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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #26 to #27: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='bundles'>Bundles</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/bundle'>bundles</a></strong></p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/parameterized+homotopy+theory'>stable</a>) <a class='existingWikiWord' href='/nlab/show/diff/parameterized+homotopy+theory'>parameterized homotopy theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+bundles+in+physics'>fiber bundles in physics</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/over-topos'>slice topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-topos'>slice (∞,1)-topos</a></p> </li> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/dependent+linear+type+theory'>linear</a>) <a class='existingWikiWord' href='/nlab/show/diff/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/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/retractive+space'>retractive space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fiber bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/fiber+infinity-bundle'>fiber ∞-bundle</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/numerable+fiber+bundle'>numerable bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere+fiber+bundle'>sphere bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+bundle'>projective bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+bundle'>principal bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+2-bundle'>principal 2-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/principal+3-bundle'>principal 3-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle+bundle'>circle bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/circle+n-bundle+with+connection'>circle n-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/double+cover'>orientation bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spinor+bundle'>spinor bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/stringor+bundle'>stringor bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gerbe'>gerbe</a>, <a class='existingWikiWord' href='/nlab/show/diff/2-gerbe'>2-gerbe</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-gerbe'>∞-gerbe</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/local+coefficient+bundle'>local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/2-vector+bundle'>2-vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-vector bundle</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/real+vector+bundle'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+vector+bundle'>complex</a>/<a class='existingWikiWord' href='/nlab/show/diff/holomorphic+vector+bundle'>holomorphic</a>, <a class='existingWikiWord' href='/nlab/show/diff/quaternionic+vector+bundle'>quaternionic</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological</a>, <a class='existingWikiWord' href='/nlab/show/diff/differentiable+vector+bundle'>differentiable</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+vector+bundle'>algebraic</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+vector+bundle'>with connection</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>line bundle</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/holomorphic+vector+bundle'>holomorphic</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+line+bundle'>algebraic</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/cubical+structure+on+a+line+bundle'>cubical structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tensor+category'>tensor category</a> <a class='existingWikiWord' href='/nlab/show/diff/Vect%28X%29'>of vector bundles</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/VectBund'>VectBund</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>direct sum</a>, <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+vector+bundles'>tensor product</a>, <a class='existingWikiWord' href='/nlab/show/diff/external+tensor+product+of+vector+bundles'>external tensor product</a>, <a class='existingWikiWord' href='/nlab/show/diff/inner+product+on+vector+bundles'>inner product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dual+vector+bundle'>dual vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+vector+bundle'>stable vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/virtual+vector+bundle'>virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/parametrized+spectrum'>bundle of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/natural+bundle'>natural bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/universal+principal+bundle'>universal principal bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/universal+principal+infinity-bundle'>universal principal ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+vector+bundle'>universal vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/universal+complex+line+bundle'>universal complex line bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subobject+classifier'>subobject classifier</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a></p> </li> </ul> <h2 id='sidebar_presentations'>Presentations</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bundle+gerbe'>bundle gerbe</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/groupal+model+for+universal+principal+infinity-bundles'>groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/microbundle'>microbundle</a></p> </li> </ul> <h2 id='sidebar_examples'>Examples</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+bundle'>empty bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/zero+bundle'>zero bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+bundle'>normal bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tautological+line+bundle'>tautological line bundle</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/basic+complex+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/diff/Hopf+fibration'>Hopf fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/canonical+bundle'>canonical line bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/prequantum+line+bundle'>prequantum circle bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/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/diff/clutching+construction'>clutching construction</a></li> </ul> </div> <h4 id='cohomology'>Cohomology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a>, <a class='existingWikiWord' href='/nlab/show/diff/coboundary'>coboundary</a>, <a class='existingWikiWord' href='/nlab/show/diff/coefficient'>coefficient</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/chain'>chain</a>, <a class='existingWikiWord' href='/nlab/show/diff/cycle'>cycle</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/characteristic+class'>characteristic class</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+characteristic+class'>universal characteristic class</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/secondary+characteristic+class'>secondary characteristic class</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+characteristic+class'>differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequence</a>/<a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>long exact sequence in cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+infinity-bundle'>fiber ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/twisted+infinity-bundle'>twisted ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/obstruction'>obstruction</a></p> </li> </ul> <h3 id='special_and_general_types'>Special and general types</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>cochain cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ordinary+cohomology'>ordinary cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/singular+cohomology'>singular cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+group+cohomology'>nonabelian group cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+group+cohomology'>Lie group cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Galois+cohomology'>Galois cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/groupoid+cohomology'>groupoid cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+groupoid+cohomology'>nonabelian groupoid cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cobordism+cohomology+theory'>cobordism cohomology theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integral+cohomology'>integral cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-theory'>K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+automorphic+form'>taf</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dolbeault+cohomology'>Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%C3%A9tale+cohomology'>etale cohomology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group+of+units'>group of units</a>, <a class='existingWikiWord' href='/nlab/show/diff/Picard+group'>Picard group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Brauer+group'>Brauer group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crystalline+cohomology'>crystalline cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/syntomic+cohomology'>syntomic cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/motivic+cohomology'>motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology+of+operads'>cohomology of operads</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hochschild+cohomology'>Hochschild cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/cyclic+homology'>cyclic cohomology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/string+topology'>string topology</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nonabelian+cohomology'>nonabelian cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+principal+infinity-bundle'>universal principal ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/groupal+model+for+universal+principal+infinity-bundles'>groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+bundle'>principal bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Atiyah+Lie+groupoid'>Atiyah Lie groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+2-bundle'>principal 2-bundle</a>/<a class='existingWikiWord' href='/nlab/show/diff/gerbe'>gerbe</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+constant+infinity-stack'>covering ∞-bundle</a>/<a class='existingWikiWord' href='/nlab/show/diff/local+system'>local system</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-vector bundle</a> / <a class='existingWikiWord' href='/nlab/show/diff/n-vector+bundle'>(∞,n)-vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quantum+anomaly'>quantum anomaly</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/orientation'>orientation</a>, <a class='existingWikiWord' href='/nlab/show/diff/spin+structure'>Spin structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/spin%E1%B6%9C+structure'>Spin^c structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+structure'>String structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/Fivebrane+structure'>Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology+with+constant+coefficients'>cohomology with constant coefficients</a> / <a class='existingWikiWord' href='/nlab/show/diff/local+system'>with a local system of coefficients</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+algebra+cohomology'>∞-Lie algebra cohomology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+cohomology'>Lie algebra cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+Lie+algebra+cohomology'>nonabelian Lie algebra cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+extension'>Lie algebra extensions</a>, <a class='existingWikiWord' href='/nlab/show/diff/Gelfand-Fuks+cohomology'>Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gerstenhaber-Schack+cohomology'>bialgebra cohomology</a></p> </li> </ul> <h3 id='special_notions'>Special notions</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Čech cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hypercohomology'>hypercohomology</a></p> </li> </ul> <h3 id='variants'>Variants</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+cohomology'>equivariant cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant homotopy theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bredon+cohomology'>Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+cohomology'>twisted cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+bundle'>twisted bundle</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/twisted+K-theory'>twisted K-theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/twisted+spin+structure'>twisted spin structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/twisted+spin%E1%B6%9C+structure'>twisted spin^c structure</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+differential+c-structure'>twisted differential c-structures</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/differential+string+structure'>twisted differential string structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/differential+fivebrane+structure'>twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology'>differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cobordism+cohomology'>differential cobordism cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+elliptic+cohomology'>differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/schreiber/show/diff/differential+cohomology+in+a+cohesive+topos' title='schreiber'>differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relative+cohomology'>relative cohomology</a></p> </li> </ul> <h3 id='extra_structure'>Extra structure</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hodge+structure'>Hodge structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orientation'>orientation</a>, <a class='existingWikiWord' href='/nlab/show/diff/orientation+in+generalized+cohomology'>in generalized cohomology</a></p> </li> </ul> <h3 id='operations'>Operations</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology+operation'>cohomology operations</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cup+product'>cup product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connecting+homomorphism'>connecting homomorphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/Bockstein+homomorphism'>Bockstein homomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+integration'>fiber integration</a>, <a class='existingWikiWord' href='/nlab/show/diff/transgression'>transgression</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology+localization'>cohomology localization</a></p> </li> </ul> <h3 id='theorems'>Theorems</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+coefficient+theorem'>universal coefficient theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K%C3%BCnneth+theorem'>Künneth theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+theorem'>de Rham theorem</a>, <a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+lemma'>Poincare lemma</a>, <a class='existingWikiWord' href='/nlab/show/diff/Stokes+theorem'>Stokes theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hodge+theory'>Hodge theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hodge+theorem'>Hodge theorem</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/nonabelian+Hodge+theory'>nonabelian Hodge theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/noncommutative+Hodge+structure'>noncommutative Hodge theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>Brown representability theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>hypercovering theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Eckmann-Hilton+duality'>Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href='/nlab/edit/cohomology+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#Properties'>Properties</a><ul><li><a href='#general'>General</a></li><li><a href='#presentation_in_simplicial_presheaves'>Presentation in simplicial presheaves</a></li></ul></li><li><a href='#examples_2'>Examples</a><ul><li><a href='#fibrations_of_topological_spaces__simplicial_sets'>Fibrations of topological spaces / simplicial sets</a></li><li><a href='#gerbes'><math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Gerbes</a></li></ul></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>An <strong>associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundle</strong> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fiber bundle</a> in an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> with typical fiber <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>F \in \mathbf{H}</annotation></semantics></math> that is classified by a <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \to \mathbf{B}\underline{Aut}(F)</annotation></semantics></math> with coefficients in the <a class='existingWikiWord' href='/nlab/show/diff/delooping'>delooping</a> of the <a class='existingWikiWord' href='/nlab/show/diff/internalization'>internal</a> <a class='existingWikiWord' href='/nlab/show/diff/automorphism+infinity-group'>automorphism ∞-group</a> of <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>. We say this is <em>associated to</em> the corresponding <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\underline{Aut}(F)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a>.</p> <p><span> More generally there should be notions of<del class='diffmod'> accociated</del><ins class='diffmod'> associated</ins></span><math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundles whose fibers are objects in an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-topos'>(∞,n)-topos</a> over <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> for some <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n \gt 1</annotation></semantics></math>.</p> <h2 id='definition'>Definition</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a>.</p> <div class='num_def'> <h6 id='definition_2'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>,</mo><mi>X</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>V,X \in \mathbf{H}</annotation></semantics></math> two objects, say a <em><math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/fiber+infinity-bundle'>fiber ∞-bundle</a> over <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></em> is a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> (an object in the <a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-topos'>slice (∞,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathbf{H}_{/X}</annotation></semantics></math>) such that there exists an <a class='existingWikiWord' href='/nlab/show/diff/effective+epimorphism+in+an+%28infinity%2C1%29-category'>effective epimorphism</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \to X</annotation></semantics></math> and an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a> square</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>U</mi><mo>×</mo><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ U \times V &\to& E \\ \downarrow && \downarrow \\ U &\to& X } \,. </annotation></semantics></math></div></div> <div class='num_def'> <h6 id='definition_3'>Definition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>G \in Grp(\mathbf{H})</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a> equipped with an <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ρ</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>. Then for <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P \to X</annotation></semantics></math> a <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a> over <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, the <em><math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ρ</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math>-associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundle</em> is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>→</mo><mi>X</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> P \times_G V \to X \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>:</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>P</mi><mo>×</mo><mi>V</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo stretchy='false'>/</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>P \times_G V := (P \times V)//G</annotation></semantics></math> is the homotopy quotient of the diagonal <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-action.</p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>Below in <em><a href='#Properties'>Properties</a></em> we see that every <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ρ</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math>-associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundle is a <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-fiber <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundle and that every <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-fiber <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundle arises as associated to an <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Aut</mi></mstyle><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{Aut}(V)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a></p> </div> <h2 id='Properties'>Properties</h2> <h3 id='general'>General</h3> <div class='num_prop' id='Classification'> <h6 id='proposition'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>V \in \mathbf{H}</annotation></semantics></math>, write <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Aut</mi></mstyle><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>Grp</mi><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{Aut}(V) \in Grp(\mathbf{H})</annotation></semantics></math> for the internal <a class='existingWikiWord' href='/nlab/show/diff/automorphism+infinity-group'>automorphism ∞-group</a> of <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>. This comes with a canonical action on <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>. Then the operation of sending an <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Aut</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{Aut}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P \to X</annotation></semantics></math> to the associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P \times_G V \to X</annotation></semantics></math> establishes an equivalence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant='bold'><mi>Aut</mi></mstyle><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>≃</mo><mo stretchy='false'>{</mo><mi>V</mi><mo>−</mo><mi>fiber</mi><mspace width='thickmathspace' /><mn>∞</mn><mo>−</mo><mi>bundles</mi><mo stretchy='false'>}</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> H^1(X, \mathbf{Aut}(V)) \simeq \{V-fiber\;\infty-bundles\} \,. </annotation></semantics></math></div></div> <p>More specifically, if <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ρ</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a> of <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on some <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>V \in \mathbf{H}</annotation></semantics></math>, then under the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence of (∞,1)-categories</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mi>Act</mi><mo>≃</mo><msub><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mrow><mo stretchy='false'>/</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> G Act \simeq \mathbf{H}_{/\mathbf{B}G} </annotation></semantics></math></div> <p>it corresponds to a <a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequence</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy='false'>/</mo><mo stretchy='false'>/</mo><mi>G</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mover><mi>ρ</mi><mo>¯</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ V &\to& V//G \\ && \downarrow^{\mathrlap{\overline{\rho}}} \\ && \mathbf{B}G } </annotation></semantics></math></div> <p>in <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>. This is the <strong>universal <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ρ</mi></mrow><annotation encoding='application/x-tex'>\rho</annotation></semantics></math>-associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-bundle</strong> in that for <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P \to X</annotation></semantics></math> any <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a> modulated by <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>g \colon X \to \mathbf{B}G</annotation></semantics></math> we have a natural equivalence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>≃</mo><msup><mi>g</mi> <mo>*</mo></msup><mover><mi>ρ</mi><mo>¯</mo></mover><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> P \times_G V \simeq g^* \overline{\rho} \,. </annotation></semantics></math></div> <p>This is discussed in (<a href='#NSS'>NSS, section I 4.1</a>).</p> <h3 id='presentation_in_simplicial_presheaves'>Presentation in simplicial presheaves</h3> <p>In (<a href='#Wendt'>Wendt</a>), section 5.5, a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentation</a> of the general situation for <a class='existingWikiWord' href='/nlab/show/diff/n-localic+%28infinity%2C1%29-topos'>1-localic</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a> is given in terms of the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a> (as discussed at <a class='existingWikiWord' href='/nlab/show/diff/presentations+of+%28infinity%2C1%29-sheaf+%28infinity%2C1%29-toposes'>models for ∞-stack (∞,1)-toposes</a>) .</p> <p>Under this presentation we have:</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>The universal <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundle <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>E</mi></mstyle><mi>F</mi><mo>→</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>Aut</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{E} F \to \mathbf{B}Aut(F)</annotation></semantics></math> is presented by the <a class='existingWikiWord' href='/nlab/show/diff/bar+construction'>bar construction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>→</mo><mi>B</mi><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mi>Aut</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>F</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>B</mi><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mi>Aut</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> F \to B(*, Aut(F), F) \to B(*, Aut(F), *) \,. </annotation></semantics></math></div></div> <p>Compare <a class='existingWikiWord' href='/nlab/show/diff/universal+principal+infinity-bundle'>universal principal ∞-bundle</a>.</p> <h2 id='examples_2'>Examples</h2> <h3 id='fibrations_of_topological_spaces__simplicial_sets'>Fibrations of topological spaces / simplicial sets</h3> <p>For the special case that <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding='application/x-tex'>\mathbf{H} = </annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> and using the <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentation</a> by the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+topological+spaces'>model structure on topological spaces</a>/<a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sets'>model structure on simplicial sets</a> the classification theorem <a class='maruku-ref' href='#Classification'>1</a> reduces to the classical statement of (<a href='#Stasheff'>Stasheff</a>, <a href='#May'>May</a>).</p> <h3 id='gerbes'><math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Gerbes</h3> <p>In the case that the fiber <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/delooping'>delooping</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>=</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>F = \mathbf{B}G</annotation></semantics></math> of an <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a> object <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, the <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\underline{Aut}(\mathbf{B}G)</annotation></semantics></math>-associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundles are called <strong><math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/infinity-gerbe'>∞-gerbes</a></strong>. See there for more details.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action'>action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+bundle'>principal bundle</a> / <a class='existingWikiWord' href='/nlab/show/diff/torsor'>torsor</a> / <a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+2-bundle'>principal 2-bundle</a> / <a class='existingWikiWord' href='/nlab/show/diff/gerbe'>gerbe</a> / <a class='existingWikiWord' href='/nlab/show/diff/bundle+gerbe'>bundle gerbe</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+3-bundle'>principal 3-bundle</a> / <a class='existingWikiWord' href='/nlab/show/diff/2-gerbe'>2-gerbe</a> / <a class='existingWikiWord' href='/nlab/show/diff/bundle+2-gerbe'>bundle 2-gerbe</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a> / <a class='existingWikiWord' href='/nlab/show/diff/infinity-gerbe'>∞-gerbe</a> / <strong>associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundle</strong></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-vector bundle</a></p> </li> </ul> <h2 id='References'>References</h2> <p>Early work on associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundles takes place in the <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_72' xmlns='http://www.w3.org/1998/Math/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>-topos <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo></mrow><annotation encoding='application/x-tex'>\simeq</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>. In</p> <ul id='Stasheff'> <li><a class='existingWikiWord' href='/nlab/show/diff/Jim+Stasheff'>Jim Stasheff</a>, <em>A classification theorem for fiber spaces</em> , Topology 2 (1963) 239-246</li> </ul> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jim+Stasheff'>Jim Stasheff</a>, <em>H-spaces and classifying spaces: foundations and recent developments</em>. Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), pp. 247–272. MR0321079 (47 #9612)</li> </ul> <p>a classification of <a class='existingWikiWord' href='/nlab/show/diff/fibration'>fibrations</a> of <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complexes</a> with given CW-complex fiber in terms of maps into a classifying CW-complex is given.</p> <p>In</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Gottlieb'>Daniel Gottlieb</a>, <em>The total space of universal fibrations.</em> Pacific J. Math. Volume 46, Number 2 (1973), 415-417.</li> </ul> <p>the total space of the universal <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/fiber+infinity-bundle'>fiber ∞-bundle</a> in the pointed context is identified with <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>Aut</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{B}Aut_*(F)</annotation></semantics></math> (the pointed <a class='existingWikiWord' href='/nlab/show/diff/automorphism+infinity-group'>automorphism ∞-group</a>).</p> <p>A generalization or more systematic account of the classification theory is then given in</p> <ul id='May'> <li><a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <em>Classifying Spaces and Fibrations</em> Mem. Amer. Math. Soc. 155 (1975) (<a href='http://www.math.uchicago.edu/~may/BOOKS/Classifying.pdf'>pdf</a>)</li> </ul> <p>This has been reproven in various guises, such as the statement of <a class='existingWikiWord' href='/nlab/show/diff/univalence+axiom'>univalence</a> in the <a class='existingWikiWord' href='/nlab/show/diff/model'>model</a> <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a> for <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a>. See the references at <em><a class='existingWikiWord' href='/nlab/show/diff/univalence+axiom'>univalence</a></em> for more on this.</p> <p>Generalizations with extra structure on the fibers are discussed in</p> <ul> <li>Claudio Pacati, Petar Pavesic, Renzo Piccinini, <em>On the classification of <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{F}</annotation></semantics></math>-fibrations</em>, Topology and its applications 87 (1998) (<a href='http://www.fmf.uni-lj.si/~pavesic/RESEARCH/On%20the%20classification%20of%20F-fibrations.pdf'>pdf</a>)</li> </ul> <p>Consideration of associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundles / <a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequences</a> in general <a class='existingWikiWord' href='/nlab/show/diff/n-localic+%28infinity%2C1%29-topos'>1-localic</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a> <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presented</a> by a <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a> (which subsumes the above case for the trivial site) is discussed in</p> <ul id='Wendt'> <li><a class='existingWikiWord' href='/nlab/show/diff/Matthias+Wendt'>Matthias Wendt</a>, <em>Classifying spaces and fibrations of simplicial sheaves</em> , Journal of Homotopy and Related Structures 6(1), 2011, pp. 1–38. (<a href='http://arxiv.org/abs/1009.2930'>arXiv</a>) (<a href='http://tcms.org.ge/Journals/JHRS/volumes/2011/volume6-1.htm'>published version</a>)</li> </ul> <p>Related discussion on the behaviour of <a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequences</a> under left <a class='existingWikiWord' href='/nlab/show/diff/Bousfield+localization'>Bousfield localization of model categories</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Matthias+Wendt'>Matthias Wendt</a>, <em>Fibre sequences and localization of simplicial sheaves</em> (<a href='http://home.mathematik.uni-freiburg.de/arithmetische-geometrie/preprints/wendt-flocal.pdf'>pdf</a>)</li> </ul> <p>Similar considerations and results are in</p> <ul id='BlomgrenChacholski'> <li>Martin Blomgren, <a class='existingWikiWord' href='/nlab/show/diff/Wojciech+Chach%C3%B3lski'>Wojciech Chacholski</a>, <em>On the classification of fibrations</em> (<a href='http://arxiv.org/abs/1206.4443'>arXiv:1206.4443</a>)</li> </ul> <p>With the advent of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos+theory'>(∞,1)-topos theory</a> all these statements and their generalizations follow from the existence of <a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifiers</a> in an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a>. For the classical case in <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo></mrow><annotation encoding='application/x-tex'>\simeq</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a><math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mrow /> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>{}^\circ</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a><math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mrow /> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>{}^\circ</annotation></semantics></math> this is discussed in</p> <ul> <li><em><a href='http://ncatlab.org/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos#ObjectClassifierInInfinityGroupoid'>object classifier in ∞Grpd</a></em>,</li> </ul> <p>which reproduces the classical results (<a href='#Stasheff'>Stasheff</a>, <a href='#May'>May</a>).</p> <p>For general <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a> the classification of associated <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-bundles is discussed in section I 4.1 of</p> <ul> <li><a class='existingWikiWord' href='/schreiber/show/diff/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications' title='schreiber'>Principal ∞-bundles -- theory, presentations and applications</a></li> </ul> <p>Models in <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a> of classifying spaces for homotopy types <math class='maruku-mathml' display='inline' id='mathml_94d8b08a17881b74fd4ae839023bc4e07f8ad840_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Aut</mi><mo stretchy='false'>(</mo><mi>F</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Aut(F)</annotation></semantics></math> go back to <a class='existingWikiWord' href='/nlab/show/diff/Dennis+Sullivan'>Sullivan</a>’s remarks on the <a class='existingWikiWord' href='/nlab/show/diff/automorphism+infinity-Lie+algebra'>automorphism L-infinity algebra</a>. Further developments are reviewed and developed in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Andrey+Lazarev'>Andrey Lazarev</a>, <em>Models for classifying spaces and derived deformation theory</em> (<a href='http://arxiv.org/abs/1209.3866'>arXiv:1209.3866</a>)</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on December 8, 2013 at 08:29:38. 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