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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Pro-sets</title></head> <body> <h1 id="prosets">Pro-sets</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#prosets_versus_locales'>Pro-sets versus locales</a></li> <ul> <li><a href='#the_classifying_locale_of_a_proset'>The classifying locale of a pro-set</a></li> <li><a href='#the_proset__of_a_locale'>The pro-set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> of a locale</a></li> <li><a href='#the_classifying_locale_functor_is_not_an_embedding'>The classifying locale functor is not an embedding</a></li> </ul> </ul> </div> <h2 id="definition">Definition</h2> <p>A <strong>pro-set</strong> is a <a class="existingWikiWord" href="/nlab/show/pro-object">pro-object</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>This should not be confused with <a class="existingWikiWord" href="/nlab/show/proset">proset</a>, an abbreviation of “preordered set.”</p> <h2 id="prosets_versus_locales">Pro-sets versus locales</h2> <h3 id="the_classifying_locale_of_a_proset">The classifying locale of a pro-set</h3> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pro(Set)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+completion">free completion</a> of <a class="existingWikiWord" href="/nlab/show/Set">Set</a> under <a class="existingWikiWord" href="/nlab/show/cofiltered+limits">cofiltered limits</a>, any functor out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> into a category with cofiltered limits extends uniquely to a cofiltered-limit-preserving functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pro(Set)</annotation></semantics></math>. In particular, we can consider the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo>→</mo><mi>Loc</mi></mrow><annotation encoding="application/x-tex">Set \to Loc</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> to the category of <a class="existingWikiWord" href="/nlab/show/locales">locales</a> which regards a set as a <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete</a> locale.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">(S_i)_i</annotation></semantics></math> is a pro-set, which we may WLOG assume to be indexed on a <a class="existingWikiWord" href="/nlab/show/directed+poset">directed poset</a>, then the corresponding locale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\lim S_i</annotation></semantics></math> is presented by the following <a class="existingWikiWord" href="/nlab/show/posite">posite</a>. Its underlying <a class="existingWikiWord" href="/nlab/show/poset">poset</a> is the <a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a> of the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">(S_i)_i</annotation></semantics></math>, i.e. its elements are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,x)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x\in S_i</annotation></semantics></math>, and we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,x)\le (j,y)</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≤</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i\le j</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">s_{i j}(x)=y</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>→</mo><msub><mi>S</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">s_{i j} \colon S_i \to S_j</annotation></semantics></math> is the transition map. The covers in the posite are generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊲</mo><msubsup><mi>s</mi> <mrow><mi>i</mi><mi>j</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,x) \lhd s_{i j}^{-1}(x)</annotation></semantics></math> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">i,j,x</annotation></semantics></math>.</p> <p>Thus, the open sets in the locale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\lim S_i</annotation></semantics></math> are the “ideals” for this coverage, i.e. sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,x)</annotation></semantics></math> which are down-closed and such that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(j,y)\in A</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><msubsup><mi>s</mi> <mrow><mi>i</mi><mi>j</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y\in s_{i j}^{-1}(x)</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(i,x)\in A</annotation></semantics></math>.</p> <h3 id="the_proset__of_a_locale">The pro-set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> of a locale</h3> <p>On the other hand, there is a naturally defined functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mi>Loc</mi><mo>→</mo><mi>Pro</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_0\colon Loc \to Pro(Set)</annotation></semantics></math> which sends a locale to its pro-set of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>. The vertices of the cofiltered diagram defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_0(X)</annotation></semantics></math> are decompositions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X = \coprod_{i\in I} U_i</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a>, and the corresponding set is the index set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>. A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U_i) \to (V_j)</annotation></semantics></math>, called a <em>refinement</em>, consists of a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">f:I\to J</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">V_{f(i)}</annotation></semantics></math>; the corresponding function is of course <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <p>This diagram is cofiltered: 1. It is nonempty, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the 1-ary coproduct of itself. 1. Given decompositions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U_i)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V_j)</annotation></semantics></math>, the decomposition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>j</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(U_i \cap V_j)_{i,j}</annotation></semantics></math> refines both of them. 1. Given parallel refinements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g:(U_i)\to (V_j)</annotation></semantics></math>, for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊆</mo><msub><mi>V</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>∩</mo><msub><mi>V</mi> <mrow><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">U_i \subseteq V_{f(i)} \cap V_{g(i)}</annotation></semantics></math>. If we define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mo stretchy="false">{</mo><mi>i</mi><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">K = \{ i | f(i) = g(i) \}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">W_i = U_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">i\in K</annotation></semantics></math>, then we have an obvious refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>W</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h\colon (W_k) \to (U_i)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>h</mi><mo>=</mo><mi>g</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">f h = g h</annotation></semantics></math>. It remains to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>W</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W_k)</annotation></semantics></math> is actually a cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mo>∩</mo><msub><mi>V</mi> <mrow><msub><mi>j</mi> <mn>2</mn></msub></mrow></msub><mo>=</mo><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo><mo stretchy="false">{</mo><msub><mi>V</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">|</mo><msub><mi>j</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>j</mi> <mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V_{j_1} \cap V_{j_2} = \bigcup \{ V_{j_1} | j_1 = j_2 \}</annotation></semantics></math> (the latter being the <a class="existingWikiWord" href="/nlab/show/join">join</a> of a <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a>), we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊆</mo><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_i \subseteq \bigcup \{ U_i | f(i) = g(i) \}</annotation></semantics></math> (another join of a subsingleton) and thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊆</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></msub><msub><mi>W</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_i \subseteq \bigcup_{k\in K} W_k</annotation></semantics></math>. Thus, since the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, so do the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">W_k</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematician</a> may be forgiven for thinking this last argument to be more confusing than necessary, since classically, either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(i)=g(i)</annotation></semantics></math> (in which case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">W_i = U_i</annotation></semantics></math>) or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>≠</mo><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(i)\neq g(i)</annotation></semantics></math> (in which case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">U_i = \emptyset</annotation></semantics></math>). Constructively, however, the more involved argument is required.</p> <p>Note that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected</a>, then it has a “minimal” such decomposition, namely its decomposition into <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>. Thus, in this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_0(X)</annotation></semantics></math> is a mere set.</p> <p>This is the <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a> version of the <a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a> or the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+an+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of an (∞,1)-topos</a>.</p> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><mi>Loc</mi><mo>→</mo><mi>Pro</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_0\colon Loc \to Pro(Set)</annotation></semantics></math> is left adjoint to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mo lspace="verythinmathspace">:</mo><mi>Pro</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Loc</mi></mrow><annotation encoding="application/x-tex">\lim\colon Pro(Set)\to Loc</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi></mrow><annotation encoding="application/x-tex">\lim</annotation></semantics></math> is given by regarding a pro-set as a diagram of discrete locales and taking its limit, it suffices to show that morphisms of pro-sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\Pi_0(X) \to S</annotation></semantics></math>, for a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, are equivalent to morphisms of locales <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>S</mi> <mi>disc</mi></msub></mrow><annotation encoding="application/x-tex">X \to S_{disc}</annotation></semantics></math>. But a locale map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>S</mi> <mi>disc</mi></msub></mrow><annotation encoding="application/x-tex">X \to S_{disc}</annotation></semantics></math> is precisely a decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> into disjoint opens indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, which exactly defines a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\Pi_0(X) \to S</annotation></semantics></math>.</p> </div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/overt+locale">overt locale</a>, then every decomposition is refined by a decomposition into <a class="existingWikiWord" href="/nlab/show/positive+elements">positive elements</a>, so we may as well consider only decompositions into positive opens. If we do this, the cofiltered category indexing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/codirected+poset">codirected poset</a>, since (in the argument above) each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U_i)</annotation></semantics></math> is covered by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ U_i | f(i) = g(i) \}</annotation></semantics></math>, which must therefore be an <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited set</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f=g</annotation></semantics></math>. Moreover, since in any refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f\colon (U_i) \to (V_j)</annotation></semantics></math>, each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">V_j</annotation></semantics></math> is covered by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>j</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ U_i | f(i)=j\}</annotation></semantics></math>, in this case the transition maps of the resulting pro-set are surjective. However, for a non-overt locale (which, recall, cannot exist <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classically</a>), it seems that the pro-set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_0(X)</annotation></semantics></math> need not be surjective in this sense.</p> <h3 id="the_classifying_locale_functor_is_not_an_embedding">The classifying locale functor is not an embedding</h3> <p>It is well-known that when restricted to the subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><mi>FinSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pro(FinSet)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/profinite+sets">profinite sets</a>, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi></mrow><annotation encoding="application/x-tex">\lim</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a> and in fact lands inside the subcategory of <a class="existingWikiWord" href="/nlab/show/topological+locales">topological locales</a>, its image being the category of <a class="existingWikiWord" href="/nlab/show/Stone+spaces">Stone spaces</a>.</p> <p>It is also true that when lifted to <a class="existingWikiWord" href="/nlab/show/progroups">progroups</a>, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mo lspace="verythinmathspace">:</mo><mi>Pro</mi><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Grp</mi><mo stretchy="false">(</mo><mi>Loc</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim\colon Pro(Grp) \to Grp(Loc)</annotation></semantics></math> into localic groups is fully faithful when restricted to <em>strict</em> or <em>surjective</em> progroups (those whose transition maps are surjective).</p> <p>However, in general the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mo lspace="verythinmathspace">:</mo><mi>Pro</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Loc</mi></mrow><annotation encoding="application/x-tex">\lim\colon Pro(Set) \to Loc</annotation></semantics></math> is not an embedding. For a counterexample, consider morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">S\to 2</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a pro-set and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo>=</mo><mo stretchy="false">{</mo><mo>⊥</mo><mo>,</mo><mo>⊤</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">2=\{\bot,\top\}</annotation></semantics></math>, regarded as a pro-set in the trivial way (and thus giving rise to a discrete locale). A morphism of pro-sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">S\to 2</annotation></semantics></math> is determined by a partition of some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub><mo>=</mo><msubsup><mi>S</mi> <mi>i</mi> <mo>⊥</mo></msubsup><mo>⊔</mo><msubsup><mi>S</mi> <mi>i</mi> <mo>⊤</mo></msubsup></mrow><annotation encoding="application/x-tex">S_i = S_i^\bot \sqcup S_i^\top</annotation></semantics></math> (modulo a suitable equivalence relation as we change <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>). But a morphism of locales <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><msub><mi>S</mi> <mi>i</mi></msub><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\lim S_i \to 2</annotation></semantics></math> consists of two ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>⊥</mo></msup></mrow><annotation encoding="application/x-tex">A^\bot</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>⊤</mo></msup></mrow><annotation encoding="application/x-tex">A^\top</annotation></semantics></math> which are disjoint and whose union <em>generates</em> the improper ideal (which consists of all pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,x)</annotation></semantics></math>). A pro-set morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">S\to 2</annotation></semantics></math> induces a locale map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><msub><mi>S</mi> <mi>i</mi></msub><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\lim S_i \to 2</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>⊥</mo></msup></mrow><annotation encoding="application/x-tex">A^\bot</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>⊤</mo></msup></mrow><annotation encoding="application/x-tex">A^\top</annotation></semantics></math> are the ideals generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>i</mi> <mo>⊥</mo></msubsup></mrow><annotation encoding="application/x-tex">S_i^\bot</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>i</mi> <mo>⊤</mo></msubsup></mrow><annotation encoding="application/x-tex">S_i^\top</annotation></semantics></math>, but in general not every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><msub><mi>S</mi> <mi>i</mi></msub><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\lim S_i \to 2</annotation></semantics></math> is induced by one <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">S\to 2</annotation></semantics></math>.</p> <p>Specifically, consider the following pro-set, which is indexed on the natural numbers with the inverse ordering:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>→</mo><mi>⋯</mi><mo>→</mo><msub><mi>S</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>S</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>S</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> \cdots \to S_i \to \cdots \to S_2 \to S_1 \to S_0 </annotation></semantics></math></div> <p>We define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>ℕ</mi><mo>×</mo><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mrow><msub><mo>∼</mo> <mi>i</mi></msub></mrow></mrow><annotation encoding="application/x-tex">S_i = (\mathbb{N} \times \{a,b\}) / {\sim_i}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∼</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sim_i</annotation></semantics></math> is the equivalence relation generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo>∼</mo> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,a) \sim_i (k,b)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">k \ge i</annotation></semantics></math>. The transition maps are the obvious projections, which are surjective. Define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>⊥</mo></msup><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>k</mi><mo><</mo><mi>i</mi><mo stretchy="false">}</mo><mspace width="1em"></mspace><mtext>and</mtext><mspace width="1em"></mspace><msup><mi>A</mi> <mo>⊤</mo></msup><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>k</mi><mo><</mo><mi>i</mi><mo stretchy="false">}</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> A^\bot = \{ (i,(k,a)) | k \lt i \} \quad\text{and}\quad A^\top = \{ (i,(k,b) | k \lt i \}.</annotation></semantics></math></div> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>⊥</mo></msup><mo>∪</mo><msup><mi>A</mi> <mo>⊤</mo></msup></mrow><annotation encoding="application/x-tex">A^\bot \cup A^\top</annotation></semantics></math> generates the improper ideal, since for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>⊂</mo><msup><mi>A</mi> <mo>⊥</mo></msup><mo>∪</mo><msup><mi>A</mi> <mo>⊤</mo></msup></mrow><annotation encoding="application/x-tex">\{ (i+1, (i,a)), (i+1,(i,b)) \} \subset A^\bot \cup A^\top</annotation></semantics></math>, which covers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mo>?</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,(i,?))</annotation></semantics></math>, which covers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mo>?</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i-1,(i,?))</annotation></semantics></math>, and so on down to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mo>?</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,(i,?))</annotation></semantics></math>. However, no <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math> can be partitioned as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub><mo>=</mo><msubsup><mi>S</mi> <mi>i</mi> <mo>⊥</mo></msubsup><mo>⊔</mo><msubsup><mi>S</mi> <mi>i</mi> <mo>⊤</mo></msubsup></mrow><annotation encoding="application/x-tex">S_i = S_i^\bot \sqcup S_i^\top</annotation></semantics></math> in such a way that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>i</mi> <mo>⊥</mo></msubsup></mrow><annotation encoding="application/x-tex">S_i^\bot</annotation></semantics></math> generates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>⊥</mo></msup></mrow><annotation encoding="application/x-tex">A^\bot</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>i</mi> <mo>⊤</mo></msubsup></mrow><annotation encoding="application/x-tex">S_i^\top</annotation></semantics></math> generates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>⊤</mo></msup></mrow><annotation encoding="application/x-tex">A^\top</annotation></semantics></math>. Thus, this defines a locale map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><msub><mi>S</mi> <mi>i</mi></msub><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\lim S_i \to 2</annotation></semantics></math> which does not arise from a pro-set morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">S\to 2</annotation></semantics></math>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on January 8, 2018 at 16:49:39. 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