{"id":6841,"date":"2023-07-20T08:00:00","date_gmt":"2023-07-20T06:00:00","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6841"},"modified":"2023-07-14T11:28:15","modified_gmt":"2023-07-14T09:28:15","slug":"exponentiable-locales-ii-the-exponentiable-locales-are-the-continuous-frames","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6841","title":{"rendered":"Exponentiable locales II: the exponentiable locales are the continuous frames"},"content":{"rendered":"\n

In a previous post<\/a>, we had seen that every exponentiable locale \u03a9 has to be a continuous frame. We will show the converse: every continuous frame \u03a9 is exponentiable in the category Loc<\/strong> of locales, thus entirely characterizing the exponentiable locales. The result is due to Martin Hyland [1]. Since then, there have been several other proofs of it, by Peter T. Johnstone [2,3] and by Christopher F. Townsend [4], at least.<\/p>\n\n\n\n

Martin Hyland proves this in a very elementary way. (Warning: I said “elementary”, not “simple”. “Elementary” means with the help of little background knowledge.) Given a continuous frame \u03a9, he builds a frame out of generators and relations, and then shows that this frame is the desired exponential object in Loc<\/strong>.<\/p>\n\n\n\n

One can understand the construction by looking at exponential objects in Top<\/strong> (see Section 5.4 of the book<\/a>.) This is entirely similar. When X<\/em> is a core-compact space, namely a space such that O<\/strong>X<\/em> is a continuous frame, one builds the exponential object YX<\/sup><\/em>, for every topological space Y<\/em>, as the space [X<\/em> \u2192 Y<\/em>] of all continuous maps from X<\/em> to Y<\/em>, with the core-open topology. The core-open topology is generated by subbasic open sets of the form [U<\/em> \u22d0\u20131<\/sup> V<\/em>] (Definition 5.4.1 in the book<\/a>), where U<\/em> ranges over the open subsets of X<\/em> and V<\/em> ranges over the open subsets of Y<\/em>; and [U<\/em> \u22d0\u20131<\/sup> V<\/em>] denotes the collection of continuous maps f<\/em> : X<\/em> \u2192 Y<\/em> such that U<\/em> \u22d0 f<\/em>\u20131<\/sup>(V<\/em>).<\/p>\n\n\n\n

In order to build the exponential locale L<\/em>\u03a9<\/sup>, where \u03a9 is substituted for X<\/em>, and L<\/em> is now an arbitrary frame, we cannot rely on having a convenient space of functions f<\/em>, so we declare that L<\/em>\u03a9<\/sup> should be generated from generators which I would write as [u<\/em> \u22d0\u20131<\/sup> v<\/em>], where u<\/em> ranges over \u03a9 and v<\/em> over L<\/em>. We cannot define<\/em> [u<\/em> \u22d0\u20131<\/sup> v<\/em>] as any kind of collection of functions, but we can axiomatize the properties that the sets [U<\/em> \u22d0\u20131<\/sup> V<\/em>] have in the topological case, and quotient the free frame generated by the generators [u<\/em> \u22d0\u20131<\/sup> v<\/em>] by the resulting axioms (or relations, as one usually says).<\/p>\n\n\n\n

In this post, I will allow myself a few shortcuts. This has no interest whatsoever, mathematically, especially since those shortcuts will eventually rely on results that require classical logic and the axiom of choice, while M. Hyland’s argument does not: his argument is constructive. (If you object: yes, I have sinned, and I repent.) But I hope that what I will do will be easier to grasp.<\/p>\n\n\n\n

The first of these shortcuts is that, if \u03a9 is a continuous frame, then it is spatial. In fact, \u03a9 is isomorphic to O<\/strong>X<\/em>, for some topological space X<\/em> (the space pt<\/strong> \u03a9 of points of \u03a9, as usual with Stone duality), so that I will simply equate \u03a9 with O<\/strong>X<\/em>. Additionally, X<\/em> is locally compact sober: this is Hofmann\u2013Lawson duality (Theorem 8.3.21 in the book<\/a>).<\/p>\n\n\n\n

This leads us to the second shortcut. Instead of imitating an axiomatization of the core-open topology, we will imitate an axiomatization of the compact-open topology: when X<\/em> is locally compact, the compact-open and core-open topologies coincide on [X<\/em> \u2192 Y<\/em>] for any space Y<\/em> (Exercise 5.4.8 in the book<\/a>), so that should be fine. The compact-open topology has subbasic open sets of the form [Q<\/em> \u2286 V<\/em>], where Q<\/em> is compact saturated in X<\/em> and V<\/em> is open in Y<\/em>, where [Q<\/em> \u2286 V<\/em>] means the collection of continuous maps f<\/em> : X<\/em> \u2192 Y<\/em> such that f<\/em>[Q<\/em>] \u2286 V<\/em>; or equivalently, such that Q<\/em> \u2286 f<\/em>\u20131<\/sup>(V<\/em>).<\/p>\n\n\n\n

Axiomatizing the compact-open topology<\/h2>\n\n\n\n

Let me list all the axioms we can think of that the subbasic open subsets [Q<\/em> \u2286 V<\/em>] defining the compact-open topology on [X<\/em> \u2192 Y<\/em>] should have, for all spaces X<\/em> and Y<\/em> such that X<\/em> is locally compact and sober.<\/p>\n\n\n\n

However, before I do that, let me list a few useful properties that I will need later on. Let us write Q<\/strong>X<\/em> for the set of compact saturated subsets of X<\/em>; we order it by reverse<\/em> inclusion \u2287, yielding the (extended) Smyth powerdomain of X<\/em>, which we have already encountered a few times, and is also mentioned in the book<\/a> (in Proposition 8.3.25, for example).<\/p>\n\n\n\n

Property Wf.<\/strong> (Wf is for `well-filtered’) Every sober space X<\/em> is well-filtered: given any filtered family (Qi<\/sub><\/em>)i<\/em> \u2208 I<\/em><\/sub> of compact saturated subsets of X<\/em> and any open subset U<\/em> of X<\/em>, if \u2229i<\/em> \u2208 I<\/em><\/sub> Qi<\/sub><\/em> \u2286 U<\/em> then Qi<\/sub><\/em> \u2286 U<\/em> for some i<\/em> in I<\/em>.<\/p>\n\n\n\n

By filtered, I mean directed in Q<\/strong>X<\/em>, namely: the family is non-empty, and for all i<\/em> and j<\/em> in I<\/em>, there is a k<\/em> in I<\/em> such that both Qi<\/sub><\/em> and Qj<\/sub><\/em> contain Qk<\/sub><\/em>. Now Property Wf is simply Proposition 8.3.5 in the book<\/a>.<\/p>\n\n\n\n

Property U.<\/strong> (U is for `union’) For every open subset U<\/em> of a locally compact space X<\/em>, there is a family (Qi<\/sub><\/em>)i<\/em> \u2208 I<\/em><\/sub> of compact saturated subsets of X<\/em>, all included in U<\/em>, such that U<\/em> = \u222ai<\/em> \u2208 I<\/em><\/sub> int(Qi<\/sub><\/em>).<\/p>\n\n\n\n

Proof.<\/em> We write U<\/em> as a union of interiors int(Qx<\/sub><\/em>) of compact saturated subsets Qx<\/sub><\/em> of U<\/em>: this is by local compactness, choosing a compact saturated neighborhood Qx<\/sub><\/em> of x<\/em> included in U<\/em> for each point x<\/em> of U<\/em>. The desired family is (Qx<\/sub><\/em>)x<\/em> \u2208 U<\/em><\/sub>. \u2610<\/p>\n\n\n\n

Property I.<\/strong> (I is for `interpolation’) Let X<\/em> be a locally compact space, Q<\/em> be a compact saturated subset of X<\/em>, and U<\/em>1<\/sub>, …, Un<\/sub><\/em> be finitely many open subsets of X<\/em>. If Q<\/em><\/em> \u2286 \u222ak<\/em>=1<\/sub>n<\/sup><\/em> Uk<\/sub><\/em> then there are compact saturated subsets Q<\/em>1<\/sub> \u2286 U<\/em>1<\/sub>, …, Q<\/em>n<\/em><\/sub> \u2286 Un<\/sub><\/em> such that Q<\/em><\/em> \u2286 \u222ak<\/em>=1<\/sub>n<\/sup><\/em> int(Qk<\/sub><\/em>).<\/p>\n\n\n\n

Proof.<\/em> For each x<\/em> in Q<\/em>, let us pick some index kx<\/sub><\/em> in {1, …, n<\/em>} such that x<\/em> \u2208 Ukx<\/sub><\/em><\/sub><\/em>. Then by local compactness there is a compact saturated neighborhood Qx<\/sub><\/em> of x<\/em> that is included in Ukx<\/sub><\/em><\/sub><\/em>. The open sets int(Qx<\/sub><\/em>) cover Q<\/em>. Since Q<\/em> is compact, finitely many cover Q<\/em> already. In other words, there is a finite subset E<\/em> of Q<\/em> such that Q<\/em> \u2286 \u222ax<\/em> \u2208 E<\/em><\/sub> int(Qx<\/sub><\/em>). For each index k<\/em> \u2208 {1, …, n<\/em>}, we let Qk<\/sub><\/em><\/em><\/em> be the union of those compact sets Qx<\/sub><\/em> such that x<\/em> \u2208 E<\/em> and kx<\/sub><\/em>=k<\/em>. Then \u222ax<\/em> \u2208 E, kx<\/sub><\/em>=k<\/em><\/em><\/sub> int(Qx<\/sub><\/em>) \u2286 int(Qk<\/sub><\/em><\/em><\/em>), so Q<\/em><\/em> \u2286 \u222ak<\/em>=1<\/sub>n<\/sup> <\/em>int(Qk<\/sub><\/em>). Each Qk<\/sub><\/em><\/em><\/em> is a finite union of compact saturated sets, and is therefore in Q<\/strong>X<\/em>. And finally, each Qk<\/sub><\/em><\/em><\/em> <\/sub>is a union of sets Qx<\/sub><\/em> included in Ukx<\/sub><\/em><\/sub><\/em> where kx<\/sub><\/em>=k<\/em>, so Q<\/em>k<\/em><\/sub> \u2286 Uk<\/sub><\/em>. \u2610<\/p>\n\n\n\n

We will use Properties Wf and I pretty soon, while we will use Property U only once, and much later on. All right, this being done, let us list the axioms that [Q<\/em> \u2286 V<\/em>] should satisfy.<\/p>\n\n\n\n

The first axiom is that [Q<\/em> \u2286 V<\/em>] is antitonic in Q<\/em> and monotonic in V<\/em>: if Q<\/em> \u2287 Q’<\/em> and V<\/em> \u2286 V’<\/em> then [Q<\/em> \u2286 V<\/em>] \u2286 [Q’<\/em> \u2286 V’<\/em>]. Indeed, every element f<\/em> of [Q<\/em> \u2286 V<\/em>] is such that f<\/em>[Q<\/em>] \u2286 V<\/em>, so f<\/em>[Q’<\/em>] \u2286 f<\/em>[Q<\/em>] \u2286 V<\/em> \u2286 V’<\/em>. If we order compact saturated subsets by reverse inclusion \u2287 and open subsets by ordinary inclusion \u2286, this axiom says that the map (Q<\/em>, V<\/em>) \u21a6 [Q<\/em> \u2286 V<\/em>] is monotonic. I will not include that axiom in my list of axioms, though, as it is a consequence of axioms (Infs,left) and (Infs,right) below; this is because any map that preserves binary infima has to be monotonic already.<\/p>\n\n\n\n

The (remaining) axioms are of the following five kinds.<\/p>\n\n\n\n

    \n
  1. (Infs,left) For all Q<\/em>1<\/sub>, …, Qn<\/sub><\/em> \u2208 Q<\/strong>X<\/em>, \u2229i<\/em>=1<\/sub>n<\/sup><\/em> [Qi<\/sub><\/em> \u2286 V<\/em>] \u2286 [\u222ai<\/em>=1<\/sub>n<\/sup><\/em> Qi<\/sub><\/em><\/em> \u2286 V<\/em>]. In fact the two sides of the inclusion are equal: it is equivalent for a function f<\/em> to map every element of each Qi<\/sub><\/em> to a point of V<\/em>, or to map every element of \u222ai<\/em>=1<\/sub>n<\/sup><\/em> Qi<\/sub><\/em><\/em> to a point of V<\/em>. But the converse inclusion [\u222ai<\/em>=1<\/sub>n<\/sup><\/em> Qi<\/sub><\/em><\/em> \u2286 V<\/em>] \u2286 \u2229i<\/em>=1<\/sub>n<\/sup><\/em> [Qi<\/sub><\/em> \u2286 V<\/em>] is an immediate consequence of Axiom 1, so we do not need to write it down.
    One can equivalently state Axiom 2 as two subaxioms, corresponding to the cases n<\/em>=0 and n<\/em>=2: X<\/em> \u2286 [\u2205 \u22d0\u20131<\/sup> V<\/em>], and [Q<\/em>1<\/sub> \u2286 V<\/em>] \u2229 [Q<\/em>2<\/sub> \u2286 V<\/em>] \u2286 [Q<\/em>1<\/sub> \u222a Q<\/em>2<\/sub> \u2286 V<\/em>].
    Alternatively, this axiom means that the map Q<\/em> \u21a6 [Q<\/em> \u2286 V<\/em>] (with V<\/em> fixed) preserves finite infima; note that, since the ordering on <\/em>Q<\/strong>X<\/em> is reverse inclusion, finite infima in Q<\/strong>X<\/em> are unions.<\/li>\n\n\n\n
  2. (Infs,right) For every finite list of open subsets V<\/em>1<\/sub>, …, Vn<\/sub><\/em> of X<\/em>, \u2229i<\/em>=1<\/sub>n<\/sup><\/em> [Q<\/em> \u2286 Vi<\/sub><\/em>] \u2286 [Q<\/em><\/em> \u2286 \u2229i<\/em>=1<\/sub>n<\/sup> Vi<\/sub><\/em>]. Similar comments apply. Equivalently, X<\/em> \u2286 [Q<\/em><\/em> \u2286 Y<\/em>] and [Q<\/em> \u2286 V<\/em>1<\/sub>] \u2229 [Q<\/em> \u2286 V<\/em>2<\/sub>] \u2286 [Q<\/em><\/em> \u2286 V<\/em>1<\/sub> \u2229 V<\/em>2<\/sub>].
    Alternatively, the map V<\/em> \u21a6 [Q<\/em> \u2286 V<\/em>] (with Q<\/em> fixed) preserves finite infima.<\/li>\n\n\n\n
  3. (Cont.,right) For every directed family (Vi<\/sub><\/em>)i<\/em> \u2208 I<\/em><\/sub> of open subsets of Y<\/em>, [Q<\/em><\/em> \u2286 \u222ai<\/em> \u2208 I<\/em><\/sub> Vi<\/sub><\/em>] \u2286 \u222ai<\/em> \u2208 I<\/em><\/sub> [Q<\/em><\/em> \u2286 Vi<\/sub><\/em>]. Again, equality holds. Now this is a statement about the compactness of Q<\/em>. For every continuous map f<\/em> from X<\/em> to Y<\/em>, if f<\/em>[Q<\/em>] \u2286 \u222ai<\/em> \u2208 I<\/em><\/sub> Vi<\/sub><\/em>, then, using the fact that f<\/em>[Q<\/em>] is compact, and that the family (Vi<\/sub><\/em>)i<\/em> \u2208 I<\/em><\/sub> is directed, f<\/em>[Q<\/em>] must be included in some Vi<\/sub><\/em>; namely, f<\/em> must be in some [Q<\/em><\/em> \u2286 Vi<\/sub><\/em>].
    In other words, the map V<\/em> \u21a6 [Q<\/em> \u2286 V<\/em>] (with Q<\/em> fixed) is Scott-continuous.<\/li>\n\n\n\n
  4. (Cont.,left) For every filtered family (Qi<\/sub><\/em>)i<\/em> \u2208 I<\/em><\/sub> of compact saturated subsets of X<\/em>, [\u2229i<\/em> \u2208 I<\/em><\/sub> Qi<\/sub><\/em> \u2286 V<\/em>] \u2286 \u222ai<\/em> \u2208 I<\/em><\/sub> <\/em>[Qi<\/sub><\/em> \u2286 V<\/em>].
    Indeed, for every continuous map f<\/em> from X<\/em> to Y<\/em>, if f<\/em>[\u2229i<\/em> \u2208 I<\/em><\/sub> Qi<\/sub><\/em>] \u2286 V<\/em>, namely if \u2229i<\/em> \u2208 I<\/em><\/sub> Qi<\/sub><\/em> \u2286 f<\/em>\u20131<\/sup>(V<\/em>), then by Property Wf (well-filteredness), we have Qi<\/sub><\/em> \u2286 f<\/em>\u20131<\/sup>(V<\/em>) for some i<\/em> in I<\/em>; namely, f<\/em> is in [Qi<\/sub><\/em> \u2286 V<\/em>] for some i<\/em> in I<\/em>.
    As usual, this axiom is really equivalent to the equality [\u2229i<\/em> \u2208 I<\/em><\/sub> Qi<\/sub><\/em> \u2286 V<\/em>] = \u222ai<\/em> \u2208 I<\/em><\/sub> <\/em>[Qi<\/sub><\/em> \u2286 V<\/em>], and states that the map Q<\/em> \u21a6 [Q<\/em> \u2286 V<\/em>] (with V<\/em> fixed) is Scott-continuous.<\/li>\n\n\n\n
  5. (Interpolation) For every finite family of open subsets V<\/em>1<\/sub>, …, Vn<\/sub><\/em> of Y<\/em>, [Q<\/em><\/em> \u2286 \u222ak<\/em>=1<\/sub>n<\/sup> Vk<\/sub><\/em>] \u2286 \u222a{\u2229k<\/em>=1<\/sub>n<\/sup><\/em> [Qk<\/sub><\/em><\/em><\/em> \u2286 Vk<\/sub><\/em>] | Q<\/em>1<\/sub>, …, Qn<\/sub><\/em> \u2208 Q<\/strong>X<\/em>, Q<\/em><\/em> \u2286 \u222ak<\/em>=1<\/sub>n<\/sup> <\/em>int(Qk<\/sub><\/em>)}. Indeed, for every f<\/em> \u2208 [Q<\/em><\/em> \u2286 \u222ak<\/em>=1<\/sub>n<\/sup> Vk<\/sub><\/em>], by definition, Q<\/em> is included in f<\/em>\u20131<\/sup>(\u222ak<\/em>=1<\/sub>n<\/sup> Vk<\/sub><\/em>) = \u222ak<\/em>=1<\/sub>n<\/sup><\/em> f<\/em>\u20131<\/sup>(Vk<\/sub><\/em>). By Property I (interpolation), there are compact saturated sets Q<\/em>1<\/sub> \u2286 f<\/em>\u20131<\/sup>(V<\/em>1<\/sub>), …, Qn<\/sub><\/em> \u2286 f<\/em>\u20131<\/sup>(V<\/em>n<\/em><\/sub>) \u2208 Q<\/strong>X<\/em> such that Q<\/em><\/em> \u2286 \u222ak<\/em>=1<\/sub>n<\/sup> <\/em>int(Qk<\/sub><\/em>), and the inclusions Qk<\/sub><\/em> \u2286 f<\/em>\u20131<\/sup>(V<\/em>k<\/em><\/sub>) mean that f<\/em> is in \u2229k<\/em>=1<\/sub>n<\/sup><\/em> [Qk<\/sub><\/em><\/em><\/em> \u2286 Vk<\/sub><\/em>]. Since f<\/em> is arbitrary, [Q<\/em><\/em> \u2286 \u222ai<\/em>=1<\/sub>n<\/sup> Vi<\/sub><\/em>] \u2286 \u222a{\u2229i<\/em>=1<\/sub>n<\/sup><\/em> [Qi<\/sub><\/em><\/em><\/em> \u2286 Vi<\/sub><\/em>] | Q<\/em>1<\/sub>, …, Qn<\/sub><\/em> \u2208 Q<\/strong>X<\/em>, Q<\/em><\/em> \u2286 \u222ai<\/em>=1<\/sub>n<\/sup> <\/em>int(Qi<\/sub><\/em>)}, as claimed.
    Let me note that (Interpolation) is equivalent to the equality [Q<\/em><\/em> \u2286 \u222ai<\/em>=1<\/sub>n<\/sup> Vi<\/sub><\/em>] = \u222a{\u2229i<\/em>=1<\/sub>n<\/sup><\/em> [Qi<\/sub><\/em><\/em><\/em> \u2286 Vi<\/sub><\/em>] | Q<\/em>1<\/sub>, …, Qn<\/sub><\/em> \u2208 Q<\/strong>X<\/em>, Q<\/em><\/em> \u2286 \u222ai<\/em>=1<\/sub>n<\/sup> <\/em>int(Qi<\/sub><\/em>)}. In order to prove the right-to-left inclusion, it is enough to note that for all Q<\/em>1<\/sub>, …, Qn<\/sub><\/em> \u2208 Q<\/strong>X<\/em> such that Q<\/em><\/em> \u2286 \u222ai<\/em>=1<\/sub>n<\/sup> <\/em>int(Qi<\/sub><\/em>), \u2229i<\/em>=1<\/sub>n<\/sup><\/em> [Qi<\/sub><\/em><\/em><\/em> \u2286 Vi<\/sub><\/em>] is included in \u2229i<\/em>=1<\/sub>n<\/sup><\/em> [Qi<\/sub><\/em><\/em><\/em> \u2286 \u222ai<\/em>=1<\/sub>n<\/sup> Vi<\/sub><\/em>] by monotonicity, hence in [\u222ai<\/em>=1<\/sub>n<\/sup><\/em> Qi<\/sub><\/em><\/em><\/em> \u2286 \u222ai<\/em>=1<\/sub>n<\/sup> Vi<\/sub><\/em>] by (Infs,left), hence in [Q<\/em> \u2286 \u222ai<\/em>=1<\/sub>n<\/sup> Vi<\/sub><\/em>] by monotonicity.
    The (Interpolation) property is also equivalent with its subcases n<\/em>=0, n<\/em>=1 and n<\/em>=2.\n