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href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#in_topological_spaces'>In topological spaces</a></li> <ul> <li><a href='#homotopy'><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>-Homotopy</a></li> <li><a href='#HomotopyTheory'>Homotopy theory 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>-spaces</a></li> <li><a href='#InfGTop'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category 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>-equivariant spaces</a></li> <li><a href='#global_equivariant_homotopy_theory'>Global equivariant homotopy theory</a></li> </ul> <li><a href='#in_more_general_model_categories'>In more general model categories</a></li> <ul> <li><a href='#in_stack_toposes'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</a></li> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#basic_facts'>Basic facts</a></li> <li><a href='#elmendorfs_theorem'>Elmendorf’s theorem</a></li> <li><a href='#equivariant_hopf_degree_theorem'>Equivariant Hopf degree theorem</a></li> <li><a href='#stabilization'>Stabilization</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#equivariance'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-Equivariance</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Equivariant homotopy theory is <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> for the case that a <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/action">acts</a> on all the <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> or other objects involved, hence the homotopy theory of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a>.</p> <p>The canonical <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of 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>-spaces are <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>-equivariant <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>, and the canonical choice of <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> between these are <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>-equivariant continuous homotopies (for trivial <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 on the interval). 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>-equivariant version of the <a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a> says that on <a class="existingWikiWord" href="/nlab/show/G-CW+complexes">G-CW complexes</a> these <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>-equivariant <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> are equivalently those maps that induce <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> on all <a class="existingWikiWord" href="/nlab/show/fixed+point">fixed point</a> spaces for all <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</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> (compact subgroups, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is allowed to be a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>). By <a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a>, this, in turn, is equivalent to the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaves">(∞,1)-presheaves</a> over the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</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>. See below at <em><a href="#HomotopyTheory">In topological spaces – Homotopy theory</a></em>.</p> <p>(Beware that <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>-homotopy theory is crucially different from (namely “finer” and “more geometric” than) the homotopy theory of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-actions">∞-actions</a> of the underlying <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</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>, and this is so even when <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/discrete+group">discrete group</a>, see <a href="#RelationToInfinityActions">below</a>).</p> <p>The union 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>-equivariant homotopy theories 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> is allowed to vary is <em><a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a></em>.</p> <p>The direct <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of equivariant homotopy theory is the theory of <a class="existingWikiWord" href="/nlab/show/spectra+with+G-action">spectra with G-action</a>. More generally there is a concept of <a class="existingWikiWord" href="/nlab/show/G-spectra">G-spectra</a> and they are the subject of <a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a>.</p> <p>The concept of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of equivariant homotopy theory is <em><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></em>:</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in the presence of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Borel+equivariant+cohomology">Borel equivariant cohomology</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AAA</mi></mphantom><mo>←</mo><mphantom><mi>AAA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AAA}\leftarrow\phantom{AAA}</annotation></semantics></math></th><th>general (<a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon</a>) <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AAA</mi></mphantom><mo>→</mo><mphantom><mi>AAA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AAA}\rightarrow\phantom{AAA}</annotation></semantics></math></th><th>non-equivariant <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> with <a class="existingWikiWord" href="/nlab/show/homotopy+fixed+point">homotopy fixed point</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>G</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}</annotation></semantics></math></td><td style="text-align: left;">trivial <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">action</a> on <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><msup><mo stretchy="false">]</mo> <mi>G</mi></msup><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}[X,A]^G\phantom{AA}</annotation></semantics></math></td><td style="text-align: left;">trivial <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">action</a> on <a class="existingWikiWord" href="/nlab/show/domain">domain</a> space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>A</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="in_topological_spaces">In topological spaces</h2> <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/discrete+group">discrete group</a>.</p> <h3 id="homotopy"><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>-Homotopy</h3> <div class="num_defn" id="GSpace"> <h6 id="definition">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/topological+G-space">topological G-space</a></strong> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> 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>.</p> </div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">I = \mathbb{R}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mover><mo>→</mo><mn>0</mn></mover><mi>I</mi><mover><mo>←</mo><mn>1</mn></mover><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">({*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*})</annotation></semantics></math> regarded as 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>-space by equipping it with the trivial <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>.</p> <div class="num_defn" id="GHomotopy"> <h6 id="definition_2">Definition</h6> <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>-<a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> between <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>-maps, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f, g : X \to Y</annotation></semantics></math>, is a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> with respect to this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></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>X</mi><mo>×</mo><mo>*</mo><mo>=</mo><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>Id</mi><mo>×</mo><mn>0</mn></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>→</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mn>1</mn></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mi>g</mi></msub></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mo>*</mo><mo>=</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \times {*} = X \\ {}^{\mathllap{Id \times 0}}\downarrow & \searrow^{f} \\ X \times I &\stackrel{\eta}{\to}& Y \\ {}^{\mathllap{1}}\uparrow & \nearrow_{g} \\ X\times {*} = X } \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#May96">May 96, p. 15</a>)</p> <h3 id="HomotopyTheory">Homotopy theory 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>-spaces</h3> <div class="num_defn" id="HomotopicalStructures"> <h6 id="definition_3">Definition</h6> <p><strong>(models for <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>-equivariant spaces)</strong></p> <p>Consider the following three <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical categories</a> that model <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>-spaces:</p> <ol> <li> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mi>Top</mi> <mi>cof</mi></msub><mo>⊂</mo><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex"> G Top_{cof} \subset G Top </annotation></semantics></math></div> <p>for the full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of <a class="existingWikiWord" href="/nlab/show/G-CW-complexes">G-CW-complexes</a>, regarded as equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> by taking the weak equivalences to be the <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/homotopy+equivalences">homotopy equivalences</a> according to def. <a class="maruku-ref" href="#GHomotopy"></a>.</p> </li> <li> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mi>Top</mi> <mi>loc</mi></msub></mrow><annotation encoding="application/x-tex"> G Top_{loc} </annotation></semantics></math></div> <p>for all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">G Top</annotation></semantics></math> equipped with weak equivalences given by those morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">(f : X \to Y) \in G Top</annotation></semantics></math> that induce for all <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mi>H</mi></msup><mo>:</mo><msup><mi>X</mi> <mi>H</mi></msup><mo>→</mo><msup><mi>Y</mi> <mi>H</mi></msup></mrow><annotation encoding="application/x-tex">f^H : X^H \to Y^H</annotation></semantics></math> on the <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>-<a class="existingWikiWord" href="/nlab/show/fixed+point">fixed point</a> spaces, in the standard <a class="existingWikiWord" href="/nlab/show/Quillen+model+structure+on+topological+spaces">Quillen model structure on topological spaces</a> (i.e. inducing <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>).</p> </li> <li> <p>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><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup><mo>,</mo><msub><mi>Top</mi> <mi>loc</mi></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> [Orb_G^{op}, Top_{loc}]_{proj} </annotation></semantics></math></div> <p>for the projective <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+functors">global model structure on functors</a> from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">O_G</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> to the <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (with its <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>).</p> </li> </ol> </div> <p>The following theorem (the <a class="existingWikiWord" href="/nlab/show/equivariant+Whitehead+theorem">equivariant Whitehead theorem</a> together with <a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a>) says that these models all present the same <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <div class="num_theorem" id="EquivariantWhiteheadAndElmendorfTheorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Elmendorf’s theorem)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy categories</a> of all three <a class="existingWikiWord" href="/nlab/show/homotopical+categories">homotopical categories</a> in def. <a class="maruku-ref" href="#HomotopicalStructures"></a> are <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>G</mi><msub><mi>Top</mi> <mi>cof</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Ho</mi><mo stretchy="false">(</mo><mi>G</mi><msub><mi>Top</mi> <mi>loc</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Ho</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup><mo>,</mo><mi>Top</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Ho(G Top_{cof}) \overset{\simeq}{\longrightarrow} Ho(G Top_{loc}) \overset{\simeq}{\longrightarrow} Ho([Orb_G^{op}, Top]) \,, </annotation></semantics></math></div> <p>where the equivalence is induced by the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> that sends 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>-space to the <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> that it <a class="existingWikiWord" href="/nlab/show/representable+functor">represents</a>.</p> <p>The first of these equivalences is the <em><a class="existingWikiWord" href="/nlab/show/equivariant+Whitehead+theorem">equivariant Whitehead theorem</a></em>, the second is <em><a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a></em>.</p> </div> <p>This is stated as (<a href="#May96">May 96, theorem VI.6.3</a>).</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>Ho</mi><mo stretchy="false">(</mo><mi>G</mi><msub><mi>Top</mi> <mi>cof</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>cof</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mfrac linethickness="0"><mrow><mtext>equivariant</mtext></mrow><mrow><mtext>Whitehead</mtext></mrow></mfrac></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mo>≃</mo></mpadded></mtd> <mtd></mtd> <mtd><mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mtext>Whitehead</mtext></mpadded></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>G</mi><msub><mi>Top</mi> <mi>loc</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>loc</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mi>Elmendorf</mi></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mo>≃</mo></mpadded></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ho</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>Orb</mi> <mi>G</mi></msub><mo>,</mo><msub><mi>Top</mi> <mi>loc</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Ho</mi><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><msub><mi>Top</mi> <mi>loc</mi></msub><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Ho(G Top_{cof}) &\underset{}{\longrightarrow}& Ho(Top_{cof}) \\ {\mathllap{\text{equivariant} \atop \text{Whitehead}}}\big\downarrow{\mathrlap{\simeq}} && {\mathllap{\simeq}}\big\downarrow{\mathrlap{\text{Whitehead}}} \\ Ho(G Top_{loc}) &\overset{}{\longrightarrow}& Ho(Top_{loc}) \\ {\mathllap{Elmendorf}}\big\downarrow{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} \\ Ho( PSh( Orb_G, Top_{loc} ) )_{proj} &\longrightarrow& Ho( \ast, Top_{loc} )_{proj} } </annotation></semantics></math></div> <h3 id="InfGTop"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category 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>-equivariant spaces</h3> <p>At <a class="existingWikiWord" href="/nlab/show/topological+%E2%88%9E-groupoid">topological ∞-groupoid</a> it is discussed that the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s may be understood as the <a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(Top)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">of (∞,1)-sheaves</a> on <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>, at the collection of morphisms of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{X \times I \to X\}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> the real line.</p> <p>The analogous statement is true for <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>-spaces: the equivariant homotopy category is the <a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> of the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">G Top</annotation></semantics></math>.</p> <p>More in detail: let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">G Top</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/site">site</a> whose objects are <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>-spaces that admit <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>-equivariant open covers, morphisms are <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>-equivariant maps and morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> is in the <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> if it admits 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>-equivariant splitting over such <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>-equivariant open covers.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sSh</mi><mo stretchy="false">(</mo><mi>G</mi><mi>Top</mi><msub><mo stretchy="false">)</mo> <mi>loc</mi></msub></mrow><annotation encoding="application/x-tex"> sSh(G Top)_{loc} </annotation></semantics></math></div> <p>for the corresponding <a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercomplete</a> local <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">model structure on simplicial sheaves</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be the unit interval, the standard <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, equipped with the trivial <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, regarded as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">G Top</annotation></semantics></math> and hence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSh</mi><mo stretchy="false">(</mo><mi>G</mi><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sSh(G Top)</annotation></semantics></math>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sSh</mi><mo stretchy="false">(</mo><mi>G</mi><mi>Top</mi><msubsup><mo stretchy="false">)</mo> <mi>loc</mi> <mi>I</mi></msubsup><mover><mo>→</mo><mo>←</mo></mover><mi>sSh</mi><mo stretchy="false">(</mo><mi>G</mi><mi>Top</mi><msub><mo stretchy="false">)</mo> <mi>loc</mi></msub></mrow><annotation encoding="application/x-tex"> sSh(G Top)_{loc}^I \stackrel{\leftarrow}{\to} sSh(G Top)_{loc} </annotation></semantics></math></div> <p>for the left <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a> at thecollection of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>X</mi><mover><mo>→</mo><mrow><mi>Id</mi><mo>×</mo><mn>0</mn></mrow></mover><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{X \stackrel{Id \times 0}{\to} X \times I\}</annotation></semantics></math>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSh</mi><mo stretchy="false">(</mo><mi>G</mi><mi>Top</mi><msubsup><mo stretchy="false">)</mo> <mi>loc</mi> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">sSh(G Top)_{loc}^I</annotation></semantics></math> is the equivariant homotopy category described above</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>sSh</mi><mo stretchy="false">(</mo><mi>G</mi><mi>Top</mi><msubsup><mo stretchy="false">)</mo> <mi>loc</mi> <mi>I</mi></msubsup><mo stretchy="false">)</mo><mo>≃</mo><mi>G</mi><msub><mi>Top</mi> <mi>loc</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(sSh(G Top)_{loc}^{I}) \simeq G Top_{loc} \,. </annotation></semantics></math></div> <p>This is (<a href="#MorelVoevodsky03">Morel-Voevodsky 03, example 3, p. 50</a>).</p> <h3 id="global_equivariant_homotopy_theory">Global equivariant homotopy theory</h3> <p>The above constructions may be unified to apply “for all groups at once”, this is the content of <em><a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a></em>.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/cohesion+of+global-+over+G-equivariant+homotopy+theory">cohesion of global- over G-equivariant homotopy theory</a></em>.</p> <h2 id="in_more_general_model_categories">In more general model categories</h2> <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 finite <a class="existingWikiWord" href="/nlab/show/group">group</a> as above. We describe the generalizaton of the above story as <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is replaced by a more general <a class="existingWikiWord" href="/nlab/show/model+category">model 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> (<a href="#Guillou06">Guillou 2006</a>).</p> <div class="num_defn"> <h6 id="definition_and_proposition">Definition and proposition</h6> <ol> <li> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a> with generating cofibrations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> and generating acyclic cofibrations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>.</p> <p>There is a cofibrantly generated model category</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><msubsup><mi>O</mi> <mi>G</mi> <mi>op</mi></msubsup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>loc</mi></msub></mrow><annotation encoding="application/x-tex"> [O_G^{op}, C]_{loc} </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> from the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</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> to <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> by taking the generating cofibrations to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mrow><msub><mi>O</mi> <mi>G</mi></msub></mrow></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>×</mo><mi>i</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi><mo>,</mo><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> I_{O_G} := \{G/H \times i\}_{i \in I, H \subset G} </annotation></semantics></math></div> <p>and the generating acyclic cofibrations to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mrow><msub><mi>O</mi> <mi>G</mi></msub></mrow></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>×</mo><mi>j</mi><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi><mo>,</mo><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> J_{O_G} := \{G/H \times j\}_{j \in J, H \subset G} \,. </annotation></semantics></math></div></li> <li> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</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> and let</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>loc</mi></msub></mrow><annotation encoding="application/x-tex"> [\mathbf{B}G^{op}, C]_{loc} </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> to <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> – the category of objects 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> 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> equipped with a set of generating (acyclic) cofibrations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>×</mo><mi>i</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi><mo>,</mo><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> I_{\mathbf{B}G} := \{G/H \times i\}_{i \in I, H \subset G} </annotation></semantics></math></div> <p>and the generating acyclic cofibrations to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>×</mo><mi>j</mi><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi><mo>,</mo><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> J_{\mathbf{B}G} := \{G/H \times j\}_{j \in J, H \subset G} \,. </annotation></semantics></math></div> <p>This defines a cofibrantly generated model category if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathbf{B}G^{op}, C]</annotation></semantics></math> has a <em>cellular fixed point functor</em> (see…).</p> </li> </ol> </div> <div class="num_defn"> <h6 id="definition_and_proposition_2">Definition and proposition</h6> <p><strong>(generalized Elmendorf’s theorem)</strong></p> <p>There is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>e</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mover><mo>→</mo><mo>←</mo></mover><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>loc</mi></msub><mo>:</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>e</mi></msup></mrow><annotation encoding="application/x-tex"> G/e \times (-) : C \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : (-)^e </annotation></semantics></math></div> <p>and a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo>:</mo><mo stretchy="false">[</mo><msubsup><mi>O</mi> <mi>G</mi> <mi>op</mi></msubsup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>loc</mi></msub><mover><mo>→</mo><mo>←</mo></mover><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>loc</mi></msub><mo>:</mo><mi>Φ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Theta : [O_G^{op}, C]_{loc} \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : \Phi \,. </annotation></semantics></math></div></div> <p>This is <a href="#Guillou06">Guillou 2006, Prop. 3.1.5</a>.</p> <h3 id="in_stack_toposes">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes</h3> <p>The assumption on the <a class="existingWikiWord" href="/nlab/show/model+category">model 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> entering the generalized Elmendorf theorem above is satisfied in particular by every left <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>:</mo><mo>=</mo><msub><mi>L</mi> <mi>A</mi></msub><mi>SPSh</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C := L_A SPSh(D) </annotation></semantics></math></div> <p>of the global projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> onany <a class="existingWikiWord" href="/nlab/show/small+category">small 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> at any <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of morphisms, i.e. for every <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model 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>.</p> <p>This is <a href="#Guillou06">Guillou 2006, Ex. 4.4</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">{</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">A = \{C(\{U_i\}) \to X\}</annotation></semantics></math> the collection of <a class="existingWikiWord" href="/nlab/show/Cech+cover">Cech cover</a>s for all <a class="existingWikiWord" href="/nlab/show/sieve">covering families</a> of a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, this are the standard <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>This way the above theorem provides a model for <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>-equivariant refinements of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es.</p> <ul> <li> <p>For instance, in <a class="existingWikiWord" href="/nlab/show/motivic+homotopy+theory">motivic homotopy theory</a> one considers cohomology in a <a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> on the <a class="existingWikiWord" href="/nlab/show/Nisnevich+site">Nisnevich site</a>, presented by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>:</mo><mo>=</mo><msub><mi>L</mi> <mi>Cech</mi></msub><mi>SPSh</mi><mo stretchy="false">(</mo><mi>Nis</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C := L_{Cech} SPSh(Nis)</annotation></semantics></math> . Its <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>-equivariant version as above should be the right context for the <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon</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/equivariant+cohomology">equivariant cohomology</a> refinement of such cohomology theories, such as <a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a>.</p> <p>This is <a href="#Guillou06">Guillou 06, Ex. 4.5</a>.</p> <p>(Actually here one localizes moreover at <a class="existingWikiWord" href="/nlab/show/hypercovers">hypercovers</a> and at <a class="existingWikiWord" href="/nlab/show/A1-homotopy+theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msup><mi>𝔸</mi> <mn>1</mn></msup> </mrow> <annotation encoding="application/x-tex">\mathbb{A}^1</annotation> </semantics> </math>-homotopies</a>.)</p> </li> </ul> <h2 id="Properties">Properties</h2> <h3 id="basic_facts">Basic facts</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/G-CW+approximation">G-CW approximation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+Whitehead+theorem">equivariant Whitehead theorem</a></p> </li> </ul> <h3 id="elmendorfs_theorem">Elmendorf’s theorem</h3> <p>By <a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a> the <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/equivariant+homotopy+theory">equivariant homotopy theory</a> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>.</p> <p>By (<a href="#Rezk14">Rezk 14</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">G Top</annotation></semantics></math> is also the <a class="existingWikiWord" href="/nlab/show/base+%28%E2%88%9E%2C1%29-topos">base (∞,1)-topos</a> of the <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> of the <a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a> <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">sliced</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>. See at <em><a class="existingWikiWord" href="/nlab/show/cohesion+of+global-+over+G-equivariant+homotopy+theory">cohesion of global- over G-equivariant homotopy theory</a></em>.</p> <h3 id="equivariant_hopf_degree_theorem">Equivariant Hopf degree theorem</h3> <p>See at <em><a href="Hopf+degree+theorem#InEquivariantHomotopyTheory">equivariant Hopf degree theorem</a></em>.</p> <h3 id="stabilization">Stabilization</h3> <p>The <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi><mo>≃</mo><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><msub><mi>Orb</mi> <mi>G</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G Top \simeq PSh_\infty(Orb_G)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/spectra+with+G-action">spectra with G-action</a> (“<a class="existingWikiWord" href="/nlab/show/naive+G-spectra">naive G-spectra</a>”).</p> <h2 id="examples">Examples</h2> <h3 id="equivariance"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-Equivariance</h3> <p><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-equivariant homotopy theory may be presented by <a class="existingWikiWord" href="/nlab/show/cyclic+sets">cyclic sets</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariance">equivariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+structure">equivariant structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+group">equivariant homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+differential+topology">equivariant differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+equivariant+homotopy+theory">proper equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cobordism+theory">equivariant cobordism theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+rational+homotopy+theory">equivariant rational homotopy theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/Borel-equivariant+rational+homotopy+theory">Borel-equivariant rational homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+motivic+homotopy+theory">equivariant motivic homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+conjecture">Sullivan conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Arf-Kervaire+invariant+problem">Arf-Kervaire invariant problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+symmetric+monoidal+category">equivariant symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Parametrized+Higher+Category+Theory+and+Higher+Algebra">Parametrized Higher Category Theory and Higher Algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Burnside+category">Burnside category</a>, <a class="existingWikiWord" href="/nlab/show/Burnside+ring">Burnside ring</a></p> </li> </ul> <p>Equivariant homotopy theory is to <a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a> as <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> is to <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>.</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Global+Homotopy+Theory+and+Cohesion">Rezk</a>-<a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a>:</strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></th><th>its <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></th><th><a class="existingWikiWord" href="/nlab/show/base+%28%E2%88%9E%2C1%29-topos">base (∞,1)-topos</a></th><th>its <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>Glo</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_\infty(Glo)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/global+equivariant+indexing+category">global equivariant indexing category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Glo</mi></mrow><annotation encoding="application/x-tex">Glo</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \simeq PSh_\infty(\ast)</annotation></semantics></math></td><td style="text-align: left;">point</td></tr> <tr><td style="text-align: left;">… <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">sliced</a> over terminal <a class="existingWikiWord" href="/nlab/show/orbispace">orbispace</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>Glo</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>𝒩</mi></mrow></msub></mrow><annotation encoding="application/x-tex">PSh_\infty(Glo)_{/\mathcal{N}}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Glo</mi> <mrow><mo stretchy="false">/</mo><mi>𝒩</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Glo_{/\mathcal{N}}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orbispaces">orbispaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>Orb</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_\infty(Orb)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/global+orbit+category">global orbit category</a></td></tr> <tr><td style="text-align: left;">… <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">sliced</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>Glo</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">PSh_\infty(Glo)_{/\mathbf{B}G}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Glo</mi> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Glo_{/\mathbf{B}G}</annotation></semantics></math></td><td style="text-align: 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>-<a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/G-spaces">G-spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>we</mi></msub><mi>G</mi><mi>Top</mi><mo>≃</mo><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><msub><mi>Orb</mi> <mi>G</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L_{we} G Top \simeq PSh_\infty(Orb_G)</annotation></semantics></math></td><td style="text-align: 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>-<a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Orb</mi> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>=</mo><msub><mi>Orb</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">Orb_{/\mathbf{B}G} = Orb_G</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Lecture notes:</p> <ul> <li id="Blumberg17"> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <em>Equivariant homotopy theory</em>, 2017 (<a href="https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf">pdf</a>, <a href="https://github.com/adebray/equivariant_homotopy_theory">GitHub</a>)</p> </li> <li id="Guillou20"> <p><a class="existingWikiWord" href="/nlab/show/Bert+Guillou">Bert Guillou</a>, <em>Equivariant Homotopy and Cohomology</em>, lecture notes, 2020 (<a href="http://www.ms.uky.edu/~guillou/F20/751Notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GuillouEquivariantHomotopyAndCohomology.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Textbooks and other accounts</p> <ul> <li id="tomDieck87"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em><a class="existingWikiWord" href="/nlab/show/Transformation+Groups">Transformation Groups</a></em>, de Gruyter 1987 (<a href="https://doi.org/10.1515/9783110858372">doi:10.1515/9783110858372</a>)</p> </li> <li id="tomDieck79"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em><a class="existingWikiWord" href="/nlab/show/Transformation+Groups+and+Representation+Theory">Transformation Groups and Representation Theory</a></em>, Lecture Notes in Mathematics 766, Springer 1979 (<a href="https://link.springer.com/book/10.1007/BFb0085965">doi:10.1007/BFb0085965</a>)</p> </li> <li id="Lueck89"> <p><a class="existingWikiWord" href="/nlab/show/Wolfgang+L%C3%BCck">Wolfgang Lück</a>, Chapter I of: <em>Transformation Groups and Algebraic K-Theory</em>, Lecture Notes in Mathematics <strong>1408</strong> (Springer 1989) (<a href="https://doi.org/10.1007/BFb0083681">doi:10.1007/BFb0083681</a>)</p> </li> <li id="May96"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a> et al., <em>Equivariant homotopy and cohomology theory</em>, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington DC (1996) [<a href="https://bookstore.ams.org/cbms-91/?startBookmarkIdx=200">ISBN: 978-0-8218-0319-6</a>, <a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MayEtAlEquivariant96.pdf" title="pdf">pdf</a>]</p> </li> <li id="Schwede12"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, appendix A.4 of of <em><a class="existingWikiWord" href="/nlab/show/Symmetric+spectra">Symmetric spectra</a></em> (2012)</p> </li> <li id="HillHopkinsRavenel21"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hill">Michael Hill</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem</em>, New Mathematical Monographs, Cambridge University Press (2021) [<a href="https://doi.org/10.1017/9781108917278">doi:10.1017/9781108917278</a>]</p> <blockquote> <p>(with an eye towards the <a class="existingWikiWord" href="/nlab/show/Arf-Kervaire+invariant+problem">Arf-Kervaire invariant problem</a>)</p> </blockquote> </li> </ul> <p>The case of <a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℤ</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> <annotation encoding="application/x-tex">\mathbb{Z}/2</annotation> </semantics> </math></a>-equivariance (such as with <a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a>):</p> <ul> <li id="Crabb80"><a class="existingWikiWord" href="/nlab/show/Michael+C.+Crabb">Michael C. Crabb</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-Homotopy Theory</em>, Lond. Math. Soc. Lecture Notes <strong>44</strong>, Cambridge University Press (1980) [<a href="https://archive.org/details/zz2homotopytheor0000crab">ark:/13960/t3jx7bg4w</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/crabb.pdf">pdf</a>]</li> </ul> <p>The generalization of the homotopy theory 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>-spaces and of <a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a> to that 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>-objects in more general <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a> is in</p> <ul> <li id="Guillou06"><a class="existingWikiWord" href="/nlab/show/Bert+Guillou">Bert Guillou</a>, <em>A short note on models for equivariant homotopy theory</em>, 2006 (<a href="https://faculty.math.illinois.edu/~bertg/EquivModels.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GuillouModelsForEquivariantHomotopyTheory.pdf" title="pdf">pdf</a>)</li> </ul> <p>and further discussed in</p> <ul> <li id="Stephan13"><a class="existingWikiWord" href="/nlab/show/Marc+Stephan">Marc Stephan</a>, <em>On equivariant homotopy theory for model categories</em>, Homology Homotopy Appl. 18(2) (2016) 183-208 (<a href="http://arxiv.org/abs/1308.0856">arXiv:1308.0856</a>)</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Some results in equivariant homotopy theory</em> (1978) [scan: <a href="https://homepages.warwick.ac.uk/~maaac/segal.html">web</a>, <a class="existingWikiWord" href="/nlab/files/Segal-EquivariantHT.pdf" title="pdf">pdf</a>]</p> <blockquote> <p>(on equivariant <a class="existingWikiWord" href="/nlab/show/iterated+loop+spaces">iterated loop spaces</a> and <a class="existingWikiWord" href="/nlab/show/configuration+spaces+of+points">configuration spaces of points</a>)</p> </blockquote> </li> <li id="Waner80"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Waner">Stefan Waner</a>, <em>Equivariant Homotopy Theory and Milnor’s Theorem</em>, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 (<a href="http://www.jstor.org/stable/1998061">JSTOR</a>)</p> </li> </ul> <p>Specifically with an eye towards equivariant <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a> (such as <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+construction">Pontryagin-Thom construction</a> for <a class="existingWikiWord" href="/nlab/show/equivariant+cohomotopy">equivariant cohomotopy</a>):</p> <ul> <li id="Wasserman69"><a class="existingWikiWord" href="/nlab/show/Arthur+Wasserman">Arthur Wasserman</a>, <em>Equivariant differential topology</em>, Topology Vol. 8, pp. 127-150, 1969 (<a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf">pdf</a>)</li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a> is in</p> <ul> <li id="Rezk14"><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em><a class="existingWikiWord" href="/nlab/show/Global+Homotopy+Theory+and+Cohesion">Global Homotopy Theory and Cohesion</a></em> (2014)</li> </ul> <p>Discussion via <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is in</p> <ul> <li id="Shulman15"><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Univalence for inverse EI diagrams</em> (<a href="http://arxiv.org/abs/1508.02410">arXiv:1508.02410</a>)</li> </ul> <p>An alternative <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>-structure:</p> <ul> <li>Mehmet Akif Erdal, Aslı Güçlükan İlhan, <em>A model structure via orbit spaces for equivariant homotopy</em> (<a href="https://arxiv.org/abs/1903.03152">arXiv:1903.03152</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 17, 2024 at 12:36:17. 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