Subobjects and Compactness in Point-Free Convergence

Abstract

We consider subobjects in the context of point-free convergence (in the sense of Goubault-Larrecq and Mynard), characterizing extremal monomorphisms in the opposite category of that of convergence lattices. It turns out that special ones are needed to capture the notion of subspace. We call them standard and they essentially depend on one element of the convergence lattice. We introduce notions of compactness and closedness for general filters on a convergence lattice, obtaining adequate notions for standard extremal monos by restricting ourselves to principal filters. The classical facts that a closed subset of a compact space is compact and that a compact subspace of a Hausdorff space is closed find generalizations in the point-free setting under the form of general statements about filters. We also give a point-free analog of the classical fact that a continuous bijection from a compact pseudotopology to a Hausdorff pseudotopology is a homeomorphism.

1. Introduction

Recall that a convergence on a set is a relation between the set of (set-theoretic) filters on and the set , denoted byif and are -related, subject to the following two axioms:for every and every . Continuity of a map is simply preservation of limits, that is,where is the image filter of under .

Let denote the category of convergence spaces and continuous maps. This is a topological category in which subspaces are defined as usual as the initial structure for the inclusion map; namely, if and is a convergence space, then the induced convergence on is defined by , where is the filter generated on by the filter on .

The category of topological spaces and continuous maps is a concretely reflective subcategory of . See e.g., [1], for a systematic treatment of and of classical topology from that viewpoint.

In [2], Goubault-Larrecq and Mynard introduce a point-free generalization of in which the functionis abstracted away to a monotone functionfrom (order-theoretic) filters on a lattice to . Note that (point axiom) is not part of the axiomatic in this point-free version of convergence spaces, though the notion can also be recovered (as so-called centered convergence lattices) in an abstract order-theoretic form.

In the context of this paper, a lattice is an ordered set with all finite joins and meets, including the empty ones, so that our lattices have a greatest element, usually denoted by and a least element usually denoted by . Lattice morphisms preserve all finite joins and meets including the empty ones, so that our lattice morphisms send the greatest element to greatest element and the least to least. Let denote the corresponding category. By a category of lattices, we mean a subcategory of .

Definition 1. Given a category of lattices, a convergence -object is a -object together with a monotone map . The objects of the category are the convergence -objects, and the morphisms are the -morphisms that are continuous in the sense that for every,where . The morphism is final (in the usual categorical sense) if we have equality in (ptfree continuity).
The category embeds coreflectively into when is the category of frames or of coframes: the powerset-functor sending to and to (in ) is then right-adjoint to the point-functor (the coreflector), where the underlying set of , the set of “points” of , is the set of -morphisms from to , hence depending on the choice of . The convergence structure on is given bywhere and . Finally, if , then is defined by .
In the same work [2], adjunctions between , , and some of their important subcategories are proved, and it is also shown that, if is the category of coframes, is a topological category. Though the point-free analog of the category of topological spaces in is not related in a straightforward way with the classical approach to point-free topology [2], ([2], section 8.5) shows how the latter can be recovered, realizing the category of locales as a reflective subcategory of the opposite of the category of strong topological coframes in the context. This new framework already proved further versatility, as the category of convergence approach spaces in the sense of [3] can also be faithfully represented in [4].
In the present work, we explore the basic concepts of subspace, compactness, closedness, and Hausdorffness in the setting of . More specifically, we characterize the extremal monomorphisms of , that is, the extremal epimorphism of , as the -extremal epimorphism that are also final in . Though this provides an adequate notion of subobject, a stricter notion is necessary to capture exactly those subobjects of the convergence lattice coming from a convergence space that represent a subspace of . We call such extremal epis standard. They essentially depend on a single element of the convergence lattice, like picking a subspace of depends on picking an element of .
In the context of convergence spaces, the notions of closedness and compactness have been generalized to families of subsets rather than single subsets, e.g., [1, 5, 6] and references therein, to the effect that a subset has the property if and only if its principal filter does. Similarly, in the point-free context, we consider compactness and closedness for filters, and the case of a principal filter provides notions for standard extremal epis. We also consider a second direction of generalization, defining compactness for morphisms, though few results are obtained in this direction. This avenue may be explored further in future work.
Throughout this work, we are going to use definitions and notations from [2] for convergence lattices, from [7, 8] for categorical notions, and from [9, 10] for lattice theory and locales. Moreover, though we work with categories of lattices, some notions involve infinite suprema or infima. Whenever they appear, the existence of such infima or suprema is implicitly assumed, so that the notion is most naturally considered within convergence lattices that are complete lattices, though the morphisms remain those of .

2. Subobjects in Categories of Convergence Lattices

As pointed out in ([10], III. 1) and ([8], 1.2.2.6 2), the extremal monomorphisms are the categorical candidates to represent the subobjects of a topological category. Because we are working in , the extremal monomorphisms there are the extremal epimorphisms in . Recall that an epimorphism is extremal if whenever for a morphism and a monomorphism , is an isomorphism.

It turns out that these extremal epimorphisms have the expected topological behavior of being final.

Theorem 1. Let be a category of lattices. The extremal epimorphisms of are exactly extremal epimorphisms of that are final in .

Proof. Assume that is a -extremal epimorphism and let be a -factorization of , where is a -monomorphism.

Note that, even if is not a category of coframes, we can always construct the final structures for sources with only one morphism ([2], Corollary 3.3). Here we can give the final structure . As is final and is a morphism, we conclude that is a -morphism. Therefore, is a -isomorphism because is an extremal epimorphism in . In particular, is an isomorphism of . Hence, is an extremal epimorphism of .
Moreover, is final. To see this, consider the next commutative triangle

Assume conversely that is an extremal epimorphism of that is final in . Let be a factorization of with a -monomorphism:

Then is in particular a -monomorphism, hence a -isomorphism, which means that as maps, and therefore, because is final in , we conclude that is a morphism.

Remark 1. Note that when is the category of frames, the extremal epimorphisms of are the onto -morphisms ([10], Proposition 1.1.3). This is also true for the category of coframes.
Note that with this notion of subobject, there may be more subobjects of a convergence space seen as a convergence lattice than subspaces of .

Proposition 1. Let be a convergence space. Given an extremal epimorphism of , the following are equivalent:(1)There is with inclusion map such that  commutes, where is an isomorphism (equivalently a monomorphism) of .(2) satisfies and is injective on .

Proof. : note that in (1) is an isomorphism whenever it is a monomorphism because is an extremal epimorphism. Assuming (1),because as is an isomorphism. Hence, and and moreover, if , that is, , then and are different elements of and thus their images under the isomorphism are different elements of . Hence, .
: let . Consider the map defined by for every . As and is a morphism of , so is . Moreover, is injective on so that is a monomorphism of and for every ,Moreover, is continuous because is final and is continuous.
Recall that an epimorphism is split if it has a right-inverse, that is, if there is a morphism such that . Split epimorphisms are extremal. Note that when is a subspace of as in Proposition 1, then is also a split epimorphism, taking , where is . However, there are split epimorphisms that do not correspond to subspaces (e.g., Example 1 below).

Definition 2. An extremal epimorphism of is standard if satisfies and is injective on .

Proposition 2. An extremal epimorphism of is standard if and only if is isomorphic to a sublattice of of the form for some with the limit given byfor every .

Proof. For every , the map defined by is a standard extremal epimorphism if is given by (10). Indeed, is onto and is the identity on , so that , and is injective on .
Conversely, given a standard extremal epimorphism of , we can show that is isomorphic to with the corresponding limit given by (10).
However, not all extremal epimorphisms are standard as shown in the following.

Proposition 3. Given a convergence lattice with at least two elements and , then is a split (hence extremal) epimorphism of .

Proof. Let be defined by and . This is continuous because so that for every proper filter , because . Hence, .

Example 1. (a split epimorphism of that is not standard). Note that if is the category of lattices and is a convergence space, then a point of can be identified with the filter , which is prime because is a lattice morphism and satisfies by the continuity condition. As prime filters of are ultrafilters, points of are ultrafilters on satisfying . There may be such nonprincipal ultrafilters on a convergence space. For instance, on an infinite Noetherian topological space , all free ultrafilters are points of (see [11] for details), which are nonstandard extremal epimorphisms, for and .
Let us now examine the action of the functor on extremal epimorphism of .

Theorem 2. If is a final epimorphism of (in particular an extremal epimorphism), then is one-to-one and initial; that is, is (homeomorphic to) a subspace of .

Proof. is one-to-one. Indeed, if are two points of and , that is, , then because is an epimorphism. If is final, then is initial. Indeed, if with continuous, then whenever , that is, in , equivalently, . We want to show that , that is, in . Since is final, , so that . Hence, it is enough to show that , which follows from

3. Variants of Compactness in Convergence Lattices

We shall define notions of adherence (the adherence defined here is the natural generalization to subsets of of what is called raw adherence and denoted by in [2]), compactness and closedness, in the point-free convergence setting. To this end, we say that two subsets and of a lattice mesh, in symbols , if for every and every . We also write . Note that if two filters and on mesh, then there is the smallest filter that contains them both.

Definition 3. Let be a convergence lattice and , we define the adherence of asNote that because , and if there is with , so that

Remark 2. As is well known, under the Axiom of Choice, denoting the set of maximal filters on , the set is always nonempty and thus the adherence only depends on maximal filters via:

Lemma 1. If is a -morphism, thenfor every .

Proof. Since for every with ,Abstracting from the case of , we say that is compact if every with satisfies . We shall generalize the notion of compactness in several directions. On one hand, compactness of an element should coincide with compactness of the corresponding standard extremal epimorphism for an appropriate notion of compactness of . On the other hand, compactness has been extended to families of subsets in a very useful fashion in the context of (See, e.g., [1, 5, 6, 12] and references therein) and this can be extended to the point-free setting. We start with the latter, as the corresponding notions will be useful in analyzing the former concept.

3.1. Compactness for Filters

Definition 4. Let be a convergence lattice and . A filter is -compact at ifWhen , we omit the prefix . When we omit “at .” Hence, is compact ifIn the case where , we omit “at ” and add the suffix “oid.” So is -compactoid means thatA related notion is that of near -compactness as introduced (in the case ) in [5]. A filter on is nearly -compact at ifNote that with these definitions, an element is compact if and only if its principal filter is a compact filter. Hence, relativizations of compactness with respect to a class of filters (yielding notions of countable compactness, Lindelöfness, etc.) or with respect to a subset of can be applied to a single element, identifying it with its principal filter. As a result, we say that a convergence lattice is -compact if is -compactoid, that is, for every proper filter .

Remark 3. Note that when is the class of all filters, then -compactness and near -compactness are equivalent. Indeed, if is nearly compact (at ) and there is a filter finer than and . By near compactness, . Since , and thus .
It is clear that the image under the functor of a compact convergence space is -compact (that is, is compact).

Definition 5. Let be a convergence lattice. A filter is closed if

Remark 4. Note that this is a generalization to filters of the notion of closed element of a convergence lattice as defined in [4, 13] (which is different from the notion of closed element introduced in [2], where the present notion is called quasi-closed), where is closed iffor every . Indeed, is a closed element if and only if its principal filter is closed in the sense of Definition 5.
Hence, a subspace of a convergence space is closed if and only if is a closed element of if and only if the principal filter of in the convergence lattice is closed in the sense of the previous definition.

Theorem 3. Let be a convergence lattice and , let be filters on , where is -compactoid and is closed. Then, is nearly -compact.

Proof. Let with . Note that . Then, because , and is -compactoid, hence,As is closed, so thatfor every and the conclusion follows.
In the case where , we obtain, in view of Remark 3:

Corollary 1. If is compactoid, , and is closed then is compact.

In particular, in the case we have the following.

Corollary 2. If is a compact convergence lattice then every closed filter on is compact.

In particular, applying this fact to principal filters, the fact that closed subspaces of a compact convergence space are closed extends to the point-free setting: every closed element of a compact convergence lattice is compact.

The point-free version of the classical fact that a continuous image of a compact set is compact will not extend straightforwardly to all morphisms in an arbitrary category of convergence lattices, but we can give a version in convergence frames.

To this end, note that a class of filters consists of a set of filters on each lattice .

Definition 6. We say that a class of filters is admissible if given a lattice morphism , whenever and whenever .
Note that, in particular, if is admissible and , then every generates a filter of .
Recall that frames are pseudocomplemented; that is, every has a pseudocomplement, which satisfies . Note that if has a pseudocomplement, then if and only if . In general, a lattice or frame morphism does not need to preserve pseudocomplements, though whenever and have pseudcomplements. On the other hand, a morphism of Heyting algebras preserves pseudocomplements.

Lemma 2. Let and be lattices and be a lattice morphism. Let and . Then,

If is pseudocomplemented and preserves pseudocomplements, then the converse is true.

Proof. If then for every , for and . In particular, if then .
Conversely, if then , that is, . If respects pseudocomplements equivalently, .

Theorem 4. Let be a morphism that preserves pseudocomplements. If is compact, then is compact.

Note that in the classical case where , , and , where , then and are Boolean algebras, hence complemented, and respects complements, hence pseudocomplements.

Proof. Let such that . If , then by Lemma 2. Thus, because is compact. In particular, , that is,Since is a frame, this implies that there is with .
Now, by (ptfree continuity). Thus,so that , because is a lattice morphism, hence sends to .
Recall that a convergence space is Hausdorff if consists of at most one point for every filter on the space. The following definition gives an abstraction to the point-free setting.

Definition 7. We say a convergence lattice is Hausdorff if for every either or is an atom of ; that is, is minimal in .
It is straightforward that the image under of any Hausdorff convergence space is a Hausdorff convergence lattice.

Theorem 5. Let be a Hausdorff convergence frame and let be a compact filter on . Then, is closed.

Proof. We have to prove thatIt will be enough to prove thatfor any and any . For any such and , by compactness of . Hence,equivalently,because is a frame. Hence,Because is Hausdorff, it follows that

Remark 5. The same result can be obtained without using the assumption that be a frame, using maximal filters as in Remark 2, hence using the Axiom of Choice.

3.2. Pseudotopological Convergence Lattices and Minimality of Compact Hausdorff Structures

We call a convergence lattice pseudotopological, if for every ,

Note that the reverse inequality in (34) is always true (of course, (34) implicitly assumes that the infimum involved exists. Hence, pseudotopological convergence lattices are most naturally considered in the context of complete lattices).

Given a convergence lattice , we define its pseudotopological modification by

In view of Remark 2, , under the ultrafilter principle, in particular under the Axiom of Choice.

Lemma 3. Let be a convergence lattice. Under the Axiom of Choice, is Hausdorff if and only if is Hausdorff.

Proof. Since , is either or an atom if this is the case for . Conversely, assume is not Hausdorff, that is, there is and with . Under the Axiom of Choice, this means that for every and thus is not Hausdorff.

Remark 6. Note that the corresponding statement that a convergence is Hausdorff if and only if its pseudotopological modification is, can be proved without invoking the ultrafilter principle, but we were not able to obtain the general point-free analog in a choice-free manner.
Another classical convergence result ([1], Corollary IX. 2.8), to the effect that a continuous bijection from a compact pseudotopology to a Hausdorff pseudotopology is a homeomorphism, finds a natural point-free generalization.

Theorem 6. Assume the Axiom of Choice. Let be a bijective morphism, be Hausdorff, and be compact. Then, is an isomorphism.

Keep in mind that if is a bijective morphism of , then is a maximal filter if and only if is maximal, and preserves all infima and suprema.

Proof. We shall prove continuity of , that is,for all .
If then and thus by compactness of . Moreover, by continuity of applied to the filter , which is equivalent to because is bijective. Hence, . Because is Hausdorff, is minimal in and therefore, .
Now take . Applying the fact that is a -isomorphism and thus preserves all infima while , we obtain

3.3. Compact Morphisms

A morphism of is compact iffor every . This terminology comes from the following.

Proposition 4. A standard extremal epimorphism of is compact if and only if its defining element is a compact element.

Proof. In view of Proposition 2, we may assume and . Hence, is compact if and only if for every , where is the filter generates on . Of course, it is equivalent to ask for for every with , for with .
The following observation can be seen as an alternative abstract version (compare Theorem 4) of the fact that the continuous image of a compact space is compact (remember that is an abstraction of a map induced by a map ).

Proposition 5. If is a -morphism and is compact, then is compact.

Proof. Let . We need to show thatIn view of Lemma 1, . Hence, by compactness of ,In particular, in the case of a standard extremal epimorphism , we can identify with for a compact element and with . That is compact means thatfor every with . Now if , then , so that, in view of ,and thus for every , that is, . In other words, is a compact filter.
There are various remaining problems to consider related to compactness in the context of point-free convergence, most notably an analog of the Tychonoff theorem (which first requires an adequate notion mimicking products) and an analog of the Čech–Stone compactification. We hope to address these issues in a future work.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.