1. Introduction
Equivalences or dualities between categories are a form of symmetry on the category theory level. While this form of symmetry is very abstract, it is also extremely fruitful since it can connect apparently distant branches of mathematics.
Extensions of Stone Duality and their applications have been quite popular in recent years. One can mention the locally compact Hausdorff case of [
1] and remove the zero-dimensionality together with the commutativity assumptions in [
2]. We also have generalisations of Gelfand–Naimark–Stone Duality to completely regular spaces (see [
3]) and its application to the characterisation of normal, Lindelöf and locally compact Hausdorff spaces in [
4]. Some extensions of Stone Duality drop the compactness assumption completely: for example, the paper [
5] extends this duality to all zero-dimensional Hausdorff spaces. From the point of view of algebraic structures, Stone Duality has been extended in [
6] to (non-distributive, in general) orthomodular lattices, which correspond to spectral presheaves, while [
7] extends it to some non-distributive (implicative, residuated, or co-residuated) lattices and applies to the semantics of substructural logics, and [
8] extends this duality to a non-commutative case of left-handed skew Boolean algebras. Moreover, Esakia Duality has been extended to implicative semilattices in [
9]. Applications of Stone Duality have been developed in [
10] (canonical extensions of lattice-ordered algebras) and [
11] (the semantics of non-distributive propositional logics). Some recent applications of Stone Duality appear also in the theory of
-algebras, see [
12].
On the other hand, one of the greatest mathematicians in history, Alexander Grothendieck, suggested in his scientific programme [
13] creating a new type of topology, called
tame topology, that would eliminate pathological phenomena such as space-filling curves. O-minimality is widely recognized as a realisation of Grothendieck’s programme. It is usually understood as studying o-minimal structures (in the sense of model theory). The fundamental monograph about o-minimal structures is [
14]. It appears that arbitrary open sets are not so important in o-minimality, but the definable open sets play the main role.
Although Grothendieck’s programme has been being realised for many decades in many special situations, the authors of [
15] are disappointed that no clear definition of the notion of tame topology suggested by A. Grothendieck was given (Cruz Morales [
16] gives some history of the evolution of Grothendieck’s ideas about the notion of space to physical and philosophical questions).
It seems that not enough attention was paid to linking Stone-like dualities to the tame topology of A. Grothendieck. This paper fills this gap by clearly proposing the theory of small spaces and locally small spaces as a realisation of Grothendieck’s postulate on a purely topological level (i.e., on the level of general topology) and making another step in developing this sort of topology (apparently a kind of generalised topology, but in fact the usual topology with additional structure).
Dropping some of the requirements for a topology leads to the notions of a unitary smopology and a small space (already used in [
17,
18,
19,
20]) as well as the notions of an arbitrary smopology and a locally small space (used, for example, in [
19,
20,
21]). In this way, we construct a kind of algebra-friendly topology. Recall that the small spaces are the underlying topological structures of the definable spaces and the locally small spaces are the underlying topological structures of the locally definable spaces. (Both definable and locally definable spaces were used by many authors. See, for example, [
14,
22,
23], implicitly even [
24]).
The main objective of this paper is to give a version of Esakia Duality for suitably chosen small spaces. We call them Heyting small spaces, following the conventions of [
25]. The first tool to analyse small spaces is the theory of spectral spaces, which is already developed enough (see the monograph [
25]). The present paper is a continuation of the paper [
20] about some versions of Stone Duality [
26] or Priestley Duality [
27] for locally small spaces. This time we focus on giving a new version of a related duality due to Leo Esakia [
28] for small spaces. Basic facts about spectral spaces may be found in [
25,
29]. On the other hand, refs. [
30,
31] are good resources about Esakia Duality and its connections to modal logics. Our version of Esakia Duality helps to understand open continuous definable mappings between definable spaces over o-minimal structures.
Recognizing the role of smopologies (implicit in [
32] (Definition 7.1.14) or [
33] (p. 12)) should be useful in many areas of mathematics such as the generalisations of o-minimality, analytic geometry, algebraic geometry and in many other contexts since families of sets closed under only finite unions are much more natural in many branches of mathematics than usual topologies. In contrast to the usual topology, where only spectral reflections (see [
25] (Chapter 11)) are available, using dualities or equivalences for suitably chosen smopologies allows transferring structural information without any losses between algebraic and topological structures. Translating between the topological and the algebraic languages made available by our version of Esakia Duality should give more understanding of locally definable spaces over structures with topologies, especially in the case of definable topologies. We want to stress that our approach considers the geometry of (first-order) definable sets but does not reduce to it.
As far as the set-theoretical axiomatics is concerned, we follow Saunders Mac Lane’s standard Zermelo–Fraenkel axioms with the Axiom of Choice plus the existence of a set which is a universe, see [
34] (p. 23). This allows speaking about proper classes of sets (in particular, about categories) while using methods from the usual mathematics developed in the axiomatic system
ZFC (See “Axiomatic assumptions” in [
35] for the full explanation of an axiomatic system using a universe).
Notation. We shall use a special notation for operations on families of sets. For example, for a family intersection
or for other operations
2. Pre-Heyting and Pre-Boolean Small Spaces
Definition 1 (cf. [
19] (Definition 2.21) and [
20] (Definition 2))
. A small space
is a pair , where X is any set and satisfies the following conditions:- (S1)
,
- (S2)
if , then .
Elements of are called smops (i.e., small open subsets of X, for reasons that become clear after reading [21] or [19]). Their complements will be called co-smops or inverse smops. The family will be called a (unitary) smopology, and the family of co-smops will be called a co-smopology. The Boolean combinations of smops will be called the constructible sets and the family of all constructible sets of a small space will be denoted by . Definition 2. A mapping between small spaces is:
- 1.
continuous (or strictly continuous) if the preimage of any smop is a smop (),
- 2.
a strict homeomorphism if f is a bijection and .
We have the category of small spaces and continuous mappings.
Example 1. (1) Each topology on X and each Boolean subalgebra of is a unitary smopology on X. (2) The spaces , , , , from [19] (Example 2.14) are examples of small spaces. (3) Since the category of small spaces and continuous mappings in our sense is concretely isomorphic ([36] (Remark 5.12)) to the category in the sense of [17], also [17,18] give examples of small spaces. Definition 3. The topology , generated by the smops, will be called the original topology. The closure, the interior and the exterior operations in the original topology will be denoted by , respectively. The topology , generated by the co-smops, will be called the inverse topology. The closure and the interior operations in the inverse topology will be denoted by , respectively. The topology , generated by the constructible sets, will be called the constructible topology. The closure and the interior operations in the constructible topology will be denoted by , respectively.
Example 2. For each small space , we can produce the following small spaces:
.
Definition 4. A small space will be called pre-Boolean if any of the equivalent conditions:
- 1.
,
- 2.
,
- 3.
are satisfied.
Fact 1. In a pre-Boolean small space, all the three above-mentioned topologies are equal.
Fact 2. For each , the space is pre-Boolean.
Fact 3. For each , the space is pre-Boolean iff is pre-Boolean.
Definition 5. For any subset in a small space , the pair is called a subspace of .
Remark 1. The original, the inverse and the constructible topologies in a subspace are topological subspace topologies of, respectively, the original, the inverse and the constructible topologies of the whole small space .
Definition 6. A small space will be called:
- 1.
pre-semi-Heyting if the closure in the original topology of any smop is a co-smop (i.e., for any ),
- 2.
pre-Heyting if the closure in the original topology of any constructible set is a co-smop (i.e., for any ).
The following proposition and theorem are inspired by Section 8.3 of [
25].
Proposition 1. Assume that is a pre-semi-Heyting small space. Then:
- (1)
for each , we have ,
- (2)
the following two mappings are well defined:
- (a)
the open regularisation mapping given by the formula ,
- (b)
the closed regularisation mapping given by the formula ,
- (3)
for each , the subspace is pre-semi-Heyting.
Proof. (1) Follows from the definition by taking complements.
(2) Obvious by the above.
(3) For , we get , a co-smop in V. □
Theorem 1 (characterisation of pre-Heyting spaces). For a small space , the following conditions are equivalent:
- (1)
is pre-Heyting,
- (2)
for each , we have ,
- (3)
for each , we have ,
- (4)
for each , the subspace is pre-Heyting,
- (5)
for each , the subspace is pre-semi-Heyting,
- (6)
for each , the subspace is pre-semi-Heyting.
Proof. By taking complements.
By taking finite unions.
For and , we have . Now is a (relative to A) co-smop.
Trivial.
Trivial.
If , then . Since by (6), we have . □
3. Spectral Small Spaces
Notation. We use the following notation for some distinguished families of subsets in a small space :
, the weakly open sets,
, the weakly closed sets,
, the inversely weakly open sets,
, the inversely weakly closed sets,
, the constructibly weakly open sets,
, the constructibly weakly closed sets.
We use the symbols ℧,
after [
37].
Remark 2. By we denote the category of spectral topological spaces (the name introduced by Hochster [29]) and spectral mappings (compare [20,25]). By the Stone Duality ([25] (Chapter 3)), this category is dually equivalent to the category of bounded distributive lattices and their homomorphisms (denoted in [20] by ). Recall that the category of Priestley spaces and Priestley mappings ([25] (1.5.15), denoted here by ) is concretely isomorphic to , so these two categories may be identified. Definition 7. A spectral small space is a small space where is a spectral topological space and is the family of all compact open sets of this space. The category of spectral small spaces and (strictly) continuous mappings is concretely isomorphic with , so these categories may be identified.
Example 3. For a spectral topological space , the corresponding spectral small space has the property, that the constructible weakly open sets are exactly the compact open sets, so smops, i.e., and (See [25] (1.3)). Example 4. A weakly open constructible set in a small space may not be a smop. Take the small space where is the family of all finite subsets of . Then the original topology is discrete and the set is a weakly open co-smop but not a smop, so . In particular, .
Example 5. A small space may have all smops (topologically) compact but not be sober. Let X be an infinite set. Then X is irreducible in the small space but not the closure of a point ( is the family of all cofinite subsets of X.)
Example 6. A small space may be (topologically) sober but not compact. Take any non-compact Hausdorff topological space.
Example 7. A small space may be sober and compact with not all smops compact. Take the interval with the natural topology as the smopology.
Proposition 2. If X is a sober small space with all smops compact, then X is a spectral small space.
Proof. Notice that , so . The other conditions for spectrality are obvious. □
Proposition 3. If a small space has all smops compact, then:
- (1)
the ideals in are in a bijective correspondence with the weakly open sets, so also with the weakly closed sets,
- (2)
the prime ideals of are in a bijective correspondence with the non-empty, irreducible, weakly closed sets.
Proof. For an ideal I in , the set is weakly open (and its complement is weakly closed). For a weakly open set , the set is an ideal in . Obviously, and . On the other hand, if is compact and , then there exist a finite family of members of I covering B, so . This means that and the mappings s and i are bijections.
By Proposition 1.1.11 of [
25], the ideal
is prime (
implies
or
) iff
is a prime weakly open set iff
is a non-empty, irreducible, weakly closed set. □
Example 8. If not all smops are compact, then it may happen that , using notation from the above proof.
- 1.
In the small space we can take .
- 2.
Consider the interval with the natural topology as the unitary smopology. Finite unions of open subintervals in the open interval form an ideal that covers but not all smops that are contained in are of this form.
8. Other Useful Categories of Small Spaces
Definition 17. Let be a partially ordered set. For any subset , we have:When , we write instead of . Definition 18. A Priestley small space is a system where is a Boolean small space and is a partial order on X satisfying the Priestley separation axiom
A Priestley mapping is a strictly (equivalently: weakly) continuous non-decreasing mapping between Priestley small spaces. We have the category of Priestley small spaces and Priestley mappings. Remark 7. The category is concretely isomorphic to the category of topological Priestley spaces and Priestley morphisms (see [25,31]) as well as to and to , so all these categories may be identified. Definition 19. An Esakia small space is a Priestley small space that satisfies the conditionAn Esakia mapping is a Priestley mapping that is a p-morphism
, i.e., it satisfies . We have the category of Esakia small spaces and Esakia mappings. Remark 8. Following [30,31], we understand as saying that y is a generalisation of x, denoted also by , while [25] uses the opposite convention. Remark 9. The category is concretely isomorphic to both the category of topological Esakia spaces and Esakia morphisms (see [25,30,31]) and to from Definition 13, hence these categories may be identified. Similarly, the category of Esakia small spaces with distinguished decent sets and Esakia mappings respecting the decent sets is concretely isomorphic to . In analogy to the category
from [
20], we have the following category.
Definition 20. An object of is a system where is a Heyting (topological) spectral space and is a constructibly dense subset. (Such is called a decent subset of X.) A morphism from to in is such a Heyting spectral mapping between Heyting spectral spaces that respects the decent set, i.e., .
In analogy to the category
from [
20], we have the following category.
Definition 21. A Heyting algebra is a system where is a bounded distributive lattice and for each there exists the largest such that . We then denote c by . A homomorphism of Heyting algebras is a homomorphism of bounded lattices satisfying additionally the conditionWe have the category of Heyting algebras and their homomorphisms. Remark 10 (The classical Esakia Duality)
. The category is dually equivalent to . (See [30] or [31] for a good exposition.) Namely, we have the covariant functors:- (1)
given by and where is a homomorphism, for and denotes the set of all prime filters in L,
- (2)
given by with the obvious set-theoretic operations and where for and .
In analogy to the category
from [
20], we have the following category.
Definition 22. Objects of are pairs where L is a Heyting algebra and is a constructibly dense subset (i.e., a decent set of prime filters). Morphisms of are such homomorphisms of Heyting algebras that .
10. Conclusions
After introducing pre-Heyting and pre-Boolean small spaces (
Section 2) and giving a characterisation of pre-Heyting small spaces (Theorem 1), we have noticed that topological spectral spaces may be seen as
sober small spaces with all smops compact (
Section 3). We have recalled the basic properties of the specialization relation in the context of small spaces (
Section 4). We have introduced the method of the standard spectralification of a
small space (
Section 5) and inferred some of its properties including preserving of being Heyting both ways (
Section 6). After introducing the category
HSS of Heyting small spaces and Heyting continuous mappings, we have given an example of such mappings: open continuous definable mappings between definable spaces over o-minimal structures (
Section 7), which means that those mappings can be described using homomorphisms of Heyting algebras. Next, we have constructed the category
ESS of Esakia small spaces and Esakia mappings, concretely isomorphic to
HSpec, and other useful categories (
Section 8). Finally, we have proven a version of Esakia Duality for Heyting small spaces (
Section 9, Theorem 2), making a new step in developing
tame topology.
As extending our version of Esakia Duality to locally small spaces needs another set of notions, we postpone this to another paper. Future research may also concern the use of the (already developed) theory of (up-)spectral spaces to understand (locally) small spaces, especially exploiting the notions of normal spectral spaces and spectral root systems.