{"id":888,"date":"2016-04-10T12:14:05","date_gmt":"2016-04-10T10:14:05","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=888"},"modified":"2022-11-19T15:25:27","modified_gmt":"2022-11-19T14:25:27","slug":"locales-sublocales","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=888","title":{"rendered":"Locales, sublocales I"},"content":{"rendered":"

Stone duality leads naturally to the idea of locale<\/em> theory.\u00a0 Quickly said, the idea is that, instead of reasoning with topological spaces, we reason with frames.\u00a0 The two concepts are not completely interchangeable, but the O<\/strong> \u22a3 pt adjunction shows that they are close.\u00a0 More: sober<\/em> spaces form a category that is equivalent to the opposite of that of spatial<\/em> frames.<\/p>\n

Topology without points<\/h2>\n

It has come to the mind of several mathematicians, from John Isbell to Peter Johnstone, and including Bernhard Banaschewski and others, that one might redo most of topology by looking at frames instead of topological spaces.<\/p>\n

The result is sometimes known as pointless topology<\/em> (a horrible pun), because frames that are not necessarily spatial can be thought of as sober spaces, except that they might lack a few points.\u00a0 In fact, some of those frames, such as the frame of regular open sets of [0, 1], have no point at all, and despite that, have a very rich structure.<\/p>\n

If you wish to learn about the theory of locales, let me recommend Picado and Pultr’s excellent book [1].\u00a0 They do a really great job of presenting the theory as simply as possible, and have proofs that are probably the simplest possible.<\/p>\n

Let me also remind you that the adjunction\u00a0O<\/strong> \u22a3 pt between topological spaces and frames is actually an adjunction between the category Top<\/strong> of topological spaces and the opposite<\/em> of the category Frm<\/strong> of frames and frame homomorphisms.\u00a0 To obtain the right pointless equivalent of Top<\/strong>, we must therefore move to Frm<\/strong>op<\/sup><\/em>, the opposite of Frm<\/strong>.\u00a0 That opposite category Loc<\/strong>=Frm<\/strong>op<\/sup><\/em>, is called the category of locales<\/em>.<\/p>\n

Hence a locale is just a frame, but a morphism f<\/em> : L<\/em> \u2192 L’<\/em> in Loc<\/strong> should be thought the other way around, as a frame homomorphism from L’<\/em> to L<\/em>.<\/p>\n

This inversion of the direction of morphisms is the source of immense confusion.\u00a0 We shall instead reason with frames.\u00a0 Picado and Pultr do the same, and call that dealing with locale theory “mostly covariantly”.<\/p>\n

Things that go well with locales<\/h2>\n

There are many things that work well in the realm of locales. You may have heard of Johnstone’s result that “Tychonoff’s theorem can be proved in the realm of locales without any recourse to the axiom of choice”.\u00a0 This is usually the first thing I hear mentioned when a discussion comes to the matter of locales.<\/p>\n

This is true, but, as I will recall later, locale products are not the same as topological products.\u00a0 Also, it is a pretty complicated theorem.\u00a0 But it certainly is a neat result.<\/p>\n

Let us look at something simpler: coproducts of locales, i.e., products of frames.\u00a0 They always exist, and the lattice of open sets of a coproduct of topological spaces is exactly the locale coproduct, i.e., the frame product, of the various lattices of open sets involved.\u00a0 In short: O<\/strong>(coproduct of spaces Xi<\/sub><\/em>) = frame product of O<\/strong>(Xi<\/sub><\/em>).\u00a0 For binary coproducts, this mostly says that an open set of X<\/em>+Y<\/em> is just a pair of an open set from X<\/em> and one from Y<\/em>.<\/p>\n

In general, Loc<\/strong> has all colimits, and O<\/strong> preserves them.\u00a0 The latter part is because O<\/strong> is a right adjoint.\u00a0 The former must be checked by hand.\u00a0 Products of frames are computed componentwise, and equalizers are obtained as subframes, i.e., as subsets of frames that are closed under finite infima and arbitrary suprema: there is no difficulty in doing that verification.<\/p>\n

Constructions that are more painful: locale limits<\/h2>\n

The situation is more painful with limits.\u00a0 Section 8.4.4 of the book<\/a> is devoted to the case of binary products.\u00a0 This is already complicated: binary products of locales, that is, binary coproducts of frames, are described as a frame of Galois connections.\u00a0 (More general frame coproducts are described by so-called C-ideals<\/em>: for frames Li<\/sub><\/em>,\u00a0i<\/em> in\u00a0I<\/em>, their coproduct is the set of relations, i.e., of subsets of \u220f Li<\/sub><\/em>, that are downwards-closed and separately Scott-closed in each subscript i<\/em>, with the componentwise ordering.)<\/p>\n

Worse is that the O<\/strong> functor does not<\/em> preserve binary products, that is, O<\/strong>(X<\/em> x Y<\/em>) may fail to be the frame coproduct of O<\/strong>(X<\/em>) and of O<\/strong>(Y<\/em>).\u00a0 For that, X<\/em> and Y<\/em> must not be core-compact, as O<\/strong>(X<\/em> x Y<\/em>) = O<\/strong>(X<\/em>) + O<\/strong>(Y<\/em>) when X<\/em> or Y<\/em> is core-compact (Exercise 8.4.23 in the book<\/a>). \u00a0I may one day brace myself and try to explain Johnstone’s counterexample showing that, in general, O<\/strong>(X<\/em> x Y<\/em>) and O<\/strong>(X<\/em>) + O<\/strong>(Y<\/em>) are not isomorphic.<\/p>\n

No, what I would like to consider today is the case of subspaces<\/em>.\u00a0 A topological subspace (together with the canonical inclusion map) is a special case of limit: it is an equalizer, and an equalizer is a special case of limit.\u00a0 Conversely, all limits can be built as equalizers of products, so products and equalizers are really the only kinds of limits we have to consider.<\/p>\n

To show that topological subspaces are exactly equalizers in Top<\/strong>, we must show that for every topological subspace A<\/em> of a topological space X<\/em>, the inclusion map m<\/em> : A<\/em> \u2192 X<\/em> equalizes a pair of arrows from X<\/em> to some space Y<\/em>.\u00a0 There is a general recipe to show that.\u00a0 Form the cokernel pair of m<\/em>, i.e., the pushout of a diagram made of twice the map m<\/em> starting from apex A<\/em>.\u00a0 This is the space Y<\/em> defined as the quotient of X<\/em>+X<\/em> (whose elements are (0, x<\/em>) and (1, x<\/em>) for x<\/em> in X<\/em>) by the equivalence relation that equates (0, a<\/em>) with (1, a<\/em>) for each a<\/em> in A<\/em>.\u00a0 We can now check that m<\/em> is the equalizer of the two maps x<\/em> \u27fc equivalence class of (0, x<\/em>) and x<\/em> \u27fc equivalence class of (1, x<\/em>).<\/p>\n

Hence finding the right notion of sublocale<\/em> \u2014 the equivalent of subspaces for locales \u2014 is just a matter of understanding what equalizers are in Loc<\/strong>, or equivalently what coequalizers are in Frm<\/strong>.\u00a0 I will not take the categorical route, however.\u00a0 Just as for products, I will try to indicate what sublocales<\/em> should be by analogy to topological subspaces.\u00a0 And just as for products, this analogy will eventually fail very badly, although it is useful for starters.<\/p>\n

Sublocales, nuclei, and congruences<\/h2>\n

There are three possible, equivalent, ways of defining sublocales: as… something called sublocales, first; as nuclei<\/em>; and as frame congruences<\/em>.\u00a0 Here are the raw definitions.<\/p>\n

    \n
  1. A sublocale<\/em> of a frame \u03a9 is a subset L<\/em> of \u03a9 that is closed under arbitrary infima (taken in \u03a9), and such that\u00a0\u03c9 \u27f9 x<\/em> is in L<\/em> for every x<\/em> in L<\/em> and every \u03c9 in \u03a9.\u00a0 Here \u27f9 is residuation (a.k.a., intuitionistic implication): a<\/em> \u27f9 b<\/em> is the largest c<\/em> such that inf (a<\/em>, c<\/em>) \u2264 b<\/em>.\u00a0 Note that a sublocale is not a subframe<\/em>, which would be a subset of \u03a9 that is closed under finite infima and arbitrary suprema, as I have already said.<\/li>\n
  2. A nucleus<\/em> on \u03a9 is a closure operator on \u03a9 that preserves binary infima.\u00a0 A closure operator is by definition a monotonic map \u03bd : \u03a9 \u2192 \u03a9 such that \u03bd(\u03c9) \u2265 \u03c9 for every \u03c9 in \u03a9, and \u03bd(\u03bd(\u03c9)) = \u03bd(\u03c9) for every \u03c9 in \u03a9.\u00a0 A nucleus additionally satisfies \u03bd(inf(\u03c9, \u03c9’)) = inf (\u03bd(\u03c9), \u03bd(\u03c9’)).\u00a0 It automatically satisfies \u03bd(\u22a4) = \u22a4, because of the law \u03bd(\u03c9) \u2265 \u03c9, so \u03bd in fact preserves all finite infima.<\/li>\n
  3. A frame congruence<\/em> on \u03a9 is an equivalence relation \u2261 that is compatible with finite infima and arbitrary suprema: if \u03c91<\/sub>\u2261\u03c9’1<\/sub> and \u03c92<\/sub>\u2261\u03c9’2<\/sub> then inf (\u03c91<\/sub>, \u03c92<\/sub>)\u2261inf(\u03c9’1<\/sub>, \u03c9’2<\/sub>), and if \u03c9i<\/sub><\/em>\u2261\u03c9’i<\/sub><\/em> for every i<\/em> in I<\/em>, then sup \u03c9i<\/sub><\/em> \u2261 sup \u03c9’i<\/sub><\/em>.<\/li>\n<\/ol>\n

    This all looked difficult to understand the first time I saw those definitions.\u00a0 So let me try to rederive those definitions, as we did in the book<\/a> with products and frame coproducts, by imagining naively that sublocales are a pointfree encoding of topological subspaces.\u00a0 In other words, imagine A<\/em> is a topological subspace of X<\/em>, and let me try to give a description of how the open subsets of A<\/em> can be encoded from the knowledge of the lattice \u03a9 of the open subsets of X<\/em>, and without mentioning points.<\/p>\n

    The easiest thing to explain is the frame congruence encoding.\u00a0 An open subset of A<\/em> is an intersection U<\/em> \u22c2 A<\/em>, and we may try to encode it as U<\/em>.\u00a0 But then, such an open subset of A<\/em> may have many encodings.\u00a0 However, all those encodings are related by the equivalence relation \u2263 (on \u03a9) defined by U<\/em> \u2263 V<\/em> if and only if U<\/em> \u22c2 A<\/em> = V<\/em> \u22c2 A<\/em>.\u00a0 We can check that this is a frame congruence.\u00a0 From a localic point of view, giving an congruence on \u03a9 is enough to describe the lattice of open subsets of A<\/em>: we take the quotient of \u03a9 by \u2263, this gives you a frame isomorphic to the frame O<\/strong>(A<\/em>) of open subsets of A<\/em>, and the quotient map q<\/em> : \u03a9 \u2192 \u03a9\/\u2263 is the localic encoding of the inclusion map m<\/em> : A<\/em> \u2192 X<\/em>, in the sense that q<\/em> = m<\/em>-1<\/sup> : \u03a9 = O<\/strong>(X<\/em>)\u00a0\u2192 \u03a9\/\u2263 = O<\/strong>(A<\/em>).<\/p>\n

    Instead of encoding A<\/em> by a congruence, we can pick a distinguished representative in each equivalence class.\u00a0 There is an obvious choice for that: define \u03bd(U<\/em>) as the largest<\/em> open subset of X<\/em> that is equivalent with U<\/em>, that is, that has the same intersection with A<\/em> as U<\/em>.\u00a0 More precisely, the union of all the open subsets that are equivalent to U<\/em> is again equivalent to U<\/em>, and is the largest such equivalent open subset.\u00a0 I’ll let you check that \u03bd is a nucleus.<\/p>\n

    Any nucleus, in fact, any closure operator\u00a0\u03bd is entirely determined by its set L<\/em> of fixed points.\u00a0 Indeed, \u03bd(\u03c9) is the least fixed point of \u03bd above \u03c9.\u00a0 You may also note that L<\/em> is nothing but the image of \u03bd, as well.<\/p>\n

    Hence we can encode the subspace A<\/em> (together with its inclusion map) as the set of open subsets U<\/em> of X<\/em> that are largest in the class of all open subsets of X<\/em> that have a given intersection with A<\/em>.\u00a0 Checking directly that it is a sublocale is not entirely obvious, and it is as easy (or as difficult) to show directly that sublocales are adequate encoding of nuclei, in general.<\/p>\n

    Let me do so now.\u00a0 To deal with residuation \u27f9<\/em>, I will use the following equivalence: (*) inf (a<\/em>, b<\/em>) \u2264 c<\/em> if and only if a<\/em> \u2264 [b\u00a0\u27f9\u00a0<\/em>c<\/em>].\u00a0 This holds not only in every frame (= complete Heyting algebra), but more generally in every Heyting algebra.\u00a0 Note that, by taking a<\/em> = [b\u00a0\u27f9\u00a0<\/em>c<\/em>], this entails the inequality: (**) inf ([b\u00a0\u27f9\u00a0<\/em>c<\/em>], b<\/em>) \u2264 c<\/em>.<\/p>\n

    Lemma.<\/strong> For every nucleus \u03bd, its set L<\/em> of fixed points is a sublocale.<\/p>\n

    Proof. Because \u03bd commutes with finite infima, L<\/em> is closed under binary infima. It is in fact closed under arbitrary<\/em> infima, for the following reason. Given a family of elements xi<\/sub><\/em> of L<\/em>, \u03bd(inf xi<\/sub><\/em>) \u2264 inf \u03bd (xi<\/sub><\/em>) by monotonicity, and since each xi<\/sub><\/em> is a fixed point, \u03bd(inf xi<\/sub><\/em>) \u2264 inf xi<\/sub><\/em>. Because \u03bd is a closure operator, \u03bd(inf xi<\/sub><\/em>) \u2265 inf xi<\/sub><\/em>, whence the equality follows.<\/p>\n

    The second property of sublocales will follow from the general identity \u03bd (a<\/em> \u27f9 \u03bd (b<\/em>)) = a<\/em> \u27f9 \u03bd (b<\/em>). Indeed, if\u00a0x<\/em> is a fixed point of \u03bd, then\u00a0\u03bd (\u03c9 \u27f9 x<\/em>) = \u03bd (\u03c9 \u27f9 \u03bd (x<\/em>)) = \u03c9 \u27f9 \u03bd (x<\/em>) = \u03c9 \u27f9 x<\/em>, showing that\u00a0\u03c9 \u27f9 x<\/em> is also a fixed point of \u03bd.<\/p>\n

    We show that identity as follows.\u00a0 Let c<\/em> = \u03bd (a<\/em> \u27f9 \u03bd (b<\/em>)). The infimum of c<\/em> and a<\/em> is below inf (c<\/em>, \u03bd (a<\/em>)), and the latter is equal to \u03bd (inf (a<\/em> \u27f9 \u03bd (b<\/em>), a<\/em>)) because \u03bd commutes with binary infima. Using the inequality (**), inf (a<\/em> \u27f9 \u03bd (b<\/em>), a<\/em>) \u2264 \u03bd (b<\/em>), so inf (c<\/em>, a<\/em>) \u2264 \u03bd (\u03bd (b<\/em>)) = \u03bd (b<\/em>).\u00a0 By (*), this implies c<\/em> \u2264 [a<\/em> \u27f9 \u03bd (b<\/em>)]. The converse inequality is because \u03bd(\u03c9) \u2265 \u03c9 for every \u03c9 in \u03a9. \u2610<\/p>\n

    Lemma.<\/strong> For every sublocale L<\/em>, the map \u03bd that sends \u03c9 to the smallest element of L<\/em> above \u03c9 is a nucleus.<\/p>\n

    Proof.\u00a0 That smallest element is the infimum of the elements of L<\/em> above \u03c9, which is in L<\/em> because L<\/em> is closed under arbitrary infima.\u00a0 The fact that \u03bd is a closure operator is clear.\u00a0 Since\u00a0\u03bd is monotonic,\u00a0\u03bd(inf(\u03c9, \u03c9’)) \u2264 inf (\u03bd(\u03c9), \u03bd(\u03c9’)) for all \u03c9, \u03c9’.<\/p>\n

    We note that inf(\u03c9, \u03c9’) \u2264 \u03bd(inf(\u03c9, \u03c9’)), so \u03c9 \u2264 [\u03c9’ \u27f9 \u03bd(inf(\u03c9, \u03c9’))] by (*).\u00a0 Note that\u00a0\u03bd(inf(\u03c9, \u03c9’)) is in L<\/em> by definition, so \u03c9’ \u27f9 \u03bd(inf(\u03c9, \u03c9’)) is also in L<\/em>, using the second clause in the definition of sublocales.\u00a0 By definition of \u03bd(\u03c9) as the smallest element of L<\/em> above \u03c9, it follows that\u00a0\u03bd(\u03c9) \u2264 [\u03c9’ \u27f9 \u03bd(inf(\u03c9, \u03c9’))].\u00a0 By (*) read in the other direction, inf (\u03bd(\u03c9), \u03c9’) is below \u03bd(inf(\u03c9, \u03c9’)).\u00a0 Permute the two arguments of the first inf: inf (\u03c9’, \u03bd(\u03c9)) is below\u00a0\u03bd(inf(\u03c9, \u03c9’)), so \u03c9’ \u2264 [\u03bd(\u03c9) \u27f9 \u03bd(inf(\u03c9, \u03c9’))], using (*) again.\u00a0 By the same argument as above, \u03bd(\u03c9’) \u2264 [\u03bd(\u03c9) \u27f9 \u03bd(inf(\u03c9, \u03c9’))], so inf (\u03bd(\u03c9), \u03bd(\u03c9’)) \u2264 \u03bd(inf(\u03c9, \u03c9’)). \u2610<\/p>\n

    Moreover, the two constructions, from a nucleus to a sublocale and conversely, are inverses of each other.<\/p>\n

    Subspaces and sublocales<\/h2>\n

    Despite the fact that I explained those constructions by imitating the construction of the lattice of open subsets of a topological subspace, sublocales (resp., nuclei, resp. congruences) are very<\/em> different from topological subspaces.<\/p>\n

    For example, even spatial locales have sublocales that are not spatial.\u00a0 This is the dark side of a coin, whose bright side is Isbell’s density theorem: every locale contains a least dense sublocale [1, 8.3].\u00a0 The latter is, in general, not spatial.\u00a0 The topological counterpart, which would say that every space contains a least dense topological subspace, is completely wrong.<\/p>\n

    The most curious result that shows how different sublocales are subspaces are is probably the following.\u00a0 The poset P<\/strong>(X<\/em>) of topological subspaces of a given topological space X<\/em> is a complete atomic Boolean lattice.\u00a0 In particular every subspace A<\/em> of X<\/em> is a complemented<\/em> element of P<\/strong>(X<\/em>): there is another subspace X<\/em>\u2014A<\/em>, whose infimum (=intersection) with A<\/em> is the bottom element (the empty subspace) and whose supremum (=union) with A<\/em> is the top element (X<\/em> itself).\u00a0 On the contrary, the poset of all sublocales is in general only a coframe<\/em>, that is, its opposite is a frame, but not all sublocales are complemented.\u00a0 In fact, all complemented sublocales are spatial, and I have already mentioned that not all sublocales of a locale are spatial in general.<\/p>\n

    The correspondence between sublocales and nuclei reverses the ordering (inclusion of sublocales becomes pointwise \u2265 on nuclei), and the former can therefore be rephrased as: nuclei form a frame<\/em>, in which not all elements are complemented.<\/p>\n

    It is a zero-dimensional<\/em> frame, that is, every element of the frame is the (directed) supremum of complemented elements.\u00a0 This is shown as follows.\u00a0 For each u<\/em> in \u03a9\u00a0 (which we may think encodes an open subset<\/em> of a topological space X<\/em>), there is a so-called open <\/em>nucleus o<\/strong>(u<\/em>), which maps every \u03c9 to u<\/em> \u27f9 \u03c9 (which intuitively encodes the subset<\/em> u<\/em> as a subspace<\/em>), and there is a so-called closed<\/em> nucleus c<\/strong>(u<\/em>), which maps every \u03c9 to sup (u<\/em>, \u03c9) (and intuitively encodes the complement of u<\/em> as a subspace).\u00a0 Those particular nuclei are complemented, and as one might expect, o<\/strong>(u<\/em>) is the complement of c<\/strong>(u<\/em>).\u00a0 Then every nucleus \u03bd can be written as the supremum of the (complemented) nuclei inf (c<\/strong>(\u03bd(u<\/em>)), o<\/strong>(u<\/em>)), u<\/em> in \u03a9.\u00a0 (There is a short, but tricky proof.)\u00a0 That shows that the frame of nuclei is zero-dimensional, although not all nuclei are complemented.<\/p>\n

    The analogous result on the topological side would say something like: “every subspace is an intersection of special subsets of the form U<\/em> \u22c3 C<\/em>, where U<\/em> is open and C<\/em> is closed”, which would be completely wrong again.\u00a0 (I realize I don’t know how to prove this, but that would seem strange anyway.)<\/p>\n

    The facts that nuclei form a frame, and that certain, so-called open and closed nuclei are complemented and generate the frame of all nuclei would probably deserve another post.\u00a0 Instead, next time, I may choose to describe a fourth<\/em> way of “representing” subspaces, and to show that it is another equivalent encoding of sublocales \/ nuclei \/ congruences.<\/p>\n

    \u2014 Jean Goubault-Larrecq<\/a>\u00a0(April 10th, 2016)\"jgl-2011\"<\/p>\n

    [1] Jorge Picado and Ale\u0161 Pultr.\u00a0 Frames and locales \u2014 topology without points.\u00a0 Birkh\u00e4user, 2010.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Stone duality leads naturally to the idea of locale theory.\u00a0 Quickly said, the idea is that, instead of reasoning with topological spaces, we reason with frames.\u00a0 The two concepts are not completely interchangeable, but the O \u22a3 pt adjunction shows … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-888","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/888","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=888"}],"version-history":[{"count":12,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/888\/revisions"}],"predecessor-version":[{"id":5950,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/888\/revisions\/5950"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=888"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}