Limits in Categories of Vietoris Coalgebras
LLIMITS IN CATEGORIES OF VIETORIS COALGEBRAS
DIRK HOFMANN, RENATO NEVES, AND PEDRO NORA
Abstract.
Motivated by the need to reason about hybrid systems, we study limits in categories ofcoalgebras whose underlying functor is a Vietoris polynomial one — intuitively, the topological analogueof a Kripke polynomial functor. Among other results, we prove that every Vietoris polynomial functoradmits a final coalgebra if it respects certain conditions concerning separation axioms and compactness.When the functor is restricted to some of the categories induced by these conditions the resultingcategories of coalgebras are even complete.As a practical application, we use these developments in the specification and analysis of non-deterministic hybrid systems, in particular to obtain suitable notions of stability, and behaviour. Introduction
Motivation and context.
Coalgebras [Rut00, Adá05, Jac12] form a powerful theory of state-basedtransition systems where definitions and results are formulated at a high level of genericity that coversseveral families of systems at once, from deterministic automata and Kripke frames to different kindsof probabilistic models. Traditionally, these formulations are elaborated in a set-based context; i.e. nofurther structure in the system’s state space than that of a set is assumed. In many cases, however, aswitch of context is needed. The projects on the coalgebraic foundations of stochastic systems, where theGiry functor and measurable spaces have a central role ( cf. [Vig05, Pan09, Dob09]), are evident examplesof this. Research on coalgebras over Stone spaces ( e.g. [KKV04, BFV10, VV14]) and coalgebras overpseudometric spaces [BBKK14] forms equally important cases. In [KKV04, BFV10, VV14], the aim isto provide a suitable coalgebraic semantics for finitary modal logics by taking advantage of a Vietorisfunctor, while in [BBKK14] is to introduce a notion of distance between states.In this paper our focus is on coalgebras over arbitrary topological spaces , because we believe thatthey provide important mechanisms to the design and analysis of hybrid systems [Tab09, Alu15, Sta01].Briefly put, hybrid systems are those that possess both discrete and continuous behaviour, a result of thecomplex interaction between digital devices, and physical processes like velocity, movement, temperature,and time. Two recurring examples are the cruise control system, basically a digital device with influenceover velocity, and the bouncing ball . In the latter, movement and velocity have a continuous nature, whilethe impact on the ground is assumed to be a discrete event that instantaneously alters the current velocity.As we will see in the following sections, such an interaction between discrete and continuous behaviour callsfor a shift from the set-based setting to richer contexts, in particular to topological ones so that suitablenotions of stability, bisimulation, and behaviour can be obtained. These are the practical motivationsfor the theoretical results that this paper provides. But we stress that coalgebras over topological spaceshave the potential for much more – the works [Vig05, Pan09, Dob09, BFV10, VV14, BBKK14, KKV04],for example, elegantly attest this. Our results are therefore applicable to a much broader context thanthat of hybrid systems.Each functor F : C → C induces a category of coalgebras CoAlg ( F ) that can be seen as a framework fora particular family of state-based transition systems, whose transition type is determined by F : C → C ( cf. [Rut00]). The powerset P : Set → Set , for example, often associated with non-deterministic behaviour,gives rise to Kripke frames.
Date : February 7, 2017.
Key words and phrases.
Coalgebra, topological space, stably compact, Vietoris space, codirected limit. a r X i v : . [ c s . L O ] F e b DIRK HOFMANN, RENATO NEVES, AND PEDRO NORA
In such a context, the systematic study of (co)limits in categories of coalgebras is a natural researchline. In fact, final coalgebras, which form a specific type of limit, are often searched for, as they encode acanonical notion of behaviour for all F -coalgebras. Another special kind of limit, equalisers of coalgebras,is extensively used in coalgebraic specification ( cf. [Rut00, Adá05]). It provides a notion of subsystem,and is essential to characterise a system induced by a set of coequations.1.2. Contributions and related work.
As mentioned before, this paper concerns coalgebras over ar-bitrary topological spaces. More concretely, coalgebras whose underlying functor is defined over thecategory
Top of topological spaces and continuous maps. Analogously to what has already been done in
Set ( e.g. [Rut00, GS01]), the aim here is to investigate the existence of limits in categories of coalgebraswhose underlying functor is Vietoris polynomial — the topological analogue of a Kripke polynomialfunctor. The former is called ‘Vietoris polynomial’ because it arises from the composition of different Vi-etoris functors [Vie22, Mic51, CT97] (the topological analogues of the powerset functor) with polynomialfunctors over
Top . To keep the nomenclature simple, we call every coalgebra whose underlying functor isVietoris polynomial a
Vietoris coalgebra .As composites of constant, (co)product, identity, and powerset functors, Kripke polynomial functorshave long since been recognised as a particularly relevant class of functors ( cf. [Rut00, BRS09, KKV04]).They are intuitive and the corresponding coalgebras subsume several types of state-based systems. More-over, they are well-behaved in regard to the existence of limits in their categories of coalgebras if thepowerset functor is submitted to certain cardinality restrictions. We will see that somewhat similar re-sults hold for Vietoris polynomial functors as well. Actually, an instance of a Vietoris functor, which wecall compact Vietoris functor , has already been studied multiple times in the coalgebraic setting ( e.g. [KKV04, BFV10, VV14, DDG16]), and will appear in a book on coalgebras that is currently in prepara-tion [AMM16]. In particular, [KKV04] shows that compact Vietoris polynomial functors in the category
Stone of Stone spaces and continuous maps admit a final coalgebra. Also, document [DDG16] presents atheorem that can be generalised to show that the compact Vietoris functor in the category
CompHaus ofcompact Hausdorff spaces and continuous maps, admits a final coalgebra. In fact, this generalised resultis also implicitly mentioned in [Eng89, page 245]. Related to this, but in a broader setting, we collect anumber of results scattered in coalgebraic and topological literature, and • add to this collection some results of our own. In particular, we generalise Hughes’ theorem (The-orem 2.14) and prove that, under certain conditions, functors between categories of coalgebrasare topological [Adá05]. Topological functors have powerful properties such as the existence ofleft and right adjoints, lifting of limits, and lifting of factorisations. • This collection of results allows us to obtain several new results about limits in categories ofVietoris coalgebras. For example, that categories of polynomial coalgebras over
Top are complete,and that categories of compact Vietoris coalgebras over
CompHaus are complete as well. Usingin particular [Zen70, Lemma B], we also show that categories of compact Vietoris coalgebras arecomplete in the category
Haus of Hausdorff spaces and continuous maps. Moreover we will seethat all categories of Vietoris coalgebras over
Top have equalisers. • We then take advantage of the limit-preserving properties of the inclusion functors
CompHaus → Top and
Haus → Top to show that every compact Vietoris polynomial functor F : Top → Top that can be restricted either to
CompHaus or Haus admits a final coalgebra.Our setting is a broader one also because we consider different instances of Vietoris functors, a partic-ular case being what we call the lower Vietoris functor , studied in a coalgebraic setting in [BKR07]. • We will show that every lower Vietoris polynomial functor behaves well in the category
StablyComp of stably compact spaces and spectral maps. In particular, that its category of coalgebras is com-plete. • In order to extend these results to more variants of Vietoris functors, we study the existence ofadjunctions between categories of coalgebras. One positive result is that, assuming the existence
IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 3 of a monomorphic natural transformation between the underlying functors, such an adjunctionexists under mild conditions.To illustrate the practical side of these developments, and, more generally, the potential of coalgebrasover
Top to the design and analysis of hybrid systems, we argue that the coalgebraic specification in
Set of the bouncing ball has some deficiencies. Among them, the incapability to reason about the system’sstability, and the non-existence of a suitable final coalgebra if non-determinism is taken into account. Wewill see that these issues can be solved, to some extent, by adopting the category
Top as the underlyingsemantic universe.1.3.
Roadmap.
The ensuing section introduces some categorial notions, provides an overview, and ex-tends some results about limits in categories of coalgebras. Then, it formally reviews the concept ofVietoris coalgebra and different instances of Vietoris functors — as already mentioned, our agenda has abroader scope than most coalgebraic literature on Vietoris functors, which mainly focuses on one specificcase.Section 3 starts with our study about polynomial coalgebras over
Top , and topological functors betweencategories of coalgebras. Then, it adds two instances of Vietoris functors (the lower and the compact)to the mix which, as expected, introduce a number of difficulties. A number of topological concepts arerecalled at this point to help us achieve some of the results mentioned above.Section 4 explores the existence of adjunctions between categories of coalgebras induced by naturaltransformations relating functors on the underlying categories. As already stated, this allows to extendthe results of the previous section to subfunctors of Vietoris polynomial ones, thus covering at once severalvariants of Vietoris functors.Section 5 illustrates an application of this work to the design of hybrid systems. Finally, Section 6suggests possible research lines for future work and concludes.We assume that the reader has basic knowledge of category theory [Mac71, AHS90], topology [Kel55,Gou13], and coalgebras [Rut00, Adá05, Jac12].2.
Preliminaries
Categorial notions.
Some categorial notions that the reader may not frequently meet will be used.This section provides a brief overview about them.
Definition 2.1.
A diagram D : I → C is said to be codirected whenever I is a codirected partiallyordered set, that is, I is non-empty and for all i, j ∈ I there is some k ∈ I with k → i and k → j . A conefor a codirected diagram is called a codirected cone . In particular, a limit of a codirected diagram iscalled codirected . Example 2.2.
Inverse sequence (or ω op ) diagrams, which have the shape depicted below, are codirected. · ←− · ←− · ←− . . . Inverse sequence diagrams have a central role in showing that a given functor admits a final coalgebra(see Theorem 2.10).
Remark . The codirected limit of a diagram D : I → Set is given by the subset (cid:40) ( x i ) i ∈ I ∈ (cid:89) i ∈ I D ( i ) | ∀ j → i ∈ I, D ( j → i )( x j ) = x i (cid:41) of the product (cid:81) i ∈ I D ( i ) . Definition 2.4.
A category C is said to be connected if it is non-empty and every two objects A, B ∈ C can be connected by a finite zig-zag of morphisms as depicted below. A ← · → · · · ← · → B DIRK HOFMANN, RENATO NEVES, AND PEDRO NORA
A diagram D : I → C is called connected diagram if I is connected, and a limit of D is called connectedlimit if D : I → C is connected. Examples 2.5.
Equalisers and codirected limits are two examples of connected limits.We will see in the following section that polynomial functors over
Top preserve connected limits, inparticular codirected ones.
Definition 2.6.
Let F : A → B be a functor. A cone C = ( C → X i ) i ∈ I in A is said to be initialwith respect to F if for every cone D = ( D → X i ) i ∈ I and every morphism h : F D → F C such that F D = F C · h , there exists a unique A -morphism ¯ h : D → C with D = C · ¯ h and h = F ¯ h .We simply say that the cone is initial whenever no ambiguities arise. Examples 2.7. (1) A cone ( f i : X → X i ) in Top is initial with respect to the forgetful functor
Top → Set if and only if X is equipped with the so called initial (weak) topology. Explicitly, thetopology generated by the subbasis f − i ( U ) ( i ∈ I, U ⊆ X i open ) . (2) In the category CompHaus of compact Hausdorff spaces and continuous maps, a monocone isinitial in
Top ( cf. [Gou13, Theorem 4.4.27]). Interestingly, the converse also holds, as a initialcone in Top whose domain is a T space is necessarily mono. Remark . In Example 2.7(1) the subbasis is actually a basis if the cone is codirected.
Theorem 2.9 ([AHS90, Proposition 13.15]) . Let F : A → B be a limit preserving faithful functor and D : I → A a diagram. A cone C for D is a limit of D if and only if the cone F C is a limit of F D and C is initial with respect to F . Limits in categories of coalgebras.
Let F : C → C be an arbitrary functor. Then, dually to thealgebraic case, one can easily show that colimits in CoAlg ( F ) exist if they do so in C ( cf. [Rut00, Adá05]).The story about limits in categories of coalgebras is, however, more complex. In this subsection we reviewsome well-known results on this topic, a special focus being given to those more relevant to the paper.We start at a generic level, with the following two theorems ( cf. [Rut00, Adá05]). Theorem 2.10.
Let C be a category with a final object and F : C → C a functor. If the category C hasa limit L of the diagram ←− F ←− F F ←− . . . and F preserves this limit, then the canonical isomorphism L → F L is a final F -coalgebra. Theorem 2.11.
Assume that F : C → C preserves limits of a certain type. Then the forgetful functor CoAlg ( F ) → C creates limits of the same type. An important consequence of the last theorem is that
CoAlg ( F ) has all types of limit that C has and thatthe functor F : C → C preserves. Unfortunately, as we will witness later, this assumption is often toostrong. Resorting to the notion of covarietor, the following results will be more helpful. Definition 2.12.
A functor F : C → C is said to be a covarietor if the canonical forgetful functor CoAlg ( F ) → C is left adjoint.This adjoint situation allows to take advantage of the theory of (co)monads regarding (co)completenessof Eilenberg-Moore (co)algebras to derive the following theorem ( cf. [Lin69]). Theorem 2.13.
Let F be a covarietor over a complete category. If CoAlg ( F ) has equalisers then CoAlg ( F ) is complete. Related to this, Hughes proved the following theorem in [Hug01, Theorem 2.4.2].
IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 5
Theorem 2.14.
Let C be regularly wellpowered, cocomplete, and possess equalisers. Moreover, assumethat it has an (Epi, RegMono)-factorisation structure, and that the functor F : C → C preserves regularmonomorphisms. Then CoAlg ( F ) has equalisers. Using Theorem 2.13, one can then easily deduce the following corollary.
Corollary 2.15.
If the conditions in the last theorem hold and, additionally, C is complete and F is acovarietor, then the category CoAlg ( F ) is complete. We refer the interested reader to other results on limits in categories of coalgebras. In particular, thework of Kurz [Kur01], which shows that
CoAlg ( F ) is complete whenever it has a suitable factorisationstructure, F is a covarietor, and C is complete; document [GS01], where the authors study the existenceof equalisers and products in categories of coalgebras over Set ; and the documents [PW98, GS01], wherethe existence of limits is studied under the assumption of F being bounded.To close this section, we provide an improvement to Hughes’ theorem. We start with notation. Definition 2.16.
For a small category I , a cone for I in a category C is given by a functor D : I → C together with a cone ( X → D ( i )) i ∈ I for D . Given a class M of cones for I , the category C is called M -wellpowered if for every functor D : I → C there is up to isomorphism only a set of cones for D in M .Our first lemma is in the spirit of [AHS90, Section 12] and shows that “cocompleteness almost impliescompleteness”. Lemma 2.17.
Let C be a cocomplete category and I a small category. Furthermore, let E be a class of C -morphisms and M be a class of cones for I in C . If C is M -wellpowered and every cone for I has a ( E, M ) -factorisation, then C has limits of shape I .Proof. We will show that the diagonal functor ∆ : C → C I has a right adjoint, using Freyd’s General Adjoint Functor Theorem (see [Mac71]). By assumption, C iscocomplete and the functor ∆ clearly preserves colimits, so we just need to show that the Solution SetCondition holds. In this context it unfolds to the following condition: for every functor D : I → C , thereis a set S of cones for D such that every cone ( f i : C → D ( i )) i ∈ I for D factors through a cone in S .Since C is M -wellpowered we have, by assumption, a set S of representants for D in M . Moreover C has a ( E, M ) -factorisation system for I , which means that the cone ( f i : C → D ( i )) i ∈ I can be factorisedas depicted below C f i (cid:47) (cid:47) e (cid:31) (cid:31) D ( i ) A g i (cid:61) (cid:61) with the cone ( g i : A → D ( i )) i ∈ I in S . (cid:3) The factorisation system assumed in this lemma may appear to be rather unconventional, but, as thefollowing remarks will show, it actually emerges from mild conditions.
Remark . Consider a category C equipped with classes E and M of morphisms so that every mor-phism in C has a ( E, M ) -factorisation and C is M -wellpowered. Under additional assumptions, suchfactorisations can be extended to cones for I . To be more concrete:(1) Assume that C has products. Then we put M = (cid:40) all cones ( f i : X → D ( i )) i ∈ I for I where (cid:104) f i (cid:105) i ∈ I : X → (cid:89) i ∈ I D ( i ) is in M (cid:41) . DIRK HOFMANN, RENATO NEVES, AND PEDRO NORA
Clearly, every cone for I is ( E, M ) -factorisable (see [AHS90, Proposition 15.19]), and C is M -wellpowered.(2) In order to relate the previous lemma with Hughes’ theorem, assume that I = { ⇒ } and that E is contained in the class of epimorphisms of C . The class of cones M = { all cones ( f i : X → D ( i )) i ∈ I for I with f in M } , makes every cone for I ( E, M ) -factorisable and the category C is M -wellpowered.Finally, we apply the results above to categories of coalgebras. Theorem 2.19.
Let F : C → C be an endofunctor over a cocomplete category C and let I be a smallcategory. If C is ( E, M ) -structured for cones for I , M -wellpowered and F sends cones in M to cones in M , then CoAlg ( F ) has limits of shape I .Proof. The assumptions guarantee that the factorisation system in C lifts to the category CoAlg ( F ) ( cf. [Adá05, Che14]). The claim then follows from Lemma 2.17. (cid:3) Let us now relate in a more precise manner the previous theorem with Hughes’ theorem.
Theorem 2.20.
Let F : C → C be an endofunctor over a cocomplete category C . If C is regularly well-powered, has an (Epi, RegMono)-factorisation structure and F : C → C preserves regular monomorphisms,then CoAlg ( F ) has equalisers.Proof. Let I = { ⇒ } and use Remark 2.18(2) to provide a ( E, M ) -factorisation system for cones for I . The category C is clearly M -wellpowered and by a simple reasoning one shows that F sends cones in M to cones in M . Now apply Theorem 2.19. (cid:3) The last result shows that Hughes’ assumption of C having equalisers is not necessary. Anotherinteresting point is the ability that we gain to reason not just about equalisers but any type of limit. Wewill take advantage of this generalisation in the next section (see Corollary 3.16).Note also that the following corollaries can be obtained almost for free. Corollary 2.21.
Let F : Set → Set be a functor that preserves monocones of a certain type. Then thecategory
CoAlg ( F ) has limits of the same type. Recall that
Top is an (Epi,initial monocones)-category and an (RegEpi,monocones)-category ( cf. [AHS90, Examples 15.3 (6)]). The following result can then be derived.
Corollary 2.22.
Let F : Top → Top be a functor that preserves either small monocones or small initialmonocones of a certain type. Then the category
CoAlg ( F ) has limits of the same type. Vietoris polynominal functors.
Although traditionally considered in
Set ( e.g. [BRS09, Jac12]),the notion of a polynomial functor can be formally defined at a more generic level. Definition 2.23.
Let C be a category with (co)products. We call a functor F : C → C polynomial if itcan be recursively defined from the grammar below F ::= F + F | F × F | A | Id where A corresponds to an object of C . Remark . Alternatively, one can define the class of polynomial functors as the smallest class offunctors F : C → C that contains the identity functor, all constant functors, and is closed under productsand sums of functors. Here, for functors F, G : C → C , the product of F and G , and the sum of F and G are, respectively, the composites C (cid:104) F,G (cid:105) −−−−→ C × C × −→ C , and C (cid:104) F,G (cid:105) −−−−→ C × C + −→ C . Note that if the functors
F, G : C → C preserve limits of a certain type the functor F × G : C → C preserves limits of the same type as well. Note also that IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 7
Proposition 2.25.
The functor (+) :
Top × Top → Top preserves connected limits.Proof.
It is well-known that the functor (+) :
Set × Set → Set preserves connected limits. Then observethat (+) :
Top × Top → Top preserves initial cones and apply Theorem 2.9. (cid:3)
Corollary 2.26.
If the functors
F, G : Top → Top preserve connected limits the functor F + G : Top → Top preserves connected limits as well.
In the set-based context, the powerset functor P : Set → Set is traditionally used in conjunction withpolynomial functors to bring non-deterministic behaviour into the scene, the resulting functor being a socalled
Kripke polynomial functor . The situation is more complex in the topological context becausea number of functors can be seen as ‘analogues’ of the powerset. Most of them have their roots in theHausdorff metric ( cf. [Pom05, Hau14]) and in Vietoris’ “Bereiche zweiter Ordnung” [Vie22]. Informally,we call them
Vietoris functors . The remainder of this section provides some details about them.Consider a compact Hausdorff space X , the classic Vietoris space V X [Vie22] consists of the setof all closed subsets of X , i.e. V X = { K ⊆ X | K is closed } equipped with the ‘hit-and-miss topology’ generated by the subbasis of sets of the form U ♦ = { A ∈ V X | A ∩ U (cid:54) = ∅ } (“ A hits U ”) ,U (cid:3) = { A ∈ V X | A ⊆ U } (“ A misses X \ U ”) , where U ⊆ X is open. Nowadays there are several well-studied variants of this archetype that give riseto endofunctors over specific subcategories of Top . The interested reader will find in [Mic51] and [CT97]more details about these constructions. For now, we concentrate on two particular cases, described below.
Examples 2.27. (1) For a topological space X , define V X = { K ⊆ X | K is compact } with thetopology generated by the sets U (cid:3) and U ♦ , with U ranging over all open subsets U ⊆ X . Then,given a continuous map f : X → Y , define V f : V X → V Y as V f ( A ) = f [ A ] . We call this variant compact Vietoris functor . It is well-known that V X is compact Hausdorff whenever X is. Infact, for compact Hausdorff spaces this construction coincides with the classic one [Vie22].(2) For a topological space X , define V X = { K ⊆ X | K is closed } with the topology generatedby the sets U ♦ , with U ranging over all open subsets U ⊆ X . Then, given a continuous map f : X → Y , define V f : V X → V Y as V f ( A ) = f [ A ] , where f [ A ] denotes the closure of f [ A ] .This variant is called lower Vietoris functor . Remark . The classic Vietoris construction, with closed sets, does not define an obvious functor on
Top . That is, adding the sets U (cid:3) to the subbasis of Example 2.27 (2) does not define a functor. To seewhy, consider the set { , , } equipped with the topology generated by the sets { , } and { , } . Forthe subspace embedding i : { , } → { , , } , ( V i ) − [ { , } (cid:3) ] = { ∅ , { }} . However, every open set of V{ , } that contains { } contains { , } .A number of projects on (coalgebraic) modal logic studied the compact Vietoris functor in the categoryof Stone spaces ( e.g. [KKV04, VV14]) and in the category of compact Hausdorff spaces [BBH12]. Thesecond case was explored by [CLP91, Pet96, BKR07] in the context of Priestley spaces. Definition 2.29.
Let V : Top → Top be the lower Vietoris functor. We call a functor F : Top → Top lower Vietoris polynomial if it can be recursively defined from the grammar below. F ::= F + F | F × F | A | Id | V Similarly, if we consider the compact Vietoris functor V : Top → Top in lieu of the lower one, then wespeak of a compact Vietoris polynomial functor.
DIRK HOFMANN, RENATO NEVES, AND PEDRO NORA On limits in categories of Vietoris coalgebras
Polynomial functors in
Top . Using standard results, we now show that for a polynomial functor F : Top → Top the associated category of coalgebras
CoAlg ( F ) is complete. A useful fact for this proofis that the category Top is (co)complete ( cf. [AHS90]). Moreover, note that
Theorem 3.1.
All polynomial functors F : Top → Top preserve connected limits.Proof.
Clearly the identity functor
Id :
Top → Top preserves all limits, and the constant functor A : Top → Top trivially preserves connected limits. The claim now follows from Remark 2.24 and Corollary2.26. (cid:3)
From the theorem above one can derive the following results in a straightforward manner.
Proposition 3.2.
All polynomial functors F : Top → Top preserve regular monomorphisms.Proof.
First note that the diagrams associated with equalisers are connected. Then, recall that a regularmonomorphism is an equaliser of a pair of morphisms. (cid:3)
Theorem 3.3.
All polynomial functors F : Top → Top are covarietors.Proof.
Since a polynomial functor F : Top → Top preserves connected limits (Theorem 3.1) it preservesthe codirected ones as well. The claim is then a direct consequence of [Bar93, Theorem 2.1]. (cid:3)
In regard to equalisers in
CoAlg ( F ) , one can easily show that the necessary requirements to apply Theorem2.14 are met. Actually, it is well-known that the category Top is regularly wellpowered ( cf. [AHS90]),and we already saw that it is (co)complete. Moreover, it has an (Epi, RegMono)-factorisation structure( cf. [AHS90]). Therefore,
Corollary 3.4. If F : Top → Top is a polynomial functor, the category
CoAlg ( F ) has equalisers.Proof. A direct consequence of Theorem 2.14 and Proposition 3.2. (cid:3)
Theorem 3.5. If F : Top → Top is a polynomial functor, the category
CoAlg ( F ) is complete.Proof. Observe that F is a covarietor (Theorem 3.3), and that the category CoAlg ( F ) has equalisers(Corollary 3.4). Then, apply Theorem 2.13. (cid:3) We will now use ‘less standard’ results to go further than the previous theorem. More concretely, wewill show that not only is
CoAlg ( F ) complete but also that there is a functor with powerful propertiesfrom CoAlg ( F ) to the analogous category of coalgebras over Set . By going further we also mean that theresults that we will introduce next may be used in categories different than
Top , prime examples are thecategory of preordered sets
Ord and the category of pseudometric spaces
PMet .The general idea is that starting with a category B with good properties and assuming the existenceof a functor A → B that lifts these properties to a category A , there will often be a functor CoAlg ( F ) → CoAlg ( F ) with the same lifting properties than A → B for functors F : A → A , F : B → B making thediagram below commute. A F (cid:47) (cid:47) U (cid:15) (cid:15) A U (cid:15) (cid:15) B F (cid:47) (cid:47) B The following definition recalls the notion of topological functor, which lifts several properties of acategory.
Definition 3.6.
A functor U : A → B is called topological if every cone C = ( X → U X i ) i ∈ I in B has a U -initial lifting, i.e. a initial cone D = ( A → X i ) i ∈ I with respect to U : A → B such that C = U D . IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 9
Remark . Every topological functor is both left and right adjoint, lifts limits and certain types offactorisations (see [Adá05]).
Proposition 3.8.
Consider two categories A , B a functor U : A → B , endofunctors F : A → A , F : B → B , and a natural transformation δ : U F → F U.
Then, there is a functor U : CoAlg ( F ) → CoAlg ( F ) defined by the equations U ( X, c ) = (
U X, δ X · U c ) , U f = U f that makes the diagram below commute.
CoAlg ( F ) (cid:47) (cid:47) U (cid:15) (cid:15) A U (cid:15) (cid:15) CoAlg ( F ) (cid:47) (cid:47) B Moreover,
Proposition 3.9.
If the functor U : A → B is faithful and the natural transformation δ : U F → F U ismono, then the induced functor U : CoAlg ( F ) → CoAlg ( F ) is faithful.Proof. Direct consequence of the natural transformation δ : U F → F U being mono and the functor U : A → B being faithful. (cid:3) Lemma 3.10.
Assume that the natural transformation δ : F U → U F is mono and U is faithful. Let ( f i : ( X, c ) → ( Y i , d i )) i ∈ I be a cone in CoAlg ( F ) , and ( f i : X → Y i ) i ∈ I be initial with respect to U : A → B .Then, the cone ( f i : ( X, c ) → ( Y i , d i )) i ∈ I is initial with respect to the functor U : CoAlg ( F ) → CoAlg ( F ) .Proof. Let ( f i : ( X, c ) → ( Y i , d i )) i ∈ I be a cone in CoAlg ( F ) and ( f i : X → Y i ) i ∈ I be initial with respectto U : A → B . Then, consider another cone ( g i : ( Z, e ) → ( Y i , d i )) i ∈ I in CoAlg ( F ) and assume that its U -image is factorised as shown by the diagram below. U ( Z, e ) h (cid:15) (cid:15) Ug i (cid:37) (cid:37) U ( X, c ) Uf i (cid:47) (cid:47) U ( Y i , d i ) The forgetful functor
CoAlg ( F ) → B yields the following factorisation of the cone ( U g i : U Z → U Y i ) i ∈ I . U Z h (cid:15) (cid:15) Ug i (cid:34) (cid:34) U X Uf i (cid:47) (cid:47) U Y i Since the cone ( f i : X → Y i ) i ∈ I is initial with respect to U : A → B , there is a unique arrow h : Z → X in A such that for all i ∈ I we have g i = f i · h, U h = h. It remains to show that the arrow h : Z → X is also a coalgebra homomorphism h : ( Z, e ) → ( X, c ) . Forthis, consider the diagram below. Z h (cid:47) (cid:47) e (cid:15) (cid:15) X f i (cid:47) (cid:47) c (cid:15) (cid:15) Y id i (cid:15) (cid:15) F Z
F h (cid:47) (cid:47)
F X
F f i (cid:47) (cid:47) F Y i By assumption, the equation
F h · δ Z · U e = δ X · U c · h holds. Then reason in the following manner. F h · δ Z · U e = δ X · U c · h ≡ F U h · δ Z · U e = δ X · U c · U h ≡ δ X · U F h · U e = δ X · U c · U h ⇒ U F h · U e = U c · U h ≡ U ( F h · e ) = U ( c · h ) ⇒ F h · e = c · h (cid:3) Theorem 3.11.
Assume that F : A → A preserves initial cones and that U F = F U . Then if the functor U : A → B is topological, the functor U : CoAlg ( F ) → CoAlg ( F ) is topological as well.Proof. Let ( f i : ( X, c ) → U ( Y i , d i )) i ∈ I be a cone in CoAlg ( F ) . Since the functor U : A → B is topological,the induced cone ( f i : X → U Y i ) i ∈ I admits a U -initial lifting ( f i : A → Y i ) i ∈ I . By assumption, the cone ( F f i : F A → F Y i ) i ∈ I is also initial. Moreover, note that the following equationshold, U (cid:18) A f i → Y i d i → F Y i (cid:19) = (cid:16) X f i → U Y i Ud i → F U Y i (cid:17) U (cid:18) F A
F f i → F Y i (cid:19) = (cid:16) F X
F f i → F U Y i (cid:17) ( i ∈ I ) and that we have the factorisation below. X Ud i · f i (cid:35) (cid:35) c (cid:15) (cid:15) F X
F f i (cid:47) (cid:47) F U Y i This provides an arrow c : A → F A such that
U c = c , and that makes the diagram below to commute. A d i · f i (cid:34) (cid:34) c (cid:15) (cid:15) F A
F f i (cid:47) (cid:47) F Y i We thus have a cone ( f i : ( A, c ) → ( Y i , d i )) i ∈ I in CoAlg ( F ) . To finish the proof recall that the cone ( f i : A → Y i ) i ∈ I is initial with respect to the functor U : A → B and apply Lemma 3.10. (cid:3) Corollary 3.12.
Let U : A → B be a topological functor and consider two functors F : A → A , F : B → B such that F : A → A preserves initial cones. Moreover assume that U F = F U . Then the category
CoAlg ( F ) is complete iff CoAlg ( F ) is complete. The forgetful functor
Top → Set is topological ( cf. [Adá05]) and it is straightforward to show thatall polynomial functors over
Top preserve initial cones. Using the previous corollary this entails that allcategories of coalgebras of a polynomial functor over
Top are complete.As hinted before, Corollary 3.12 has stronger consequences than Theorem 3.5: it considers all functorsin
Top that preserve initial cones (and not just the polynomial ones) and it does not make any assumptionabout the category A being Top . In fact, the only assumption about the category A is that it hasa topological functor A → B . We invite the reader to examine in [Adá05] several examples of suchcategories. IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 11
Some notes about Vietoris functors.
The last corollary is a positive result of our study of limitsin categories of polynomial coalgebras. On the other hand, the addition of Vietoris functors to the mixbrings a whole new level of difficulty that calls for a number of topological concepts, an investigation ofVietoris functors and some of their preservation properties. The study of such properties is the main goalof this section.
Lemma 3.13.
Let X be a topological space and B a base for the topology of X .(1) The set { B ♦ | B ∈ B} is a subbase for the lower Vietoris space V X ( cf. Example 2.27(2)).(2) If B is closed under finite unions, then the set { B ♦ | B ∈ B} ∪ { B (cid:3) | B ∈ B} is a subbase for thecompact Vietoris space V X ( cf. Example 2.27(1)).Proof.
Let S be a set of open subsets of X . First note that, for both the lower and the compact Vietorisspace, (cid:16)(cid:91) S (cid:17) ♦ = (cid:91) (cid:8) S ♦ | S ∈ S (cid:9) . This proves the first statement. To see that the second one is also true, observe that (cid:16)(cid:91) S (cid:17) (cid:3) = (cid:91) (cid:26)(cid:16)(cid:91) F (cid:17) (cid:3) | F ⊆ S finite (cid:27) since we only consider compact subsets of X . (cid:3) Lemma 3.14.
Both the compact and the lower Vietoris functor V : Top → Top preserve initial codirectedcones.Proof.
Let ( f i : X → X i ) i ∈ I be an initial codirected cone in Top . Then the set (cid:8) f − i ( U ) | i ∈ I, U ⊆ X i open (cid:9) is a base for the topology of X (Remark 2.8). Moreover, the base is closed under finite unions. Therefore,by the lemma above, the proof follows from the equations (( f i ) − ( U )) (cid:3) = ( V f i ) − ( U (cid:3) ) (( f i ) − ( U )) ♦ = ( V f i ) − (cid:0) U ♦ (cid:1) , for all i ∈ I and U ⊆ X i open, which are straightforward to show. (cid:3) Theorem 3.15.
The lower Vietoris functor preserves initial codirected monocones. The compact Vietorisfunctor preserves initial codirected monocones of Hausdorff spaces.Proof.
First note that for a topological space X the lower Vietoris space V X is T , and if X is Hausdorffthe compact Vietoris space V X is Hausdorff as well ( cf. [Mic51]). Then recall that a initial cone in Top whose domain is T (or T ) is necessarily mono and apply Lemma 3.14. (cid:3) Together with Proposition 3.2 it follows:
Corollary 3.16.
Every compact polynomial functor and every lower polynomial functor F : Top → Top preserves regular monomorphisms.Proof.
We already saw that all polynomial functors preserve regular monomorphisms (Proposition 3.2),and that the lower Vietoris functor preserves them as well (Theorem 3.15). Moreover, we saw that thecompact Vietoris functor preserves initial monomorphisms (Lemma 3.14) and it is straightforward toshow that it preserves monomorphisms. (cid:3)
From Theorem 3.15 and Corollary 2.19 we obtain the following results.
Corollary 3.17.
For every lower Vietoris polynomial functor F : Top → Top the category
CoAlg ( F ) hascodirected limits. For every compact Vietoris polynomial functor F : Top → Top the category
CoAlg ( F ) has codirected limits of Hausdorff spaces. Corollary 3.18.
For every Vietoris polynomial functor F : Top → Top the category
CoAlg ( F ) hasequalisers. Proof.
Direct consequence of Theorem 2.20 and Corollary 3.16. (cid:3)
Remark . The assumption above about codirectedness is essential: neither the compact nor the lowerVietoris functor V : Top → Top preserve monocones in general. Take, for instance, a compact Hausdorffspace X with at least two elements. Then A = { ( x, x ) | x ∈ X } is a closed subset of X × X , and A isdifferent from B = X × X . However, with π : X × X → X and π : X × X → X denoting the projectionmaps, V π ( A ) = V π ( B ) = X = V π ( A ) = V π ( B ); which shows that the cone ( V π : V ( X × X ) → V X, V π : V ( X × X ) → V X ) is not mono.Theorem 3.15 shows some good behaviour with respect to codirected initial monocones. However, noneof the functors of Examples 2.27 preserves codirected limits in Top . Examples 3.20. (1) We consider I = N with the natural order, and the functor D : N → Set whichsends n ≤ m to the inclusion map { , . . . n } (cid:44) → { , . . . , m } . Clearly, the set of natural numbers N is a colimit of this directed diagram. Then, the composite Set ( − , N ) · D op : N op → Set yieldsa codirected diagram with limit
Set ( N , N ) , the limit projections p n : Set ( N , N ) → Set ( D ( n ) , N ) being given by restriction. Equipping all sets with the indiscrete topology, we obtain a codirectedlimit in Top . The compact Vietoris functor does not send this limit to a monocone since ( V p n ) n ∈ N cannot distinguish between the sets Set ( N , N ) and { f : N → N | { n ∈ N | f ( n ) (cid:54) = 0 } is finite } . (2) The next example is based on the “empty inverse limit” of [Wat72]. Here I is the set of all finitesubsets of R , with order being containment ⊇ . For F ∈ I , let D ( F ) be the discrete space ofall injective functions F → N , and the map D ( G ⊇ F ) is given by restriction. Note that eachconnecting map D ( G ⊇ F ) is surjective. Then the limit of this diagram in Top is empty since anelement of this limit would define an injective function R → N . The lower Vietoris functor sendsthe limit cone for D to a monocone but not to a limit cone since the limit of V D has at leasttwo elements: ( ∅ ) F ∈ I and ( D ( F )) F ∈ I . Using the indiscrete topology instead of the discrete oneshows that the lower Vietoris functor does not preserve codirected limits of diagrams of compactspaces and closed maps.(3) In the example above we can use other topologies to show that the lower or the compact Vietorisfunctor does not preserve certain codirected limits. As an example, we consider here N equippedwith the topology {↑ n | n ∈ N } ∪ { ∅ } ; where ↑ n = { k ∈ N | n ≤ k } . Note that N is T and every non-empty collection of open subsetsof N has a largest element with respect to inclusion ⊆ . The latter implies that, for every finiteset F , every subset of N F is compact. To see this, let C ⊆ N F and assume that C is covered bybasic open subsets of N F : C ⊆ (cid:91) i ∈ I ↑ i × · · · × ↑ i n . We already know that for every k ∈ F the family ( ↑ j k ) j ∈ I contains a largest element with respectto inclusion. This allows us to construct a finite subcover in the following manner: for each k ∈ F let S k be an element of the cover whose k -projection is the largest element of the family ( ↑ j k ) j ∈ I .Then, C ⊆ (cid:91) k ∈ F S k and the family ( S k ) k ∈ F is a finite subcover. We conclude that C is compact.With I being as in the previous example, we consider now D ( F ) as a subspace of N F . Then,for every G ⊇ F , the map D ( G ⊇ F ) : D ( G ) → D ( F ) is continuous. Hence, this constructiondefines a codirected diagram D : I → Top where each D ( F ) is T , compact, and locally compact; IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 13 and the limit of this diagram is empty. With the same argument as above, neither the lower northe compact Vietoris functor preserve this limit.3.3.
Vietoris polynomial functors.
Section 3.1 studied limits in categories of polynomial coalgebras,essentially by analysing the preservation of connected limits in
Top and by providing sufficient conditionsfor the existence of topological functors between categories of coalgebras. In the current section our focusis on Vietoris coalgebras. In fact, Examples 3.20 already showed that it is highly problematic to considerall topological spaces, because the lower and the compact Vietoris functors do not preserve codirectedlimits in
Top . Hence, we will restrict our attention to different subcategories of
Top where more positiveresults appear.
Definition 3.21.
A topological space X is called stably compact whenever X is T , locally compact,well-filtered and every finite intersection of compact saturated subsets is compact [Jun04]. A continuousmap between stably compact spaces is called spectral whenever the inverse image of compact saturatedsubsets is compact. Stably compact spaces and spectral maps form a category which we denote by StablyComp . Remark . Note that every stably compact space is compact. More information on this type of spacecan be found in [GHK +
03] and [Jun04].
Theorem 3.23.
The category
StablyComp is complete and regularly wellpowered. The inclusion functor
StablyComp → Top preserves limits and finite coproducts.Proof.
It is straightforward to check that the finite coproduct of stably compact spaces is stably compact( cf. [Gou13, Proposition 9.2.1]). The other claims follow from monadicity of
StablyComp → Top which isshown in [Sim82]. We note that [Sim82] uses the designation well-compacted instead of stably compact. (cid:3)
Further properties of
StablyComp can be easily derived if ones uses a order-theoretic perspective.
Definition 3.24. A partially orderered compact space is a triple ( X, ≤ , τ ) consisting of a set X , apartial order ≤ on X and a compact topology τ on X so that the set { ( x, y ) ∈ X × X | x ≤ y } is closed with respect to the product topology. Remark . Every partially ordered compact space ( X, ≤ , τ ) is necessarily Hausdorff as the antisym-metry property of the relation ≤ implies that the diagonal { ( x, x ) | x ∈ X } is closed in X × X .The category StablyComp is isomorphic to the category
PosComp of partially orderered compact spacesand monotone continuous maps ( cf. [GHK + PosComp → StablyComp commuteswith the underlying forgetful functors to
Set , sending a partially ordered compact space ( X, ≤ , τ ) tothe stably compact space with the same underlying set and the topology defined by the upper-opensets of ( X, ≤ , τ ) . Its inverse functor uses the specialisation order of a topological space, defined by x ≤ y ⇐⇒ x ∈ { y } . It maps a stably compact space ( X, τ ) into a space ( X, τ (cid:48) , ≤ ) where the relation ≤ is the specialisation ordering and τ (cid:48) the patch topology of ( X, τ ) , i.e. the topology generated by thecomplements of compact saturated subsets and also the opens in ( X, τ ) . Remark . The canonical forgetful functor
PosComp → CompHaus has a left adjoint which equips acompact Hausdorff space with the discrete order. Using the isomorphism above, the adjunction
PosComp (cid:62) forgetful (cid:40) (cid:40) discrete (cid:104) (cid:104)
CompHaus reads in the language of stably compact spaces as
StablyComp (cid:62) patch (cid:40) (cid:40) inclusion (cid:104) (cid:104)
CompHaus . In the sequel we will freely jump between both perspectives.
Theorem 3.27.
The category
PosComp is cocomplete and the epimorphisms of
PosComp are preciselythe surjective morphisms.Proof.
Cocompleteness of
PosComp follows from [Tho09, Corollary 2]. Combining several results of[Nac65], it is shown in [HN16] that every epimorphism in
PosComp is surjective. (cid:3)
Clearly, (Surjections,Substructure) is a factorisation structure for morphisms in
PosComp . Since thesurjections are precisely the epimorphisms in
PosComp , we conclude that
PosComp is (Epi,RegMono)-structured, and thus also the category
StablyComp . Moreover, the regular monomorphisms in
StablyComp are precisely the topological subspace embeddings.Let us turn our attention back to the study of Vietoris functors with the isomorphism
StablyComp (cid:39)
PosComp in mind. The lower Vietoris functor on
Top restricts to a functor V : StablyComp → StablyComp ( cf. [Sch93]). Its counterpart on PosComp can be described in the following manner.
Proposition 3.28.
Under the isomorphism
StablyComp (cid:39)
PosComp , the lower Vietoris functor V : StablyComp → StablyComp corresponds to the functor
PosComp → PosComp which sends a partially ordered compact space X to the space of all lower-closed subsets of X , with orderinclusion ⊆ , and compact topology generated by the sets { A ⊆ X | A lower-closed and A ∩ U (cid:54) = ∅ } ( U ⊆ X upper-open) , (3.i) { A ⊆ X | A lower-closed and A ∩ K = ∅ } ( K ⊆ X upper-closed) . Given a map f : X → Y in PosComp , the functor returns the map that sends a lower-closed subset A ⊆ X to the down-closure ↓ f [ A ] of f [ A ] .Proof. Let ( X, ≤ , τ ) be a partially ordered compact space with corresponding stably compact space ( X, σ ) . Clearly, the underlying set of V ( X, σ ) is the set of all lower-closed subsets of X . We will showthat the patch topology of V ( X, σ ) coincides with the topology defined by (3.i). First note that every setof the form { A ⊆ X | A lower-closed and A ∩ U (cid:54) = ∅ } ( U ⊆ X upper-open) , is open in V ( X, σ ) and therefore is also in the patch topology. For K ⊆ X upper-closed, the complementof the set { A ⊆ X | A lower-closed and A ∩ K = ∅ } is equal to K ♦ . Using Alexander’s Subbase Theorem, it is straightforwad to verify that K ♦ is compactin V ( X, σ ) . Since the specialisation order of V ( X, σ ) is subset inclusion, K ♦ is also saturated. Hence,the topology defined by (3.i) is coarser than the patch topology of V ( X, σ ) . Since it is also Hausdorff,by [Jun04, Lemma 2.2], both topologies coincide ( cf. [Eng89]). In particular, the construction of theproposition defines indeed a partially ordered compact space.In regard to maps in PosComp , [Nac65, Proposition 4 on page 44] tells that for every map f : X → Y in PosComp and every lower-closed subset A ⊆ X , the down-closure ↓ f [ A ] of f [ A ] is closed in Y , andtherefore coincides with the closure of f [ A ] in the stably compact topology of Y . (cid:3) IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 15
Recall that the lower Vietoris functor preserves codirected initial monocones (see Theorem 3.15).Hence, for every codirected diagram D : I → StablyComp with limit cone ( p i : L D → D ( i )) i ∈ I , thecanonical comparison map h : V L D → L V D , K (cid:55)→ ( p i [ K ]) i ∈ I is an embedding. To show that V : StablyComp → StablyComp preserves these limits, we are left with thetask of proving that h is also surjective. To do so, we use the fact that StablyComp inherits a nice char-acterisation of codirected limits from the category
CompHaus . A first hint of the latter characterisationis in [Bou42], but, to the best of our knowledge, is rarely used in the literature. Actually, we were notable to find a proof in the literature, except for [Hof99]; so we sketch a proof below.
Theorem 3.29.
Let D : I → CompHaus be a codirected diagram and C = ( p i : L → D ( i )) i ∈ I a cone for D . The following conditions are equivalent:(1) The cone C is a limit of D .(2) The cone C is mono and, for every i ∈ I , the image of p i contains the intersection of the imagesof all D ( j → i ) , in symbols im p i ⊇ (cid:92) j → i im D ( j → i ) . Proof.
Assume first that ( p i : L → D ( i )) i ∈ I satisfies the two conditions and let ( f i : X → D ( i )) i ∈ I be acone for D . Let x ∈ X , and, for every i ∈ I , put A i = p − i ( f i ( x )) . Clearly, A i is closed, moreover, A i isnon-empty since im f i ⊆ (cid:92) j → i im D ( j → i ) = im p i Since the family ( A i ) i ∈ I is codirected and L is compact, there is some z ∈ (cid:84) i ∈ I A i . We put f ( x ) = z , thisway we define a map f : X → L with p i · f = f i , for all i ∈ I . Since ( p i : L → D ( i )) i ∈ I is a monocone,we conclude that ( p i : L → D ( i )) i ∈ I is a limit of D . Conversely, if ( p i : L → D ( i )) i ∈ I is a limit, then itis clearly a monocone. Let now i ∈ I and x ∈ (cid:84) j → i im D ( j → i ) . We may assume that i is final in I .For each i ∈ I , we put A i = { ( x i ) i ∈ I ∈ (cid:89) i ∈ I D ( i ) | x i = x and, for all i → j ∈ I, x j = D ( i → j )( x i ) } . Then A i is non-empty, and it is a closed subset of (cid:81) i ∈ I D ( i ) since it is an equaliser of continuous mapsbetween Hausdorff spaces. Furthermore, for i → j ∈ I , A i ⊆ A j . Hence there is some z ∈ (cid:84) i ∈ I A i ; byconstruction, z ∈ L and p i ( z ) = x . (cid:3) Remark . For every cone ( p i : C → D ( i )) i ∈ I the inequality im p i ⊆ (cid:84) j → i im D ( j → i ) holds. Hence,in the theorem above, the reverse inequality, distinguishes monocones from limit cones. Proposition 3.31.
Let A be a codirected set of closed subsets of a partially ordered compact space X .Then, ↓ (cid:84) A ∈A A = (cid:84) A ∈A ↓ A .Proof. Clearly, ↓ (cid:84) A ∈A A ⊆ (cid:84) A ∈A ↓ A . To show that the reverse inequality holds, consider z ∈ (cid:84) A ∈A ↓ A .Then, for every A ∈ A , the set ↑ z ∩ A is non-empty, and closed because { z } is compact ( cf. [Nac65,Proposition 4 on page 44]). Morevover, since A is codirected, the set {↑ z ∩ A | A ∈ A} has the finiteintersection property. Therefore, by compactness, it follows that ↑ z ∩ (cid:84) A ∈A A (cid:54) = ∅ , which implies that z ∈ ↓ (cid:84) A ∈A A . (cid:3) Proposition 3.32.
Let D : I → PosComp be a codirected diagram, ( p i : L D → D ( i )) i ∈ I a limit for D and ( L V D → V D ( i )) i ∈ I a limit for V D : I → PosComp . Then the function h : V L D → L V D defined by K (cid:55)→ ( ↓ p i [ K ]) i ∈ I is surjective.Proof. Let ( K i ) i ∈ I ∈ L V D . For every i ∈ I , K i ⊆ D ( i ) is closed, hence, K i ∈ PosComp . For every i ∈ I and j → i ∈ I , take K ( i ) as K i and K ( j → i ) as the continuous and monotone map of type K j → K i given by the restriction of D ( j → i ) to K j . This way, by Remark 2.3, we obtain a codirected diagram K : I → PosComp such that for every j → i ∈ I , ↓ K ( j → i )[ K ( j )] = [ K ( i )] .Let ( p i : L K → K ( i )) i ∈ I be a limit for K . By construction, L K ⊆ L D is lower-closed. Thus, L K ∈ V L D . We claim that h ( L k ) = ( K i ) i ∈ I . Let i ∈ I . Since the following diagram of forgetful functors PosComp (cid:33) (cid:33) (cid:47) (cid:47)
CompHaus (cid:124) (cid:124)
Set commutes and the functor
PosComp → CompHaus preserves limits, from Theorem 3.29 we obtain p i [ L K ] = (cid:92) j → i K ( j → i )[ K j ] . Therefore, by Propostion 3.31, ↓ p i [ L K ] = ↓ (cid:92) j → i K ( j → i )[ K ( j )] = (cid:92) j → i ↓ K ( j → i )[ K ( j )] = K i . (cid:3) As expected, we obtain the following results.
Corollary 3.33.
The lower Vietoris functor V : StablyComp → StablyComp preserves codirected limits.Proof.
A direct consequence of the previous proposition. (cid:3)
Corollary 3.34.
All lower Vietoris polynomial functors F : StablyComp → StablyComp preserve codi-rected limits.Proof.
Analogous to that of Theorem 3.1. (cid:3)
Theorem 3.35.
For every lower Vietoris polynomial functor F : StablyComp → StablyComp , the category
CoAlg ( F ) is complete.Proof. Firstly, observe that Theorems 3.23, 3.27 and Corollary 3.16 guarantee the hypothesis of Theo-rem 2.14, therefore the category
CoAlg ( F ) has equalisers. Then the assertion follows from Corollary 3.34and [Bar93, Theorem 2.1]. (cid:3) In regard to final coalgebras, there is still room to improve the theorem above. Indeed, the inclusionfunctor I : StablyComp → Top is well-behaved with respect to limits, in particular it preserves and reflectsthem ( cf. [Sim82]); this allows us to derive the following theorem.
Theorem 3.36.
Every lower Vietoris polynomial functor in
Top that can be restricted to
StablyComp admits a final coalgebra.
The lower and the compact Vietoris functors on
Top are seemingly unrelated, notwithstanding, thesefunctors are closely related when restricted, respectively, to
StablyComp and
CompHaus . From thedescription of the lower Vietoris functor V on PosComp we obtain that the compact Vietoris functor V : CompHaus → CompHaus is the composite
CompHaus discrete −−−−−→
PosComp V −→ PosComp forgetful −−−−−→
CompHaus . Being right adjoint, the functor
PosComp forgetful −−−−−→
CompHaus preserves limits, but also the inclusionfunctor
CompHaus → PosComp does so. As an interesting consequence, studying preservation of limitsby the lower Vietoris functor in
StablyComp (cid:39)
PosComp encompasses studying preservation of limits bythe compact Vietoris in
CompHaus . In particular, the following results come for free.
Corollary 3.37.
The compact Vietoris V : CompHaus → CompHaus preserves codirected limits.
Corollary 3.38.
All compact Vietoris polynomial functors F : CompHaus → CompHaus preserve codi-rected limits.
IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 17
By taking advantage of the fact that a compact subspace of an Hausdorff space is a compact Hausdorffspace, [Zen70] proves this property of the compact Vietoris functor even for Hausdorff spaces.
Theorem 3.39.
The compact Vietoris functor V : Haus → Haus preserves codirected limits.
The following results then emerge in a straightforward manner.
Theorem 3.40.
All compact Vietoris polynomial functors F : Haus → Haus preserve codirected limits.Proof.
Follows from the previous theorem and the fact that all polynomial functors F : Haus → Haus preserve codirected limits. (cid:3)
Corollary 3.41.
Let F : Haus → Haus be a compact Vietoris polynomial functor. The associated categoryof coalgebras
CoAlg ( F ) is complete.Proof. Being an epireflective subcategory of
Top , the category
Haus is complete and cocomplete, andregularly wellpowered. Furthermore,
Haus is (Epi,RegMono)-structured; but note that f : X → Y in Haus is a regular monomorphism if and only if f is a closed embedding. It is straightforward to provethat the compact Vietoris functor preserves closed embeddings; therefore, by Theorem 2.14, CoAlg ( F ) has equalisers. As an alternative, Haus is also (Surjection, Embedding)-structured; and now use Corol-lary 2.19 and Corollary 3.16 to conclude that
CoAlg ( F ) has equalisers. Then the assertion follows fromTheorem 3.40 and [Bar93, Theorem 2.1]. (cid:3) Theorem 3.42.
Let F : Top → Top be a Vietoris polynomial functor that can be restricted to
Haus .Then, the category
CoAlg ( F ) has a final coalgebra.Proof. A consequence of the fact that I : Haus → Top preserves and reflects limits ( cf. [AHS90]). (cid:3)
To close this section we will relate its results with the works [KKV04, BKR07]. Recall that the formerconsiders compact Vietoris polynomial functors over
Stone . The latter consider coalgebras for the lowerVietoris functor in the category
Spec of spectral spaces and spectral maps.The categories
Stone and
Spec have a close relation with some of the categories we considered so far,in particular
CompHaus and
StablyComp . By taking advantage of this relation we will see that the factthat every compact Vietoris functor F : Stone → Stone admits a final coalgebra (as shown in [KKV04])is actually a consequence of Corollary 3.38, and the fact that every lower Vietoris polynomial functor F : Spec → Spec admits a final coalgebra is a direct consequence of Theorem 3.35.
Remark . Recall that a Stone space X is a compact Hausdorff space with a basis of clopen sets. Thisis equivalent to saying that X is compact Hausdorff and that the cone of continuous maps ( X → tothe discrete two-point-space is initial. Lemma 3.44.
Let ( X → X i ) i ∈ I be a initial cone in CompHaus where X i is a Stone space for every i ∈ I .Then X is a Stone space as well.Proof. Follows from the fact that each space X i defines a initial cone of continuous maps ( X i → andthat initial cones are closed under composition. (cid:3) Corollary 3.45.
The canonical forgetful functor
Stone → CompHaus creates limits. Hence, the category
Stone is complete, and the functor
Stone → CompHaus preserves and reflects limits.
Theorem 3.46.
Every compact Vietoris polynomial functor F : Stone → Stone preserves codirectedlimits.Proof.
Observe that every compact Vietoris polynomial functor F : Stone → Stone is also a functor F : CompHaus → CompHaus and that the diagram below commutes. The claim then follows directly from the fact that the functor
Stone → CompHaus preserves and reflects limits.
Stone (cid:15) (cid:15) F (cid:47) (cid:47) Stone (cid:15) (cid:15)
CompHaus F (cid:47) (cid:47) CompHaus (cid:3)
Corollary 3.47.
Every compact Vietoris polynomial functor F : Stone → Stone admits a final coalgebra.
Analogous results can be achieved for the category
Spec . To see this let us start a remark akin toRemark 3.43.
Remark . Recall that a spectral space X is a stably compact space with a basis of compact opensubsets. This is equivalent to saying that X is stably compact and that the cone of spectral maps ( X → to the Sierpiński space is initial. Lemma 3.49.
Let ( X → X i ) i ∈ I be a initial cone in StablyComp where X i is a spectral space for every i ∈ I . Then X is a spectral space as well.Proof. Follows from the fact that each space X i defines a initial cone of continuous maps ( X i → tothe Sierpiński space and that initial cones are closed under composition. (cid:3) Corollary 3.50.
The canonical forgetful functor
Spec → StablyComp creates limits. Hence, the category
Spec is complete, and the functor
Spec → StablyComp preserves and reflects limits.
Theorem 3.51.
Every lower Vietoris polynomial functor F : Spec → Spec preserves codirected limits.Proof.
Observe that every lower Vietoris polynomial functor F : Spec → Spec is also a functor F : StablyComp → StablyComp and that the diagram below commutes. The claim then follows directly fromthe fact that the functor
Spec → StablyComp preserves and reflects limits.
Spec (cid:15) (cid:15) F (cid:47) (cid:47) Spec (cid:15) (cid:15)
StablyComp F (cid:47) (cid:47) StablyComp (cid:3)
Corollary 3.52.
Every lower Vietoris polynomial functor F : Spec → Spec admits a final coalgebra. Limits via adjunction
In this section we extend the results of the previous section to subfunctors of (Vietoris) polynomialfunctors, by making use of adjunction. To achieve this we introduce a number of conditions whichguarantee that a functor
CoAlg ( F ) → CoAlg ( G ) induced by a natural transformation F → G has a rightadjoint: note that if the functor CoAlg ( F ) → CoAlg ( G ) is also fully faithful, then we can easily show that CoAlg ( F ) is “as complete as” CoAlg ( G ) . A key property we use here is a straightforward generalisationof the notion of taut natural transformation originally introduced in [Möb83] and [Man02].We start with the definition below. Definition 4.1.
Every natural transformation σ : F → G induces a functor I : CoAlg ( F ) → CoAlg ( G ) ,defined by I ( X, c ) = (
X, σ X · c ) , If = f. Note that the functor I : CoAlg ( F ) → CoAlg ( G ) is faithful. Moreover, IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 19
Proposition 4.2. If σ : F → G is a monomorphic natural transformation, then the functor I : CoAlg ( F ) → CoAlg ( G ) is also full.Proof. Take a homomorphism f : I ( X, c ) → I ( Y, d ) . By assumption, the equation Gf · σ X · c = σ Y · d · f holds. Then, use naturality and the fact that σ Y : F Y → GY is a monomorphism to show that F f · c = d · f . (cid:3) We will now show that, under some assumptions on the natural transformation σ : F → G , the functorabove has a right adjoint. Assumption 4.3.
In the remainder of this section the letter C denotes a category with an ( E, M ) -factorisation structure where M is included in the class of monomorphisms. We assume that C is M -wellpowered, that σ : F → G is a natural transformation between endofunctors on C where everycomponent σ X is in M , and that G sends morphisms in M to morphisms in M . Theorem 4.4.
Under Assumption 4.3 with C cocomplete, the functor I : CoAlg ( F ) → CoAlg ( G ) is leftadjoint.Proof. We will show that the assumptions of the General Adjoint Functor Theorem hold. Since C iscocomplete, the category CoAlg ( F ) is cocomplete as well. Moreover, I : CoAlg ( F ) → CoAlg ( G ) preservescolimits, as U I : CoAlg ( F ) → C preserves colimits, and the forgetful functor U : CoAlg ( G ) → C reflectsthem. It remains to verify the Solution Set Condition. For this, take a coalgebra d : Y → GY . Let S be a set of representatives of the collection of all C -objects Q admitting an M -morphism Q → Y , andlet S be the set of all F -coalgebras based on an object in S . Let now ( X, c ) be an F -coalgebra and f : ( X, σ X · c ) → ( Y, d ) be a homomorphism of G -coalgebras. By hypothesis, f : X → Y factorises as f = m · e X e −→ Q m −−→ Y with e ∈ E and m ∈ M . Since σ Q : F Q → GQ and Gm : GQ → GY are in M , there is a diagonal q : Q → F Q so that the right hand square and the lower-left square in GX Ge (cid:47) (cid:47) GQ Gm (cid:47) (cid:47) GYF X σ X (cid:79) (cid:79) F e (cid:47) (cid:47)
F Q σ Q (cid:79) (cid:79) X c (cid:79) (cid:79) e (cid:47) (cid:47) Q m (cid:47) (cid:47) q (cid:79) (cid:79) Y d (cid:79) (cid:79) commute; the upper-left square commutes since σ is a natural transformation. This proves that f :( X, σ X · c ) → ( Y, d ) factorises via the image of an object in S . (cid:3) Corollary 4.5.
The category
CoAlg ( F ) has all (co)limits of a certain type if CoAlg ( G ) does so. Corollary 4.6.
Let F : Top → Top be a compact Vietoris polynomial functor that can be restricted to
Haus . Every subfunctor of F admits a final coalgebra.Remark . The Corollary above applies to various interesting variants of the compact Vietoris functorthat were not yet mentioned. In particular, • the one that discards the empty set, • analogously to the finitary powerset functor, the one that takes infinite sets out of comission, and • the one which considers only compact and connected subsets ( cf. [Dud72]).All these variants are subfunctors of the compact Vietoris functor. In conjunction with the polynomialones, they form a family of subfunctors of compact Vietoris polynomial functors. Corollary 4.8.
Let F : Top → Top be a lower Vietoris polynomial functor that can be restricted to
StablyComp . Every subfunctor of F admits a final coalgebra. The proof of Theorem 4.4 gives us also a hint on how to construct a coreflection of a G -coalgebra ( Y, d ) :take the “largest M -subcoalgebra of ( Y, d ) ”. In the sequel we make this idea more precise. To do so,motivated by [Möb83] and [Man02], we introduce the following notion. Definition 4.9.
A natural transformation σ : F → G is M -taut if each naturality square induced by amorphism in M is a pullback square; that is, for every morphism m : X → Y in M , the diagram belowis a pullback square. F X
F m (cid:47) (cid:47) σ X (cid:15) (cid:15) F Y σ Y (cid:15) (cid:15) GX Gm (cid:47) (cid:47) GY Recall from [AHS90, Definition 7.79] that, for monomorphisms m : M → X and m : M → X in acategory, m is smaller than m (written as m ≤ m ) whenever there is some m : M → M with m · m = m . Note that m is necessarily a monomorphism. Assuming that C has pullbacks, take a G -coalgebra ( Y, d ) and consider the pullback square(4.i) S i (cid:15) (cid:15) (cid:47) (cid:47) F Y σ Y (cid:15) (cid:15) Y d (cid:47) (cid:47) GY in C . Note that i : S → Y is in M , by [AHS90, Proposition 14.15]. Lemma 4.10. (1) For every F -coalgebra ( X, c ) and every homomorphism m : ( X, σ X · c ) → ( Y, d ) with m ∈ M , m is smaller than i : S → Y .(2) Assume now that the natural transformation σ : F → G is M -taut and let m : ( Q, q ) → ( Y, d ) bea homomorphism in CoAlg ( G ) where m ∈ M and m ≤ i . Then there is a F -coalgebra structure q (cid:48) : Q → F Q on Q with σ Q · q (cid:48) = q .Proof. An easy calculation, and [AHS90, Proposition 14.9] show that the first claim is true. In regard tothe second one, let ¯ m : Q → S be the arrow in C with i · ¯ m = m . Then, since in the diagram Y d (cid:47) (cid:47) GYS i (cid:63) (cid:63) (cid:47) (cid:47) F Y σ Y (cid:60) (cid:60) GQ Gm (cid:79) (cid:79) Q ¯ m (cid:79) (cid:79) c (cid:74) (cid:74) m (cid:50) (cid:50) (cid:47) (cid:47) F Q
F m (cid:79) (cid:79) σ Q (cid:61) (cid:61) the right hand parallelogram is a pullback square and the outer diagram and the top parallelogramcommute. This provides the desired arrow Q → F Q . (cid:3) For a G -coalgebra ( Y, d ) , the class of all subcoalgebras m : ( X, c ) → ( Y, d ) with m ∈ M is preorderedunder the smaller-than relation. Since C is M -wellpowered, this class is equivalent to an ordered set; andby a slight abuse of language we will speak of the ordered set of M -subobjects of ( Y, d ) . IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 21
Recall from Proposition 4.2 that the induced functor I : CoAlg ( F ) → CoAlg ( G ) is fully faithful since σ : F → G is a monomorphic natural transformation. Hence, we can consider CoAlg ( F ) as a fullsubcategory of CoAlg ( G ) . From the results above we obtain: Theorem 4.11.
In addition to Assumption 4.3, assume that C has pullbacks, σ is M -taut and, for every G -coalgebra ( Y, d ) , the ordered set of M -subobjects is complete. Then, for every G -coalgebra ( Y, d ) , thecoreflection of ( Y, d ) is given by the supremum ( ¯ Q, ¯ q ) → ( Y, d ) of all G -homomorphisms ( Q, q ) → ( Y, d ) with ( Q, q ) in CoAlg ( F ) and m : Q → Y in M smaller than i : S → Y (defined by the pullback square (4.i) ).Remark . If C has coproducts, then also CoAlg ( G ) has coproducts which guarantees completeness ofthe ordered set of M -subobjects of ( Y, d ) . In fact, let ( m i : ( X i , c i ) → ( Y, d )) i ∈ I a family of subcoalgebrasof ( Y, d ) with m i ∈ M , for every i ∈ I . Then the supremum m : ( X, c ) → ( Y, d ) of this family is given bythe ( E, M ) -factorisation of the canonical map f : (cid:96) i ∈ I ( X i , c i ) → ( Y, d ) induced by this family. (cid:96) i ∈ I ( X i , c i ) e (cid:47) (cid:47) f (cid:39) (cid:39) ( X, c ) m (cid:47) (cid:47) ( Y, d )( X, c i ) k i (cid:79) (cid:79) m i (cid:52) (cid:52) Vietoris coalgebras at work
Moving to the more practical side, recall the bouncing ball system mentioned in the introduction. For-mally, it consists of a ball that is dropped at a certain height ( p ) , and with an initial velocity ( v ) . Dueto the gravitional effect ( g ) , it falls into the ground and then bounces back up, losing, for example,half of its kinetic energy. As the documents [NBHM16, NB16] show, such a behaviour can be describedcoalgebraically, with the help of the functor defined below. Definition 5.1.
Let T denote the topological space R ≥ . Then define H : Top → Top as the functorsuch that for any topological space X , and any continuous map g : X → Y , H X = { ( f, d ) ∈ X T × D | f · (cid:102) d = f } , H g = g T × id where D is the one-point compactification of T and (cid:102) d = min( _ , d ) .Intuitively, the functor H : Top → Top captures continuous behaviour as considered in hybrid systems, i.e. the continuous evolutions of physical processes, such as the movement of a plane, or the temperatureof a room. Document [NB16] provides the following specification for the bouncing ball described above.
Definition 5.2.
Use
S, O as shorthand to R ≥ × R , and R , respectively. The bouncing ball is given bythe Set -coalgebra (cid:104) nxt , out (cid:105) : S → S × U H O (cid:104) nxt , out (cid:105) ( p, v ) = (cid:0) (0 , u ) , ( mov ( p, v, _ ) , d ) (cid:1) where variable u corresponds to the (abrupt) change of velocity due to the collision with the ground,function mov ( p, v, _ ) : T → O describes the ball’s movement between jumps, and d denotes the timethat the ball takes to reach the ground. In symbols, u = ( v + gd ) × − . , mov ( p, v, t ) = p + vt + gt , d = √ gp + v + vg . Recall that for each set A the functor ( _ × A ) : Set → Set has a final coalgebra ( cf. [Rut00]), thusproviding a notion of behaviour for the ball. To be more concrete, the coalgebra (cid:0) S, (cid:104) nxt , out (cid:105) (cid:1) hasa canonical homomorphism [( _ )] : S → ( U H O ) ω to the final coalgebra (cid:0) ( U H O ) ω , (cid:104) tl , hd (cid:105) (cid:1) , where tl : ( U H O ) ω → ( U H O ) ω , and hd : ( U H O ) ω → U H O are the ‘tail’ and ‘head’ functions, respectively. Themap (cid:104) tl , hd (cid:105) computes the behaviour of the ball for a given height and velocity. For example, the firstthree elements of [((0 , yield the following plots. . . . . time p o s . . . . time p o s . . . . time p o s In order to bring non-determinism into the scene, suppose, for example, that when the ball hits theground it loses part of its kinetic energy non-deterministically. In this context, one may consider thecoalgebra (cid:104) nxt , out (cid:105) : S → P S × U H O (cid:104) nxt , out (cid:105) ( p, v ) = (cid:0) U, ( mov ( p, v, _ ) , d ) (cid:1) with U = (cid:8) (cid:0) , ( v + gd ) × c (cid:1) ∈ S | c ∈ [ − . , − . (cid:9) . However, the functor ( P × U H O ) : Set → Set has nofinal coalgebra ( cf. [Rut00]), and thus there is no canonical notion of behaviour for the non-deterministicbouncing ball specified above. We will show that the issue can be fixed by shifting to
Top . For this, thefollowing result is useful.
Proposition 5.3.
Let V : Top → Top be the compact Vietoris functor. The family τ = ( τ X,Y ) of maps τ X,Y : ( V X ) × Y → V ( X × Y )( S, y ) (cid:55)→ S × { y } defines a natural transformation Top × Top × (cid:47) (cid:47) V× Id (cid:15) (cid:15) Top × Top V (cid:15) (cid:15) Top × Top × (cid:47) (cid:47) Top . (cid:67) (cid:75) τ Proof.
Let X and Y be topological spaces. For all S ∈ V X and y ∈ Y , since S is compact, the product S × { y } is also compact, which entails that S × { y } ∈ V ( X × Y ) . Then, continuity of the map τ X,Y is adirect consequence of the equalities below. τ − X,Y (cid:104)(cid:16) (cid:91) i ∈ I U i × V i (cid:17) ♦ (cid:105) = (cid:91) i ∈ I ( U i ) ♦ × V i τ − X,Y (cid:104)(cid:16) (cid:91) i ∈ I U i × V i (cid:17) (cid:3) (cid:105) = (cid:91) (cid:110)(cid:16) (cid:91) i ∈ F U i (cid:17) (cid:3) × (cid:92) i ∈ F V i | F ⊆ I finite (cid:111) The proof that all naturality squares commute is straightforward. (cid:3)
Remark . When the compact Vietoris functor is equipped with the natural transformation above itbecomes a strong functor. The latter concept was introduced in [Koc72] and is widely adopted in monadicprogramming.With the natural transformation above, it becomes straightforwad to consider the non-deterministicbouncing ball in a topological setting. Actually, it can be shown to be a coalgebra (cid:104) nxt , out (cid:105) : S → V S × H O First, the map out : S → H O was already shown to be continuous in [NBHM16]. Then, observe that themap nxt : S → V S can be rewritten as a composite S (cid:104) f,g (cid:105) (cid:47) (cid:47) V S × S S τ (cid:47) (cid:47) V ( S × S S ) V ev (cid:47) (cid:47) V S IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 23 f ( p, v ) = { } × [0 . , . g ( p, v ) = λ ( x, y ) ∈ S. (0 , ( v + gd ) × − y ) which proves our claim. One more result is needed. Theorem 5.5.
The functor H : Top → Top can be restricted to the category of Hausdorff spaces.Proof.
Let X be a locally compact space and Y an Hausdorff space. Then, the function space Y X equipped with the compact-open topology is Hausdorff ( cf. [Kel55]). The claim now follows from Haus-dorff spaces being closed under products, and subspaces. (cid:3) As discussed in the previous sections, every compact Vietoris polynomial functor that can be restrictedto the category of Hausdorff spaces has a final coalgebra, which, according to Theorem 5.5, is the case for
V × H O : Top → Top . Intuitively, the elements of the final ( V × H O ) -coalgebra can be seen as compactlybranching trees, i.e. trees where the set of sons of each node is compact. This is similar to the propertyimposed to finitely branching trees, which occur in the final coalgebras involving the finite powersetfunctor ( cf. [Rut00]). Interestingly, the functor V × H O : Top → Top admits an alternative representation:superimpose the evolutions of each level of the tree. To illustrate this, the non-deterministic bouncingball yields the following plots for the first two bounces, with the pair (5 , as the initial state. . . . . . . time p o s . . . . . . time p o s . . . . . . time p o s The notion of stability [Sta01] is another important aspect in the development of hybrid systems.Roughly put, the term ‘stability’ refers to a system’s stability in regard to its behaviour against per-turbations; the system is called stable if small changes in its state (or input) only produce small changesin its behaviour — such a notion is directly related to that of distance between behaviours, which wasalready studied in a coalgebraic setting [BBKK14].In a
Set -based context it is difficult to reason about the stability of a system, because its state space,which is assumed to be just a set, lacks sufficient structure. In the topological setting, however, the issuecan be better handled. To start with, observe that topological spaces already carry a notion of proximity,given by the open sets. Moreover, note that the notion of a stable system is closely related to that ofa continuous map, as discussed, for example, in [Sta01]. This relation can be precisely described in acoalgebraic context: take a functor F : Top → Top , and assume that
CoAlg ( F ) has a final coalgebra ( ν F , ω F ) . Then, for any F -coalgebra ( S, c ) there is a continuous map [( _ )] : S → ν F such that for eachstate s ∈ S , [( s )] is the associated behaviour. Since the map is continuous, ‘close’ states must have ‘close’behaviours, which coincides with our notion of system stability. This suggests the following coalgebraicdefinition of stability. Definition 5.6.
Let F : Top → Top be a functor that admits a final coalgebra. Then a (not necessarilycontinuous) map c : X → F X is called stable if it is a member of
CoAlg ( F ) . In other words, if it is acontinuous map. Examples 5.7.
The bouncing balls (cid:104) nxt , out (cid:105) : S → S × H O , and (cid:104) nxt , out (cid:105) : S → V S × H O arecontinuous maps, and, consequently, stable systems. In this case calling either of the bouncing ballsstable, is to say that a small change in their initial position and velocity does not drastically alter their(possible) trajectories over time. Finally, note that the systems considered here jump between states discretely, as opposed to their outputswhich are, essentially, evolutions in time of specific values. One possible way to accomodate the evolutionof states as well is to consider coalgebras in
CoAlg ( H ) . We will use the results of the previous sections toshow that this category is also complete. Definition 5.8.
Let F : C → C be a functor over a category C with (co)products. We call F exponentpolynomial if it can be recursively defined from the grammar below, with letters A and B denoting,respectively, an arbitrary object and an exponentiable object of C . F ::= F + F | F × F | A | Id | F B Since all exponential functors ( _ ) B : C → C are right adjoints, the following results come almost for free. Proposition 5.9.
All exponent polynomial functors F : Top → Top preserve connected limits.
Corollary 5.10.
The categories of coalgebras of all exponent polynomial functors over
Top are complete.
Theorem 5.11.
The category of coalgebras
CoAlg ( H ) is complete.Proof. The previous corollary assures that the category
CoAlg (cid:0) ( _ ) T × D (cid:1) is complete. Then, observethat the functor H : Top → Top is a subfunctor of ( _ ) T × D : Top → Top , and apply Theorem 4.4. (cid:3)
The previous theorem takes advantage of the adjoint situation below.
CoAlg (cid:0) ( _ ) T × D (cid:1) (cid:62) (cid:40) (cid:40) (cid:104) (cid:104) CoAlg ( H ) Then, with the theorem below, and using the results of the previous sections, we obtain a specificmethod to construct coreflections of (( _ ) T × D ) -coalgebras. Theorem 5.12.
The ‘inclusion’ natural transformation ι : H → ( _ ) T × D is mono-taut.Proof. Consider a monomorhism m : X → Y in Top . We will show that the diagram below is a pullbacksquare. H X H m (cid:47) (cid:47) ι X (cid:15) (cid:15) H Y ι Y (cid:15) (cid:15) X T × D m T × id (cid:47) (cid:47) Y T × D Thus, take two morphisms f : Z → X T × D , g : Z → H Y , and assume that the equation below holds. ( m T × id ) · f = ι Y · g Let z ∈ Z and put ( x, y ) = f ( z ) and ( a, b ) = g ( z ) . Then, by the definition of H , a = a · (cid:102) b since im g ⊆ H Y . Using ( m T × id ) · f = ι Y · g , one gets m · x = m · x · (cid:102) y ; and from this, one obtains x = x · (cid:102) y since m : X → Y is a monomorphism. This shows that the condition im f ⊆ H X holds. Then, since themap ι X : H X → X T × D is an embedding, and im f ⊆ im ι X , there must be a unique arrow h : Z → H X such that ι X · h = f . It remains to show that g = H m · h . This is a direct consequence of the diagramabove being commutative, and the map ι Y : H Y → Y T × D mono. (cid:3) Conclusions and future work
Even if most coalgebraic literature takes
Set as the base category, state-based transition systems oftencall for a shift to other categories, where mechanisms that suitably handle their intricacies are available.Such was the case in [Pan09, Dob09], two research lines on the topic of stochastic systems, and in[KKV04, BFV10, VV14], where the category of Stone spaces and continuous maps plays a key role insetting an appropriate coalgebraic semantics for finitary modal logics.In our case the base category adopted was
Top . As discussed in the previous section, this was becausethe
Set -based context proved to be insufficient for the design of (non-deterministic) hybrid systems,
IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 25 namely in what concerns canonical representations of behaviour and stability. The shift to the topologicalsetting provided, almost for free, a notion of stability (in the spirit of [Sta01]), and showed that a numberof non-deterministic hybrid systems in
Top have an associated final coalgebra, even if in
Set they do not.Both results were achieved using this paper’s theoretical developments. But again, we stress that thelatter can be applied to other contexts as well.The relevance of Vietoris coalgebras for different topics is further witnessed by the common existenceof important limits in their categories of coalgebras. We saw that every compact Vietoris polynomialfunctor admits a final coalgebra if it can be restricted to the category
Haus while every lower Vietorispolynomial functor admits a final coalgebra if it can be restricted to
StablyComp . Moreover, we saw thatseveral variants of such functors also inherit these results and that all categories of Vietoris coalgebrashave equalisers.However, several theoretical questions concerning limits in categories of Vietoris coalgebras still re-main open. For example, we studied codirected limit preservation by Vietoris functors under differenttopological contexts (see Section 3), showing cases in which they were preserved, and cases in which theywere not. But we are still not precisely sure what is the ‘weakest’ context in which they are preserved.Another example concerns the existence of products in categories of Vietoris coalgebras. Recall alsoour study of topological functors between categories of coalgebras. Among other things, it provides afull characterisation of situations in which it is possible to systematically lift well-known results aboutcoalgebras over
Set to coalgebras over other categories. We saw that this is indeed the case betweencoalgebras of polynomial functors over
Set and their counterparts in
Top , but we are also interested inother situations. Two prime examples that we will explore in future work pertain coalgebras over thecategory
Ord and coalgebras over the category
PMet . These coalgebras have significant relevance withinthe coalgebraic community ( e.g. [BBKK14, BK11, BKV13]) and we believe that our study can contributeto the topic.On a note closer to practice, the use of topologies to specify and analyse (non-deterministic) hybridsystems brings a number of benefits, which were just barely grasped in this paper. Our main goal is tofurther explore them in the near future. The plan is to do so in a coalgebraic component-based approach[Bar03, HJ11], where simple hybrid systems can be composed to form more complex ones. The resultsthat this paper reports provide an interesting step in this direction.
Acknowledgements
We are grateful to Lawrence Moss for many fruitful discussions and for his helpful references on thesubject. This work is financed by the ERDF – European Regional Development Fund through the Oper-ational Programme for Competitiveness and Internationalisation – COMPETE 2020 Programme and byNational Funds through the Portuguese funding agency, FCT – Fundação para a Ciência e a Tecnologiawithin project POCI-01-0145-FEDER-016692. We also gratefully acknowledge partial financial assistanceby Portuguese funds through CIDMA (Center for Research and Development in Mathematics and Appli-cations), and the Portuguese Foundation for Science and Technology (“FCT – Fundação para a Ciênciae a Tecnologia”), within the project UID/MAT/04106/2013. Finally, Renato Neves and Pedro Nora arealso supported by FCT grants SFRH/BD/52234/2013 and SFRH/BD/95757/2013, respectively.
References [Adá05] Jiří Adámek. Introduction to coalgebra.
Theory and Applications of Categories , 14(8):157–199, 2005.[AHS90] Jiří Adámek, Horst Herrlich, and George E. Strecker.
Abstract and concrete categories: The joy of cats . Pureand Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1990. Republished in: Reprints inTheory and Applications of Categories, No. 17 (2006) pp. 1–507.[Alu15] Rajeev Alur.
Principles of Cyber-Physical Systems . MIT Press, 2015.[AMM16] Jiří Adámek, Stefan Milius, and Lawrence S. Moss.
Initial Algebras and Terminal Coalgebras . In preparation,2016. [Bar93] Michael Barr. Terminal coalgebras in well-founded set theory.
Theoretical Computer Science , 114(2):299–315,June 1993.[Bar03] Luís S. Barbosa. Towards a calculus of state-based software components.
Journal of Universal Computer Science ,9:891–909, 2003.[BBH12] Guram Bezhanishvili, Nick Bezhanishvili, and John Harding. Modal compact Hausdorff spaces.
Journal ofLogic and Computation , 25(1):1–35, July 2012, eprint: http://logcom.oxfordjournals.org/content/early/2012/07/07/logcom.exs030.full.pdf+html .[BBKK14] Paolo Baldan, Filippo Bonchi, Henning Kerstan, and Barbara König. Behavioral metrics via functor lifting. InVenkatesh Raman and S. P. Suresh, editors, , volume 29 of
LIPIcs , pages 403–415. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2014.[BFV10] Nick Bezhanishvili, Gaëlle Fontaine, and Yde Venema. Vietoris bisimulations.
Journal of Logic and Computation ,20(5):1017–1040, 2010.[BK11] Adriana Balan and Alexander Kurz. Finitary functors: From set to preord and poset. In Andrea Corradini,Bartek Klin, and Corina Cîrstea, editors,
Algebra and Coalgebra in Computer Science - 4th InternationalConference, CALCO 2011, Winchester, UK, August 30 - September 2, 2011. Proceedings , volume 6859 of
Lecture Notes in Computer Science , pages 85–99. Springer, 2011.[BKR07] M. M. Bonsangue, A. Kurz, and I. M. Rewitzky. Coalgebraic representations of distributive lattices with oper-ators.
Topology and its Applications , 154(4):778–791, February 2007.[BKV13] Adriana Balan, Alexander Kurz, and Jiri Velebil. Positive fragments of coalgebraic logics. In Reiko Heckel andStefan Milius, editors,
Algebra and Coalgebra in Computer Science - 5th International Conference, CALCO2013, Warsaw, Poland, September 3-6, 2013. Proceedings , volume 8089 of
Lecture Notes in Computer Science ,pages 51–65. Springer, 2013.[Bou42] Nicolas Bourbaki.
Éléments de mathématique. 3. Pt. 1: Les structures fondamentales de l’analyse. Livre 3:Topologie générale. Chap. 3: Groupes topologiques. Chap. 4: Nombres réels.
Paris: Hermann & Cie., 1942.[BRS09] Marcello M. Bonsangue, Jan J. M. M. Rutten, and Alexandra Silva. An algebra for kripke polynomial coalgebras.In
Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, LICS 2009, 11-14 August2009, Los Angeles, CA, USA , pages 49–58. IEEE Computer Society, 2009.[Che14] Liang-Ting Chen.
On a purely categorical framework for coalgebraic modal logic . PhD thesis, University ofBirmingham, 2014.[CLP91] R. Cignoli, S. Lafalce, and A. Petrovich. Remarks on Priestley duality for distributive lattices.
Order , 8(3):299–315, 1991.[CT97] Maria Manuel Clementino and Walter Tholen. A characterization of the Vietoris topology. In
Proceedings of the12 th Summer Conference on General Topology and its Applications (North Bay, ON, 1997) , volume 22, pages71–95, 1997.[DDG16] Fredrik Dahlqvist, Vincent Danos, and Ilias Garnier. Giry and the Machine.
Electronic Notes in TheoreticalComputer Science , 325:85–110, October 2016. The Thirty-second Conference on the Mathematical Foundationsof Programming Semantics (MFPS XXXII).[Dob09] Ernst-Erich Doberkat.
Stochastic Coalgebraic Logic . Monographs in Theoretical Computer Science. An EATCSSeries. Springer, 2009.[Dud72] Roman Duda. One result on inverse limits and hyperspaces.
General Topology and its Relations to ModernAnalysis and Algebra , pages 99–102, 1972.[Eng89] Ryszard Engelking.
General topology , volume 6 of
Sigma Series in Pure Mathematics . Heldermann Verlag,Berlin, second edition, 1989. Translated from the Polish by the author.[GHK +
80] Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S.Scott.
A compendium of continuous lattices . Springer-Verlag, Berlin, 1980.[GHK +
03] Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S.Scott.
Continuous lattices and domains , volume 93 of
Encyclopedia of Mathematics and its Applications . Cam-bridge University Press, Cambridge, 2003.[Gou13] Jean Goubault-Larrecq.
Non-Hausdorff Topology and Domain Theory—Selected Topics in Point-Set Topology ,volume 22 of
New Mathematical Monographs . Cambridge University Press, March 2013.[GS01] Peter H. Gumm and Tobias Schröder. Products of coalgebras.
Algebra Universalis , 46(1):163–185, 2001.[Hau14] Felix Hausdorff.
Grundzüge der Mengenlehre . Leipzig: Veit & Comp. VIII and 476 pages, 1914.[HJ11] Ichiro Hasuo and Bart Jacobs. Traces for coalgebraic components.
Mathematical Structures in Computer Sci-ence , 21(2):267–320, 2011.[HN16] Dirk Hofmann and Pedro Nora. Enriched Stone-type dualities. Technical report, April 2016, arXiv:1605.00081[math.CT] .[Hof99] Dirk Hofmann.
Natürliche Dualitäten und das verallgemeinert Stone-Weierstraß Theorem . PhD thesis, Univer-sity of Bremen, 1999.
IMITS IN CATEGORIES OF VIETORIS COALGEBRAS 27 [Hug01] Jess Hughes.
A study of categories of algebras and coalgebras . PhD thesis, Carnegie Mellon University, 2001.[Jac12] Bart Jacobs. Introduction to coalgebra. towards mathematics of states and observations., 2012.[Jun04] Achim Jung. Stably compact spaces and the probabilistic powerspace construction. In J. Desharnais andP. Panangaden, editors,
Domain-theoretic Methods in Probabilistic Processes , volume 87 of entcs , pages 5–20. Elsevier, November 2004. 15pp.[Kel55] John Kelley.
General Topology . Van Nostrand, 1955. Reprinted by Springer-Verlag, Graduate Texts in Mathe-matics, 27, 1975.[KKV04] Clemens Kupke, Alexander Kurz, and Yde Venema. Stone coalgebras.
Theoretical Computer Science , 327(1-2):109–134, October 2004.[Koc72] Anders Kock. Strong functors and monoidal monads.
Archiv der Mathematik , 23(1):113–120, 1972.[Kur01] Alexander Kurz.
Logics for coalgebras and applications to computer science . BoD–Books on Demand, 2001.[Lin69] F. E. J. Linton. Coequalizers in categories of algebras. In
Seminar on Triples and Categorical Homology Theory ,pages 75–90. Springer, Berlin, 1969. Republished in: Reprints in Theory and Applications of Categories, No. 18(2008) pp. 61-72.[Mac71] Saunders MacLane.
Categories for the working mathematician . Springer-Verlag, New York, 1971. GraduateTexts in Mathematics, Vol. 5.[Man02] Ernest G. Manes. Taut monads and T -spaces. Theoretical Computer Science , 275(1-2):79–109, March 2002.[Mic51] Ernest Michael. Topologies on spaces of subsets.
Transactions of the American Mathematical Society , 71(1):152–182, January 1951.[Möb83] Axel Möbus. Alexandrov compactification of relational algebras.
Archiv der Mathematik , 40(6):526–537, De-cember 1983.[Nac65] Leopoldo Nachbin.
Topology and order . Translated from the Portuguese by Lulu Bechtolsheim. Van NostrandMathematical Studies, No. 4. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1965.[NB16] Renato Neves and Luís Soares Barbosa. Hybrid automata as coalgebras. In Augusto Sampaio and Farn Wang,editors,
Theoretical Aspects of Computing - ICTAC 2016 - 13th International Colloquium, Taipei, Taiwan,ROC, October 24-31, 2016, Proceedings , volume 9965 of
Lecture Notes in Computer Science , pages 385–402,2016.[NBHM16] Renato Neves, Luis S. Barbosa, Dirk Hofmann, and Manuel A. Martins. Continuity as a computational effect.
Journal of Logical and Algebraic Methods in Programming , 85(5):1057–1085, August 2016.[Pan09] Prakash Panangaden.
Labelled Markov Processes . Imperial College Press, 2009.[Pet96] Alejandro Petrovich. Distributive lattices with an operator.
Studia Logica , 56(1-2):205–224, 1996. Special issueon Priestley duality.[Pom05] Dimitrie Pompeiu. Sur la continuité des fonctions de variables complexes.
Annales de la Faculté des Sciences del’Université de Toulouse pour les Sciences Mathématiques et les Sciences Physiques. 2ième Série , 7(3):265–315,1905.[PW98] John Power and Hiroshi Watanabe. An axiomatics for categories of coalgebras.
Electronic Notes in TheoreticalComputer Science , 11:158–175, 1998.[Rut00] J.J.M.M. Rutten. Universal coalgebra: a theory of systems.
Theoretical Computer Science , 249(1):3 – 80, 2000.Modern Algebra.[Sch93] Andrea Schalk.
Algebras for Generalized Power Constructions . PhD thesis, Technische Hochschule Darmstadt,1993.[Sim82] Harold Simmons. A couple of triples.
Topology and its Applications , 13(2):201–223, March 1982.[Sta01] Thomas Stauner.
Systematic development of hybrid systems . PhD thesis, Technische Uuniversität München,2001.[Tab09] Paulo Tabuada.
Verification and Control of Hybrid Systems - A Symbolic Approach . Springer, 2009.[Tho09] Walter Tholen. Ordered topological structures.
Topology and its Applications , 156(12):2148–2157, July 2009.[Vie22] Leopold Vietoris. Bereiche zweiter Ordnung.
Monatshefte für Mathematik und Physik , 32(1):258–280, December1922.[Vig05] Ignacio Darío Viglizzo.
Coalgebras on measurable spaces . PhD thesis, Department of Mathematics, IndianaUniversity, 2005.[VV14] Yde Venema and Jacob Vosmaer. Modal logic and the vietoris functor. In Guram Bezhanishvili, editor,
LeoEsakia on Duality in Modal and Intuitionistic Logics , pages 119–153. Springer Netherlands, Dordrecht, 2014.[Wat72] William C. Waterhouse. An empty inverse limit.
Proceedings of the American Mathematical Society , 36(2):618,February 1972.[Zen70] Phillip Zenor. On the completeness of the space of compact subsets.
Proceedings of the American MathematicalSociety , 26(1):190–192, 1970.
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