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href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a 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theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#remarks'>Remarks</a></li> <ul> <li><a href='#relationship_to_fundamental_group'>Relationship to fundamental group</a></li> <li><a href='#topologizing_the_fundamental_groupoid'>Topologizing the fundamental groupoid</a></li> <li><a href='#_with_a_chosen_set_of_basepoints'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> with a chosen set of basepoints</a></li> <li><a href='#in_higher_category_theory'>In higher category theory</a></li> <li><a href='#simplicial_version'>Simplicial version</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <strong>fundamental groupoid</strong> of a space <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 <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> whose objects are the points 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> and whose morphisms are <a class="existingWikiWord" href="/nlab/show/paths">paths</a> in <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>, identified up to endpoint-preserving <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>.</p> <p>In parts of the literature the fundamental groupoid, and more generally the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a>, is called the <strong>Poincaré groupoid</strong>.</p> <h2 id="definition">Definition</h2> <p>The <strong>fundamental groupoid</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> of a topological space <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 <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> whose set of objects is <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> and whose morphisms from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are the equivalence classes of homotopy of homotopy relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\partial I</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\gamma]</annotation></semantics></math> of continuous maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma : [0,1] \to X</annotation></semantics></math> whose endpoints map to <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> (which the homotopies are required to fix). Composition is by concatenation (and reparametrization) of representative maps. Under the <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>-<a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> this becomes an associative and unital composition with respect to which every morphism has an inverse; hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> is a groupoid.</p> <p>The use of the fundamental groupoid of a manifold for describing the monodromy principle on the extension of local morphisms is discussed in the paper by Brown/Mucuk listed below.</p> <h2 id="remarks">Remarks</h2> <h3 id="relationship_to_fundamental_group">Relationship to fundamental group</h3> <p>For any <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> in <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> the first homotopy group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</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> based at <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> arises as the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> 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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Aut</mi> <mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_1(X,x) = Aut_{\Pi_1(X)}(x) \,. </annotation></semantics></math></div> <p>So the fundamental groupoid gets rid of the choice of basepoint for the fundamental group, and this is valuable for some applications. The set of connected components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> is precisely the 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> of path-components 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>. (This is not to be confused with the set of connected components 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>, sometimes denoted by the same symbol. Of course they are the same when <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 locally path-connected.)</p> <h3 id="topologizing_the_fundamental_groupoid">Topologizing the fundamental groupoid</h3> <p>The fundamental groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> can be made into a <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a> (i.e. a <a class="existingWikiWord" href="/nlab/show/internal+groupoid">groupoid internal</a> to <a class="existingWikiWord" href="/nlab/show/Top">Top</a>) when <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/path-connected+space">path-connected</a>, <a class="existingWikiWord" href="/nlab/show/locally+path-connected+space">locally path-connected</a> and <a class="existingWikiWord" href="/nlab/show/semi-locally+simply+connected+space">semi-locally simply connected</a>. This is a special case of (<a href="#Browno6">Brown 06, 10.5.8</a>). This construction is closely linked with the construction of a <a class="existingWikiWord" href="/nlab/show/universal+covering+space">universal covering space</a> for a path-connected pointed space. The object space of this groupoid is just the space <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>When <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 not semi-locally simply connected, the set of arrows of the fundamental groupoid inherits the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient topology</a> from the path space such that the fibres of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Mor</mi><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(s,t):Mor(\Pi_1(X)) \to X\times X</annotation></semantics></math> are not discrete, which is an obstruction to the above-mentioned source fibre's being a covering space. However the composition is no longer continuous. When <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 not locally path-connected, <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> also inherits a non-discrete topology (the quotient topology 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> by the relation of path connections).</p> <p>In circumstances like these more sophisticated methods are appropriate, such as <a class="existingWikiWord" href="/nlab/show/shape+theory">shape theory</a>. This is also related to the <a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a>, which is in general a <a class="existingWikiWord" href="/nlab/show/progroup">progroup</a> or a <a class="existingWikiWord" href="/nlab/show/localic+group">localic group</a> rather than an ordinary group.</p> <h3 id="_with_a_chosen_set_of_basepoints"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> with a chosen set of basepoints</h3> <p>An improvement on the fundamental group and the total fundamental groupoid relevant to the <a class="existingWikiWord" href="/nlab/show/van+Kampen+theorem">van Kampen theorem</a> for computing the fundamental group or groupoid is to use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X,A)</annotation></semantics></math>, defined for a set <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> to be the full subgroupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\cap X</annotation></semantics></math>, thus giving a set of base points which can be chosen according to the geometry at hand. Thus 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 the union of two open sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>,</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">U,V</annotation></semantics></math> with intersection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> then we can take <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> large enough to meet each path-component of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>,</mo><mi>V</mi><mo>,</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">U,V,W</annotation></semantics></math>; note that by the above definition we can write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(U,A)</annotation></semantics></math>, etc. 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> has an action of a group <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> then <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> acts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X,A)</annotation></semantics></math> if <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> is a union of orbits of the action. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X,A)</annotation></semantics></math> can represent some symmetry of a given situation.</p> <p>The notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X,A)</annotation></semantics></math> was introduced in 1967 by <a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a> to give a version of the Seifert-van Kampen Theorem which allowed the determination of the fundamental group of a connected space which is the union of connected subspaces with nonconnected intersection, such as the circle, a space which is, after all, THE basic example in topology.</p> <p>Grothendieck writes in his 1984 <span class="newWikiWord">Esquisse d'un Programme<a href="/nlab/new/Esquisse+d%27un+Programme">?</a></span> (English translation):</p> <p>“ ..,people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups `a la van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points,..”.</p> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X,X)</annotation></semantics></math> recovers the full fundamental groupoid, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X,\{x\})</annotation></semantics></math> is simply the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math>.</p> <p>Basically, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X,A)</annotation></semantics></math> allows for the <em>computation of homotopy 1-types</em>; the theory was developed in <em>Elements of Modern Topology</em> (1968), now available as <em>Topology and Groupoids</em> (2006). These accounts show the use of the algebra of groupoids in 1-dimensional <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, for example for <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a>s, and, in the later edition, for <a class="existingWikiWord" href="/nlab/show/orbit+spaces">orbit spaces</a>. spring</p> <h3 id="in_higher_category_theory">In higher category theory</h3> <p>See <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a>.</p> <h3 id="simplicial_version">Simplicial version</h3> <p>See <a class="existingWikiWord" href="/nlab/show/simplicial+fundamental+groupoid">simplicial fundamental groupoid</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+2-groupoid">fundamental 2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a href="smooth+homotopy#HomotopyOfSmoothPathsRelativeToTheirEndpoints">homotopy of smooth paths relative to their endpoints</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+fundamental+groupoid">simplicial fundamental groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+fundamental+groupoid">equivariant fundamental groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <h2 id="references">References</h2> <p>A detailed treatment is available in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, <em>Topologie Algébrique</em>, Chapitres 1 à 4, Springer (1998, 2016) <a href="https://doi.org/10.1007/978-3-662-49361-8">doi:10.1007/978-3-662-49361-8</a>, ISBN 978-3-662-49361-8.</li> </ul> <p>Monograph:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Philip+J.+Higgins">Philip J. Higgins</a>, §6 of: <em>Categories and Groupoids</em>, Mathematical Studies <strong>32</strong>, van Nostrand New York (1971), Reprints in Theory and Applications of Categories <strong>7</strong> (2005) 1-195 [<a href="http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html">tac:tr7</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/7/tr7-2a4.pdf">pdf</a>]</li> </ul> <p>Review and Exposition:</p> <ul> <li id="Moller11"> <p><a class="existingWikiWord" href="/nlab/show/Jesper+M%C3%B8ller">Jesper Møller</a>, <em>The fundamental group and covering spaces</em> (2011) [<a href="https://arxiv.org/abs/1106.5650">arXiv:1106.5650</a>, <a href="http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Moller-FundamentalGroup.pdf" title="pdf">pdf</a>]</p> </li> <li> <p>Alberto Santini, <em>Topological groupoids</em> (2011) [<a href="http://web.math.ku.dk/~moller/students/alberto.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Santini-Groupoids.pdf" title="pdf">pdf</a>]</p> <blockquote> <p>(about <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, notably <a class="existingWikiWord" href="/nlab/show/fundamental+groupoids">fundamental groupoids</a> – not about <a class="existingWikiWord" href="/nlab/show/topological+groupoids">topological groupoids</a>)</p> </blockquote> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Topology – 2</a></em></p> </li> </ul> <p>See also:</p> <ul> <li> <p>R. Brown, Groupoids and Van Kampen’s theorem, <em>Proc. London Math. Soc</em>. (3) 17 (1967) 385-401.</p> </li> <li> <p>R. Brown and G. Danesh-Naruie, The fundamental groupoid as a topological groupoid, <em>Proc. Edinburgh Math. Soc.</em> 19 (1975) 237-244.</p> </li> <li> <p>R. Brown and O. Mucuk, The monodromy groupoid of a Lie groupoid, <em>Cah. Top. G'eom. Diff. Cat</em>. 36 (1995) 345-369.</p> </li> <li id="Brown06"> <p><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>Topology and Groupoids</em>, Booksurge (2006). (See particularly 10.5.8, using lifted topologies to topologise <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">(\pi_1 X)/N</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> is a normal, totally disconnected subgroupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\pi_1 X</annotation></semantics></math>, and <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> admits a universal cover). (<a href="http://pages.bangor.ac.uk/~mas010/topgpds.html">more info</a>)</p> </li> </ul> <p>Discussion from the point of view of <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Luis+Javier+Hern%C3%A1ndez-Paricio">Luis Javier Hernández-Paricio</a>, <em>Fundamental pro-groupoids and covering projections</em>, Fund. Math. (1998), (<a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf">pdf</a>)</p> </li> <li> <p>The use of many base points is discussed at this (<a href="http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one">mathoverflow page</a>).</p> </li> </ul> <p>Discussion of the fundamental groupoid (for good <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> and for <a class="existingWikiWord" href="/nlab/show/noetherian+schemes">noetherian schemes</a>) as the <span class="newWikiWord">costack<a href="/nlab/new/costack">?</a></span> (via the <a class="existingWikiWord" href="/nlab/show/Seifert-van+Kampen+theorem">Seifert-van Kampen theorem</a>) characterized as being 2-<a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> is in</p> <ul> <li id="Pirashvili14a"> <p><a class="existingWikiWord" href="/nlab/show/Ilia+Pirashvili">Ilia Pirashvili</a>, <em>The fundamental groupoid as a terminal costack</em> (<a href="https://arxiv.org/abs/1406.4419">arXiv:1406.4419</a>)</p> </li> <li id="Pirashvili14b"> <p><a class="existingWikiWord" href="/nlab/show/Ilia+Pirashvili">Ilia Pirashvili</a>, <em>The Étale Fundamental Groupoid as a Terminal Costack</em> (<a href="https://arxiv.org/abs/1412.5473">arXiv:1412.5473</a>)</p> </li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/dynamical+systems">dynamical systems</a>:</p> <ul> <li>Emmanuel Paul, Jean-Pierre Ramis, <em>Dynamics on Wild Character Varieties</em>, <em>SIGMA</em> <strong>11</strong> (2015) 068 [<a href="https://arxiv.org/abs/1508.03122">arXiv:1508.03122</a>, <a href="https://doi.org/10.3842/SIGMA.2015.068">doi:10.3842/SIGMA.2015.068</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 18, 2023 at 05:19:02. 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