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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 3rd 2011
    • (edited Jan 3rd 2011)

    added to locale a section relation to toposes stating localic reflection

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 3rd 2011

    locale, not ocale

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 3rd 2011

    thanks, fixed.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 3rd 2011

    Added to locale remakrs on how the frame of subobjects Sub (*)τ 1Sub_{\mathcal{E}}(*) \simeq \tau_{\leq -1} is the subcategory of (1)(-1)-truncated objects and how this is the beginning of a pattern continued at n-localic (infinity,1)-topos.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2011

    Have added to locale in the section Category of locales two theorems from the Elephant on the externalization of internal locales (needed in the discussion of Bohrification)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2011

    At locale in the section relation to topological spaces I have tried to make some of the statements more pronounced.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2011

    I have made more explicit in the discussion of localic reflection at locale that Sh:LocaleToposSh : Locale \to Topos is fully faithful in fact as a 2-functor.

    The discussion of this point in the entry would deserve a bit of expansion. Maybe I find time to expand on it later.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2011

    I realized that we never stated the definition of the 2-category (or (1,2)-category) of locales. So I have now added it to the Definition-section at Locale.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeJul 25th 2011

    You mean at locale; Locale already had it!

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2011
    • (edited Jul 25th 2011)

    Oh. dear, I should take a break :-) I did look for it, but didn’t see it! Thanks.

    Anyway, I have now added to Locale, too, the precise definition, and some more pointers.

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 6th 2011

    Added to locale the observation that frames are the same as lex total posets, by way of introducing connections between locales and toposes.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeDec 6th 2011

    Nice, thanks! Should we have a page lex-total category with Street’s theorem on it?

    • CommentRowNumber13.
    • CommentAuthorCharles Rezk
    • CommentTimeMay 20th 2019

    The second paragraph of Locale says:

    For example, there is a locale of all surjections from natural numbers (thought of as forming the discrete space NN) to real numbers (forming the real line RR): the locale of real numbers. This local has no points, since there are no such surjections, but it contains many nontrivial open subspaces; these open subspaces are generated by a family parametrised by n:Nn\colon N and x:Rx\colon R; the basic open associated to nn and xx may be described as {f:NR|f(n)=x}\{f\colon N \twoheadrightarrow R \;|\; f(n) = x\}.

    Can someone explain exactly what this means, or at least give a reference?

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 20th 2019

    Capitalised ’Locale’ is redirected to Loc, the category. So you mean the lower case version locale.

    Does the description at locale of real numbers help? It seems to be largely the work of Toby.

    • CommentRowNumber15.
    • CommentAuthorCharles Rezk
    • CommentTimeMay 20th 2019

    Yes I mean locale. The locale of real numbers doesn’t help at all. In any case I would guess that the particular sets (locales?) NN and RR are basically irrelevant for this example. (Except for having |N|<|R||N|\lt|R|, which is supposed to imply there are no points.) I just want to know what definition is underlying this example.

    • CommentRowNumber16.
    • CommentAuthorCharles Rezk
    • CommentTimeMay 20th 2019
    • (edited May 20th 2019)

    Here’s my speculation, though it doesn’t seem to work right. Let XX and YY be sets, and let 𝒫\mathcal{P} be the poset of “finite partial graphs” in X×YX\times Y, i.e., finite subsets ΓX×Y\Gamma\subseteq X\times Y such that the composite ΓX×YX\Gamma\to X\times Y \to X is injective. Let 𝒪\mathcal{O} be the collection of upward-closed subsets of 𝒫\mathcal{P}: this set is supposed to be the collection of “opens” in the locale.

    Given any function f:XYf\colon X\to Y let U f𝒪U_f\in \mathcal{O} be the set of all Γ𝒫\Gamma\in \mathcal{P} such that ΓΓ f\Gamma\cup \Gamma_f is not the graph of any function: it’s the set of finite partial graphs inconsistent with ff. Note that U f𝒫U_f\neq \mathcal{P} since U f\varnothing\notin U_f, and I can show that U fU_f is non-trivially an intersection of non-maximal elements of 𝒪\mathcal{O} if and only if ff is not surjective. That is, surjective functions give rise to points in the locale 𝒪\mathcal{O}. What I can’t prove is that these are the only points in the locale.

    Edit. Nevermind that doesn’t make sense anyway. But I expect they mean some kind of example along these lines.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeMay 20th 2019

    The locale of surjections XYX\to Y is the classifying locale of the propositional geometric theory of surjections XYX\to Y. There are basic propositions “f(x)=yf(x)=y” (an atomic formula) for each xXx\in X and yYy\in Y, and then axioms like xX yYf(x)=y\vdash_{x\in X} \bigvee_{y\in Y} f(x)=y and (f(x)=y)(f(x)=y) xX,yyY(f(x)=y) \wedge (f(x)=y') \vdash_{x\in X,y\neq y'\in Y} \bot to make it a function and yY xXf(x)=y\vdash_{y\in Y} \bigvee_{x\in X} f(x)=y to make it surjective. So the opens of the locale are generated by these propositions “f(x)=yf(x)=y” under finite meet and arbitrary join, modulo those axioms. This can be re-expressed in terms of partial functions in a way kind of like you say, but I don’t remember (and don’t have time to work out) exactly how it goes. But with this description, the points of the locale are, by the universal property of classifying locales (or equivalently, by the universal property of a frame presented by generators and relations), precisely surjections XYX\to Y.

    • CommentRowNumber18.
    • CommentAuthorCharles Rezk
    • CommentTimeMay 23rd 2019

    Thanks Mike. I would still love a reference for this. It looks to me that what I was describing above is the locale of “partial graphs in X×YX\times Y”; at least, the points are precisely the partial graphs. I assume the additional axioms are imposed by choosing appropriate sublocales.

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2019

    It’s C1.2.8 in Sketches of an Elephant.

    • CommentRowNumber20.
    • CommentAuthorCharles Rezk
    • CommentTimeJun 7th 2019

    Thanks Mike.

    Here’s a brief sketch of the construction (where I’ve unwound most of the terminology Johnstone uses here).

    Fix discrete spaces XX and YY, and let M= XYM=\prod_X Y with the usual product topology. For a finite subset SYS\subseteq Y write M S={fM|Sf(X)}M_S=\{f\in M\;|\; S\subseteq f(X)\}, the subset of functions which map onto SS. Let 𝒪\mathcal{O} be the collection of open subsets UMU\subseteq M such that

    for all open VMV\subseteq M and finite SYS\subseteq Y, we have that VM SUV\cap M_S\subseteq U implies VUV\subseteq U.

    The claims are that:

    1. 𝒪\mathcal{O} is a complete lattice satisfying the infinite distributive law, i.e., it corresponds to a locale.

    2. If XX is infinite then 𝒪\varnothing\in \mathcal{O} and M𝒪M\in \mathcal{O}, e.g., 𝒪\mathcal{O} is a non-trivial locale when XX is infinite and YY is non-empty.

    3. The points of 𝒪\mathcal{O} correspond exactly to surjective functions XYX\to Y.

    In particular, if |X|<|Y|\infty\leq |X| \lt |Y| then 𝒪\mathcal{O} is a non-trivial pointless locale. (Johnstone in the example assumes X=X=\mathbb{N}, but as far as I can tell the construction is entirely general.)

    It is easy to see that 𝒪\mathcal{O} is closed under pairwise intersections of open sets and contains MM, so has pairwise meets and a top element. To show it is a locale, you need to know that for every open set UU there is a smallest open set j(U)j(U) which contains UU and is an element of 𝒪\mathcal{O}. The recipe for jj is to define for any open set UU,

    h(U):= VM SUV, h(U) := \bigcup_{V\cap M_S\subseteq U} V,

    where the union ranges over all open VUV\subseteq U and finite SYS\subseteq Y satisfying the condition. Then iterate hh some possibly transfinite number of times to get j(U)j(U). Then j()j(\varnothing) is the bottom element and U i=j(U i)\bigvee U_i = j(\bigcup U_i). The key observation is that hh preserves pairwise intersection and has exactly elements of 𝒪\mathcal{O} as its fixed points, whence jj has these same properties.

    More conceptually: 𝒪\mathcal{O} is the limit in Locale of the family of subspaces M SM_S of MM indexed by the poset of all finite SYS\subseteq Y; compare with the limit in Top, which is just the subspace SM S\bigcap_S M_S of surjective functions. (Johnntone says “intersection of sublocales” here, but I think it is also an example of a limit in Locale this context, as is the intersection subspace in Top.)

    Unrelated remark: Apparently the TeX engine on this thing displays the same character for \varnothing\backslash varnothing (\varnothing) and \emptyset\backslash emptyset (\emptyset), which is unfortunate since \emptyset is so ugly.

    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeJun 8th 2019

    Thanks for writing that out! It would be useful to add it to some nlab page, maybe locale?

    I generally prefer \emptyset to \varnothing; I find the latter uglier than the former. But if you really want the latter you may be able to get it with unicode &#8709; (∅).

    • CommentRowNumber22.
    • CommentAuthorTobyBartels
    • CommentTimeSep 25th 2019

    Web page for Picado & Pultr 2012.

    diff, v76, current

    • CommentRowNumber23.
    • CommentAuthorTobyBartels
    • CommentTimeSep 25th 2019

    I added a bit about the locale of surjections discussed in comments #13–#21, only at the level of detail of Mike #17 (actually less detail than that), not Charles #20.

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