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Quasi-uniform and syntopogenous structures on categories
Topology and its Applications 263 (2019) 16–25
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Topology and its Applications www.elsevier.com/locate/topol
Quasi-uniform and syntopogenous structures on categories ✩ David Holgate, Minani Iragi ∗ Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville 7535, South Africa
a r t i c l e
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Article history: Received 30 April 2018 Received in revised form 7 August 2018 Accepted 26 September 2018 Available online 31 May 2019 MSC: 18A20 54A15 54B30 54E15
a b s t r a c t We introduce and study an abstract notion of Császár’s syntopogenous structure which provides a convenient setting to investigate a quasi-uniformity on a category. We show that a quasi-uniformity is a family of categorical closure operators. In particular, it is shown that every idempotent closure operator is quasi-uniformity. Various notions of completeness of objects and precompactness with respect to a quasi-uniformity defined in a natural way are also studied. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Keywords: Closure operator Categorical topogenous structure Syntopogenous structure Quasi-uniform structure and completeness
1. Introduction The unified approach to the categorical closure ([4]), interior ([14]) and neighbourhood operators ([8]) provided by the recently introduced notion of categorical topogenous structures ([7,9]) has shed a light on the study of a concept of quasi-uniformity on an abstract category. In a category C with an (E, M)-factorization system, this paper introduces a concept of syntopogenous structure ([3]) that is used to study a quasiuniformity ([6]). Classical notions and results on quasi-uniform spaces are expressed in a more general categorical setting. This leads to new results that are applied to specific examples in Topology and Algebra. Although the syntopogenous structure on C appears as an appropriate family of order relations on the sub✩
This work was supported by the South African National Research Foundation.
* Corresponding author. E-mail addresses: [email protected] (D. Holgate), [email protected] (M. Iragi). https://doi.org/10.1016/j.topol.2019.05.024 0166-8641/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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object lattice, subX, for any object X of C, a quasi-uniformity on C is thought of as a suitably axiomatized family of endomaps on subX for any X ∈ C. The definitions obtained include the fact that every morphism in C must be continuous with respect to the structure. A subconglomerate of the conglomerate of all syntopogenous structures which is isomorphic to the conglomerate of all quasi-uniform structures on C is defined and studied. This leads to the observation that a quasi-uniformity is a family of closure operators. Since every interpolative topogenous order is shown to be a syntopogenous structure and the class of these orders is known (see [9]) to be essentially equivalent to the conglomerate of all idempotent closure operators, it will be proved that every idempotent closure operator is a quasi-uniformity on C. We shall then define and study (different kinds of) complete and precompact objects of C with respect to a quasi-uniformity. It is a pleasure for the authors to thank Professor Hans-Peter Künzi and Doctor Francesco G. Russo for organising this wonderful workshop. 2. Preliminaries For categorical matters, we refer the reader to [1]. The basic facts on categorical closure operators and topogenous structures used here can be found in [5] and [7] respectively. Throughout the paper, we consider a category C and a fixed class M of C-monomorphisms which is closed under composition and contains all isomorphisms of C. The category C is assumed to be M-complete so that pullbacks of M-morphisms along C-morphisms and arbitrary M-intersections of M-morphisms exist and are again in M. For any C-object X, subX is the class of M-morphisms with codomain X. The elements of subX are called subobjects of X. They are ordered by n ≤ m ⇔ ∃ j | m ◦ j = n. If m ≤ n and n ≤ m then they are isomorphic. We shall simply write m = n in this case. A number of properties can be derived from the above assumptions on M, namely; subX is a complete lattice with 0X : OX −→ X the least element and 1X : X −→ X the largest, there is a class E of C-morphisms such that (E, M) is factorization structure for morphisms in C (see [5]), each C-morphism f : X −→ Y induces an image/preimage adjunction f (m) ≤ n ⇔ m ≤ f −1 (n) for all n ∈ subY , m ∈ subX with f (m) given by the M-component of the (E, M)-factorization of f ◦ m and f −1 (n) the pullback of n along f . We shall sometimes find it important to assume that for any C-morphism f : X −→ Y , the inverse image f −1 commutes with the joins of subobjects so that it has a right adjoint f∗ given by f∗ (m) = {n ∈ subY | f −1 (n) ≤ m}. Thus f −1 (n) ≤ m ⇔ n ≤ f∗ (m), f −1 (f∗ (m)) ≤ m (with f −1 (f∗ (m)) = m if f ∈ M) and n ≤ f∗ (f −1 (n)) (with f∗ (f −1 (n)) = n if f ∈ E and E stable under pullbacks). Definition 2.1. A closure operator c on C with respect to M is given by a family of maps {cX : subX −→ subX | X ∈ C} such that: (C1) m ≤ cX (m) for all m ∈ subX; (C2) m ≤ n ⇒ cX (m) ≤ cX (n) for all m, n ∈ subX; (C3) every morphism f : X −→ Y is c-continuous, that is: f (cX (m)) ≤ cY (f (m)) for all m ∈ subX. We denote by CLOS the conglomerate of all closure operators on C with respect to M ordered as follows: c ≤ c if cX (m) ≤ cX (m) for all m ∈ subX and X ∈ C.
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Definition 2.2. A topogenous order on C is a family { X | X ∈ C} of relations, each X on subX, such that: (T 1) m X n ⇒ m ≤ n for every m, n ∈ subX, (T 2) m ≤ n X p ≤ q ⇒ m X q for every m, n, p, q ∈ subX, and (T 3) every morphism f : X −→ Y in C is -continuous, m Y n ⇒ f −1 (m) X f −1 (n) for every m, n ∈ subY . Given two topogenous orders and on C, ⊆ if and only if m X n ⇒ m X n for all m, n ∈ subX. The resulting ordered conglomerate of all topogenous orders on C is denoted by T ORD. A topogenous order is said to be (1) -preserving if (∀i ∈ I : m X ni ) ⇒ m X ni , and (2) interpolative if m X n ⇒ (∃ p) | m X p X n for all X ∈ C. The ordered conglomerate of all -preserving and interpolative topogenous orders is denoted by −T ORD and IN T ORD respectively. −IN T ORD will denote the conglomerate of all interpolative meet preserving topogenous orders. Proposition 2.1. by
−T ORD is order isomorphic to CLOS. The inverse morphisms of each other are given c X (m) =
{p | m X p} and m cX n ⇔ cX (m) ≤ n for all X ∈ C.
Proposition 2.1 will play a vital role in understanding the relationship between closure and quasiuniformity on C. 3. Quasi-uniform structure or family of closure operators Definition 2.2 allows us to generalize in a natural way the syntopogenous structure on C. Definition 3.1. A syntopogenous structure on C with respect to M is a family S = {SX | X ∈ C} such that each SX = { iX | i ∈ IX } is a family of relations on subX satisfying: (S1) iX is a relation on subX satisfying (T 1) and (T 2). (S2) { iX | i ∈ IX } is a directed set with respect to inclusion. (S3) X = i∈IX iX is an interpolative topogenous order. We can extend the ordering of topogenous orders to syntopogenous structures in the following way: S ≤ S if for all X ∈ C and iX ∈ SX , there is jX ∈ S X such that iX ⊆ jX . The resulting conglomerate will be denoted by SY N T . We are interested in particular classes of syntopogenous structures as these will play an important role in what follows. Definition 3.2. We shall say that S ∈ SY N T is: (1) co-perf ect if each iX ∈ SX is -preserving for all X ∈ C. (2) interpolative if for every iX ∈ SX , m X n ⇒ (∃ p) | m iX p iX n for all X ∈ C. (3) simple if SX = { X } is an interpolative topogenous order for any X ∈ C.
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The ordered conglomerate of all co-perfect (resp. interpolative) syntopogenous structures will be denoted by CSY N T (resp. IN T SY N T ). It is readily seen from Definition 3.2 that IN T ORD is the class of simple syntopogenous structures. A routine check shows that IN T ORD is closed under arbitrary unions in SY N T . Consequently, it is a coreflective sub-quasicategory of SY N T . The observation that an (entourage) quasi-uniformity on a set X can be equivalently expressed as a family of maps f : X −→ P(X) which can easily be extended to a family of maps f : P(X) −→ P(X), motivates the point in our definition that a quasi-uniformity on C will be given as a family of endomaps on subX for each X ∈ C. Let X ∈ C, subX being a complete lattice can be seen as a category, monotone maps f [−], g[−] from subX to subX are the functors and there is a natural transformation α : f [−] ⇒ g[−] exactly when f (m) ≤ g(m) for all m ∈ subX. It turns out that F(subX), the endofunctor category on subX is ordered by ≤ “pointwise”. Definition 3.3. A quasi-unif ormity on C with respect to M is a family U = {UX | X ∈ C} with UX a full subcategory of F(subX) for each X such that: (U 1) For any U ∈ UX , 1 ≤ U (where 1 is the identity map on subX). (U 2) For any U ∈ UX , there is U ∈ UX such that U ◦ U ≤ U . (U 3) For any U ∈ UX and U ≤ U , U ∈ UX . (U 4) For any U, U ∈ UX , U ∧ U ∈ UX . (U 5) For any C-morphism f : X −→ Y and U ∈ UY , there is U ∈ UX such that f (U (m)) ≤ U (f (m)) for all m ∈ subX. We shall denote by QU N IF the conglomerate of all quasi-uniform structures on C. It is ordered as follows: U ≤ V if for all X ∈ C and U ∈ UX , there is V ∈ VX such that V ≤ U . We shall often describe a quasi-uniformity by defining a base for it. Definition 3.4. A full subcategory BX of F(subX) for all X ∈ C is a base for a quasi-uniformity on C if it satisfies all the axioms in Definition 3.3 except (U 3). Definition 3.5. A base for quasi-uniformity on C is transitive if for all X ∈ C and U ∈ BX , U ◦ U = U . A quasi-uniformity with a transitive base is called a transitive quasi-unif ormity. The ordered conglomerate of all transitive quasi-uniformities on C will be denoted by T QU N IF . ∗ ∗ | X ∈ C} where UX = {U ∗ | U ∈ UX } with Definition 3.6. Let U be a quasi-uniformity on C, then U ∗ = {UX ∗ U (m) = {n | m ≤ U (n)} is a base of a quasi-uniformity on C called the conjugate quasi-unif ormity of U. ∗ We say that a quasi-uniformity U on C is a uniformity provided that UX = UX for every X ∈ C, equivalently if for every U ∈ UX and X ∈ C, there is V ∈ U such that m ≤ U (n) ⇔ n ≤ V (m) for any m, n ∈ subX. While our intention is to study a quasi-uniformity on a category, we have found it more fruitful to express many of our proofs in terms of a syntopogenous structure. The next theorem that describes the clear relationship between the two structures shall lead us to the achievement of this goal. Let S ∈ SY N T , B S will denote the base for a quasi-uniformity induced by S. If B is a base for a quasi-uniformity, S B denotes the syntopogenous structure induced by B.
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Theorem 3.1. QU N IF is order isomorphic to CSY N T . The inverse morphisms of each other U −→ S U and S −→ U S are given by B U SX = { U X | U ∈ BX } where m X n ⇔ U (m) ≤ n, and S = {U | X ∈ SX } where U (m) = {n | m X n} BX
for all X ∈ C and m, n ∈ subX. S
Proof. It is easily seen that U −→ S U and S −→ U S are well defined and order preserving. Now, S B = S X X = X , m U n ⇔ U (m) ≤ n ⇔ {p | m X n} ≤ n ⇔ m X n for all since for any X ∈ SX , U X X B U X ∈ C and B S = B since U (m) = {n | U (m) ≤ n} = U (m) for any U ∈ B and m ∈ subX. 2 Since S ⊆ −T ORD for each S ∈ CSY N T , it follows from the above theorem and Proposition 2.1 that a quasi-uniformity on C is a family of closure operators. It is proved in ([9], Corollary 4.2.3) that −IN T ORD is isomorphic to the conglomerate of idempotent closure operators and from Theorem 3.1, CSY N T ∼ = QU N IF . Thus every idempotent closure operator on C is a quasi-uniformity. Let us denote by IN T CSY N T the conglomerate of all interpolative co-perfect syntopogenous structures on C. We next prove that interpolative co-perfect syntopogenous structures are exactly the transitive quasi-uniformities. Proposition 3.2. IN T CSY N T is order isomorphic to T QU N IF . U Proof. Let B be a transitive base and m U X n for any U ∈ BX and X ∈ C, then U (m) = U (U (m)) X U U U (m) ≤ n, hence m X U (m) X n. Conversely if S ∈ IN T CSY N T and X ∈ SX , m X n ⇒ (∃ p) | m X p X n ⇒ U X (m) ≤ p X n ⇒ U X (m) ≤ n. This implies that {l | U X (m) X l} ≤ {q | m q} that is U X (U X (m)) ≤ U X (m) and U X (m) ≤ U X (U X (m)) follows from (U 1) so that U X (m) = U X (U X (m)). 2
We often write X ∈ SX instead of iX ∈ SX to ease the notation. We now have a bridge from quasi-uniformity to syntopogenous structure and back. This allows us to always compare the results we obtain for the two structures and work at the side where the proofs are easier to manipulate. For the rest of this section we assume that for any C-morphism f : X −→ Y , f −1 commutes with the joins of subobjects. Definition 3.7. Let f : X −→ Y be a C-morphism and S be a syntopogenous structure on C. Then f is S-initial if for every X ∈ SX there is Y ∈ SY such that m X n ⇒ f (m) Y f∗ (n) for all m, n ∈ subX. Proposition 3.3. Let f : X −→ Y be a C-morphism and S ∈ CSY N T . Then f is S-initial if and only if for every U ∈ BX there is U ∈ BY such that f −1 (U (f (m))) ≤ U (m) for all m ∈ subX. Proof. Assume that for any U ∈ BX there is U ∈ BY such that f −1 (U (f (m))) ≤ U (m) for all m ∈ subX and m X n for some X ∈ SX and m, n ∈ subX. Then there is U ∈ BX such that U = X and there is U ∈ BY such that f −1 (U (f (m)) ≤ U (m) ⇒ f −1 (U (f (m)) ≤ U (m) ≤ n ⇒ f −1 (U (f (m)) ≤ n ⇔ U (f (m)) ≤ f∗ (n) ⇔ f (m) U Y f∗ (n). Conversely if S ∈ CSY N T and f is S-initial, then for all U ∈ BX and m, n subX, U (m) ≤ n ⇔ (∃ X ∈ SX ) | m X n ⇔ (∃ Y ∈ SY ) | f (m) Y f∗ (n) ⇔ (∃ U ∈ BY ) | U (f (m)) ≤ f∗ (n) ⇔ f −1 (U (f (m))) ≤ n. Thus f −1 (U (f (m))) ≤ U (m) which completes the proof. 2
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Definition 3.8. A source (fi : X −→ Xi )i∈I is S-initial if for any X ∈ SX there are i ∈ I and Xi ∈ SXi such that m X n ⇒ f (m) Xi (fi )∗ (n). A similar reasoning to the one in Proposition 3.3 produces the following. Proposition 3.4. Let (fi : X −→ Xi )i∈I be a source in C and S ∈ CSY N T . Then (fi )i∈I is S-initial if and only if for any U ∈ BX there are i ∈ I and U ∈ BXi such that (fi )−1 (U (fi (m))) ≤ U (m) for all m ∈ subX. Example 3.1. By Theorem 3.1, which establishes an equivalence between co-perfect syntopogenous structures and quasi-uniformities on a category, it will be enough to define a co-perfect syntopogenous structure when exhibiting examples of a categorical quasi-uniformity. Already our first example shows that the classical quasi-uniformity is a particular co-perfect syntopogenous structure and hence a particular categorical quasi-uniformity. (a) Let C be Qunif , the category of quasi-uniform spaces and quasi-uniformly continuous maps ([6]) with M the class of quasi-uniform embeddings. Every (X, D) ∈ Qunif gives is a co-perfect syntopogenous structure, D S(X,D) = { D (X, D) | D ∈ D} on Qunif where A (X, D) B ⇔ D[A] ⊆ B and A, B ⊆ X. On the other hand, it is well known from [3] that the co-perfect syntopogenous structure that describes a quasi-uniformity is the one satisfying the property that, for all A ∈ A, A B ⇒ A B (A ⊆ 2X ). A quasi-uniformly continuous map f : (X, D) −→ (Y, D ) is S-initial provided D is the initial quasi-uniformity induced by f on X. (b) In the category TopGrp of topological groups and continuous group homomorphisms, let (E, M) be the (surjective, injective)-factorization structure. For any (X, ·) ∈ TopGrp, let η(e) be the neighbourhood U filter of the identity element e. Then SX = { U X | U ∈ η(e)} with A X B ⇔ U · A ⊆ B is a co-perfect syntopogenous structure on TopGrp. We have proved that a single co-perfect syntopogenous structure is equivalent to an idempotent closure operator. Below we present a number of examples derived from idempotent closure operators. (c) Let C = Grp be the category of groups and group homomorphisms with (epi, mono)-factorization system. For any A, B ≤ G, SG = { G | G ∈ Grp} with A G B ⇔ A ≤ N ≤ B with N G is a co-perfect simple syntopogenous structure on Grp. (d) Let C = Top the category of topological spaces and continuous maps with M the class of embeddings and (of course) E the class of surjective continuous maps. For A, B ⊆ X, (i) S(X, T ) = { (X, T ) | (X, T ) ∈ Top} with A X B ⇔ A ⊆ C ⊆ B for some closed C ⊆ X is a single co-perfect syntopogenous structure. A continuous map f : (X, TX ) −→ (Y, TY ) is S-initial iff TX is the initial topology induced by f on X. Another single co-perfect syntopogenous structure on Top that is not obtained from a idempotent closure operator is (ii) S(X, T ) = { (X, T ) | (X, T ) ∈ Top} with A X B ⇔ A ⊆ O ⊆ B for some O ∈ T . 4. The S-Cauchy filters Since for any X ∈ C, subX is a complete lattice, by a filter on X we shall always understand a filter on subX. Lemma 4.1. Let X ∈ C and F be a filter on X. Then F is an ultrafilter on X if and only if m ∧ n > 0X for all n ∈ F implies that m ∈ F.
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Let f : X −→ Y be a C-morphism and F a filter on X then f (F) is the filter defined by m ∈ f (F) ⇔ f −1 (m) ∈ F. It is an ultrafilter on Y provided F is an ultrafilter on X and f ∈ E with E being pullback stable along M-morphisms. If F is an ultrafilter on Y and f reflects 0 then f −1 (F) is also an ultrafilter on X where f −1 (F) is the filter defined by m ∈ f −1 (F) ⇔ f (m) ∈ F. For any X ∈ C, we shall denote by subo X, the class of all m ∈ subX such that m > 0X and for all S m ∈ subo X, νX (m) = {n | m X n for some X ∈ S}. We note that ν S (m) does not form a filter in general unless S is co-perfect. Definition 4.1. Let S ∈ CSY N T , X ∈ C and F be a filter on X. We say that F converges to a subobject m ∈ S subo X with respect to S and write F −→ m if for any X ∈ SX , m X n ⇒ n ∈ F for n ∈ subX. F is S-Cauchy if for any X ∈ SX there is m ∈ subo X such that m X n ⇒ n ∈ F. S
S One easily checks that for any S ∈ CSY N T and X ∈ C, νX (m) −→ m for all m ∈ subo X and that every S-convergent filter on X is S-Cauchy.
Definition 4.2. Let S ∈ SY N T and X ∈ C. A filter F on X is said to be S-Cauchy if it is Cauchy with respect to any S ∗ ∈ CSY N T that is coarser than S. F converges with respect to S if it converges with respect to any S ∗ ∈ CSY N T such that S ∗ ≤ S. For the rest of this paper, we assume that C has products, the class E is stable under pullbacks along M-morphisms and any f ∈ C reflects 0. Proposition 4.2. Let f : X −→ Y be a C-morphism, F a filter on X and S ∈ CSY N T . S
S
(a) If F −→ m then f (F) −→ f (m). The converse implication holds if f is S-initial. (b) If F is S-Cauchy on X then so is f (F) on Y . The converse holds if f is S-initial and belongs to E. S
Proof. (a) Assume that F −→ m with m ∈ subo X. Then f (m) ∈ subo Y . Let f (m) Y p for any Y ∈ SY and p ∈ subY . Then by (S4) there is X ∈ SX such that m X f −1 (p) and f −1 (p) ∈ F ⇔ p ∈ f (F). Conversely if f (F) −→ f (m) and f (m) ∈ subo Y . Then m ∈ subo X. Let m X n for any X ∈ SX . Then by S-initiality of f , there is Y ∈ SY such that f (m) Y f∗ (n) and f∗ (n) ∈ f (F). Thus f −1 (f∗ (n)) ∈ F which implies that n ∈ F. (b) Let F be S-Cauchy and Y ∈ SY . Then by (S4) there is X ∈ SX such that f (p) Y q ⇒ p X f −1 (q) and there is m ∈ subo X such that m X n ⇒ n ∈ F. Thus f (m) ∈ subo Y and f (m) Y l ⇒ m X f −1 (l) ⇒ f −1 (l) ∈ F ⇔ l ∈ F. Conversely if f (F) is S-Cauchy and X ∈ SX , then there is Y ∈ SY such that m X n ⇒ f (m) Y f∗ (n) and there is p ∈ subo Y such that p Y q ⇒ q ∈ f (F) for any p ∈ subY . Now f −1 (p) ∈ subo X since f ∈ E and f −1 (p) X n ⇒ p = f (f −1 (p)) Y f∗ (n) ⇒ f∗ (n) ∈ f (F) ⇔ f −1 (f∗ (n)) ∈ F ⇒ n ∈ F. 2 Proposition 4.3. Let X = is an S-initial source. S
i∈I
Xi be a product in C and F a filter on X. Assume that (pi : X −→ Xi )i∈I S
(1) F −→ m if and only if pi (F) −→ pi (m) for each i. (2) If for each i ∈ I the projections belong to E, then F is S-Cauchy if and only if pi (F) is S-Cauchy for each i. Definition 4.3. Let U be a quasi-uniformity on C and F filter on X. We shall say that F converges to m U (m ∈ subo X) with respect to U and write F −→ m if for any U ∈ UX , U (m) ∈ F. F is U-cauchy if for each U ∈ UX there is m ∈ subo X such that U (m) ∈ F.
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If U and V are quasi-uniform structures on C such that U ≤ V, then every V-Cauchy filter is U-Cauchy. S
US
Proposition 4.4. Let S ∈ CSY N T and F a filter on X. Then F −→ m ⇔ F −−→ m S
Proof. Let m ∈ subo X and F −→ m. If U ∈ UX then, by Theorem 3.1 there is X ∈ SX such that X = U X US
and m X n ⇒ n ∈ F. Now, m X U (m) and U (m) ∈ F. Conversely if F −−→ m and m X n then S U (m) ≤ n for some U ∈ UX and U (m) ∈ F. Hence n ∈ F. 2 Theorem 4.5. Let S ∈ CSY N T and F be a filter on X. Then F is U S -Cauchy if and only if it is S-Cauchy. S Proof. Let F be U S -Cauchy and X ∈ SX . Since S ∈ CSY N T , by Theorem 3.1 there is U ∈ UX that determines X and there is m ∈ subo X such that U (m) ∈ F. Now, m X n ⇔ U (m) ≤ n ⇒ n ∈ F. S Conversely if F is S-Cauchy and U ∈ UX then there is X ∈ SX such that U X = X and there is m ∈ subo X such that m X n ⇒ n ∈ F. Now m X U (m) since S ∈ CSY N T and U (m) ∈ F. 2
Left (resp. right) Cauchy filters in quasi-uniform spaces have been investigated in ([10]) and these lead to left (resp. right) complete quasi-uniform spaces which are closely related to the Smyth complete quasiuniform spaces ([13]). It seems that our co-perfect syntopogenous structure provides a simple and natural way of stating these notions in categorical language and link them to those obtained above. Let us also note that a categorical approach to convergence has already been studied in the literature (see e.g [11]). Definition 4.4. A filter F on an object X ∈ C will be called: (1) S-round if for every n ∈ F, there are m ∈ F and X ∈ SX such that m X n. (2) W eakly S-Cauchy (or simply wS-Cauchy) if for every X ∈ SX and for every n ∈ F, there is m ∈ subo X with m ≤ n such that m X p ⇒ p ∈ F for any p ∈ subX. (3) Lef t S-Cauchy if for all X ∈ SX there is n ∈ F such that m X p ⇒ p ∈ F for any m ≤ n and m ∈ subo X. S (m) = {n | for some X ∈ SX , m X n} is S-round and S-Cauchy For any X ∈ C and m ∈ subo X, νX filter on X. A filter that is S-round and wS-Cauchy shall simply be called S-round Cauchy filter. The above definition can also be characterized in terms of a quasi-uniformity by using Theorem 3.1.
Proposition 4.6. A filter F on X ∈ C is: S such that U (m) ≤ n. (1) S-round if and only f for every n ∈ F, there is m ∈ F and there is U ∈ UX S (2) wS-Cauchy if and only if for every U ∈ UX and for every n ∈ F, there is m ∈ subo X with m ≤ n such that U (m) ∈ F. S (3) left S-Cauchy if and only if for every U ∈ UX there is n ∈ F such that U (m) ∈ F for any m ≤ n and m ∈ subo X.
Definition 4.5. A filter F on X ∈ C is right U-Cauchy if for any U ∈ UX there is n ∈ F such that U ∗ (m) ∈ F for any m ≤ n and m ∈ subo X. Proposition 4.7. Consider the following for a filter F on X ∈ C. (1) F is S-Cauchy. (2) F is left S-Cauchy.
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(3) F is wS-Cauchy. Then (2) ⇒ (1) and (2) ⇒ (3) Proof. (2) ⇒ (1) is clear from the definition. (2) ⇒ (3) If F is left S-Cauchy, X ∈ SX and n ∈ F then there is l ∈ F such that m X p ⇒ p ∈ F for any m ≤ l and m ∈ subo X. Since n, l ∈ F, l∧n > 0X and putting m = l∧n, we get that m X p ⇒ p ∈ F, that is F is wS-Cauchy. 2 5. Completeness The different notions of Cauchy filters developed in the previous section allow us to define distinct notions of completeness. Definition 5.1. Let S ∈ SY N T . A C-object X is said to be strongly S-complete if every S-Cauchy filter on X converges with respect to S. It can be seen from Definition 4.2 that a C object X is strongly S-complete if and only if it is strongly S ∗ -complete with S ∗ ∈ CSY N T such that S ∗ ≤ S. We can also conclude from Proposition 4.4 and Theorem 4.5 that for S ∈ CSY N T , X is strongly S-complete if and only if every U S -Cauchy filter on X is U S -convergent. Our next two propositions follow from Propositions 4.2 and 4.3 respectively. Proposition 5.1. Let f : X −→ Y be a E-morphism that is S-initial. Then X is strongly S-complete if and only if Y is strongly S-complete. Theorem 5.2. Let X = i∈I Xi be a product in C and F a filter on X. Assume that (pi : X −→ Xi )i∈I is an S-initial source. Let for each i ∈ I the projections belong to E. Then X is strongly S-complete if and only if Xi is strongly S-complete for each i. Definition 5.2. Let S ∈ CSY N T . An object X ∈ C will be said to be S-complete if for every S-round S Cauchy filter on X there is a unique m ∈ subo X such that F = νX (m). X is lef t S-complete if every left S-Cauchy filter on X is S-convergent. It follows from Proposition 4.7 that every strongly S-complete object is left S-complete. Definition 5.3. Let S ∈ CSY N T . An object X ∈ C will be said to be right U-complete if every right U-Cauchy filter on X is right U-convergent. Example 5.1. (1) In the Example 3.1(a), for any (X, D) ∈ Qunif , every filter on X that converges in the topology induced by D is S-convergent (Proposition 4.4) while every Cauchy filter in (X, D) is S-Cauchy (Theorem 4.5). By Proposition 4.6, left (resp. right) K-Cauchy filters in (X, D)([10]) are left (resp. right) S-Cauchy filters and the Cauchy filters on (X, D) introduced by Smyth in ([12]) are wS-Cauchy. Thus every convergence complete (resp. Smyth complete, left and right K-complete) quasi-uniform space is a strongly S-complete (resp. wS-complete, left and right S-complete) object. Let subo X be the class of single element sets of X. (2) For the Example 3.1(d)(ii), a filter on (X, T ) is S-round Cauchy if it is a completely prime filter having a base of open sets. Thus S-complete objects coincide with sober spaces. (3) In the Example 3.1(b)(i), complete abelian groups (see [2]) are strongly S-complete objects.
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Definition 5.4. Let X ∈ C and S ∈ CSY N T . X is said to be S-precompact if every ultrafilter on X is an S-Cauchy filter. It is clear that if X is S-precompact and S ≤ S, then X is S -precompact. Proposition 5.3. Let X ∈ C and S ∈ CSY N T . X is S-precompact if and only if every ultrafilter on X is U S -Cauchy filter. S . Then by Theorem 3.1, Proof. Assume that X is S-precompact and F be an ultrafilter on X. Let U ∈ UX there is X ∈ SX such that U = U and there is m ∈ subo X such that m X n ⇒ n ∈ F. Since m X U (m), U (m) ∈ F. Conversely let every ultrafilter on X be U S -Cauchy. If F is an ultrafilter on X S such that U and X ∈ SX , then there is U ∈ UX X = X and there is m ∈ subo X such that U (m) ∈ F. Now m X n ⇔ U (m) ≤ n ⇒ n ∈ F. 2
Propositions 4.2 and 4.3 allow to prove the following. Proposition 5.4. Let f : X −→ Y be a E-morphism that is S-initial. Then X is S-precompact if and only if Y is S-precompact. Theorem 5.5. Let X = i∈I Xi be a product in C. Assume that (pi : X −→ Xi )i∈I is an S-initial source. Let pi belong to E for each i, then X is S-precompact if and only if Xi is S-precompact for each i. Definition 5.5. An object X is said to be hereditarily U-precompact (resp. hereditarily U ∗ -precompact) if every ultrafilter on X is left U-Cauchy (resp. right U-Cauchy). The next proposition is immediate from the above definition. Proposition 5.6. For an object X ∈ C, (1) If X is U-precompact, then it is U ∗ -precompact. The converse holds if U is a uniformity. (2) If X is hereditarily U-precompact then it is U-precompact. (3) If X is S-precompact and strongly S-complete then every ultrafilter on X is S-convergent. References [1] J. Adámek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories: The Joy of Cats, Repr. Theory Appl. Categ., vol. 17, Wiley, New York, 2006, pp. 1–507, reprint of the 1990 original. [2] N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971. [3] Á. Császár, Foundations of General Topology, A Pergamon Press Book, The Macmillan Co., New York, 1963. [4] D. Dikranjan, E. Giuli, Closure operators I, Topol. Appl. 27 (2) (1987) 129–143. [5] D. Dikrajan, W. Tholen, Categorical Structure of Closure Operators with Applications to Topology, Algebra and Discrete Mathematics, Mathematics and its Applications, vol. 346, Kluwer Academic Publishers Group, Dordrecht, 1995. [6] P. Fletcher, W.F. Lindgren, Quasi-Uniform Spaces, Lectures Notes in Pure Appl. Math., vol. 77, Dekker, New York, 1982. [7] D. Holgate, M. Iragi, A. Razafindrakoto, Topogenous and nearness structures on categories, Appl. Categ. Struct. (24) (2016) 447–455. [8] D. Holgate, J. Šlapal, Categorical neighborhood operators, Topol. Appl. 158 (17) (2011) 2356–2365. [9] M. Iragi, Topogenous Structures on Categories, MSc Thesis, University of the Western Cape, 2016. [10] S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, Acta Math. Hung. 73 (1–2) (1996) 159–178. [11] J. Šlapal, Net spaces in categorical topology, Ann. N.Y. Acad. Sci. 806 (1) (1996) 393–412. [12] M.B. Smyth, Completeness of quasi-uniform and syntopological spaces, J. Lond. Math. Soc. (2) 49 (2) (1994) 385–400. [13] P. Sünderhauf, The Smythe-Completion of a Quasi-Uniform Space, Semantics of Programming Languages and Model Theory, Gordon and Breach Science Publishers, Inc., Newark, NJ, 1993. [14] S.J.R. Vorster, Interior operators in general categories, Quaest. Math. 23 (4) (2000) 405–416.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Quasi-uniform and syntopogenous structures on categories ✩ David Holgate, Minani Iragi ∗ Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville 7535, South Africa
a r t i c l e
i n f o
Article history: Received 30 April 2018 Received in revised form 7 August 2018 Accepted 26 September 2018 Available online 31 May 2019 MSC: 18A20 54A15 54B30 54E15
a b s t r a c t We introduce and study an abstract notion of Császár’s syntopogenous structure which provides a convenient setting to investigate a quasi-uniformity on a category. We show that a quasi-uniformity is a family of categorical closure operators. In particular, it is shown that every idempotent closure operator is quasi-uniformity. Various notions of completeness of objects and precompactness with respect to a quasi-uniformity defined in a natural way are also studied. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Keywords: Closure operator Categorical topogenous structure Syntopogenous structure Quasi-uniform structure and completeness
1. Introduction The unified approach to the categorical closure ([4]), interior ([14]) and neighbourhood operators ([8]) provided by the recently introduced notion of categorical topogenous structures ([7,9]) has shed a light on the study of a concept of quasi-uniformity on an abstract category. In a category C with an (E, M)-factorization system, this paper introduces a concept of syntopogenous structure ([3]) that is used to study a quasiuniformity ([6]). Classical notions and results on quasi-uniform spaces are expressed in a more general categorical setting. This leads to new results that are applied to specific examples in Topology and Algebra. Although the syntopogenous structure on C appears as an appropriate family of order relations on the sub✩
This work was supported by the South African National Research Foundation.
* Corresponding author. E-mail addresses: [email protected] (D. Holgate), [email protected] (M. Iragi). https://doi.org/10.1016/j.topol.2019.05.024 0166-8641/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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object lattice, subX, for any object X of C, a quasi-uniformity on C is thought of as a suitably axiomatized family of endomaps on subX for any X ∈ C. The definitions obtained include the fact that every morphism in C must be continuous with respect to the structure. A subconglomerate of the conglomerate of all syntopogenous structures which is isomorphic to the conglomerate of all quasi-uniform structures on C is defined and studied. This leads to the observation that a quasi-uniformity is a family of closure operators. Since every interpolative topogenous order is shown to be a syntopogenous structure and the class of these orders is known (see [9]) to be essentially equivalent to the conglomerate of all idempotent closure operators, it will be proved that every idempotent closure operator is a quasi-uniformity on C. We shall then define and study (different kinds of) complete and precompact objects of C with respect to a quasi-uniformity. It is a pleasure for the authors to thank Professor Hans-Peter Künzi and Doctor Francesco G. Russo for organising this wonderful workshop. 2. Preliminaries For categorical matters, we refer the reader to [1]. The basic facts on categorical closure operators and topogenous structures used here can be found in [5] and [7] respectively. Throughout the paper, we consider a category C and a fixed class M of C-monomorphisms which is closed under composition and contains all isomorphisms of C. The category C is assumed to be M-complete so that pullbacks of M-morphisms along C-morphisms and arbitrary M-intersections of M-morphisms exist and are again in M. For any C-object X, subX is the class of M-morphisms with codomain X. The elements of subX are called subobjects of X. They are ordered by n ≤ m ⇔ ∃ j | m ◦ j = n. If m ≤ n and n ≤ m then they are isomorphic. We shall simply write m = n in this case. A number of properties can be derived from the above assumptions on M, namely; subX is a complete lattice with 0X : OX −→ X the least element and 1X : X −→ X the largest, there is a class E of C-morphisms such that (E, M) is factorization structure for morphisms in C (see [5]), each C-morphism f : X −→ Y induces an image/preimage adjunction f (m) ≤ n ⇔ m ≤ f −1 (n) for all n ∈ subY , m ∈ subX with f (m) given by the M-component of the (E, M)-factorization of f ◦ m and f −1 (n) the pullback of n along f . We shall sometimes find it important to assume that for any C-morphism f : X −→ Y , the inverse image f −1 commutes with the joins of subobjects so that it has a right adjoint f∗ given by f∗ (m) = {n ∈ subY | f −1 (n) ≤ m}. Thus f −1 (n) ≤ m ⇔ n ≤ f∗ (m), f −1 (f∗ (m)) ≤ m (with f −1 (f∗ (m)) = m if f ∈ M) and n ≤ f∗ (f −1 (n)) (with f∗ (f −1 (n)) = n if f ∈ E and E stable under pullbacks). Definition 2.1. A closure operator c on C with respect to M is given by a family of maps {cX : subX −→ subX | X ∈ C} such that: (C1) m ≤ cX (m) for all m ∈ subX; (C2) m ≤ n ⇒ cX (m) ≤ cX (n) for all m, n ∈ subX; (C3) every morphism f : X −→ Y is c-continuous, that is: f (cX (m)) ≤ cY (f (m)) for all m ∈ subX. We denote by CLOS the conglomerate of all closure operators on C with respect to M ordered as follows: c ≤ c if cX (m) ≤ cX (m) for all m ∈ subX and X ∈ C.
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Definition 2.2. A topogenous order on C is a family { X | X ∈ C} of relations, each X on subX, such that: (T 1) m X n ⇒ m ≤ n for every m, n ∈ subX, (T 2) m ≤ n X p ≤ q ⇒ m X q for every m, n, p, q ∈ subX, and (T 3) every morphism f : X −→ Y in C is -continuous, m Y n ⇒ f −1 (m) X f −1 (n) for every m, n ∈ subY . Given two topogenous orders and on C, ⊆ if and only if m X n ⇒ m X n for all m, n ∈ subX. The resulting ordered conglomerate of all topogenous orders on C is denoted by T ORD. A topogenous order is said to be (1) -preserving if (∀i ∈ I : m X ni ) ⇒ m X ni , and (2) interpolative if m X n ⇒ (∃ p) | m X p X n for all X ∈ C. The ordered conglomerate of all -preserving and interpolative topogenous orders is denoted by −T ORD and IN T ORD respectively. −IN T ORD will denote the conglomerate of all interpolative meet preserving topogenous orders. Proposition 2.1. by
−T ORD is order isomorphic to CLOS. The inverse morphisms of each other are given c X (m) =
{p | m X p} and m cX n ⇔ cX (m) ≤ n for all X ∈ C.
Proposition 2.1 will play a vital role in understanding the relationship between closure and quasiuniformity on C. 3. Quasi-uniform structure or family of closure operators Definition 2.2 allows us to generalize in a natural way the syntopogenous structure on C. Definition 3.1. A syntopogenous structure on C with respect to M is a family S = {SX | X ∈ C} such that each SX = { iX | i ∈ IX } is a family of relations on subX satisfying: (S1) iX is a relation on subX satisfying (T 1) and (T 2). (S2) { iX | i ∈ IX } is a directed set with respect to inclusion. (S3) X = i∈IX iX is an interpolative topogenous order. We can extend the ordering of topogenous orders to syntopogenous structures in the following way: S ≤ S if for all X ∈ C and iX ∈ SX , there is jX ∈ S X such that iX ⊆ jX . The resulting conglomerate will be denoted by SY N T . We are interested in particular classes of syntopogenous structures as these will play an important role in what follows. Definition 3.2. We shall say that S ∈ SY N T is: (1) co-perf ect if each iX ∈ SX is -preserving for all X ∈ C. (2) interpolative if for every iX ∈ SX , m X n ⇒ (∃ p) | m iX p iX n for all X ∈ C. (3) simple if SX = { X } is an interpolative topogenous order for any X ∈ C.
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The ordered conglomerate of all co-perfect (resp. interpolative) syntopogenous structures will be denoted by CSY N T (resp. IN T SY N T ). It is readily seen from Definition 3.2 that IN T ORD is the class of simple syntopogenous structures. A routine check shows that IN T ORD is closed under arbitrary unions in SY N T . Consequently, it is a coreflective sub-quasicategory of SY N T . The observation that an (entourage) quasi-uniformity on a set X can be equivalently expressed as a family of maps f : X −→ P(X) which can easily be extended to a family of maps f : P(X) −→ P(X), motivates the point in our definition that a quasi-uniformity on C will be given as a family of endomaps on subX for each X ∈ C. Let X ∈ C, subX being a complete lattice can be seen as a category, monotone maps f [−], g[−] from subX to subX are the functors and there is a natural transformation α : f [−] ⇒ g[−] exactly when f (m) ≤ g(m) for all m ∈ subX. It turns out that F(subX), the endofunctor category on subX is ordered by ≤ “pointwise”. Definition 3.3. A quasi-unif ormity on C with respect to M is a family U = {UX | X ∈ C} with UX a full subcategory of F(subX) for each X such that: (U 1) For any U ∈ UX , 1 ≤ U (where 1 is the identity map on subX). (U 2) For any U ∈ UX , there is U ∈ UX such that U ◦ U ≤ U . (U 3) For any U ∈ UX and U ≤ U , U ∈ UX . (U 4) For any U, U ∈ UX , U ∧ U ∈ UX . (U 5) For any C-morphism f : X −→ Y and U ∈ UY , there is U ∈ UX such that f (U (m)) ≤ U (f (m)) for all m ∈ subX. We shall denote by QU N IF the conglomerate of all quasi-uniform structures on C. It is ordered as follows: U ≤ V if for all X ∈ C and U ∈ UX , there is V ∈ VX such that V ≤ U . We shall often describe a quasi-uniformity by defining a base for it. Definition 3.4. A full subcategory BX of F(subX) for all X ∈ C is a base for a quasi-uniformity on C if it satisfies all the axioms in Definition 3.3 except (U 3). Definition 3.5. A base for quasi-uniformity on C is transitive if for all X ∈ C and U ∈ BX , U ◦ U = U . A quasi-uniformity with a transitive base is called a transitive quasi-unif ormity. The ordered conglomerate of all transitive quasi-uniformities on C will be denoted by T QU N IF . ∗ ∗ | X ∈ C} where UX = {U ∗ | U ∈ UX } with Definition 3.6. Let U be a quasi-uniformity on C, then U ∗ = {UX ∗ U (m) = {n | m ≤ U (n)} is a base of a quasi-uniformity on C called the conjugate quasi-unif ormity of U. ∗ We say that a quasi-uniformity U on C is a uniformity provided that UX = UX for every X ∈ C, equivalently if for every U ∈ UX and X ∈ C, there is V ∈ U such that m ≤ U (n) ⇔ n ≤ V (m) for any m, n ∈ subX. While our intention is to study a quasi-uniformity on a category, we have found it more fruitful to express many of our proofs in terms of a syntopogenous structure. The next theorem that describes the clear relationship between the two structures shall lead us to the achievement of this goal. Let S ∈ SY N T , B S will denote the base for a quasi-uniformity induced by S. If B is a base for a quasi-uniformity, S B denotes the syntopogenous structure induced by B.
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Theorem 3.1. QU N IF is order isomorphic to CSY N T . The inverse morphisms of each other U −→ S U and S −→ U S are given by B U SX = { U X | U ∈ BX } where m X n ⇔ U (m) ≤ n, and S = {U | X ∈ SX } where U (m) = {n | m X n} BX
for all X ∈ C and m, n ∈ subX. S
Proof. It is easily seen that U −→ S U and S −→ U S are well defined and order preserving. Now, S B = S X X = X , m U n ⇔ U (m) ≤ n ⇔ {p | m X n} ≤ n ⇔ m X n for all since for any X ∈ SX , U X X B U X ∈ C and B S = B since U (m) = {n | U (m) ≤ n} = U (m) for any U ∈ B and m ∈ subX. 2 Since S ⊆ −T ORD for each S ∈ CSY N T , it follows from the above theorem and Proposition 2.1 that a quasi-uniformity on C is a family of closure operators. It is proved in ([9], Corollary 4.2.3) that −IN T ORD is isomorphic to the conglomerate of idempotent closure operators and from Theorem 3.1, CSY N T ∼ = QU N IF . Thus every idempotent closure operator on C is a quasi-uniformity. Let us denote by IN T CSY N T the conglomerate of all interpolative co-perfect syntopogenous structures on C. We next prove that interpolative co-perfect syntopogenous structures are exactly the transitive quasi-uniformities. Proposition 3.2. IN T CSY N T is order isomorphic to T QU N IF . U Proof. Let B be a transitive base and m U X n for any U ∈ BX and X ∈ C, then U (m) = U (U (m)) X U U U (m) ≤ n, hence m X U (m) X n. Conversely if S ∈ IN T CSY N T and X ∈ SX , m X n ⇒ (∃ p) | m X p X n ⇒ U X (m) ≤ p X n ⇒ U X (m) ≤ n. This implies that {l | U X (m) X l} ≤ {q | m q} that is U X (U X (m)) ≤ U X (m) and U X (m) ≤ U X (U X (m)) follows from (U 1) so that U X (m) = U X (U X (m)). 2
We often write X ∈ SX instead of iX ∈ SX to ease the notation. We now have a bridge from quasi-uniformity to syntopogenous structure and back. This allows us to always compare the results we obtain for the two structures and work at the side where the proofs are easier to manipulate. For the rest of this section we assume that for any C-morphism f : X −→ Y , f −1 commutes with the joins of subobjects. Definition 3.7. Let f : X −→ Y be a C-morphism and S be a syntopogenous structure on C. Then f is S-initial if for every X ∈ SX there is Y ∈ SY such that m X n ⇒ f (m) Y f∗ (n) for all m, n ∈ subX. Proposition 3.3. Let f : X −→ Y be a C-morphism and S ∈ CSY N T . Then f is S-initial if and only if for every U ∈ BX there is U ∈ BY such that f −1 (U (f (m))) ≤ U (m) for all m ∈ subX. Proof. Assume that for any U ∈ BX there is U ∈ BY such that f −1 (U (f (m))) ≤ U (m) for all m ∈ subX and m X n for some X ∈ SX and m, n ∈ subX. Then there is U ∈ BX such that U = X and there is U ∈ BY such that f −1 (U (f (m)) ≤ U (m) ⇒ f −1 (U (f (m)) ≤ U (m) ≤ n ⇒ f −1 (U (f (m)) ≤ n ⇔ U (f (m)) ≤ f∗ (n) ⇔ f (m) U Y f∗ (n). Conversely if S ∈ CSY N T and f is S-initial, then for all U ∈ BX and m, n subX, U (m) ≤ n ⇔ (∃ X ∈ SX ) | m X n ⇔ (∃ Y ∈ SY ) | f (m) Y f∗ (n) ⇔ (∃ U ∈ BY ) | U (f (m)) ≤ f∗ (n) ⇔ f −1 (U (f (m))) ≤ n. Thus f −1 (U (f (m))) ≤ U (m) which completes the proof. 2
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Definition 3.8. A source (fi : X −→ Xi )i∈I is S-initial if for any X ∈ SX there are i ∈ I and Xi ∈ SXi such that m X n ⇒ f (m) Xi (fi )∗ (n). A similar reasoning to the one in Proposition 3.3 produces the following. Proposition 3.4. Let (fi : X −→ Xi )i∈I be a source in C and S ∈ CSY N T . Then (fi )i∈I is S-initial if and only if for any U ∈ BX there are i ∈ I and U ∈ BXi such that (fi )−1 (U (fi (m))) ≤ U (m) for all m ∈ subX. Example 3.1. By Theorem 3.1, which establishes an equivalence between co-perfect syntopogenous structures and quasi-uniformities on a category, it will be enough to define a co-perfect syntopogenous structure when exhibiting examples of a categorical quasi-uniformity. Already our first example shows that the classical quasi-uniformity is a particular co-perfect syntopogenous structure and hence a particular categorical quasi-uniformity. (a) Let C be Qunif , the category of quasi-uniform spaces and quasi-uniformly continuous maps ([6]) with M the class of quasi-uniform embeddings. Every (X, D) ∈ Qunif gives is a co-perfect syntopogenous structure, D S(X,D) = { D (X, D) | D ∈ D} on Qunif where A (X, D) B ⇔ D[A] ⊆ B and A, B ⊆ X. On the other hand, it is well known from [3] that the co-perfect syntopogenous structure that describes a quasi-uniformity is the one satisfying the property that, for all A ∈ A, A B ⇒ A B (A ⊆ 2X ). A quasi-uniformly continuous map f : (X, D) −→ (Y, D ) is S-initial provided D is the initial quasi-uniformity induced by f on X. (b) In the category TopGrp of topological groups and continuous group homomorphisms, let (E, M) be the (surjective, injective)-factorization structure. For any (X, ·) ∈ TopGrp, let η(e) be the neighbourhood U filter of the identity element e. Then SX = { U X | U ∈ η(e)} with A X B ⇔ U · A ⊆ B is a co-perfect syntopogenous structure on TopGrp. We have proved that a single co-perfect syntopogenous structure is equivalent to an idempotent closure operator. Below we present a number of examples derived from idempotent closure operators. (c) Let C = Grp be the category of groups and group homomorphisms with (epi, mono)-factorization system. For any A, B ≤ G, SG = { G | G ∈ Grp} with A G B ⇔ A ≤ N ≤ B with N G is a co-perfect simple syntopogenous structure on Grp. (d) Let C = Top the category of topological spaces and continuous maps with M the class of embeddings and (of course) E the class of surjective continuous maps. For A, B ⊆ X, (i) S(X, T ) = { (X, T ) | (X, T ) ∈ Top} with A X B ⇔ A ⊆ C ⊆ B for some closed C ⊆ X is a single co-perfect syntopogenous structure. A continuous map f : (X, TX ) −→ (Y, TY ) is S-initial iff TX is the initial topology induced by f on X. Another single co-perfect syntopogenous structure on Top that is not obtained from a idempotent closure operator is (ii) S(X, T ) = { (X, T ) | (X, T ) ∈ Top} with A X B ⇔ A ⊆ O ⊆ B for some O ∈ T . 4. The S-Cauchy filters Since for any X ∈ C, subX is a complete lattice, by a filter on X we shall always understand a filter on subX. Lemma 4.1. Let X ∈ C and F be a filter on X. Then F is an ultrafilter on X if and only if m ∧ n > 0X for all n ∈ F implies that m ∈ F.
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Let f : X −→ Y be a C-morphism and F a filter on X then f (F) is the filter defined by m ∈ f (F) ⇔ f −1 (m) ∈ F. It is an ultrafilter on Y provided F is an ultrafilter on X and f ∈ E with E being pullback stable along M-morphisms. If F is an ultrafilter on Y and f reflects 0 then f −1 (F) is also an ultrafilter on X where f −1 (F) is the filter defined by m ∈ f −1 (F) ⇔ f (m) ∈ F. For any X ∈ C, we shall denote by subo X, the class of all m ∈ subX such that m > 0X and for all S m ∈ subo X, νX (m) = {n | m X n for some X ∈ S}. We note that ν S (m) does not form a filter in general unless S is co-perfect. Definition 4.1. Let S ∈ CSY N T , X ∈ C and F be a filter on X. We say that F converges to a subobject m ∈ S subo X with respect to S and write F −→ m if for any X ∈ SX , m X n ⇒ n ∈ F for n ∈ subX. F is S-Cauchy if for any X ∈ SX there is m ∈ subo X such that m X n ⇒ n ∈ F. S
S One easily checks that for any S ∈ CSY N T and X ∈ C, νX (m) −→ m for all m ∈ subo X and that every S-convergent filter on X is S-Cauchy.
Definition 4.2. Let S ∈ SY N T and X ∈ C. A filter F on X is said to be S-Cauchy if it is Cauchy with respect to any S ∗ ∈ CSY N T that is coarser than S. F converges with respect to S if it converges with respect to any S ∗ ∈ CSY N T such that S ∗ ≤ S. For the rest of this paper, we assume that C has products, the class E is stable under pullbacks along M-morphisms and any f ∈ C reflects 0. Proposition 4.2. Let f : X −→ Y be a C-morphism, F a filter on X and S ∈ CSY N T . S
S
(a) If F −→ m then f (F) −→ f (m). The converse implication holds if f is S-initial. (b) If F is S-Cauchy on X then so is f (F) on Y . The converse holds if f is S-initial and belongs to E. S
Proof. (a) Assume that F −→ m with m ∈ subo X. Then f (m) ∈ subo Y . Let f (m) Y p for any Y ∈ SY and p ∈ subY . Then by (S4) there is X ∈ SX such that m X f −1 (p) and f −1 (p) ∈ F ⇔ p ∈ f (F). Conversely if f (F) −→ f (m) and f (m) ∈ subo Y . Then m ∈ subo X. Let m X n for any X ∈ SX . Then by S-initiality of f , there is Y ∈ SY such that f (m) Y f∗ (n) and f∗ (n) ∈ f (F). Thus f −1 (f∗ (n)) ∈ F which implies that n ∈ F. (b) Let F be S-Cauchy and Y ∈ SY . Then by (S4) there is X ∈ SX such that f (p) Y q ⇒ p X f −1 (q) and there is m ∈ subo X such that m X n ⇒ n ∈ F. Thus f (m) ∈ subo Y and f (m) Y l ⇒ m X f −1 (l) ⇒ f −1 (l) ∈ F ⇔ l ∈ F. Conversely if f (F) is S-Cauchy and X ∈ SX , then there is Y ∈ SY such that m X n ⇒ f (m) Y f∗ (n) and there is p ∈ subo Y such that p Y q ⇒ q ∈ f (F) for any p ∈ subY . Now f −1 (p) ∈ subo X since f ∈ E and f −1 (p) X n ⇒ p = f (f −1 (p)) Y f∗ (n) ⇒ f∗ (n) ∈ f (F) ⇔ f −1 (f∗ (n)) ∈ F ⇒ n ∈ F. 2 Proposition 4.3. Let X = is an S-initial source. S
i∈I
Xi be a product in C and F a filter on X. Assume that (pi : X −→ Xi )i∈I S
(1) F −→ m if and only if pi (F) −→ pi (m) for each i. (2) If for each i ∈ I the projections belong to E, then F is S-Cauchy if and only if pi (F) is S-Cauchy for each i. Definition 4.3. Let U be a quasi-uniformity on C and F filter on X. We shall say that F converges to m U (m ∈ subo X) with respect to U and write F −→ m if for any U ∈ UX , U (m) ∈ F. F is U-cauchy if for each U ∈ UX there is m ∈ subo X such that U (m) ∈ F.
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If U and V are quasi-uniform structures on C such that U ≤ V, then every V-Cauchy filter is U-Cauchy. S
US
Proposition 4.4. Let S ∈ CSY N T and F a filter on X. Then F −→ m ⇔ F −−→ m S
Proof. Let m ∈ subo X and F −→ m. If U ∈ UX then, by Theorem 3.1 there is X ∈ SX such that X = U X US
and m X n ⇒ n ∈ F. Now, m X U (m) and U (m) ∈ F. Conversely if F −−→ m and m X n then S U (m) ≤ n for some U ∈ UX and U (m) ∈ F. Hence n ∈ F. 2 Theorem 4.5. Let S ∈ CSY N T and F be a filter on X. Then F is U S -Cauchy if and only if it is S-Cauchy. S Proof. Let F be U S -Cauchy and X ∈ SX . Since S ∈ CSY N T , by Theorem 3.1 there is U ∈ UX that determines X and there is m ∈ subo X such that U (m) ∈ F. Now, m X n ⇔ U (m) ≤ n ⇒ n ∈ F. S Conversely if F is S-Cauchy and U ∈ UX then there is X ∈ SX such that U X = X and there is m ∈ subo X such that m X n ⇒ n ∈ F. Now m X U (m) since S ∈ CSY N T and U (m) ∈ F. 2
Left (resp. right) Cauchy filters in quasi-uniform spaces have been investigated in ([10]) and these lead to left (resp. right) complete quasi-uniform spaces which are closely related to the Smyth complete quasiuniform spaces ([13]). It seems that our co-perfect syntopogenous structure provides a simple and natural way of stating these notions in categorical language and link them to those obtained above. Let us also note that a categorical approach to convergence has already been studied in the literature (see e.g [11]). Definition 4.4. A filter F on an object X ∈ C will be called: (1) S-round if for every n ∈ F, there are m ∈ F and X ∈ SX such that m X n. (2) W eakly S-Cauchy (or simply wS-Cauchy) if for every X ∈ SX and for every n ∈ F, there is m ∈ subo X with m ≤ n such that m X p ⇒ p ∈ F for any p ∈ subX. (3) Lef t S-Cauchy if for all X ∈ SX there is n ∈ F such that m X p ⇒ p ∈ F for any m ≤ n and m ∈ subo X. S (m) = {n | for some X ∈ SX , m X n} is S-round and S-Cauchy For any X ∈ C and m ∈ subo X, νX filter on X. A filter that is S-round and wS-Cauchy shall simply be called S-round Cauchy filter. The above definition can also be characterized in terms of a quasi-uniformity by using Theorem 3.1.
Proposition 4.6. A filter F on X ∈ C is: S such that U (m) ≤ n. (1) S-round if and only f for every n ∈ F, there is m ∈ F and there is U ∈ UX S (2) wS-Cauchy if and only if for every U ∈ UX and for every n ∈ F, there is m ∈ subo X with m ≤ n such that U (m) ∈ F. S (3) left S-Cauchy if and only if for every U ∈ UX there is n ∈ F such that U (m) ∈ F for any m ≤ n and m ∈ subo X.
Definition 4.5. A filter F on X ∈ C is right U-Cauchy if for any U ∈ UX there is n ∈ F such that U ∗ (m) ∈ F for any m ≤ n and m ∈ subo X. Proposition 4.7. Consider the following for a filter F on X ∈ C. (1) F is S-Cauchy. (2) F is left S-Cauchy.
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(3) F is wS-Cauchy. Then (2) ⇒ (1) and (2) ⇒ (3) Proof. (2) ⇒ (1) is clear from the definition. (2) ⇒ (3) If F is left S-Cauchy, X ∈ SX and n ∈ F then there is l ∈ F such that m X p ⇒ p ∈ F for any m ≤ l and m ∈ subo X. Since n, l ∈ F, l∧n > 0X and putting m = l∧n, we get that m X p ⇒ p ∈ F, that is F is wS-Cauchy. 2 5. Completeness The different notions of Cauchy filters developed in the previous section allow us to define distinct notions of completeness. Definition 5.1. Let S ∈ SY N T . A C-object X is said to be strongly S-complete if every S-Cauchy filter on X converges with respect to S. It can be seen from Definition 4.2 that a C object X is strongly S-complete if and only if it is strongly S ∗ -complete with S ∗ ∈ CSY N T such that S ∗ ≤ S. We can also conclude from Proposition 4.4 and Theorem 4.5 that for S ∈ CSY N T , X is strongly S-complete if and only if every U S -Cauchy filter on X is U S -convergent. Our next two propositions follow from Propositions 4.2 and 4.3 respectively. Proposition 5.1. Let f : X −→ Y be a E-morphism that is S-initial. Then X is strongly S-complete if and only if Y is strongly S-complete. Theorem 5.2. Let X = i∈I Xi be a product in C and F a filter on X. Assume that (pi : X −→ Xi )i∈I is an S-initial source. Let for each i ∈ I the projections belong to E. Then X is strongly S-complete if and only if Xi is strongly S-complete for each i. Definition 5.2. Let S ∈ CSY N T . An object X ∈ C will be said to be S-complete if for every S-round S Cauchy filter on X there is a unique m ∈ subo X such that F = νX (m). X is lef t S-complete if every left S-Cauchy filter on X is S-convergent. It follows from Proposition 4.7 that every strongly S-complete object is left S-complete. Definition 5.3. Let S ∈ CSY N T . An object X ∈ C will be said to be right U-complete if every right U-Cauchy filter on X is right U-convergent. Example 5.1. (1) In the Example 3.1(a), for any (X, D) ∈ Qunif , every filter on X that converges in the topology induced by D is S-convergent (Proposition 4.4) while every Cauchy filter in (X, D) is S-Cauchy (Theorem 4.5). By Proposition 4.6, left (resp. right) K-Cauchy filters in (X, D)([10]) are left (resp. right) S-Cauchy filters and the Cauchy filters on (X, D) introduced by Smyth in ([12]) are wS-Cauchy. Thus every convergence complete (resp. Smyth complete, left and right K-complete) quasi-uniform space is a strongly S-complete (resp. wS-complete, left and right S-complete) object. Let subo X be the class of single element sets of X. (2) For the Example 3.1(d)(ii), a filter on (X, T ) is S-round Cauchy if it is a completely prime filter having a base of open sets. Thus S-complete objects coincide with sober spaces. (3) In the Example 3.1(b)(i), complete abelian groups (see [2]) are strongly S-complete objects.
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Definition 5.4. Let X ∈ C and S ∈ CSY N T . X is said to be S-precompact if every ultrafilter on X is an S-Cauchy filter. It is clear that if X is S-precompact and S ≤ S, then X is S -precompact. Proposition 5.3. Let X ∈ C and S ∈ CSY N T . X is S-precompact if and only if every ultrafilter on X is U S -Cauchy filter. S . Then by Theorem 3.1, Proof. Assume that X is S-precompact and F be an ultrafilter on X. Let U ∈ UX there is X ∈ SX such that U = U and there is m ∈ subo X such that m X n ⇒ n ∈ F. Since m X U (m), U (m) ∈ F. Conversely let every ultrafilter on X be U S -Cauchy. If F is an ultrafilter on X S such that U and X ∈ SX , then there is U ∈ UX X = X and there is m ∈ subo X such that U (m) ∈ F. Now m X n ⇔ U (m) ≤ n ⇒ n ∈ F. 2
Propositions 4.2 and 4.3 allow to prove the following. Proposition 5.4. Let f : X −→ Y be a E-morphism that is S-initial. Then X is S-precompact if and only if Y is S-precompact. Theorem 5.5. Let X = i∈I Xi be a product in C. Assume that (pi : X −→ Xi )i∈I is an S-initial source. Let pi belong to E for each i, then X is S-precompact if and only if Xi is S-precompact for each i. Definition 5.5. An object X is said to be hereditarily U-precompact (resp. hereditarily U ∗ -precompact) if every ultrafilter on X is left U-Cauchy (resp. right U-Cauchy). The next proposition is immediate from the above definition. Proposition 5.6. For an object X ∈ C, (1) If X is U-precompact, then it is U ∗ -precompact. The converse holds if U is a uniformity. (2) If X is hereditarily U-precompact then it is U-precompact. (3) If X is S-precompact and strongly S-complete then every ultrafilter on X is S-convergent. References [1] J. Adámek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories: The Joy of Cats, Repr. Theory Appl. Categ., vol. 17, Wiley, New York, 2006, pp. 1–507, reprint of the 1990 original. [2] N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971. [3] Á. Császár, Foundations of General Topology, A Pergamon Press Book, The Macmillan Co., New York, 1963. [4] D. Dikranjan, E. Giuli, Closure operators I, Topol. Appl. 27 (2) (1987) 129–143. [5] D. Dikrajan, W. Tholen, Categorical Structure of Closure Operators with Applications to Topology, Algebra and Discrete Mathematics, Mathematics and its Applications, vol. 346, Kluwer Academic Publishers Group, Dordrecht, 1995. [6] P. Fletcher, W.F. Lindgren, Quasi-Uniform Spaces, Lectures Notes in Pure Appl. Math., vol. 77, Dekker, New York, 1982. [7] D. Holgate, M. Iragi, A. Razafindrakoto, Topogenous and nearness structures on categories, Appl. Categ. Struct. (24) (2016) 447–455. [8] D. Holgate, J. Šlapal, Categorical neighborhood operators, Topol. Appl. 158 (17) (2011) 2356–2365. [9] M. Iragi, Topogenous Structures on Categories, MSc Thesis, University of the Western Cape, 2016. [10] S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, Acta Math. Hung. 73 (1–2) (1996) 159–178. [11] J. Šlapal, Net spaces in categorical topology, Ann. N.Y. Acad. Sci. 806 (1) (1996) 393–412. [12] M.B. Smyth, Completeness of quasi-uniform and syntopological spaces, J. Lond. Math. Soc. (2) 49 (2) (1994) 385–400. [13] P. Sünderhauf, The Smythe-Completion of a Quasi-Uniform Space, Semantics of Programming Languages and Model Theory, Gordon and Breach Science Publishers, Inc., Newark, NJ, 1993. [14] S.J.R. Vorster, Interior operators in general categories, Quaest. Math. 23 (4) (2000) 405–416.