I finished my last post by saying that there were relationships between filter spaces and equilogical spaces. This was shown and elaborated upon by Reinhold Heckmann [1]. As he notes, this had in fact been mentioned by Martin Hyland [2]. Reinhold Heckmann gives a complete account of the matter, but goes through assemblies over algebraic complete lattices and modest sets, and I would like to give a simpler account here, if that is possible.
From equilogical spaces to filter spaces
Let (X, ≡) be an equilogical space. Since X is a topological space, there is a natural notion of convergence on X, let me write it →. Instead of forming the quotient X/≡ in Top, one can instead for it in the category of filter spaces Filt. This way, we get a filter space from any equilogical space. That was quick! And we can check that this even provides a functor from Equ to Filt.
Before we try to go the other way around, and build equilogical spaces from filter spaces, let me describe this quotient a bit more explicitly.
Filt is not only Cartesian-closed, but also (complete and) cocomplete. The coproducts are built in the obvious way, and the coequalizers can be thought of as quotients, just like in Top. We build them as follows. Let q : X → X/≡ be the map that sends every point x in X to its equivalence class [x]. Given any filter F of subsets of the quotient set X/≡, we can form the smallest filter of subsets of X that contains all the subsets A of X whose direct image [A]={[x] | x in A} is in F. Because q is surjective, this is also the largest filter F’ of subsets of X such that q[F’] is included in F. (Exercise!) This filter F’ is the inverse image filter of F by q. Now say that F converges to [x] in X/≡ if and only if the inverse image filter of F by q converges to some point equivalent to x in X.
This quotient has the usual universal property: q itself is a continuous map between filter spaces, and every continuous map g from X to a filter space Y such that g(x)=g(x’) for all x≡x’ factors uniquely through q.
In the case that interests us, X is a topological space, and convergence in X/≡ is described as follows: F converges to [x] in X/≡ if and only if there is a point x’≡x such that for every open neighborhood U of x’ in X, the direct image [U] of U is in F.
In general, X/≡ will not be topological, even when X is a topological space. To understand this, let me stress that quotients are computed differently in Filt and in Top. To make things clearer, let me write X/Filt≡ for the quotient taken in Filt, and X/Top≡ for the quotient taken in Top. While convergence in X/Filt≡ is described above, F converges to [x] in X/Top≡ if and only if for every ≡-saturated open neighborhood U of x, [U] is in F. (We did not require U to be ≡-saturated in X/Filt≡.)
For an example, let Nω be the dcpo of all natural numbers plus a fresh infinity element ω added, and take X to be the topological coproduct Nω+Nω. Let ≡ equate the two copies of ω, namely (0, ω) and (1, ω). The topological quotient X/Top≡ is the dcpo N2 of Figure 5.1, p.121. The filter F of all neighborhoods of the equivalence class [(0, ω)] (= [(1, ω)]) converges to it in X/Top≡ (by definition), but it does not converge to it in X/Filt≡. The problem is that for every element x’ that is equivalent to (0, ω) (let us say (0, ω) itself, by symmetry), there is an open neighborhood U of x’ in Nω+Nω., say {42, 43, …, ω} whose direct image [U] is not in F; in fact, you can check that [U] has empty interior in N2. I’ll let you extend the argument and show that F has no limit whatsoever in X/Filt≡. As a final exercise, show that whichever equilogical space (X, ≡) is taken, convergence in X/Filt≡ implies convergence in X/Top≡; only the converse implication can fail (as in the example just given).
From filter spaces to equilogical spaces
Conversely, let (Y, →) be a filter space. Let me show you how one can build an equilogical space from it. (Something will go slightly wrong in the process, let me see whether you can spot it.) As mentioned earlier, it is equivalent to find an algebraic complete lattice with a PER on it.
For the complete lattice, the natural choice is Filt0(Y), the poset of all filters on Y under inclusion. This is very much related to the poset Filt(Y) we considered in Filters, part I, except we also include the trivial filter now. R. Heckmann writes Φ(Y) for this complete lattice.
This is a complete lattice indeed. Any intersection of filters is again a filter, and suprema of a family of filters is given by the filter generated by their union. As a special case, directed suprema are just unions, since a directed union of filters is already a filter.
We now check that Filt0(Y) is algebraic. For any subset A of Y, and generalizing the notation (x) we used in Filters, part II, write (A) for the filter of all supersets of A. (A) is a finite element of Filt0(Y): if a directed union of filters contains the filter (A), that is if A is an element of the union of the filters, then A is in one of them, which must therefore contain (A). And every filter F is the supremum of the filters (A) when A ranges over the elements of F, and this family is directed since an upper bound of (A) and (B) with A, B in F is given by (A ∩ B).
Finally, we define a partial equivalence relation ≃ on Filt0(Y). Typically, we would like two filters to be equivalent if and only if they converge to the same limit in Y, and those elements in dom ≃ should be those that converge to exactly one limit. This runs into the problem that there may just not be any such filter. For example, in Z—, the poset of non-positive integers with the Scott topology, all filters that converge to a point converge to all points below it, hence no filter converges to a unique point.
Instead, we shall consider so-called focused filters. (I just coined the term). Say that a point y in Y is a focus point of a filter F if and only if F converges to y and for every element A of F, y is in A. A filter with a focus point is called focused. This makes sure we have an ample supply of focused filters: all the principal ultrafilters (y) are focused, and y is a focus point. (I’ll give further justification for focusing below.)
Following Heckmann, we now declare that F ≃ G, for two filters F and G, if and only if both F and G are focused, and have a common focus point.
That is it, we now have an algebraic complete lattice Filt0(Y) with a PER ≃. We have already seen that this was an equivalent way of describing an equilogical space. Explicitly, this is the topological space Foc(Y) of all focused filters of subsets of Y, with the topology induced from the Scott topology on Filt0(Y), and with the equivalence relation ≡ defined as meaning “have the same focus point”. I’ll also write Foc(Y) for the resulting equilogical space.
The topology on Foc(Y) is very simple, albeit a bit surprising. Since the sets ↑(A) form a base for the Scott topology on Filt0(Y), their intersections with Foc(Y) form a base of the latter. These are the sets ☐A of all focused filters F that contain A—where A is an arbitrary subset of Y.
The whole construction is functorial, too. On morphisms f: X → Y (i.e., continuous maps between filter spaces), Foc(f) is the map that sends every focused filter F on X to its image filter f[F]; note that if F has focus point x, then f(x) is a focus point of f[F].
Foc and the quotient functor /Filt, mapping every equilogical space (X, ≡) to X/Filt≡, are not quite inverse to each other. But they are adjoints: Foc is the right adjoint, and /Filt is the left adjoint. Well, almost.
There is a problem. Have you seen where?
I’ve lied at some point.
(Spoiler below.)
You would need the relation ≃ introduced above to be a PER, right?
It is clearly symmetric.
Have you checked that it was transitive?
In general it is not… unless focus points are unique. In that case, F ≃ G if and only if both F and G are focused, and have the same focus point; then transitivity is clear. But look at the special case of filters in topological spaces. Imagine F is a focused filter, with two foci x and y. F contains all the open neighborhoods of x, and they should all contain y. So we must have x≤y, where ≤ is the specialization quasi-ordering of X. Symmetrically, y≤x. We can conclude that x=y only when X is T0…
T0 filter spaces
To correct this, define Foc(Y) only for those filter spaces (Y, →) that are T0. This should mean exactly what we intend: a filter space (Y, →) is T0 if and only if every filter has at most one focus point.
This is justified by the fact that our construction Foc(Y) will now make sense, but also by the fact that the topological filter spaces that are T0 according to this definition are exactly those that are T0 as plain topological spaces.
There are apparently many competing notions of T0 for filter spaces. The one above is equivalent to the following [1, Section 2.4]: (Y, →) is T0 if and only if ({x, y}) cannot both converge to x and y, for any pair of distinct points x, y in Y.
We now have another problem… which is that the quotient functor /Filt may produce filter spaces that fail to be T0. One might think of restricting equilogical spaces to those whose quotient in Filt is T0, but the theory starts to be messy. As Heckmann says [1, Corollary 4.2], this works for “certain full subcategories” of equilogical spaces. (Which they are is defined precisely in the preceding Theorem 4.1.) We correct this by temporarily abandoning the idea of T0 separation altogether.
Assemblies and modest sets
The Foc construction only works for T0 filter spaces. If (Y, →) is not assumed to be T0, we can still build a binary relation between focused filters and their focus points. This way, we will not get an equilogical space, or rather we will not obtain an algebraic complete lattice Filt0(Y) with a PER ≃. Rather we will obtain:
- an algebraic complete lattice Filt0(Y), and
- a set Y, together with
- a binary relation E between elements of the latter and elements of the former, such that E(y), the set of elements related by E to y, is non-empty for every y in Y.
Such a structure is called an assembly. More formally, an assembly is a triple (Y, L, E) of a set Y (the carrier), an algebraic complete lattice L, and a binary relation E between Y and L such that E(y) is non-empty for every y in Y.
Among all assemblies, we retrieve (up to isomorphism) the equilogical spaces by requiring E(x) and E(y) to be disjoint when x≠y. Such assemblies are called modest sets, an equivalent definition of algebraic complete lattice with a PER. The PER ≃ on L is defined as declaring equivalent two elements of L if and only if they are related to a common point y, namely if and only if they belong to the same set E(y). The set E(y) can be seen as equivalence classes, and building the quotient dom ≃/≃ gives you Y back, up to iso.
The relationship between equilogical spaces, modest sets, and algebraic complete lattices with a PER had already been set up in [3].
We can now extend the /Filt construction beyond equilogical spaces, and to all assemblies. Given an assembly (Y, L, E), we define a filter space structure on Y (we do not need to take a quotient here, since Y plays the role of the quotient set X/≡, directly) by declaring that a filter F on Y converges to y if and only if there is an element x in the lattice L such that for every open neighborhood U of x in L, E-1(U) is in F. Here E-1(U) is defined as the set of points y in Y such that E(y) intersects U, and plays the role we formerly assigned to the direct image [U].
With all that, Heckmann shows that the modified /Filt functor from assemblies to filter spaces is left adjoint to Foc. This adjunction now restricts to well-chosen subcategories [1, Theorem 4.1]:
- Hyland’s category of filter spaces satisfying the extra condition that if F → x then F ∩ (x) → x (a weaker condition than that required for convergence spaces), and the full subcategory of assemblies that are join-closed (for each y in Y, E(y) is closed under taking arbitrary joins formed in L) and order-convex (for each y in Y, if a ≤ b ≤ c and a and c are in E(y), then b is in E(y), too);
- The subcategory of those Hyland filter spaces that are T0, and the join-closed order-convex modest sets;
- The subcategory of convergence spaces and the full subcategory of join-closed, meet-closed, order-convex assemblies;
- The subcategory of T0 convergence spaces and the full subcategory of join-closed, meet-closed, order-convex modest sets.
However, and although modest sets and algebraic complete lattices with PERs are the same thing, characterizing join-closure, meet-closure and order-convexity directly on the latter is harder than on modest sets.
Well, that’s it. I will probably end this whole series on filters right here. That was starting to be rather technical. I may tell you what Frédéric Mynard and I have been up in January 2014, which he presented at the Summer Topology Conference 2014, some day. I’ll probably switch to another subject for the next post, though.
— Jean Goubault-Larrecq (September 29th, 2014)
[1] Reinhold Heckmann. On the Relationship between Filter Spaces and Equilogical Spaces. 1998. Available on the Web.
[2] J. Martin E. Hyland. Continuity in Spatial Toposes. A. Dold and B. Eckmann, eds., Applications of Sheaves, Springer Verlag Lecture Notes in Mathematics 753, pp. 442-465, 1977.
[3] Andrej Bauer, Lars Birkedal, and Dana S. Scott. Equilogical Spaces. Theoretical Computer Science 315(1), 5 May 2004, 35-59. (Submitted as early as 1998, as far as I know.)