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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/2419/#Item_5" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <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><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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_1' xmlns='http://www.w3.org/1998/Math/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></li></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/preorder'>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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_2' xmlns='http://www.w3.org/1998/Math/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/filtered+limit'>cofiltered limits</a>, any functor out of <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_3' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_4' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_5' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_6' xmlns='http://www.w3.org/1998/Math/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/locale'>locales</a> which regards a set as a <a class='existingWikiWord' href='/nlab/show/discrete+object'>discrete</a> locale.</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_7' xmlns='http://www.w3.org/1998/Math/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/direction'>directed poset</a>, then the corresponding locale <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_8' xmlns='http://www.w3.org/1998/Math/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/partial+order'>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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_9' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_10' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_11' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_12' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_13' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_14' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_15' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_16' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_17' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_18' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> of pairs <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_20' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math> and all <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_23' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_24' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_26' xmlns='http://www.w3.org/1998/Math/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+space'>connected components</a>. The vertices of the cofiltered diagram defining <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_27' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_28' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_29' xmlns='http://www.w3.org/1998/Math/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+subspace'>open subsets</a>, and the corresponding set is the index set <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>. A morphism <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_31' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_32' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_33' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_34' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_35' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_36' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_37' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_38' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_39' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_40' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> we have <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_42' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_43' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_44' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_45' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_46' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_47' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_48' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>Since <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_50' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_51' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_52' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math> cover <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, so do the <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_55' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_56' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_57' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_58' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_59' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_60' xmlns='http://www.w3.org/1998/Math/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+topological+space'>locally connected</a>, then it has a “minimal” such decomposition, namely its decomposition into <a class='existingWikiWord' href='/nlab/show/connected+space'>connected components</a>. Thus, in this case <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_61' xmlns='http://www.w3.org/1998/Math/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+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of an (∞,1)-topos</a>.</p> <div class='un_theorem'> <h6 id='theorem'>Theorem</h6> <p>The functor <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_62' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_63' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_64' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_65' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, are equivalent to morphisms of locales <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_67' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_68' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> into disjoint opens indexed by <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, which exactly defines a map <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_71' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/locally+positive+locale'>overt locale</a>, then every decomposition is refined by a decomposition into <a class='existingWikiWord' href='/nlab/show/positive+element'>positive elements</a>, so we may as well consider only decompositions into positive opens. If we do this, the cofiltered category indexing <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_73' xmlns='http://www.w3.org/1998/Math/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/direction'>codirected poset</a>, since (in the argument above) each <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_74' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_75' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>, so that <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_77' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_78' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_79' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_80' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_81' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_82' xmlns='http://www.w3.org/1998/Math/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+space'>profinite sets</a>, the functor <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>lim</mi></mrow><annotation encoding='application/x-tex'>\lim</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/full+and+faithful+functor'>fully faithful</a> and in fact lands inside the subcategory of <a class='existingWikiWord' href='/nlab/show/spatial+locale'>topological locales</a>, its image being the category of <a class='existingWikiWord' href='/nlab/show/Stone+space'>Stone spaces</a>.</p> <p>It is also true that when lifted to <a class='existingWikiWord' href='/nlab/show/progroup'>progroups</a>, the functor <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_84' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_85' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_86' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is a pro-set and <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_88' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_89' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_90' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>). But a morphism of locales <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_92' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mo>⊥</mo></msup></mrow><annotation encoding='application/x-tex'>A^\bot</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_94' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_95' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_96' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_97' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mo>⊥</mo></msup></mrow><annotation encoding='application/x-tex'>A^\bot</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_99' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_100' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_101' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_102' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_103' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_104' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_105' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_106' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_107' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_108' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_109' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_110' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> we have <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_112' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_113' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_114' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_115' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_116' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_117' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_118' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mo>⊥</mo></msup></mrow><annotation encoding='application/x-tex'>A^\bot</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_120' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_121' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_122' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_00f78a1a84591cef323f54da6f2ebe58a01489d7_123' xmlns='http://www.w3.org/1998/Math/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> </p> </div>  <div class="revisedby"> <p> Revision on April 2, 2012 at 04:51:52 by <a href="/nlab/author/Toby+Bartels" style="color: #005c19">Toby Bartels</a> See the <a href="/nlab/history/pro-set" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/2419/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/show/pro-set" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)</span><span class="backintime"><a href="/nlab/revision/pro-set/3" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (3 more)</span><a href="/nlab/show/pro-set" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/pro-set/4" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/pro-set" accesskey="S" class="navlink" id="history" rel="nofollow">History (4 revisions)</a><a href="/nlab/rollback/pro-set?rev=4" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/pro-set/4/cite" style="color: black">Cite</a> <a href="/nlab/source/pro-set/4" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>