{"id":6400,"date":"2023-03-20T11:00:58","date_gmt":"2023-03-20T10:00:58","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6400"},"modified":"2023-03-23T11:11:16","modified_gmt":"2023-03-23T10:11:16","slug":"localic-products-and-till-plewes-game","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6400","title":{"rendered":"Localic products and Till Plewe’s game"},"content":{"rendered":"\n

I have talked about Till Plewe’s topological game [2] last time<\/a>, showing how Matthew de Brecht used it to show that the space S<\/em>0<\/sub> is not consonant [1]. However, the original purpose of that game was to show that certain products are computed differently in the category of topological spaces and in the category of locales.<\/p>\n\n\n\n

Localic products, topological products<\/h2>\n\n\n\n

Let me recall how products are built in the category Top<\/strong> of topological spaces and continuous maps. The product X<\/em> \u00d7 Y<\/em> of two topological spaces has open subsets W<\/em> built as arbitrary unions of open rectangles U<\/em> \u00d7 V<\/em>, where U<\/em> ranges over O<\/strong>X<\/em> and V<\/em> ranges over O<\/strong>Y<\/em>.<\/p>\n\n\n\n

Any such open set W<\/em> gives rise to a Galois connection (\u03b1W<\/sub><\/em>, \u03b3W<\/sub><\/em>) between the frames O<\/strong>X<\/em> and O<\/strong>Y<\/em>: \u03b1W<\/sub><\/em>(U<\/em>) is defined as the largest open subset V<\/em> of Y<\/em> such that U<\/em> \u00d7 V<\/em> \u2286 W<\/em>, for every U<\/em> \u2208 O<\/strong>X<\/em>, and \u03b3W<\/sub><\/em>(V<\/em>) is the largest open subset U<\/em> of X<\/em> such that U<\/em> \u00d7 V<\/em> \u2286 W<\/em>, for every V<\/em> \u2208 O<\/strong>Y<\/em>. It is easy to see that \u03b1W<\/sub><\/em> and \u03b3W<\/sub><\/em> are antitonic, and that V<\/em> \u2286 \u03b1W<\/sub><\/em>(U<\/em>) if and only if U<\/em> \u2286 \u03b3W<\/sub><\/em>(V<\/em>) (if and only if U<\/em> \u00d7 V<\/em> \u2286 W<\/em>); those are the properties defining a Galois connection.<\/p>\n\n\n\n

In general, given two frames \u03a91<\/sub> and \u03a92<\/sub> (taking the place of O<\/strong>X<\/em> and of O<\/strong>Y<\/em> respectively), a Galois connection between them is a pair of antitonic maps (\u03b1, \u03b3), from \u03a91<\/sub> to \u03a92<\/sub> and from \u03a92<\/sub> to \u03a91<\/sub> respectively, satisfying the equivalence v<\/em> \u2264 \u03b1(u<\/em>) if and only if u<\/em> \u2264 \u03b3(v<\/em>). Those Galois connections form a frame Gal(\u03a91<\/sub>, \u03a92<\/sub>); the ordering consists in comparing the \u03b1 parts using the pointwise ordering, or equivalently the \u03b3 parts using the pointwise ordering. It does not matter whether we use \u03b1 or \u03b3, because any one of them determines the other one uniquely, and if one grows, the other grows as well.<\/p>\n\n\n\n

Gal(\u03a91<\/sub>, \u03a92<\/sub>) is the coproduct of \u03a91<\/sub> and \u03a92<\/sub> in the category Frm<\/strong> of frames and frame homomorphisms. Therefore it is their product<\/em> in the opposite category Loc<\/strong>=Frm<\/strong>op<\/sup> of locales.<\/p>\n\n\n\n

This is how I introduce localic products in Section 8.4.4 of the book<\/a>.<\/p>\n\n\n\n

A funny thing is that the elements of Gal(\u03a91<\/sub>, \u03a92<\/sub>) are suprema of rectangles, just like in the case of topological spaces. Those rectangles have a strange definition, although it is exactly what you would expect if you look at the Galois connections defined from regular open rectangles U<\/em> \u00d7 V<\/em> in the product of two topological spaces, and abstract away from spaces in order to obtain a formula that makes sense on all frames. Explicitly, given u<\/em> \u2208 \u03a91<\/sub> and v<\/em> \u2208 \u03a92<\/sub>, the (u<\/em> \u00d7 v<\/em>)-rectangle<\/em> u<\/em> \u2297 v<\/em> is the Galois connection (\u03b1uv<\/sub><\/em>, \u03b3uv<\/sub><\/em>) where \u03b1uv<\/sub><\/em> maps \u22a5 to \u22a4, every element below u<\/em> and different from \u22a5 to v<\/em>, and all other elements to \u22a5. (The \u03b3uv<\/sub><\/em> part is obtained in a unique way from \u03b1, and\u2014as you might guess\u2014by a symmetric formula. In the book, \u03b1uv<\/sub><\/em> is written as \u03b1u\u00d7<\/sub><\/em>v<\/sub><\/em> and \u03b3uv<\/sub><\/em> is written as \u03b3u\u00d7v<\/sub><\/em>, but I now find this small \u00d7 sign visually disturbing. Note that I am also using the notation u<\/em> \u2297 v<\/em>, just like Plewe [2]; so u<\/em> \u2297 v<\/em> is the same thing as the Galois connection (\u03b1uv<\/sub><\/em>, \u03b3uv<\/sub><\/em>).)<\/p>\n\n\n\n

The point is that every element (\u03b1, \u03b3) of Gal(\u03a91<\/sub>, \u03a92<\/sub>) is the supremum of all the rectangles below it. Additionally, this supremum is pointwise<\/em>: for every u’<\/em>, \u03b1(u’<\/em>) is the supremum of the family \u03b1uv<\/sub><\/em>(u’<\/em>) where (\u03b1uv<\/sub><\/em>, \u03b3uv<\/sub><\/em>) ranges over the rectangles below (\u03b1, \u03b3) (and similarly for \u03b3(v’<\/em>)). This is remarkable, since general suprema of Galois connections are not computed pointwise; only infima are computed pointwise.<\/p>\n\n\n\n

We have seen that every open set W<\/em> in the topological product X<\/em> \u00d7 Y<\/em> defines a Galois connection between O<\/strong>X<\/em> and O<\/strong>Y<\/em>. In other words, O<\/strong>(X<\/em> \u00d7 Y<\/em>) sits inside Gal(O<\/strong>X<\/em>, O<\/strong>Y<\/em>), through the mapping that sends W<\/em> to (\u03b1W<\/sub><\/em>, \u03b3W<\/sub><\/em>). Let me call \u03b8 this mapping. Explicitly:<\/p>\n\n\n\n

Definition.<\/strong> \u03b8 : O<\/strong>(X<\/em> \u00d7 Y<\/em>) \u2192 Gal(O<\/strong>X<\/em>, O<\/strong>Y<\/em>) maps every open subset W<\/em> of X<\/em> \u00d7 Y<\/em> to (\u03b1W<\/sub><\/em>, \u03b3W<\/sub><\/em>), where:<\/p>\n\n\n\n