Another form of Stone duality

I thought I would devote my blog this month to the Domains workshop, but a sudden health problem prevented me to go there.  I’ll have to read the papers instead…  In the meantime, I would like to talk about a curious construction Frédéric Mynard and I obtained at the end of the paper [1].

A bit of (personal) history

We started working on [1] in 2013, when Frédéric Mynard invited me at Georgia Southern University, Statesboro, GA, USA, where he was working at the time.  Our initial goal was to provide some form of Stone duality for convergence spaces.  We found what it should be pretty quickly (see Section 2 of [1]).  That consists in a pair of adjoint functors that relate the category Conv of convergence spaces and the opposite of a certain category of lattices with a monotonic `lim’ operation, which roughly sends any filter to its set of limits (convergence lattices).  As for Stone duality, one functor in the adjunction completely forgets about the points of the given convergence space, and the other one must reconstruct the points from a convergence lattice.  We were tempted to call the paper `pointfree convergence’, or even `pointless convergence’, but we were too shy to make the pun.

Since there is a whole zoo of refinements of convergence spaces—limit spaces, Kent spaces, Antoine spaces, pretopological spaces, pseudotopological spaces, and what have you—we thought of pushing that work and obtain corresponding Stone-like dualities for limit spaces (that works fine) and for various other refinements, all of those dualities being related by adequate squares of adjunctions.

We started to run into trouble with pretopological spaces and pseudotopological spaces, and even worse, with topological spaces.  One of the reasons it took us five years to eventually submit the paper is that we were never satisfied with what we had obtained in those cases.  In a nutshell, nothing worked as nicely as Stone duality does.  Eventually, we found something neat for pretopological spaces and adherence spaces (see Sections 5 through 7 in [1]), although that is somehow elaborate.  We had something for pseudo topologies as well, but not as neat as the rest, and we have not included it.  The curious part is the kind of Stone duality we obtained this way for topological spaces… which happened not to be Stone duality at all, but a different construction.  This is what I want to explain here.

Topological coframes

What we have obtained is an adjunction between Top, the category of topological spaces and continuous maps, and a new category CFtop, defined as follows.

Its objects are the topological coframes, namely the coframes L, together with a sublattice C(L) of L, whose elements are all complemented in L.  The elements of C(L) are called the closed elements of L.

A coframe is the opposite of a frame, in other words, it is a complete lattice in which arbitrary infima distribute over finite suprema (a ∨ infi bi = infi (a ∨ bi)).  In a coframe, every element a has a unique pseudocomplement a*, defined as the least element b such that a ∨ b = ⊤, and an element a is complemented if and only if a ∧ a*=⊥.  A sublattice is a subset that is closed under finite infima and finite suprema (in particular, it contains ⊤ and ⊥.)

How do such topological coframes occur?  Very simply: for every topological space X, let L be P(X), the powerset of X, and define C(L) as the sublattice of closed subsets of X.  That obviously defines a topological coframe.  For example, every element of C(L) is complemented… because every element of L is complemented anyway.  What might seem strange is our choice of conditions on L and C(L): why didn’t we simply require L to be a complete Boolean algebra, for example?  and why didn’t we require C(L) to be closed under arbitrary infima, not just finite infima?

I will mostly not explain why here: the main reasons are so that there neat adjunctions with adherence coframes, which are themselves in a Stone-like duality with adherence spaces.  Why C(L) should consist of complemented elements, and why L should be a coframe, will be made clearer when we relate topological coframes and frames, near the end of this post.

Let me just say that, in analogy with Stone duality, the lattice of sublocales of a given locale, which plays the rôle of P(X) here, is a coframe in general, not a complete Boolean algebra.  This is also how we discovered sieves, only to realize later that they were isomorphic to sublocales.

In any case, the construction of the topological coframe P(X) (with its sublattice of closed sets) from X defines a functor PTop → (CFtop)op. For that to make sense, we of course need to say what the morphisms are in CF.  Those are simply the coframe homomorphisms ϕ : L → L‘ (preserving arbitrary infima and finite suprema) that map closed elements of L to closed elements of L’.  If fX → Y is any continuous map, then P(f) is the inverse image map f-1P(Y) → P(X).  That certainly maps closed elements of P(Y) (closed subsets of Y) to closed elements of P(X).

The left adjoint to P

In order to obtain a nice duality, we need to find a right adjoint to P.  Given a topological coframe L (with sublattice of closed elements C(L)), we define the points of L as its join-prime elements: those elements u different from ⊥ such that, if u is below the supremum of two elements, then it must be below one of them.

It is interesting to discover what the join-primes are in P(X): those are the non-empty sets A such that A cannot be included in a union of two sets without being included in one of them.  It is pretty easy to see that those sets are exactly the one-element sets {x}, which one can equate with x.

We let pt L be the set of points (=join-primes) of L.  We must topologize pt L, and here is how we do it.  For every a in L, build the set a of all points x of L below a.  Then we declare that the topology of pt L is the coarsest topology that makes c closed for each c ∈ C(L).

It can be shown (Lemma 103 in [1]) that the closed sets of pt L are exactly the sets of the form c, where c is in ∧C(L), the subcoframe of L consisting of infima of elements of C(L).

Then P ⊣ pt is an adjunction (Proposition 104), whose counit εLL → P(pt L) sends every a to a, and whose unit ηX : X → pt(P(X)) maps x to {x}, and is in fact an isomorphism.  (Note that in the classical Stone adjunction, the unit at X is an isomorphism if and only if X is sober—hence there is no non-trivial notion of sobriety in our new Stone-like adjunction.)

Relating topological coframes and frames

One wonders at the relation between topological coframes, and the frames that form the core of Stone duality.  This is the topic of the final section (Section 8.5) of [1].

Every topological coframe L yields a coframe ∧C(L), hence a frame (∧C(L))op.  That defines a functor ∧CopCFtop → Frm (Fact 117 in [1]).

The converse direction is more elaborate.  There is a functor Sl : Frm → CFtop, which maps every frame Ω to its coframe of sublocales.  (This is why we defined topological coframes as coframes.)  The closed elements of Sl(Ω) are simply chosen to be the closed sublocales c(a), a in Ω, (in the usual localic sense of `closed’:  c(a) is the sublocale of all elements above a).

The fact that Sl is a functor depends on the following universal property of the coframe of sublocales: for every frame homomorphism φ : Ω → Ω’ such that φ(a) is complemented for every a in Ω, there is a unique frame homomorphism φ* : Sl(Ω)op → Ω’ such that φ*(c(a)) = φ(a) for every a in Ω.  (This is why we required C(L) to consist of complemented elements only.)  To define the Sl functor on frame homomorphisms φ : Ω → Ω’, we first build the composition c o φ : Ω → Sl(Ω’).  We observe that this is a frame homomorphism again, and one which maps every element of Ω to a complemented element of Sl(Ω’)—because closed sublocales are always complemented.  Hence we can form (c o φ)* : Sl(Ω) → Sl(Ω’), and that is Sl(φ) by definition.

The two functors ∧Cop and Sl do not form an adjunction in general, but the composition ∧Cop o Sl forms an isomorphism of categories.  In other words, Frm appears as a retract of CFtop.  That retraction even restricts to a coreflection between Frm and the full subcategory of strong topological coframes, namely those topological coframes whose closed elements are closed under arbitrary infima.

Conclusion

I am not sure yet that topological coframes will prove useful, but their definition and properties are certainly intriguing.  For example, given a topological space X, we can form its coframe LH(X) of closed subsets.  The pseudocomplement of C in L is the closure cl (X — C) of its complement, and C itself is complemented if and only if C is clopen.  We can then define a topological coframe H(X) by defining C(H(X)) as the set of clopens of X; this is the smallest possible choice for C(H(X)).

This yields a new topological space pt(H(X)).  What is it?  Its points are the join-primes of H(X), namely the irreducible closed subsets of X, but pt(H(X)) is not the sobrification of X: its closed subsets are the sets {C irreducible closed in XCD}, where D ranges over the intersections of clopens.  All such sets D are closed, hence the topology of pt(H(X)) is coarser than the topology of the sobrification of X.  Again, I am not sure this is any useful, but who knows.

 

  1. Jean Goubault-Larrecq and Frédéric Mynard.  Convergence without points.  arXiv 1807.03226, July 2018.

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