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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 #14 to #15: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <ins class='diffins'><blockquote> <p>This entry may need to be merged with <em><a class='existingWikiWord' href='/nlab/show/diff/simplicial+cochain'>simplicial cochain</a></em>.</p> </blockquote></ins><ins class='diffins'> </ins><div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <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> <h4 id='homotopy_theory'>Homotopy theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a></strong></p> <p>flavors: <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/diff/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+theory'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+theory'>directed</a>…</p> <p>models: <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/diff/localic+homotopy+theory'>localic</a>, …</p> <p>see also <strong><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>higher homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/spherical+object'>spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofibration+category'>cofibration category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Waldhausen+category'>Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+an+%28infinity%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/diff/homotopy'>left homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+universal+bundle'>universal bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+at+geometric+homotopies'>homotopy localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/homotopy+group'>homotopy group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+a+topos'>fundamental group of a topos</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+category'>fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> </ul> </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='#homotopycommutativity'>Homotopy commutativity</a></li></ul></li><li><a href='#examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#References'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The collection <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>S</mi> <mo>•</mo></msup><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S^\bullet,R]</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-valued functions on a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mo>•</mo></msup></mrow><annotation encoding='application/x-tex'>S^\bullet</annotation></semantics></math> is a commutative <a class='existingWikiWord' href='/nlab/show/diff/cosimplicial+algebra'>cosimplicial algebra</a>. Under the <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold–Kan correspondence</a> it maps to its <a class='existingWikiWord' href='/nlab/show/diff/Moore+complex'>Moore cochain complex</a> <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msup><mi>S</mi> <mo>•</mo></msup><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^\bullet([S^\bullet,R])</annotation></semantics></math> which is a <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a> under the <a class='existingWikiWord' href='/nlab/show/diff/cup+product'>cup product</a>: this is the <strong>cochain complex of the simplicial set</strong>.</p> <p>Notably, this cochain complex is an <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a> (an <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra</a> over the <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+operad'>E-∞ operad</a>). In <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>cohomology</a> it becomes a <a class='existingWikiWord' href='/nlab/show/diff/graded+algebra'>graded-commutative algebra</a>.</p> <h2 id='definition'>Definition</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> be commutative <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> a set, write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo><mo>=</mo><msup><mi>R</mi> <mi>S</mi></msup></mrow><annotation encoding='application/x-tex'>[S,R] = R^S</annotation></semantics></math></div> <p>for the <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-valued functions on <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>: the set of maps from <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> (using either <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> notation or <a class='existingWikiWord' href='/nlab/show/diff/exponential+object'>exponential object</a> notation).</p> <p>This is in particular naturally</p> <ul> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> (using the addition in <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>);</p> </li> <li> <p>and even an <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/module'>module</a></p> </li> <li> <p>and even an <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/algebra'>algebra</a>.</p> </li> <li> <p>and even a commutative <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> algebra (since <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is assumed to be commutative ring).</p> </li> </ul> <p>Similarly, for <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>S = (S_\bullet) : \Delta^{op} \to Set</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> write <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>S</mi> <mo>•</mo></msub><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S_\bullet,R] </annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/cosimplicial+algebra'>cosimplicial algebra</a> obtained by taking <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-valued functions in each degree. This is naturally</p> <ul> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>cosimplicial group</a></p> </li> <li> <p>and even a cosimplicial <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module</p> </li> <li> <p>and even a <a class='existingWikiWord' href='/nlab/show/diff/cosimplicial+algebra'>cosimplicial algebra</a> over <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> .</p> </li> </ul> <p>Equivalently, if we write <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo stretchy='false'>[</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>R [S_\bullet]</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+object'>simplicial</a> <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module which is in degree <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> the free <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module on the set <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>S_n</annotation></semantics></math>, we have a canonical <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>S</mi> <mo>•</mo></msub><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><mi>R</mi><mi>Mod</mi></mrow></msub><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>[</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>]</mo><mo>,</mo><mi>R</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> [S_\bullet,R] \simeq Hom_{R Mod}(R[S_\bullet], R) \,. </annotation></semantics></math></div> <p>This latter point of view is often preferred in the literature when <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo stretchy='false'>[</mo><msub><mi>S</mi> <mo>•</mo></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>R[S_\bullet]</annotation></semantics></math> is regarded as the collection of <em>chains</em> on <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>S</mi> <mo>•</mo></msub><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S_\bullet,R]</annotation></semantics></math> as that of <em>cochains</em> .</p> <p>More precisely, we should speak of chains and cochains after applying the <a class='existingWikiWord' href='/nlab/show/diff/Moore+complex'>Moore complex</a> functor. Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msub><mi>S</mi> <mo>•</mo></msub><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> C^\bullet(S,R) := C^\bullet([S_\bullet,R]) </annotation></semantics></math></div> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/Moore+complex'>Moore cochain complex</a> obtained from the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>cosimplicial group</a> <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>S</mi> <mo>•</mo></msub><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S_\bullet,R]</annotation></semantics></math>. This is the <strong>cochain complex</strong> of the simplicial set <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>. Using the <a class='existingWikiWord' href='/nlab/show/diff/cup+product'>cup product</a>, this is even a <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a>.</p> <h2 id='properties'>Properties</h2> <div class='un_prop'> <h6 id='proposition'>Proposition</h6> <p>The functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo><mo>:</mo><mi>SSet</mi><mo>→</mo><mo stretchy='false'>[</mo><msup><mi>Δ</mi> <mo lspace='0em' rspace='thinmathspace'>op</mo></msup><mo>,</mo><mi>R</mi><mi>Mod</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> [-,R] : SSet \to [\Delta^\op,R Mod] </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+functor'>symmetric</a> <a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>lax monoidal functor</a>.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>For instance Prop 3.8 in (<a href='#May03'>May03</a>) .</p> </div> <p>…</p> <h3 id='homotopycommutativity'>Homotopy commutativity</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a> of cochains <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^\bullet(S,R)</annotation></semantics></math> is not, in general, (graded) commutative. But it is homotopy commutative in that it is an <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra over an operad</a> for an <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+operad'>E-∞ operad</a>.</p> <div class='un_theorem'> <h6 id='theorem'>Theorem</h6> <p>The cochain functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo><mo>:</mo><mi>SSet</mi><mo>→</mo><mi>dgAlg</mi></mrow><annotation encoding='application/x-tex'> C^\bullet[-,R] : SSet \to dgAlg </annotation></semantics></math></div> <p>naturally factors through <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebras over</a> an <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+operad'>E-∞ operad</a>, notably the <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+operad'>EilenbergZilber operad</a> as well as the <a class='existingWikiWord' href='/nlab/show/diff/Barratt-Eccles+operad'>Barratt-Eccles operad</a>.</p> <p>In both these cases the complex of binary operations in these operads has a 0-cycle whose action <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>)</mo><mo>⊗</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^\bullet(S,R) \otimes C^\bullet(S,R) \to C^\bullet(S,R)</annotation></semantics></math> is the usual <a class='existingWikiWord' href='/nlab/show/diff/cup+product'>cup product</a>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>The statement for the Eilenberg–Zilber operad goes back to <a href='HinSch87'>HinSch87</a> . A good review is in (<a href='#May03'>May03</a>) . The statement for the Barrat–Eccles operad is in (<a href='#BerFre01'>BerFre01</a>) .</p> </div> <h2 id='examples'>Examples</h2> <ul> <li>For <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Δ</mi> <mi>Top</mi></msub><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>\Delta_{Top} : \Delta \to Top</annotation></semantics></math> the canonical topological simplices, the simplicial set <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mrow><msubsup><mi>Δ</mi> <mi>Top</mi> <mo>•</mo></msubsup></mrow></msup></mrow><annotation encoding='application/x-tex'>X^{\Delta^\bullet_{Top}}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/singular+simplicial+complex'>singular simplicial complex</a> of <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. It cochain dg-algebra is the one that computes the <a class='existingWikiWord' href='/nlab/show/diff/singular+cohomology'>singular cohomology</a> of <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>chain homology</a></li> </ul> <h2 id='References'>References</h2> <p>Basics are for instance in <em>Application 1.1.3</em> of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Charles+Weibel'>Charles Weibel</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/An+Introduction+to+Homological+Algebra'>An Introduction to Homological Algebra</a></em></li> </ul> <p>An explicit description of the cochains that express the homotopy symmetry of the cup product is given from page 30 on of the old</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Norman+Steenrod'>Norman Steenrod</a>, <em><a class='existingWikiWord' href='/nlab/files/SteenrodOnCohomologyOperations.pdf' title='Cohomology operations and obstructions to extending continuous functions'>Cohomology operations and obstructions to extending continuous functions</a></em> , Colloquium lectures (1957)</li> </ul> <p>The modern <a class='existingWikiWord' href='/nlab/show/diff/operad'>operad</a>-theoretic statement that for <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo></mrow><annotation encoding='application/x-tex'>S \in</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>SSet</a> a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a>, the cochain complex <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^\bullet([S,R])</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a> apparently goes back to</p> <ul id='HinSch87'> <li><a class='existingWikiWord' href='/nlab/show/diff/Vladimir+Hinich'>V. Hinich</a> and <span class='newWikiWord'>V. Schechtman<a href='/nlab/new/V.+Schechtman'>?</a></span>, <em>On homotopy limits of homotopy algebras</em>, in <em>K-theory, arithmetic and geometry</em>, Lecture notes Vol. 1289, Berlin 1987 pp. 240–264</li> </ul> <p>A particularly clear exposition is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Michael+Mandell'>Michael Mandell</a>, <em>Cochain multiplication</em>, Amer. J. Math. 124 (2002)</li> </ul> <p>This in turn is nicely reviewed and spelled out in section 3 of</p> <ul id='May03'> <li><a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <em>Operads and sheaf cohomology</em> (2003). <a href='https://www.math.uchicago.edu/~may/PAPERS/Esheaf.pdf'>PDF</a>. (unpublished private notes – but maybe we get permission to upload them here?)</li> </ul> <p>These describe actions of the <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+operad'>Eilenberg-Zilber operad</a> on <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msup><mi>S</mi> <mo>•</mo></msup><mo>,</mo><mi>R</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^\bullet([S^\bullet,R])</annotation></semantics></math>.</p> <p>An action of instead the <a class='existingWikiWord' href='/nlab/show/diff/Barratt-Eccles+operad'>Barratt-Eccles operad</a> is described in</p> <ul id='BerFre01'> <li><a class='existingWikiWord' href='/nlab/show/diff/Clemens+Berger'>Clemens Berger</a>, <a class='existingWikiWord' href='/nlab/show/diff/Benoit+Fresse'>Benoit Fresse</a> <em>Combinatorial operad actions on cochains</em> (<a href='http://arxiv.org/abs/math/0109158'>arXiv:0109158</a>)</li> </ul> <p>An action of a <a class='existingWikiWord' href='/nlab/show/diff/sequence+operad'>sequence operad</a>, which is isomorphic to the <span class='newWikiWord'>surjection operad<a href='/nlab/new/surjection+operad'>?</a></span> of Berger and Fresse was constructed by</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/James+McClure'>James E. McClure</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jeff+Smith'>Jeffrey H. Smith</a>, <em>Multivariable cochains operations and little <math class='maruku-mathml' display='inline' id='mathml_ac9e164f21df977a8f871c78db9d67fa92bfa1b1_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-cubes_</em> (<a href='https://arxiv.org/abs/math/0106024'>arXiv:math/0106024</a>)</li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on February 1, 2021 at 02:32:05. 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