼ is theequivalence relation on | X | defined in the proof of Lemma 2.1. (cid:3) We now intend to apply [Men00, Corollary 8.4.2]; by condition (f) and Proposition 2.2, X is an(infinitely) extensive weakly locally cartesian closed category, hence we are only missing the chaoticsituation described right after [Men00, Lemma 7.3.3]. This comes from the observation that thetopos Set is a mono-localization of X , since the topological functor | - | : X → Set is faithful, preservesfinite limits and has a full embedding as a right adjoint [AHS90, Proposition 21.12]. Therefore, byLemma 2.1, Lemma 2.2 and [Men00, Corollary 8.4.2], we conclude:
Theorem 2.3. X - Equ is a quasitopos. The case X =( T , V ) - Cat
We briefly introduce the ( T , V )- Cat setting, and refer the reader to the reference [CT03] for details(see also [HST14]).Although introduced in a more general setting, we are interested here in the case when • V = ( V , ⊗ , k ) is a commutative unital quantale (see for instance [HST14, II-Section 1.10]) whichis also a Heyting algebra (so that the operation infimum ∧ also has a right adjoint), and • T = ( T, m, e ) :
Set → Set is a monad satisfying the Beck-Chevalley condition ( T preserves weakpullbacks and the naturality squares of m are weak pullbacks [CHJ14]) that is laxly extended tothe pre-ordered category V - Rel , which has as objects sets and as morphisms V -relations r : X −→7 Y ,i.e. V -valued maps r : X × Y → V . N GENERALIZED EQUILOGICAL SPACES 13
Hence we assume that there exists a functor T : V - Rel → V - Rel extending T , by abuse of notationdenoted by the same letter, that commutes with involution: T ( r ◦ ) = ( T r ) ◦ , for each r : X −→7 Y ∈ V - Rel , where r ◦ ( y, x ) = r ( x, y ), for each ( x, y ) ∈ X × Y .The functor T turns m and e into oplax transformations, meaning that the naturality diagramsbecome: X e X (cid:47) (cid:47) (cid:95) r (cid:15) (cid:15) ≤ T X (cid:95)
T r (cid:15) (cid:15) T X m X (cid:111) (cid:111) (cid:95) T r (cid:15) (cid:15) ≥ Y e Y (cid:47) (cid:47) T Y T Y, m Y (cid:111) (cid:111) for each V -relation r : X −→7 Y .Hence we have a lax monad on V - Rel [CH04] and ( T , V )- Cat is defined as the category of Eilenberg-Moore lax algebras for that lax monad: objects are pairs (
X, a ), where X is a set and a : T X −→7 X is a V -relation, which is reflexive and transitive. X e X (cid:47) (cid:47) X (cid:44) (cid:44) T X (cid:95) a (cid:15) (cid:15) T X (cid:31) T a (cid:111) (cid:111) m X (cid:15) (cid:15) ≤≤ X T X (cid:31) a (cid:111) (cid:111) Such pairs are called ( T , V ) -categories ; a morphism from ( X, a ) to (
Y, b ) is a map f : X → Y commuting the diagram below. T X (cid:95) a (cid:15) (cid:15) T f (cid:47) (cid:47) ≤ T Y (cid:95) b (cid:15) (cid:15) X f (cid:47) (cid:47) Y Such a map is called ( T , V ) -functor .We are also going to restrict ourselves to the case that the extension T to V - Rel is determined bya T -algebra structure map ξ : T V → V , so we are in the setting of topological theories [Hof07] (seealso [CT14]), hence V has a ( T , V )-category structure given by the composite T V ξ (cid:47) (cid:47) V (cid:31) hom (cid:47) (cid:47) V , where hom : V × V → V is the left adjoint of ⊗ , so u ⊗ v ≤ w ⇐⇒ u ≤ hom( v, w ) , for each u, v, w in the quantale V .The forgetful functor | - | : ( T , V )- Cat → Set is topological [CH03, CT03], and before we provideexamples of categories given by ( T , V )- Cat , we verify that, for suitable monad T and quantale V satisfying the conditions assumed so far in this section, ( T , V )- Cat satisfies all conditions (a) to (f) from the two previous sections. In each item, we highlight the properties that are needed in orderto achieve the respective condition, adding the references where that is proved. (a) ( T , V ) - Cat is pre-ordered enriched.
For ( T , V )-categories ( X, a ) and (
Y, b ), consider the followingrelation on the set of ( T , V )-functors from ( X, a ) to (
Y, b ): f ≤ g ⇐⇒ ∀ x ∈ X, k ≤ b ( e Y ( f ( x )) , g ( x )) . This determines a pre-order, first defined in [CT03], which is compatible with composition of ( T , V )-functors. One can also check that a ( T , V )-category ( X, a ) is separated if, and only if, the followingpre-order on X is anti-symmetric: x ≤ x ⇐⇒ k ≤ a ( e X ( x ) , x )(see [HST14, III-Proposition 3.3.1]). (b) | - | -initial ( T , V ) -functors reflect the order. Let (
X, a ) , ( Y, b ) be ( T , V )-categories, x, x : → ( X, a ) be ( T , V )-functors, where = ( {∗} , e ◦ {∗} ) (the discrete structure on the singleton [HST14, III-Section 3.2]), and f : ( X, a ) → ( Y, b ) an | - | -initial ( T , V )-functor such that f · x ’ f · x ; | - | -initialityof f means that a ( x , x ) = b ( T f ( x ) , f ( x )), for each x ∈ T X , x ∈ X . We calculate: k ≤ b ( e Y ( f · x ( ∗ )) , f · x ( ∗ )) (definition of f · x ≤ f · x ) ≤ b ( T f · e X ( x ( ∗ )) , f · x ( ∗ )) (composition is associative, e is natural) ≤ a ( e X ( x ( ∗ )) , x ( ∗ )) ( f is | - | -initial) , so x ≤ x and in the same fashion x ≤ x , thus x ’ x . (c) ( T , V ) - Cat has enough injectives.
The tensor product ⊗ of V induces a functor ⊗ : ( T , V )- Cat × ( T , V )- Cat → ( T , V )- Cat , with (
X, a ) ⊗ ( Y, b ) = ( X × Y, c ) , where, for each w ∈ T ( X × Y ), ( x, y ) ∈ X × Y , c ( w , ( x, y )) = a ( T π X ( w ) , x ) ⊗ b ( T π Y ( w ) , y ) , and π X , π Y are the projections from X × Y onto X and Y , respectively. The following facts canbe found in [CH09, Hof11, CCH15]: for each ( T , V )-category ( X, a ), a : T X −→7 X defines a ( T , V )-functor a : X op ⊗ X → V , where X op = ( T X, m X · ( T a ) ◦ · m X ); the ⊗ -exponential mate y X : X → V X op of a is fully faithful;the ( T , V )-category P X = V X op is injective and if ( X, a ) is separated, so is
P X . N GENERALIZED EQUILOGICAL SPACES 15 (d)
Injectives are exponentiable.
Conditions under which injectivity implies exponentiability in( T , V )- Cat are studied in [CHR18]. We recall them next. Consider the maps V ⊗ V ⊗ (cid:47) (cid:47) V and X ( − ,u ) (cid:47) (cid:47) X ⊗ V , (1)with ( V , hom ξ ) , ( X, a ) ∈ ( T , V )- Cat . Define also for a V -relation r : X −→7 Y and u ∈ V , the V -relation r ⊗ u : X −→7 Y given by ( r ⊗ u )( x, y ) = r ( x, y ) ⊗ u, (2)for each ( x, y ) ∈ X × Y . As a final condition, assume that, for all u, v, w ∈ V , w ∧ ( u ⊗ v ) = { u ⊗ v | u ≤ u, v ≤ v, u ⊗ v ≤ w } , (3)which is equivalent to exponentiability of injective V -categories (see [HR13, Theorem 5.3]). Then[CHR18, Theorem 5.4] says the following: Theorem 3.1.
Suppose that: the maps ⊗ and ( − , u ) in (1) are ( T , V ) -functors; for every injective ( T , V ) -category ( X, a ) and every u ∈ V , T ( a ⊗ u ) = T a ⊗ u , with those V -relations defined as in (2);and (3) holds. Then every injective ( T , V ) -category is exponentiable in ( T , V ) - Cat . (e) The reflector from ( T , V ) - Cat to ( T , V ) - Cat sep preserves finite products.
This is proved in[CHR18, Proposition 5.4]. (f) ( T , V ) - Cat is infinitely extensive.
This is proved in [MST06] under the condition that T is ataut functor [Man02], what comes for free from the assumption that T preserves weak pullbacks.To give examples of categories satisfying all the conditions above, we consider: • the identity monad I = (Id , , ) on Set laxly extended to the identity lax monad on V - Rel ; • the ultrafilter monad U with the Barr extension to V - Rel [HST14, IV-Corollary 2.4.5], with V integral and completely distributive (see, for instance, [HST14, II-Section 1.11]); • the list monad (or free monoid monad) L = ( L, m, e ) (see [HST14, II-Examples 3.1.1(2)]), withthe extension L : V - Rel → V - Rel that sends each r : X −→7 Y to the V -relation Lr : LX −→7 LY givenby Lr (( x , . . . , x n ) , ( y , . . . , y m )) = r ( x , y ) ⊗ · · · ⊗ r ( x n , y n ) , if n = m ⊥ , if n = m ; • the monad M = ( − × M, m, e ), for a monoid ( M, · , M ), with m X : X × M × M → X × M givenby m X ( x, a, b ) = ( x, a · b ) and e X : X → X × M given by e X ( x ) = ( x, M ) (see [HST14, V-Section1.4]). The extension − × M : V - Rel → V - Rel sends the V -relation r : X −→7 Y to the V -relation r × M : X × M −→7 Y × M with r × M (( x, a ) , ( y, b )) = r ( x, y ) , if a = b , ⊥ , if a = b .As well as the quantales: = ( {⊥ , >} , ∧ , > ), P + = ([0 , ∞ ] op , + , P max = ([0 , ∞ ] op , max , = ( {⊥ , u, v, >} , ∧ , > ) (the diamond lattice [HST14, II-Exercise 1.H]) and ∆ (the quantale ofdistribution functions [HR13]). We assemble the table: (cid:64)(cid:64)(cid:64)(cid:64) V T I U L M ( M , )- CatP + Met AppP max
UltMet NA - App2 BiRel BiTop ∆ ProbMet (4) • Ord is the category of pre-ordered spaces, • Met is the category of Lawvere generalized metric spaces [Law02], • UltMet is the full subcategory of
Met of ultra-metric spaces [HST14, III-Exercise 2.B], • BiRel is the one of sets and birelations [HST14, III-Examples 1.1.1(3)], • ProbMet is the category of probabilistic metric spaces [HR13], • Top is the usual category of topological spaces and continuous functions, • App is that of Lowen’s approach spaces [Low97], and • NA - App is the full subcategory of
App of non-Archimedean approach spaces studied in details in[CVO17], and denoted in [Hof14] by
UApp , • BiTop is the category of bitopological spaces and bicontinuous maps [HST14, III-Exercise 2.D], • MultiOrd is the category of multi-ordered sets [HST14, V-Section 1.4], and • ( M , )- Cat can be interpreted as the category of M -labelled ordered sets [HST14, V-Section 1.4].For instance, an object of Ord - Equ is a pre-ordered set ( X, ≤ ) together with an equivalence relation ≡ X on X ; separatedness of ( X, ≤ ) means that ≤ is anti-symmetric, so the objects of Ord - Equ sep are partially ordered sets equipped with equivalence relations on their underlying sets. Further,a partial equilogical separated object in
Ord - PEqu sep is a complete lattice (injective ordered set)together with an equivalence relation on the underlying set. In the same fashion, the objects of thecategory
Mdst ( Ord sep , inj ) are triples ( A, ( X, ≤ ) , E A ), with A a set, E A : A → P X a function, and( X, ≤ ) a complete lattice.Furthermore, from Section 2 we conclude that, together with Top , all the other categories inTable 4 are weakly locally cartesian closed and their exact completions are locally cartesian closed
N GENERALIZED EQUILOGICAL SPACES 17 categories; moreover, their categories of equilogical objects, which are equivalent to their regularcompletions, are quasitoposes that fully embed the original categories.Concerning four of those categories, we also have adjunctions
Top (cid:31) (cid:127) (cid:47) (cid:47) (cid:15) (cid:15)
App a (cid:111) (cid:111) a (cid:15) (cid:15) Ord (cid:31) (cid:63) a (cid:79) (cid:79) (cid:31) (cid:127) a (cid:47) (cid:47) Met , (cid:111) (cid:111) (cid:31) (cid:63) (cid:79) (cid:79) where both solid and dotted diagrams commute, the hook-arrows are full embeddings and thetwo full embeddings Ord , → App coincide (see [HST14, III-Section 3.6]). One can see that thoseadjunctions extend to the respective categories of equilogical objects,
Equ (cid:31) (cid:127) (cid:47) (cid:47) (cid:15) (cid:15)
App - Equ a (cid:111) (cid:111) a (cid:15) (cid:15) Ord - Equ (cid:31) (cid:63) a (cid:79) (cid:79) (cid:31) (cid:127) a (cid:47) (cid:47) Met - Equ (cid:111) (cid:111) (cid:31) (cid:63) (cid:79) (cid:79) and we describe them now.(1)
Ord - Equ to Met - Equ . Each ordered equilogical object h ( X, ≤ ) , ≡ X i is taken to D ( X, d ≤ ) , ≡ X E ,where the metric d ≤ is given by d ≤ ( x, x ) = , if x ≤ x ∞ , otherwise,for each x, x ∈ X . The left adjoint of this inclusion assigns h ( X, ≤ d ) , ≡ X i to h ( X, d ) , ≡ X i , with x ≤ d x if and only if d ( x, x ) < ∞ , for each x, x ∈ X . Hence the category Ord - Equ is fully embeddedin
Met - Equ as the metric equilogical objects h ( X, d ) , ≡ X i for which there exists an order ≤ on X such that d = d ≤ .(2) Ord - Equ to Equ . Each h ( X, ≤ ) , ≡ X i is taken to D ( X, τ ≤ ) , ≡ X E , where τ ≤ is the Alexandrofftopology: open sets are generated by the down-sets ↓ x , x ∈ X . For its right adjoint, to anequilogical space h ( X, τ ) , ≡ X i is assigned h ( X, ≤ τ ) , ≡ X i , where ≤ τ is the specialization order of( X, τ ): for each x, x ∈ X , x ≤ x if and only if ˙ x → x , where ˙ x denotes the principal ultrafilter on x , and → denotes the convergence relation between ultrafilters and points of X determined by τ ;observe that this is the induced order described in item (a) above. Hence the category Ord - Equ isfully embedded in
Equ as the equilogical spaces h ( X, τ ) , ≡ X i for which there exists an order ≤ on X such that τ = τ ≤ , and those are exactly the Alexandroff spaces: arbitrary intersections of opensets are open (see [HST14, II-Example 5.10.5, III-Example 3.4.3(1)]). (3) Met - Equ to App - Equ . A metric equilogical object h ( X, d ) , ≡ X i becomes an approach equilogicalone h ( X, δ d ) , ≡ X i , where the approach distance is given by δ d ( x , A ) = inf { d ( x, x ) | x ∈ A } , foreach x ∈ X , A ∈ P X [HST14, III-Examples 2.4.1(1)]. The right adjoint of this embedding assigns h ( X, d δ ) , ≡ X i to h ( X, δ ) , ≡ X i , where d δ ( x, x ) = sup { δ ( x , A ) | x ∈ A ∈ P X } , for each x, x ∈ X .Hence Met - Equ is identified within
App - Equ as the approach equilogical objects h ( X, δ ) , ≡ X i suchthat δ = δ d , for some metric d on X , that is, ( X, δ ) is a metric approach space [Low97, Chapter 3].(4)
Equ to App - Equ . Each equilogical space h ( X, τ ) , ≡ X i becomes an approach equilogical one h ( X, δ τ ) , ≡ X i with the approach distance given by δ τ ( x , A ) = , if A ∈ x , for some x ∈ U X with x → x ∞ , otherwise , for each x ∈ X , A ∈ P X , where U X denotes the set of ultrafilters on X [HST14, III-Examples2.4.1(2)]. The left adjoint of this embedding is slightly more elaborate: for an approach equilogicalobject h ( X, δ ) , ≡ X i , consider the convergence relation → between ultrafilters in U X and points of X given by x → x ⇐⇒ sup { δ ( x, A ) | A ∈ x } < ∞ ;this convergence defines a pseudo-topological space [Cho48], to which we apply the reflector de-scribed in [HST14, III-Exercise 3.D], obtaining an equilogical space h ( X, τ δ ) , ≡ X i . Hence Equ isidentified within
App - Equ as the approach equilogical objects h ( X, δ ) , ≡ X i such that δ = δ τ , forsome topology τ on X , that is, ( X, δ ) is a topological approach space [Low97, Chapter 2].
Open question.
The conditions (a) to (f) of Sections 1 and 2 were derived from the successfulattempt of generalizing the structures/constructions to ( T , V )- Cat , for suitable T and V . Requiringthose conditions on an arbitrary category with a topological functor over Set produced the samedesired results. However, we do not know an example of a category satisfying those conditionswhich cannot be described as ( T , V )- Cat . Acknowledgments
This work was done during the preparation of the author’s PhD thesis, under the supervision ofMaria Manuel Clementino, whom the author thanks for proposing the investigation and advisingthe whole study. I also thank Fernando Lucatelli Nunes for fruitful discussions.
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