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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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="representation_theory">Representation theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> <h2 id="geometric_representation_theory">Geometric representation theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Beilinson-Bernstein+localization">Beilinson-Bernstein localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="bundles">Bundles</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/bundles">bundles</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> </ul> <h2 id="sidebar_context">Context</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">linear</a>) <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> </ul> <h2 id="sidebar_classes_of_bundles">Classes of bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/numerable+bundle">numerable bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+bundle">sphere bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex</a>/<a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">with connection</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+line+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/cubical+structure+on+a+line+bundle">cubical structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a><a class="existingWikiWord" href="/nlab/show/Vect%28X%29">of vector bundles</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product</a>, <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a>, <a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+of+spectra">bundle of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+bundle">natural bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a></p> </li> </ul> <h2 id="sidebar_universal_bundles">Universal bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal complex line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> </ul> <h2 id="sidebar_presentations">Presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+bundle">empty bundle</a>, <a class="existingWikiWord" href="/nlab/show/zero+bundle">zero bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+line+bundle+on+the+2-sphere">basic line bundle on the 2-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a></li> </ul> </div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#in_the_category_of_sets'>In the category of sets</a></li> <ul> <li><a href='#twosorted_definition'>Two-sorted definition</a></li> <li><a href='#singlesorted_definition'>Single-sorted definition</a></li> <li><a href='#TrivialisationInSets'>Trivialisation</a></li> <li><a href='#functoriality_change_of_structure_group'>Functoriality (change of structure group)</a></li> </ul> <li><a href='#in_general'>In general</a></li> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#in_topological_spaces'>In topological spaces</a></li> <li><a href='#in_sheaves'>In sheaves</a></li> <li><a href='#GroupExtensions'>Group extensions</a></li> </ul> <li><a href='#LocalTrivialization'>Local trivialization</a></li> </ul> <li><a href='#generalizations'>Generalizations</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>torsor</em> (in the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a>) is, roughly speaking, a <a class="existingWikiWord" href="/nlab/show/group">group</a> that has forgotten its identity element; given any (non-empty) torsor with respect to 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>, we recover a group isomorphic to <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> by making what is known as a <em>trivialisation</em> of the torsor, which roughly corresponds to choosing an identity element. That we wish to keep track of the choice is precisely the reason for working with torsors.</p> <p>Something analogous is present in the theory of <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">fibrations</a>, where it can be important to make a choice of lifts (‘cloven fibrations’).</p> <p>The notion of a torsor can be <a class="existingWikiWord" href="/nlab/show/internalisation">internalised</a> to any category with products, and more generally to any category in which the notion of an internal group can be made good sense of. We discuss this general notion below, after first discussing the notion in the category of sets.</p> <h2 id="in_the_category_of_sets">In the category of sets</h2> <h3 id="twosorted_definition">Two-sorted definition</h3> <p>Given 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>, 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>-torsor is often defined to be a <a class="existingWikiWord" href="/nlab/show/free+action">free</a> and <a class="existingWikiWord" href="/nlab/show/transitive+action">transitive action</a> 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> on an <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>. However, there are advantages to giving the definition in a purely equational form (as in an <a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a>), which we can come close to achieving as follows:</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/group">group</a>. A <em><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>-torsor</em> is an <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> together with an <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>T</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">a: G \times T \rightarrow T</annotation></semantics></math> 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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/shear+map">shear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>×</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>:</mo><mi>G</mi><mo>×</mo><mi>T</mi><mo>→</mo><mi>T</mi><mo>×</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">a \times p_2: G \times T \rightarrow T \times T</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> is the canonical <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>×</mo><mi>T</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">G \times T \rightarrow T</annotation></semantics></math>.</p> </div> </p> <p>Since we can state that the shear map is an isomorphism by positing an inverse map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>T</mi><mo>×</mo><mi>T</mi><mo>→</mo><mi>T</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">f: T \times T \to T \times G</annotation></semantics></math> and giving equations that state it is an inverse, the only non-equational part of the definition is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> be inhabited. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is not required to be <a class="existingWikiWord" href="/nlab/show/inhabited">inhabited</a> (that is, possibly <a class="existingWikiWord" href="/nlab/show/empty+set">empty</a>), we call it a <em><a class="existingWikiWord" href="/nlab/show/pseudo-torsor">pseudo-torsor</a></em>.</p> <p>Asking that the <a class="existingWikiWord" href="/nlab/show/shear+map">shear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>×</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a \times p_2</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> is the same as to say that the action is both <a class="existingWikiWord" href="/nlab/show/free+action">free</a> and <a class="existingWikiWord" href="/nlab/show/transitive+action">transitive</a>, hence <a class="existingWikiWord" href="/nlab/show/regular+action">regular</a> if <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a>: free-ness corresponds to <a class="existingWikiWord" href="/nlab/show/injectivity">injectivity</a> of the <a class="existingWikiWord" href="/nlab/show/shear+map">shear map</a>, and transitivity corresponds to <a class="existingWikiWord" href="/nlab/show/surjectivity">surjectivity</a> of the <a class="existingWikiWord" href="/nlab/show/shear+map">shear map</a>.</p> <p> <div class='num_remark' id='RemarkTorsorIsomorphicToStructureGroup'> <h6>Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is non-empty, we shall prove <a href="#TrivialisationInSets">below</a> that it follows from the definition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is isomorphic to the underlying set 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>. There are many such isomorphisms, and where torsors are used it is often important to choose/fix one. Such a choice is known as a <em>trivialisation</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>. See <a href="#TrivialisationInSets">below</a> for more details.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>As a consequence of Remark <a class="maruku-ref" href="#RemarkTorsorIsomorphicToStructureGroup"></a>, a torsor with respect to some group can be thought of as a <a class="existingWikiWord" href="/nlab/show/heap">heap</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>An <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a> of dimension <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> over a <a class="existingWikiWord" href="/nlab/show/field">field</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> is a torsor for the additive group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>: this acts by <em>translation</em>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/unit+of+measurement">unit of measurement</a> is (typically) an element in an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^\times</annotation></semantics></math>-torsor, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^\times</annotation></semantics></math> the multiplicative group of non-zero <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>s: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> any unit and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">r \in \mathbb{R}</annotation></semantics></math> any non-vanishing real number, also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">r u</annotation></semantics></math> is a unit. And for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">u_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">u_2</annotation></semantics></math> two units, one is expressed in terms of the other by a unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r \neq 0</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mn>1</mn></msub><mo>=</mo><mi>r</mi><msub><mi>u</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">u_1 = r u_2</annotation></semantics></math>. For instance for units of <a class="existingWikiWord" href="/nlab/show/mass">mass</a> we have the unit of <a class="existingWikiWord" href="/nlab/show/kilogram">kilogram</a> and that of gram and there is a unique number, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1000</mn></mrow><annotation encoding="application/x-tex">r = 1000</annotation></semantics></math> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>kg</mi><mo>=</mo><mn>1000</mn><mi>g</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> kg = 1000 g. </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_remark' id='ExampleTorsorFromAGroup'> <h6>Example</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/group">group</a>. The action 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> on itself equips the underlying set 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> with the structure of 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>-torsor.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>We shall see in Remark <a class="maruku-ref" href="#RemarkGroupStructureFromTrivialisation"></a> that all torsors actually arise as in Example <a class="maruku-ref" href="#ExampleTorsorFromAGroup"></a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>Given two isomorphic objects <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> in any category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, all isomorphisms between <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> form a torsor (both for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(X)</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(Y)</annotation></semantics></math>, which are mutually isomorphic but not canonically). This is an insight used in (M. Kontsevich, <em>Operads and motives in deformation quantization</em>, Lett.Math.Phys.48:35-72 (1999) arXiv:<a href="https://arxiv.org/abs/math/9904055">math/9904055</a> <a href="https://doi.org/10.1023/A:1007555725247">doi</a>) explaining period matrices from the point of view of a coordinate ring of an affine torsor.</p> </div> </p> <h3 id="singlesorted_definition">Single-sorted definition</h3> <p>It is possible to define torsors using a single-sorted <a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a>. This is entirely analogous to how <a class="existingWikiWord" href="/nlab/show/affine+spaces">affine spaces</a> can be defined either as sets with a free and transitive action of a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, or, equivalently, as sets equipped with operations that take arbitrary affine combinations with coefficients in a given ring.</p> <p>More precisely, a <strong>torsor</strong> (also known as a <a class="existingWikiWord" href="/nlab/show/heap">heap</a> when stated in a single-sorted form) is a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> equipped with a ternary operation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo lspace="verythinmathspace">:</mo><msup><mi>T</mi> <mn>3</mn></msup><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t\colon T^3 \to T</annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo>,</mo><mspace width="2em"></mspace><mi>t</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">t(a,a,b)=t(b,a,a)=b,\qquad t(t(a,b,c),d,e)=t(a,b,t(c,d,e)).</annotation></semantics></math></div> <p>A <strong>homomorphism of torsors</strong> is a map of sets that preserves this operation.</p> <p>The equivalence with the two-sorted definition is demonstrated as follows.</p> <p>Given 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>-torsor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, we send it to the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> equipped with the ternary operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">t(a,b,c)=g(a,b)c</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(a,b)</annotation></semantics></math> is the unique element 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mi>b</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">g(a,b)b=a</annotation></semantics></math>.</p> <p>Given a torsor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T,t)</annotation></semantics></math>, we send it to the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LTrans</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(LTrans(T),T)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LTrans</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">LTrans(T)</annotation></semantics></math> is a subgroup of the group of bijections on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> comprising precisely the bijections of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>↦</mo><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c\mapsto t(a,b,c)</annotation></semantics></math> for some elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">a,b\in T</annotation></semantics></math>. The 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> acts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> by evaluation: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gt</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">gt=g(t)</annotation></semantics></math>.</p> <p>Mapping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>LTrans</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T,t)\mapsto (LTrans(T),T)\mapsto (T,t)</annotation></semantics></math> gives back the same torsor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T,t)</annotation></semantics></math> that we started from.</p> <p>Mapping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>LTrans</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,T)\mapsto (T,t)\mapsto (LTrans(T),t)</annotation></semantics></math> produces a torsor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LTrans</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(LTrans(T),t)</annotation></semantics></math> that is naturally isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,T)</annotation></semantics></math> via the isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>LTrans</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="2em"></mspace><mi>g</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>t</mi><mo>↦</mo><mi>gt</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="2em"></mspace><mi>t</mi><mo>↦</mo><mi>t</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">(G,T)\to(LTrans(T),T),\qquad g\mapsto (t\mapsto gt),\qquad t\mapsto t.</annotation></semantics></math></div> <h3 id="TrivialisationInSets">Trivialisation</h3> <p> <div class='num_prop' id='PropositionTorsorIsomorphicAsSetToStructureGroup'> <h6>Proposition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/group">group</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>T</mi><mo>̲</mo></munder><mo>=</mo><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>T</mi><mo>→</mo><mi>T</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\underline{T} = \left( T, a: G \times T \rightarrow T \right)</annotation></semantics></math> be 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>-torsor. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is non-empty, it is isomorphic to the underlying set 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>.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/element">element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>. 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fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Ix7Wp-nP28oigEJfryYsqjYAgds=-glyph-2-1" x="76.734" y="89.661"></use> <use xlink:href="#Ix7Wp-nP28oigEJfryYsqjYAgds=-glyph-2-2" x="80.337481" y="89.661"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Ix7Wp-nP28oigEJfryYsqjYAgds=-glyph-3-1" x="85.784" y="89.661"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#Ix7Wp-nP28oigEJfryYsqjYAgds=-glyph-2-3" x="94.01775" y="89.661"></use> </g> </svg> </div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>T</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{T}</annotation></semantics></math> is 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>-torsor, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>×</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a \times p_2</annotation></semantics></math> is an isomorphism. The proposition thus follows immediately from the fact that pullbacks of isomorphisms are isomorphisms (as proven at <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>).</p> </div> </p> <p>In fact we can improve the above result to show that any <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>-torsor is isomorphic, not only as a set, but as a left <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>-set, to <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> itself. Thus every <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>-torsor has a trivialization:</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/group">group</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>T</mi><mo>̲</mo></munder><mo>=</mo><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>T</mi><mo>→</mo><mi>T</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\underline{T} = \left( T, a: G \times T \rightarrow T \right)</annotation></semantics></math> be 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>-torsor. A <em>trivialisation</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>T</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{T}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> 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>-torsors between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> and the underlying <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>-torsor 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>.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>The proof of Proposition <a class="maruku-ref" href="#PropositionTorsorIsomorphicAsSetToStructureGroup"></a> lets us see that any choice of element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> gives rise to a trivialisation.</p> <p>Note also that a trivialization equips <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> with a <em>choice</em> of group structure making it isomorphic to <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>, since we can use it to transfer the group structure 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>.</p> </div> </p> <h3 id="functoriality_change_of_structure_group">Functoriality (change of structure group)</h3> <p>Torsors can be transported, or in other words pushed forward, along group homomorphisms, as we shall now show. (See also at <em><a href="topological+G-space#ChangeOfGroupsAndFixedLoci">G-space – change of structure group</a></em>).</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">h : G_1 \rightarrow G_2</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/homomorphism">group homomorphism</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>T</mi><mo>̲</mo></munder><mo>=</mo><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\underline{T} = \left( T, a \right)</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math>-torsor. Observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>↦</mo><mrow><mo>(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>⋅</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>a</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(g_1, g_2, t \right) \mapsto \left( g_2 \cdot h(g_1)^{-1}, a(g_1, t) \right)</annotation></semantics></math> defines an action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>→</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">G_1 \times G_2 \times T \rightarrow G_2 \times T</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">G_2 \times T</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\left(G_2 \times T \right) / G_1</annotation></semantics></math>, the quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">G_2 \times X</annotation></semantics></math> with respect to this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math>-action, defines a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-torsor with respect to the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> induced by left multiplication, namely that given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mrow><mo>[</mo><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo stretchy="false">)</mo><mo>]</mo></mrow><mo>)</mo></mrow><mo>↦</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex">\left( g, \left[ (g', t') \right] \right) \mapsto \left[ \left(g g', t' \right) \right]</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mi>p</mi><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[ p \right]</annotation></semantics></math>, for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">p \in G_2 \times T</annotation></semantics></math>, denotes the <a class="existingWikiWord" href="/nlab/show/orbit">orbit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> with respect to the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">G_2 \times T</annotation></semantics></math>.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>We shall demonstrate that the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo>×</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>→</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">i : \left( G_2 \times T \right) \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow G_2 \times \left( \left( G_2 \times T \right) / G_1 \right)</annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mrow><mo>[</mo><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo stretchy="false">)</mo><mo>]</mo></mrow><mo>)</mo></mrow><mo>↦</mo><mrow><mo>(</mo><mi>g</mi><msup><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mrow><mo>[</mo><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo stretchy="false">)</mo><mo>]</mo></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left( (g, t), \left[ (g', t') \right] \right) \mapsto \left( g\left(g'\right)^{-1}, \left[ (g', t') \right] \right)</annotation></semantics></math></div> <p>induces a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo>¯</mo></mover><mo>:</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>×</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>→</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">\overline{i} : \left( \left( G_2 \times T \right) / G_1 \right) \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow G_2 \times \left( \left( G_2 \times T \right) / G_1 \right).</annotation></semantics></math></div> <p>That is, it respects, in its left factor, the passage to the quotient by the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math>.</p> <p>Suppose indeed that we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">(g, t) \in G_2 \times T</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><mi>g</mi><mo>″</mo><mo>,</mo><mi>t</mi><mo>″</mo><mo>)</mo></mrow><mo>∈</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\left( g_1, g'', t'' \right) \in G_1 \times G_2 \times T</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mi>g</mi><mo>″</mo><mo>⋅</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>a</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><mi>t</mi><mo>″</mo><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">(g, t) = \left( g'' \cdot h(g_1)^{-1}, a(g_1, t'') \right)</annotation></semantics></math>. We make the following observations.</p> <ol> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>″</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>g</mi><mo>″</mo><msup><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g'' h(g_1)^{-1} \left(g'\right) ^{-1} = g'' \left( g' h(g_1) \right)^{-1}</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo stretchy="false">)</mo><mo>]</mo></mrow><mo>=</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mi>h</mi><msup><mrow><mo>(</mo><msubsup><mi>g</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>a</mi><mo stretchy="false">(</mo><msubsup><mi>g</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mi>t</mi><mo>′</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>=</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>a</mi><mrow><mo>(</mo><msubsup><mi>g</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>)</mo></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[ (g', t') \right] = \left[ \left( g' h\left(g_1^{-1}\right)^{-1}, a(g_1^{-1}, t') \right) \right] = \left[ \left( g' h(g_1), a\left(g_1^{-1}, t'\right) \right) \right]</annotation></semantics></math></li> </ol> <p>Putting these together, we obtain that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>i</mi><mrow><mo>(</mo><mrow><mo>(</mo><mi>g</mi><mo>″</mo><mo>,</mo><mi>t</mi><mo>″</mo><mo>)</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><mi>i</mi><mrow><mo>(</mo><mrow><mo>(</mo><mi>g</mi><mo>″</mo><mo>,</mo><mi>t</mi><mo>″</mo><mo>)</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>a</mi><mrow><mo>(</mo><msubsup><mi>g</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mi>g</mi><mo>″</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>a</mi><mrow><mo>(</mo><msubsup><mi>g</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mi>g</mi><msup><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>i</mi><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} i\left( \left( g'', t'' \right), \left[ \left( g', t' \right) \right] \right) &= i\left( \left( g'', t'' \right), \left[ \left( g' h(g_1), a\left( g_1^{-1}, t' \right) \right) \right] \right) \\ &= \left( g'' h(g_1)^{-1} \left(g' \right)^{-1}, \left[ \left( g' h(g_1), a\left( g_1^{-1}, t' \right) \right) \right] \right) \\ &= \left( g \left(g' \right)^{-1}, \left[ \left( g', t' \right) \right] \right) \\ &= i\left( g, \left[ \left( g', t' \right) \right] \right), \end{aligned} </annotation></semantics></math></div> <p>as required.</p> <p>We now observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{i}</annotation></semantics></math> defines an inverse to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>′</mo><mo>×</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a' \times p_{2}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">a'</annotation></semantics></math> is the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\left( G_2 \times T \right) / G_1</annotation></semantics></math> induced by left multiplication, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> is the projection map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>→</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">G_2 \times \left( \left( G_2 \times T \right) / G_1 \right) \rightarrow \left( \left( G_2 \times T \right) / G_1 \right)</annotation></semantics></math>. In one direction, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo>∈</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">g, g' \in G_2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t \in T</annotation></semantics></math>, we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo>¯</mo></mover><mrow><mo>(</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mi>g</mi><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex">\overline{i}\left( \left[ \left(g' g, t \right) \right], \left[ \left( g, t \right) \right] \right) = \left( g' g g^{-1}, \left[ \left( g, t \right) \right] \right) = \left( g', \left[ \left( g, t \right) \right] \right),</annotation></semantics></math></div> <p>as required. In the other direction, suppose that we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>∈</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>×</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>G</mi> <mn>2</mn></msub><mo>×</mo><mi>T</mi><mo>)</mo></mrow><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right) \in \left( \left( G_2 \times T \right) / G_1 \right) \times \left( \left( G_2 \times T \right) / G_1 \right)</annotation></semantics></math>. Since the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>×</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>:</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>×</mo><mi>T</mi><mo>→</mo><mi>T</mi><mo>×</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">a \times p_2 : G_1 \times T \rightarrow T \times T</annotation></semantics></math> is an isomorphism, there is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">g_1 \in G_1</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>′</mo><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t' = a(g_1, t)</annotation></semantics></math>, and we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>=</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[ \left(g, t \right) \right] = \left[ \left( g h(g_1)^{-1}, t' \right) \right]</annotation></semantics></math>. We then have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>′</mo><mo>×</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a' \times p_2</annotation></semantics></math> applied to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo>¯</mo></mover><mrow><mo>(</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>=</mo><mover><mi>i</mi><mo>¯</mo></mover><mrow><mo>(</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi>g</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\overline{i}\left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right) = \overline{i}\left( \left[ \left(g h(g_1)^{-1}, t' \right) \right], \left[ \left( g', t' \right) \right] \right) = \left( g h(g_1)^{-1} \left(g'\right)^{-1}, \left[ \left( g', t' \right) \right] \right)</annotation></semantics></math></div> <p>is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>)</mo></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>t</mi><mo>′</mo><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex">\left( \left[ \left(g h(g_1)^{-1} \left( g' \right)^{-1} g', t' \right) \right], \left[ \left( g', t' \right) \right] \right) = \left( \left[ \left(g h(g_1)^{-1}, t' \right) \right], \left[ \left( g', t' \right) \right] \right) = \left( \left[ \left(g, t \right) \right], \left[ \left( g', t' \right) \right] \right),</annotation></semantics></math></div> <p>as required.</p> <p></p> </div> </p> <p> <div class='num_defn'> <h6>Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">h : G_1 \rightarrow G_2</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/homomorphism">group homomorphism</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>T</mi><mo>̲</mo></munder><mo>=</mo><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\underline{T} = \left( T, a \right)</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_1</annotation></semantics></math>-torsor. We refer to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-torsor constructed from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>T</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{T}</annotation></semantics></math> using <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> as in Proposition <a class="maruku-ref" href="#PropositionTorsorChangeOfStructureGroup"></a> as the torsor obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>T</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{T}</annotation></semantics></math> by <em>change of structure group</em>, and denote it <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mo>*</mo></msub><mrow><mo>(</mo><munder><mi>T</mi><mo>̲</mo></munder><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">h_{*}\left(\underline{T}\right)</annotation></semantics></math>.</p> </div> </p> <h2 id="in_general">In general</h2> <p>More generally, one may consider torsors over some base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> (in other words, working in the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> of sheaves over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>). In this case the term <strong><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>-torsor</strong> is often used more or less a synonym for the term <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>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, but torsors are generally understood in contexts much wider than the term “principal bundle” is usually taken to apply. And a principal bundle is strictly speaking a torsor that is required to be <em>locally trivial</em> . Thus, while the terminology ‘principal bundle’ is usually used in the setting of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> or <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s, the term <em>torsor</em> is traditionally used in the more general contex of <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topologies</a> (faithfully flat and étale topology in particular), <a class="existingWikiWord" href="/nlab/show/topos">topoi</a> and for generalizations in various category-theoretic setups. While in the phrase ‘<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>-principal bundle’ <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 usually a (topological) <a class="existingWikiWord" href="/nlab/show/group">group</a> or <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, when we say ‘<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>-torsor’, <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 usually a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> or <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of group(oid)s, or <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 plain <a class="existingWikiWord" href="/nlab/show/category">category</a> (not necessarily even a groupoid).</p> <p>A <strong><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>-torsor</strong>, without any base space given, can also simply be an inhabited transitive free <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>-<a class="existingWikiWord" href="/nlab/show/action">set</a>, which is the same as a principal <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>-bundle over the <a class="existingWikiWord" href="/nlab/show/point">point</a>. The notion may also be defined in any category with products: a torsor over a <a class="existingWikiWord" href="/nlab/show/group+object">group object</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> is a <a class="existingWikiWord" href="/nlab/show/well-supported+object">well-supported object</a> <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> together 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>E</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\alpha: G \times E \to E</annotation></semantics></math> such that the arrow</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>π</mi> <mn>1</mn></msub><mo>,</mo><mi>α</mi><mo stretchy="false">⟩</mo><mo>:</mo><mi>G</mi><mo>×</mo><mi>E</mi><mo>→</mo><mi>E</mi><mo>×</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\langle \pi_1, \alpha \rangle: G \times E \to E \times E</annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <h3 id="definition">Definition</h3> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/group">group</a> object in some <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, that in the following is assumed, for simplicity, to be a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>. The <a class="existingWikiWord" href="/nlab/show/object">object</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> we sometimes call <a class="existingWikiWord" href="/nlab/show/space">space</a>s. Examples to keep in mind are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> (in which case <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 an ordinary <a class="existingWikiWord" href="/nlab/show/group">group</a>) or <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (in which case it is a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>) or <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> (in which case it is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>).</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A left <strong><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>-torsor</strong> is an <a class="existingWikiWord" href="/nlab/show/inhabited+object">inhabited object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> equipped 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>-<a class="existingWikiWord" href="/nlab/show/action">action</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>P</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\rho: G \times P \to P</annotation></semantics></math> (subject to the usual laws for actions) such that the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>ρ</mi><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo>:</mo><mi>G</mi><mo>×</mo><mi>P</mi><mo>→</mo><mi>P</mi><mo>×</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\langle \rho, \pi_2 \rangle: G \times P \to P \times P</annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> </p> <p>More generally, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> be an object. Then the <a class="existingWikiWord" href="/nlab/show/slice+category">slice</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">C/B</annotation></semantics></math> is finitely complete, and the pullback functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>×</mo><mi>B</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">- \times B: C \to C/B</annotation></semantics></math> preserves finite limits. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo>:</mo><mi>G</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\pi_2: G \times B \to B</annotation></semantics></math> acquires a group structure in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">C/B</annotation></semantics></math>.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A left <strong><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>-torsor over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></strong> is 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>-torsor in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">C/B</annotation></semantics></math>.</p> </div> </p> <p>Thus, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">B = 1</annotation></semantics></math> is a point, a torsor over a point is the same as an ordinary torsor in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, but sometimes the additional “over a point” is convenient for the sake of emphasis.</p> <p>We restate this definition equivalently in more nuts-and-bolts terms. The ambient category is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, as before.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A left <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>-torsor over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">B \in C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mover><mo>→</mo><mi>π</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">P\stackrel{\pi}{\to} B</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> together with a left group <a class="existingWikiWord" href="/nlab/show/action">action</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>G</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>P</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex"> \rho : G\times_B P \to P </annotation></semantics></math></div> <p>which in terms of <a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a>s we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>→</mo><mi>g</mi><mo>.</mo><mi>p</mi></mrow><annotation encoding="application/x-tex"> (g,p)\to g.p </annotation></semantics></math></div> <p>such that the induced morphism of <a class="existingWikiWord" href="/nlab/show/product">product</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mi>G</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>P</mi><mo>→</mo><mi>P</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>P</mi></mrow><annotation encoding="application/x-tex"> \phi := (\rho, p_2) : G\times_B P \to P\times_B P </annotation></semantics></math></div> <p>which on elements acts as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>g</mi><mo>.</mo><mi>p</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (g,p)\to (g.p,p) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>As we explain <a href="#LocalTrivialization">below</a>, a torsor is in some tautological sense <strong>locally trivial</strong>, but some care must be taken in interpreting this. One sense is that there is a cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> (so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">U \to 1</annotation></semantics></math> is epi, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is inhabited) such that the torsor, when pulled back to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, becomes trivial (i.e., isomorphic to <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> as <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>-torsor). But this is a very general notion of “cover”. A more restrictive sense frequently encountered in the literature is that “cover” means a coproduct of subterminal objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">U_i \hookrightarrow 1</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U = \sum_i U_i</annotation></semantics></math> is inhabited (e.g., an open cover of a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> seen as the terminal object of the sheaf topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(B)</annotation></semantics></math>), and “torsor” would then refer to the local triviality condition for some such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. This is the more usual sense when referring to principal bundles as torsors. Or, “cover” could refer to a covering sieve in a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>.</p> <p>(The condition on the action can be translated to give transitivity etc. in the case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a point (left as a standard exercise).)</p> <p></p> </div> </p> <h3 id="examples">Examples</h3> <h4 id="in_topological_spaces">In topological spaces</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, so that all objects are <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s and groups <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> are <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>s.</p> <p>A topological <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>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\pi: P \to B</annotation></semantics></math> is an example of a torsor over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>. This becomes a definition of principal bundle if we demand local triviality with respect to some open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> (see the remarks <a href="#LocalTrivialization">below</a>).</p> <h4 id="in_sheaves">In sheaves</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = Sh(S)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> over a <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> <p>The canonical example for a torsor in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/trivial+torsor">trivial torsor</a> over a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of groups, <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>.</p> <h4 id="GroupExtensions">Group extensions</h4> <p>Every <a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A \to \hat G \to G</annotation></semantics></math> canonically equips <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> with the structure of an <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>-torsor over <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>. See <a href="http://ncatlab.org/nlab/show/group+extension#Torsors">Group extensions as torsors</a> for details</p> <h3 id="LocalTrivialization">Local trivialization</h3> <p>In other categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> besides <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>, we cannot just “pick a point” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> even if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P \to 1</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>, so this argument cannot be carried out, and indeed trivializations may not exist. However, it is possible to construct a local trivialization of a torsor, following a general philosophy from <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> that a statement is “locally true” in a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> if it becomes true when reinterpreted in a slice after pulling back <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>C</mi><mo stretchy="false">/</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">C \to C/U</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is inhabited. (This in some sense is the basis of <a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a>.)</p> <p>In the present case, we may take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>=</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">U = P</annotation></semantics></math>. Although we cannot “pick a point” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> (= global section of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P \to 1</annotation></semantics></math>), we can pick a point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> if we reinterpret it by pulling back to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">C/P</annotation></semantics></math>. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo>:</mo><mi>P</mi><mo>×</mo><mi>P</mi><mo>→</mo><mn>1</mn><mo>×</mo><mi>P</mi><mo>≅</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\pi_2: P \times P \to 1 \times P \cong P</annotation></semantics></math> does have a global section regarded as an arrow in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">C/P</annotation></semantics></math>. In fact, there is a “generic point”: the diagonal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>P</mi><mo>×</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\Delta: P \to P \times P</annotation></semantics></math>. Then, we may mimic the argument above, and consider the pullback diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi><mo>×</mo><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi><mo>×</mo><mi>P</mi><mo>×</mo><mi>P</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mrow><mo stretchy="false">⟨</mo><mi>ρ</mi><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo>×</mo><mi>id</mi></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>P</mi></mtd> <mtd><munder><mo>→</mo><mrow><mi>id</mi><mo>×</mo><mi>Δ</mi></mrow></munder></mtd> <mtd><mi>P</mi><mo>×</mo><mi>P</mi><mo>×</mo><mi>P</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ G \times P & \to & G \times P \times P \\ \downarrow & & \downarrow \mathrlap{\langle \rho, \pi_2 \rangle \times id} \\ P \times P & \underset{id \times \Delta}{\to} & P \times P \times P } </annotation></semantics></math></div> <p>living in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">C/P</annotation></semantics></math>. As argued above, the vertical arrow on the left is an isomorphism; in fact, it is the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>ρ</mi><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo>:</mo><mi>G</mi><mo>×</mo><mi>P</mi><mo>→</mo><mi>P</mi><mo>×</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\langle \rho, \pi_2 \rangle: G \times P \to P \times P</annotation></semantics></math> we started with!</p> <p>Thus, 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>-torsor in a category with products can be tautologically interpreted in terms 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>-actions on objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> which become trivialized upon pulling back to the slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">C/P</annotation></semantics></math>.</p> <h2 id="generalizations">Generalizations</h2> <ul> <li> <p>Instead of a torsor over a group, one can consider a torsor over a <a class="existingWikiWord" href="/nlab/show/category">category</a>. See <a class="existingWikiWord" href="/nlab/show/torsor+with+structure+category">torsor with structure category</a>.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a>, faithfully flat <a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a>s are considered a generalization of (affine) torsors in algebraic geometry.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> / <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> / <a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a> / <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a> / <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> / <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+along+a+torsor">descent along a torsor</a>, <a class="existingWikiWord" href="/nlab/show/Schneider%27s+descent+theorem">Schneider's descent theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a>, <a class="existingWikiWord" href="/nlab/show/quantum+homogeneous+space">quantum homogeneous space</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+principal+bundle">noncommutative principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/quantum+heap">quantum heap</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/physical+unit">physical unit</a></p> </li> </ul> <h2 id="references">References</h2> <p>Elementary exposition:</p> <ul> <li id="Baez"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>Torsors made easy</em>, (<a href="http://math.ucr.edu/home/baez/torsors.html">web</a>)</p> <blockquote> <p>(discussion for <a class="existingWikiWord" href="/nlab/show/discrete+groups">discrete groups</a>)</p> </blockquote> </li> </ul> <p>Textbook accounts:</p> <ul> <li id="Milne80"> <p><a class="existingWikiWord" href="/nlab/show/James+Milne">James Milne</a>, Prop. III.4.1 in: <em><a class="existingWikiWord" href="/nlab/show/%C3%89tale+Cohomology">Étale Cohomology</a></em>, Princeton Mathematical Series <strong>33</strong> (1980) [<a href="https://www.jstor.org/stable/j.ctt1bpmbk1">jstor:j.ctt1bpmbk1</a>, <a href="https://press.princeton.edu/books/hardcover/9780691082387/etale-cohomology-pms-33-volume-33">ISBN:9780691082387</a>]</p> <blockquote> <p>(discussion for <a class="existingWikiWord" href="/nlab/show/algebraic+groups">algebraic groups</a> in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>)</p> </blockquote> </li> </ul> <p>For more see the references at <em><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a></em> (which are torsors in the generality <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a>).</p> <p>A general <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theoretic</a> account is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, Section B3.2 of: <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a></em> .</li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></em>.</p> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>/<a class="existingWikiWord" href="/nlab/show/univalent+mathematics">univalent mathematics</a>:</p> <ul> <li id="SymmetryBook"> <p><a class="existingWikiWord" href="/nlab/show/Marc+Bezem">Marc Bezem</a>, <a class="existingWikiWord" href="/nlab/show/Ulrik+Buchholtz">Ulrik Buchholtz</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Cagne">Pierre Cagne</a>, <a class="existingWikiWord" href="/nlab/show/Bj%C3%B8rn+Ian+Dundas">Bjørn Ian Dundas</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+R.+Grayson">Daniel R. Grayson</a>, Def. 4.8.1 in: <em><a class="existingWikiWord" href="/nlab/show/Symmetry">Symmetry</a></em> [<a href="https://unimath.github.io/SymmetryBook/book.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+W%C3%A4rn">David Wärn</a>, <em>Eilenberg-MacLane spaces and stabilisation in homotopy type theory</em> [<a href="https://arxiv.org/abs/2301.03685">arXiv:2301.03685</a>]</p> </li> </ul> <p>Some further <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> articles discussing torsors:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tomasz+Brzezi%C5%84ski">Tomasz Brzeziński</a>, <em>On synthetic interpretation of quantum principal bundles</em>, AJSE D - Mathematics 35(1D): 13-27, 2010 <a href="http://arxiv.org/abs/0912.0213">arXiv:0912.0213</a></p> </li> <li> <p>D. H. Van Osdol, <em>Principal homogeneous objects as representable functors</em>, Cahiers Topologie Géom. Différentielle 18 (1977), no. 3, 271–289, <a href="http://www.numdam.org/item?id=CTGDC_1977__18_3_271_0">numdam</a></p> </li> <li> <p>K. T. S. Mohapeloa, <em>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-colimit characterization of internal categories of torsors</em>, J. Pure Appl. Algebra 71 (1991), no. 1, 75–91, <a href="http://dx.doi.org/10.1016/0022-4049%2891%2990041-Y">doi</a></p> </li> <li> <p>Thomas Booker, Ross Street, <em>Torsors, herds and flocks</em> (<a href="http://arxiv.org/abs/0912.4551">arXiv:0912.4551</a>)</p> </li> <li> <p>J. Duskin, <em>Simplicial methods and the interpretation of ‘triple’ cohomology</em>, Memoirs AMS <strong>3</strong>, issue 2, n° 163, 1975. MR393196</p> </li> <li> <p>A. Vistoli, <em>Grothendieck topologies, fibered categories and descent theory</em>, in: <a class="existingWikiWord" href="/nlab/show/FGA+explained">FGA explained</a>, 1–104, Math. Surveys Monogr., 123, AMS 2005, <a href="http://front.math.ucdavis.edu/0412.5512">math.AG/0412512</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Introduction to the language of stacks and gerbes</em>, <a href="http://arxiv.org/abs/math/0212266">math.AT/0212266</a>.</p> </li> </ul> <p>Much further material is also in Giraud’s book on nonabelian cohomology.</p> <p>In a <a class="existingWikiWord" href="/nlab/show/model+theory">model theoretic</a> context of <a class="existingWikiWord" href="/nlab/show/definable+set">definable set</a>s, principal homogeneous spaces are studied in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anand+Pillay">Anand Pillay</a>, <em>Remarks on Galois cohomology and definability</em>, The Journal of Symbolic Logic <strong>62</strong>:2 (1997) 487-492 <a href="https://doi.org/10.2307/2275542">doi</a></li> </ul> <p>See also</p> <ul> <li>MathOverflow, <em><a href="http://mathoverflow.net/questions/25863/torsors-for-monoids/25886">torsors-for-monoids</a></em>, <a href="https://mathoverflow.net/questions/46678/torsors-for-finite-group-schemes">torsors-for-finite-group-schemes</a></li> </ul> <p>On viewing torsors as <a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> (namely, <a class="existingWikiWord" href="/nlab/show/Cauchy+complete">Cauchy complete</a>/<a class="existingWikiWord" href="/nlab/show/copower">copowered</a> categories enriched in a <a class="existingWikiWord" href="/nlab/show/group">group</a> seen as a <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete category</a>):</p> <ul> <li><a href="https://golem.ph.utexas.edu/category/2013/06/torsors_and_enriched_categorie.html">Torsors and Enriched Categories</a>, <a class="existingWikiWord" href="/nlab/show/Simon+Willerton">Simon Willerton</a>, <a class="existingWikiWord" href="/nlab/show/n-category+caf%C3%A9">n-category café</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 24, 2025 at 05:52:31. 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