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In other words, it is a formal <a class="existingWikiWord" href="/nlab/show/cofiltered+limit">cofiltered limit</a> of groups.</p> <h2 id="surjective_progroups_versus_localic_groups">Surjective progroups versus localic groups</h2> <p>Of course, the category <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> is <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a>, but in general a progroup represented by some cofiltered diagram of groups is not equivalent to the actual limit of that diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grp</mi></mrow><annotation encoding="application/x-tex">Grp</annotation></semantics></math>. However, <a class="existingWikiWord" href="/nlab/show/profinite+groups">profinite groups</a> (i.e. cofiltered systems of <em>finite</em> groups) can be identified with actual limits of finite groups if we take those limits, not in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grp</mi></mrow><annotation encoding="application/x-tex">Grp</annotation></semantics></math>, but in the larger category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TopGrp</mi></mrow><annotation encoding="application/x-tex">TopGrp</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a>. The resulting topological groups are precisely those with <a class="existingWikiWord" href="/nlab/show/Stone+space">Stone</a> topologies.</p> <p>This is not true for pro-systems of non-finite groups, even if we restrict to those with surjective transition maps. The following counterexample is due to Higman and Stone, and is reproduced in (<a href="#Moerdijk">Moerdijk</a>). Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\omega_1</annotation></semantics></math> be the set of countable ordinals, with the reverse of its usual ordering, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\alpha\in\omega_1</annotation></semantics></math> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">S_\alpha</annotation></semantics></math> be the set of strictly increasing functions <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><mi>α</mi><mo stretchy="false">]</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">[0,\alpha]\to \mathbb{R}</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo><</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha\lt \beta</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>β</mi></msub><mo>→</mo><msub><mi>S</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">S_\beta \to S_\alpha</annotation></semantics></math> be the restriction. Then each such transition map is surjective, but the inverse limit is empty. The sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">S_\alpha</annotation></semantics></math> are not groups, but if we take the free <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> on each of them, we obtain a nontrivial pro-group with surjective transition maps whose limit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grp</mi></mrow><annotation encoding="application/x-tex">Grp</annotation></semantics></math>, hence also in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TopGrp</mi></mrow><annotation encoding="application/x-tex">TopGrp</annotation></semantics></math>, is trivial.</p> <p>However, we do get an embedding on pro-groups with surjective transition maps if instead of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> we take the limit in the category <a class="existingWikiWord" href="/nlab/show/Loc">Loc</a> of <a class="existingWikiWord" href="/nlab/show/locales">locales</a>.</p> <div class="un_prop" id="EquivalentCharacterizations"> <h6 id="proposition">Proposition</h6> <p>The following are equivalent for a <a class="existingWikiWord" href="/nlab/show/localic+group">localic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>: 1. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a cofiltered limit of <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>s (considered as discrete localic groups) 1. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a cofiltered limit of discrete groups with <a class="existingWikiWord" href="/nlab/show/surjection">surjective</a> transition maps. 1. The <a class="existingWikiWord" href="/nlab/show/open+subset">open</a> <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/neighborhood">neighborhood</a> <a class="existingWikiWord" href="/nlab/show/topological+base">base</a> at the identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">e\in G</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This can be found in (<a href="#Moerdijk">Moerdijk</a>).</p> </div> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>A localic group with these properties is called <strong>prodiscrete</strong>.</p> </div> <p>We may as well assume that any surjective progroup is indexed on a directed <a class="existingWikiWord" href="/nlab/show/poset">poset</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(G_i)_{i\in I}</annotation></semantics></math> is such an inverse system, then the localic group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi>lim</mi> <mi>i</mi></msub><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">G=\lim_i G_i</annotation></semantics></math> is presented by the following <a class="existingWikiWord" href="/nlab/show/posite">posite</a>. The elements of the underlying poset are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,i)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x\in G_i</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo><mo>≤</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,i)\le (y,j)</annotation></semantics></math> when <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>f</mi> <mi>ij</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f_{ij}(x)=y</annotation></semantics></math>. The coverings are given as follows: for any <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>, the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,i)</annotation></semantics></math> is covered by the family of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(z,k)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≤</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">k\le j</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≤</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(z,k)\le (x,i)</annotation></semantics></math>.</p> <div class="un_defn"> <h6 id="definition_3">Definition</h6> <p>A <strong>surjective progroup</strong>, also called a <strong>strict progroup</strong>, is a progroup whose cofiltered diagram consists of <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a>s.</p> </div> <p>One can show that a progroup is isomorphic to a surjective one, in the category of pro-groups, if and only if it satisfies the <strong>Mittag-Leffler condition</strong>: for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">G_i</annotation></semantics></math> the images of the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>j</mi></msub><mo>→</mo><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">G_j\to G_i</annotation></semantics></math> are eventually constant.</p> <p>By a fundamental fact about <a class="existingWikiWord" href="/nlab/show/locales">locales</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is prodiscrete and represented as the limit of a system with surjective transition maps, then the legs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">G\to G_i</annotation></semantics></math> of the limiting cone are also surjective (i.e. they are represented by injective <a class="existingWikiWord" href="/nlab/show/frame">frame</a> homomorphisms). This is false for limits of topological spaces.</p> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>The category of prodiscrete localic groups is equivalent to the category of surjective progroups.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>In view of the <a href="#EquivalentCharacterizations">above proposition</a> it suffices to show that for surjective progroups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G_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>H</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H_j)</annotation></semantics></math>, with prodiscrete localic limits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>LocGrp</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo><mo>≅</mo><munder><mi>lim</mi> <mi>j</mi></munder><msub><mo lspace="0em" rspace="thinmathspace">colim</mo> <mi>i</mi></msub><msub><mi>Hom</mi> <mi>Grp</mi></msub><mo stretchy="false">(</mo><msub><mi>G</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>H</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">Hom_{LocGrp}(G,H) \cong \lim_j \colim_i Hom_{Grp}(G_i,H_j).</annotation></semantics></math></div> <p>But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mi>lim</mi> <mi>j</mi></msub><msub><mi>H</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">H = \lim_j H_j</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>LocGrp</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>lim</mi> <mi>j</mi></msub><msub><mi>Hom</mi> <mi>LocGrp</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><msub><mi>H</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{LocGrp}(G,H) \cong \lim_j Hom_{LocGrp}(G,H_j)</annotation></semantics></math>. Thus it suffices to show that any map from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> (such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">H_j</annotation></semantics></math>) factors through some essentially unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">G_i</annotation></semantics></math>.</p> <p>But if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">f\colon G\to K</annotation></semantics></math> is such a map, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(f)</annotation></semantics></math> is an open normal subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. And if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i\colon G\to G_i</annotation></semantics></math> are the projections, then the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(p_i)</annotation></semantics></math> are a neighborhood base at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>, so we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊆</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(p_i)\subseteq ker(f)</annotation></semantics></math> for some <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>, hence <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> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G/ker(p_i)</annotation></semantics></math>. Finally, this last is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">G_i</annotation></semantics></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i\colon G\to G_i</annotation></semantics></math> is an open surjection of locales.</p> </div> <p>Any <a class="existingWikiWord" href="/nlab/show/localic+group">localic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a> consisting of continuous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-sets, i.e. discrete locales with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action. In general, the resulting <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LocGrp</mi><mo>→</mo><mi>Topos</mi></mrow><annotation encoding="application/x-tex"> LocGrp \to Topos </annotation></semantics></math></div> <p>is not an <a class="existingWikiWord" href="/nlab/show/embedding">embedding</a> into <a class="existingWikiWord" href="/nlab/show/Topos">Topos</a>, but it can be shown to be so when restricted to prodiscrete localic groups. One can also characterize the toposes that are sheaves on a prodiscrete localic group as the <a class="existingWikiWord" href="/nlab/show/Galois+topos">Galois toposes</a>.</p> <p>Most of these results have corresponding facts for pro-<a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> and prodiscrete localic groupoids. However, in full generality, the category of (even surjective) pro-groupoids does not embed into localic groupoids, since the category of <a class="existingWikiWord" href="/nlab/show/pro-sets">pro-sets</a> (= categorically discrete pro-groupoids) does not embed into locales (= categorically discrete localic groupoids).</p> <h2 id="references">References</h2> <ul id="Moerdijk"> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Prodiscrete groups and Galois toposes</em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 5, 2011 at 23:01:26. 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