{"id":1518,"date":"2018-07-18T20:00:33","date_gmt":"2018-07-18T18:00:33","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1518"},"modified":"2022-05-16T14:30:49","modified_gmt":"2022-05-16T12:30:49","slug":"another-form-of-stone-duality","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1518","title":{"rendered":"Another form of Stone duality"},"content":{"rendered":"

I thought I would devote my blog this month to the Domains<\/a> workshop, but a sudden health problem prevented me to go there. \u00a0I’ll have to read the papers instead… \u00a0In the meantime, I would like to talk about a curious construction Fr\u00e9d\u00e9ric Mynard<\/a> and I obtained at the end of the paper [1].<\/p>\n

A bit of (personal) history<\/h2>\n

We started working on [1] in 2013, when Fr\u00e9d\u00e9ric Mynard invited me at Georgia Southern University, Statesboro, GA, USA, where he was working at the time. \u00a0Our initial goal was to provide some form of Stone duality for convergence spaces. \u00a0We found what it should be pretty quickly (see Section 2 of [1]). \u00a0That consists in a pair of adjoint functors that relate the category\u00a0Conv<\/strong> 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<\/em> lattices<\/em>).<\/em> \u00a0As for Stone duality, one\u00a0functor in the adjunction completely forgets about the points of the given convergence space, and the other one must reconstruct the points\u00a0from a convergence lattice. \u00a0We were tempted to call the paper `pointfree convergence’, or even `pointless convergence’, but we were too shy to make the pun.<\/p>\n

Since there is a whole zoo of refinements of convergence spaces\u2014limit spaces, Kent spaces, Antoine spaces, pretopological spaces, pseudotopological spaces, and what have you\u2014we 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.<\/p>\n

We started to run into trouble with pretopological spaces and pseudotopological spaces, and even worse, with topological spaces. \u00a0One 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. \u00a0In a nutshell, nothing worked as nicely as Stone duality does. \u00a0Eventually, we found something neat for pretopological spaces and adherence spaces (see Sections 5 through 7 in [1]), although that is somehow elaborate. \u00a0We had something for pseudo topologies as well, but not as neat as the rest, and we have not included it. \u00a0The 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. \u00a0This is what I want to explain here.<\/p>\n

Topological coframes<\/h2>\n

What we have obtained is an adjunction between\u00a0Top<\/strong>, the category of topological spaces and continuous maps, and a new category CF<\/strong>top<\/sup>, defined as follows.<\/p>\n

Its objects are the\u00a0topological coframes<\/em>, namely the coframes\u00a0L<\/em>, together with a sublattice\u00a0C(L<\/em>) of\u00a0L<\/em>, whose elements are all complemented in\u00a0L<\/em>. \u00a0The elements of\u00a0C(L<\/em>) are called the\u00a0closed elements<\/em> of\u00a0L<\/em>.<\/p>\n

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<\/i>\u00a0\u2228 infi<\/sub><\/em>\u00a0bi<\/sub><\/em> = infi<\/sub><\/em> (a<\/i>\u00a0\u2228 bi<\/sub><\/em>)). \u00a0In a coframe, every element a<\/em>\u00a0has a unique pseudocomplement a*<\/em>, defined as the least element b<\/i>\u00a0such that a<\/i>\u00a0\u2228 b<\/i>\u00a0=\u00a0\u22a4, and an element a<\/i>\u00a0is complemented if and only if a<\/i>\u00a0\u2227 a*<\/em>=\u22a5. \u00a0A sublattice is a subset that is closed under finite infima and finite suprema (in particular, it contains\u00a0\u22a4 and\u00a0\u22a5.)<\/p>\n

How do such topological coframes occur? \u00a0Very simply: for every topological space\u00a0X<\/em>, let\u00a0L<\/em> be\u00a0P<\/strong>(X<\/em>), the powerset of\u00a0X<\/em>, and define\u00a0C(L<\/em>) as the sublattice of closed subsets of\u00a0X<\/em>. \u00a0That obviously defines a topological coframe. \u00a0For example, every element of C(L<\/em>) is complemented… because every element of L<\/em> is complemented anyway. \u00a0What might seem strange is our choice of conditions on\u00a0L<\/em> and\u00a0C(L<\/em>): why didn’t we simply require\u00a0L<\/em> to be a complete Boolean algebra, for example? \u00a0and why didn’t we require\u00a0C(L<\/em>) to be closed under arbitrary infima, not just finite infima?<\/p>\n

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. \u00a0Why C(L<\/em>) should consist of complemented elements, and why\u00a0L<\/em> should be a coframe, will be made clearer when we relate topological coframes and frames, near the end of this post.<\/p>\n

Let me just say that, in analogy with Stone duality, the lattice of sublocales<\/a> of a given locale, which plays the r\u00f4le of\u00a0P<\/strong>(X<\/em>) here, is a coframe in general, not a complete Boolean algebra. \u00a0This is also how we discovered sieves<\/a>, only to realize later that they were isomorphic to sublocales.<\/p>\n

In any case, the construction of the topological coframe\u00a0P<\/strong>(X<\/em>) (with its sublattice of closed sets) from\u00a0X<\/em> defines a functor\u00a0P<\/strong> :\u00a0Top<\/strong>\u00a0\u2192 (CF<\/strong>top<\/sup>)op<\/sup>. For that to make sense, we of course need to say what the morphisms are in\u00a0CF<\/strong>. \u00a0Those are simply the coframe homomorphisms \u03d5 :\u00a0L<\/em>\u00a0\u2192\u00a0L<\/em>‘ (preserving arbitrary infima and finite suprema) that map closed elements of L<\/em>\u00a0to closed elements of\u00a0L’<\/em>. \u00a0If\u00a0f<\/em> :\u00a0X<\/em>\u00a0\u2192\u00a0Y<\/em> is any continuous map, then\u00a0P<\/strong>(f<\/em>) is the inverse image map f<\/em>-1<\/sup> :\u00a0P<\/strong>(Y<\/em>)\u00a0\u2192\u00a0P<\/strong>(X<\/em>). \u00a0That certainly maps closed elements of\u00a0P<\/strong>(Y<\/em>) (closed subsets of\u00a0Y<\/em>) to closed elements of\u00a0P<\/strong>(X<\/i>).<\/p>\n

The left adjoint to\u00a0P<\/strong><\/h2>\n

In order to obtain a nice duality, we need to find a right adjoint to\u00a0P<\/strong>. \u00a0Given a topological coframe\u00a0L<\/em> (with sublattice of closed elements C(L<\/em>)), we define the points of\u00a0L<\/em> as its join-prime elements: those elements\u00a0u<\/em> different from\u00a0\u22a5 such that, if\u00a0u<\/em> is below the supremum of two elements, then it must be below one of them.<\/p>\n

It is interesting to discover\u00a0what the join-primes are in\u00a0P<\/strong>(X<\/em>): those are the non-empty sets\u00a0A<\/em> such that\u00a0A<\/em> cannot be included in a union of two sets without being included in one of them. \u00a0It is pretty easy to see that those sets are exactly the one-element sets {x<\/em>}, which one can equate with\u00a0x<\/em>.<\/p>\n

We let\u00a0pt<\/strong>\u00a0L<\/em> be the set of points (=join-primes) of\u00a0L<\/em>. \u00a0We must topologize\u00a0pt<\/strong>\u00a0L<\/em>, and here is how we do it. \u00a0For every a<\/i>\u00a0in\u00a0L<\/em>, build the set a<\/i>\u2022<\/sup> of all points x<\/em> of\u00a0L<\/em> below a<\/i>. \u00a0Then we declare that the topology of\u00a0pt<\/strong>\u00a0L<\/em> is the coarsest topology that makes c<\/i>\u2022<\/sup>\u00a0closed for each\u00a0c<\/em>\u00a0\u2208 C(L<\/em>).<\/p>\n

It can be shown (Lemma 103 in [1]) that the closed sets of\u00a0pt<\/strong>\u00a0L<\/em> are exactly the sets of the form c<\/i>\u2022<\/sup>, where\u00a0c<\/em> is in\u00a0\u2227C(L<\/em>), the subcoframe of\u00a0L<\/em> consisting of infima of elements of C(L<\/em>).<\/p>\n

Then\u00a0P<\/strong>\u00a0\u22a3\u00a0pt<\/strong> is an adjunction (Proposition 104), whose counit\u00a0\u03b5L<\/sub><\/em> :\u00a0L<\/em>\u00a0\u2192\u00a0P<\/strong>(pt<\/strong>\u00a0L<\/em>) sends every\u00a0a<\/em> to a<\/i>\u2022<\/sup>, and whose unit\u00a0\u03b7X<\/sub><\/em> : X<\/em> \u2192\u00a0pt<\/strong>(P<\/strong>(X<\/em>)) maps\u00a0x<\/em> to {x<\/em>}, and is in fact an isomorphism. \u00a0(Note that in the classical Stone adjunction, the unit at X<\/em>\u00a0is an isomorphism if and only if\u00a0X<\/em> is sober\u2014hence there is no non-trivial notion of sobriety in our new Stone-like adjunction.)<\/p>\n

Relating topological coframes and frames<\/h2>\n

One wonders at the relation between topological coframes, and the frames that form the core of Stone duality. \u00a0This is the topic of the final section (Section 8.5) of [1].<\/p>\n

Every topological coframe\u00a0L<\/em> yields a coframe\u00a0\u2227C(L<\/em>), hence a frame (\u2227C(L<\/em>))op<\/sup>. \u00a0That defines a functor\u00a0\u2227Cop<\/sup> :\u00a0CF<\/strong>top<\/sup>\u00a0\u2192 Frm<\/strong>\u00a0(Fact 117 in [1]).<\/p>\n

The fact that\u00a0Sl<\/strong> is a functor depends on the following universal property of the coframe of sublocales: for every frame homomorphism\u00a0\u03c6 : \u03a9\u00a0\u2192\u00a0\u03a9’ such that\u00a0\u03c6(a<\/em>) is complemented for every\u00a0a<\/em> in\u00a0\u03a9, there is a unique frame homomorphism\u00a0\u03c6* :\u00a0Sl<\/strong>(\u03a9)op<\/sup>\u00a0\u2192\u00a0\u03a9’ such that\u00a0\u03c6*(c<\/strong>(a<\/em>)) =\u00a0\u03c6(a<\/em>) for\u00a0every\u00a0a<\/em> in\u00a0\u03a9. \u00a0(This is why we required C(L<\/em>) to consist of complemented elements only.) \u00a0To define the\u00a0Sl<\/strong> functor on frame\u00a0homomorphisms \u03c6 : \u03a9\u00a0\u2192\u00a0\u03a9’, we first build the composition\u00a0c<\/strong> o\u00a0\u03c6 : \u03a9\u00a0\u2192\u00a0Sl<\/strong>(\u03a9’). \u00a0We observe that this is a frame homomorphism again, and one which maps every element of\u00a0\u03a9 to a complemented element of\u00a0Sl<\/strong>(\u03a9’)\u2014because closed sublocales are always complemented. \u00a0Hence we can form (c<\/strong> o\u00a0\u03c6)* :\u00a0Sl<\/strong>(\u03a9) \u2192\u00a0Sl<\/strong>(\u03a9’), and that is\u00a0Sl<\/strong>(\u03c6) by definition.<\/p>\n

The two functors\u00a0\u2227Cop<\/sup>\u00a0and\u00a0Sl<\/strong> do not form an adjunction in general, but the composition\u00a0\u2227Cop<\/sup>\u00a0o\u00a0Sl<\/strong> forms an isomorphism of categories. \u00a0In other words,\u00a0Frm<\/strong> appears as a retract of\u00a0CF<\/strong>top<\/sup>. \u00a0That retraction even restricts to a coreflection between\u00a0Frm<\/strong> and the full subcategory of strong<\/em> topological coframes, namely those topological coframes whose closed elements are closed under arbitrary infima.<\/p>\n

Conclusion<\/h2>\n

I am not sure yet that topological coframes will prove useful, but their definition and properties are certainly intriguing. \u00a0For example, given a topological space\u00a0X<\/em>, we can form its coframe L<\/em> =\u00a0H<\/strong>(X<\/em>)\u00a0of closed subsets. \u00a0The pseudocomplement of\u00a0C<\/em> in\u00a0L<\/em> is the closure cl (X<\/em> \u2014\u00a0C<\/em>) of its complement, and\u00a0C<\/em> itself is complemented if and only if\u00a0C<\/em> is clopen. \u00a0We can then define a topological coframe H<\/strong>(X<\/em>)\u00a0by defining C(H<\/strong>(X<\/em>)) as the set of clopens of\u00a0X<\/em>; this is the smallest possible choice for\u00a0C(H<\/strong>(X<\/em>)).<\/p>\n

This yields a new topological space\u00a0pt<\/strong>(H<\/strong>(X<\/em>)). \u00a0What is it? \u00a0Its points are the join-primes of\u00a0H<\/strong>(X<\/em>), namely the irreducible closed subsets of\u00a0X<\/em>, but\u00a0pt<\/strong>(H<\/strong>(X<\/em>)) is not the sobrification of\u00a0X<\/em>: its closed subsets are the sets {C<\/em> irreducible closed in\u00a0X<\/em> |\u00a0C<\/em> \u2286 D<\/em>}, where\u00a0D<\/em> ranges over the intersections of clopens. \u00a0All such sets\u00a0D<\/em> are closed, hence the topology of\u00a0pt<\/strong>(H<\/strong>(X<\/em>)) is coarser than the topology of the sobrification of\u00a0X<\/em>. \u00a0Again, I am not sure this is any useful, but who knows.<\/p>\n

\u00a0<\/p>\n

    \n
  1. Jean Goubault-Larrecq and Fr\u00e9d\u00e9ric Mynard. \u00a0Convergence without points. \u00a0arXiv 1807.03226<\/a>, July 2018.<\/li>\n<\/ol>\n

    \u2014 Jean Goubault-Larrecq<\/a>(July 18th, 2018)\"jgl-2011\"<\/p>\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    I thought I would devote my blog this month to the Domains workshop, but a sudden health problem prevented me to go there. \u00a0I’ll have to read the papers instead… \u00a0In the meantime, I would like to talk about a … 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-1518","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1518","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=1518"}],"version-history":[{"count":8,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1518\/revisions"}],"predecessor-version":[{"id":5312,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1518\/revisions\/5312"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1518"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

The converse direction is more elaborate. \u00a0There is a functor\u00a0Sl<\/strong> : Frm<\/strong>\u00a0\u2192\u00a0CF<\/strong>top<\/sup>, which maps every frame\u00a0\u03a9 to its coframe of sublocales<\/a>. \u00a0(This is why we defined topological coframes as\u00a0coframes<\/em>.) \u00a0The closed elements of\u00a0Sl<\/strong>(\u03a9) are simply chosen to be the\u00a0closed<\/em> sublocales\u00a0c<\/strong>(a<\/em>), a<\/em> in \u03a9, (in the usual localic sense of `closed’:\u00a0\u00a0c<\/strong>(a<\/em>) is the sublocale of all elements above\u00a0a<\/em>).<\/p>\n