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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+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/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/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/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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> <h4 id="topos_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#ViaTheDoldKanCorrespondence'>Via the Dold-Kan correspondence</a></li> <li><a href='#on_moore_complexes_of_cosimplicial_algebras'>On Moore complexes of cosimplicial algebras</a></li> <li><a href='#InSingularCohomology'>In singular cohomology</a></li> <li><a href='#CupProductInWhiteheadGeneralizedCohomologyTheory'>In Whitehead-generalized cohomology</a></li> </ul> <li><a href='#in_abelian_sheaf_cohomology'>In abelian sheaf cohomology</a></li> <ul> <li><a href='#in_abelian_ech_cohomology'>In abelian Čech cohomology</a></li> <li><a href='#in_echdeligne_cohomology_ordinary_differential_cohomology'>In Čech-Deligne cohomology (ordinary differential cohomology)</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Recall from the discussion at <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> that every notion of cohomology (e.g. <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>, etc) is given by Hom-spaces in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. Cohomology on an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> with coefficients in an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in \mathbf{H}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H(X,A) := \pi_0 \mathbf{H}(X,A) \,. </annotation></semantics></math></div> <p>The <em>cup product</em> is an operation on cocycles with coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_2</annotation></semantics></math> that is induced from a pairing of coefficients given by some morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>⟶</mo><msub><mi>A</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex"> A_1 \times A_2 \longrightarrow A_3 </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. In applications this is often a pairing operation with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_1 = A_2</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">A \times A \to A'</annotation></semantics></math>, and typically it is the product morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \times A \to A</annotation></semantics></math> for a <a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a> structure on the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. (See at <em><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></em>).</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>:</mo><mi>X</mi><mo>→</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">g_1 : X \to A_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>→</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">g_2 : X \to A_2</annotation></semantics></math> are two cocycles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,A_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,A_2)</annotation></semantics></math>, respectively, then their cup product with respect to this pairing is the cocycle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>g</mi> <mn>2</mn></msub></mrow></mover><msub><mi>A</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex"> g_1 \cdot g_2 \;\coloneqq\; X \stackrel{(id,id)}{\longrightarrow} X \times X \stackrel{g_1 \times g_2}{\to} A_1 \times A_2 \to A_3 </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>A</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,A_3)</annotation></semantics></math> obtained by combining the pairing with precomposition by the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta_X = (id_X, id_X)</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <h3 id="ViaTheDoldKanCorrespondence">Via the Dold-Kan correspondence</h3> <p>When the coefficient object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">A \in </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is “sufficiently abelian” in that under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> it is represented by a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> then using the <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidalness</a> of the Dold-Kan correspondence (see at <em><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></em>) one obtains a chain complex model for the cup product which makes the origin of the typical grading shift manifest.</p> <p>Write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msub><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Ch_{\bullet \geq 0}, \otimes)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>, in non-negative degrees;, regarded as a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with the standard <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>sAb</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(sAb, \otimes)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+groups">simplicial abelian groups</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with the degreewise <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>sAb</mi><mo>⟶</mo><mi>KanCplx</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">U \;\colon\; sAb \longrightarrow KanCplx \to sSet</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> to the underlying <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> (which happens to land in <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msub><mover><mo>⟶</mo><mo>≃</mo></mover><mi>sAb</mi></mrow><annotation encoding="application/x-tex">\Gamma \;\colon \; Ch_{\bullet \geq 0} \stackrel{\simeq}{\longrightarrow} sAb</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> given by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DK</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msub><munderover><mo>⟶</mo><mo>≃</mo><mi>Γ</mi></munderover><mi>sAb</mi><mover><mo>⟶</mo><mi>F</mi></mover><mi>KanCplx</mi><mo>↪</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">DK \;\colon\; Ch_{\bullet \geq 0} \underoverset{\simeq}{\Gamma}{\longrightarrow} sAb \stackrel{F}{\longrightarrow} KanCplx \hookrightarrow sSet</annotation></semantics></math> for the composite.</p> </li> </ul> <p>Now:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a>, the lax monoidal structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Γ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma_{A,B} \;\colon\; \Gamma(A) \otimes \Gamma(B) \to \Gamma(A \otimes B)</annotation></semantics></math> being induced <a href="oplax+monoidal+functor#OplaxAdjointToLax">dually</a> by the <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a>;</p> </li> <li> <p>for the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \otimes B</annotation></semantics></math> there are canonical natural <a class="existingWikiWord" href="/nlab/show/bilinear+maps">bilinear maps</a> of underlying sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> p_{A,B} \;\colon\; U(A)\times U(B) \longrightarrow U(A \otimes B) </annotation></semantics></math></div></li> </ul> <p>Using all this, then for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>V</mi> <mo>•</mo></msub><mo>⊗</mo><msub><mi>W</mi> <mo>•</mo></msub><mo>⟶</mo><msub><mi>Z</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> f \;\colon\; V_\bullet \otimes W_\bullet \longrightarrow Z_\bullet </annotation></semantics></math></div> <p>a given <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a>, this induces a map of the corresponding Kan complexes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∪</mo> <mi>DK</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>×</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \cup_{DK} \;\colon\; DK(V_\bullet) \times DK(W_\bullet) \longrightarrow DK(Z_\bullet) </annotation></semantics></math></div> <p>as the following <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∪</mo> <mi>DK</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>×</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>p</mi> <mrow><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>U</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>⊗</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></mover><mi>U</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo>⊗</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><mi>U</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cup_{DK} \;\colon\; DK(V_\bullet)\times DK(W_\bullet) = U(\Gamma(V_\bullet)) \times U(\Gamma(W_\bullet)) \stackrel{p_{\Gamma(V_\bullet), \Gamma(W_\bullet)} }{\to} U(\Gamma(V_\bullet)\otimes \Gamma(W_\bullet)) \stackrel{U(\gamma)}{\longrightarrow} U(\Gamma(V_\bullet \otimes W_\bullet)) \stackrel{U(\Gamma(f))}{\to} U(\Gamma(Z_\bullet)) = DK(Z_\bullet) \,. </annotation></semantics></math></div> <p>With this in hand then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>, the cup product on its <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">DK(V_\bullet)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">DK(W_\bullet)</annotation></semantics></math> is induced by just homming <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> into this morphism:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>×</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>W</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mo>∪</mo> <mi>DK</mi></msub><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, DK(V_\bullet)) \times \mathbf{H}(X, DK(W_\bullet)) \simeq \mathbf{H}(X, DK(V_\bullet) \times DK(W_\bullet)) \stackrel{\mathbf{H}(X,\cup_{DK})}{\longrightarrow} \mathbf{H}(X, DK(Z_\bullet)) \,. </annotation></semantics></math></div> <p>For example if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub><mo>=</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>n</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>W</mi> <mo>•</mo></msub><mo>=</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> V_\bullet = \mathbb{Z}[n_1] \,, \;\;\; W_\bullet = \mathbb{Z}[n_2] </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> concentrated in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">n_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_2</annotation></semantics></math>, respectively, on the group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mi>ℤ</mi><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><msub><mi>n</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> DK(V_\bullet) \simeq B^{n_1} \mathbb{Z} \simeq K(\mathbb{Z},n_1) </annotation></semantics></math></div> <p>is the corresponding <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a> which classifies <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> (<a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a>) with integral coefficients in the given degree. By the nature of the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a> one has</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub><mo>⊗</mo><msub><mi>W</mi> <mo>•</mo></msub><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V_\bullet \otimes W_\bullet \simeq \mathbb{Z}[n_1 + n_2] \,. </annotation></semantics></math></div> <p>Hence we may take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mo>•</mo></msub><mo>≔</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Z_\bullet \coloneqq \mathbb{Z}[n_1 + n_2]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">f = id</annotation></semantics></math> and we get a cup product</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∪</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mi>H</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cup \;\colon\; H^{n_1}(X, \mathbb{Z}) \times H^{n_2}(X, \mathbb{Z}) \to H^{n_1 + n_2}(X, \mathbb{Z}) \,. </annotation></semantics></math></div> <h3 id="on_moore_complexes_of_cosimplicial_algebras">On Moore complexes of cosimplicial algebras</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>•</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (A^\bullet)</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/cosimplicial+algebra">cosimplicial algebra</a>, its dual <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore cochain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>N</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N^\bullet(A)</annotation></semantics></math> naturally inherits the structure of a <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> under the cup product.</p> <blockquote> <p>The general formula is literally the same as that for the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">A^\bullet</annotation></semantics></math> is functions on the singular complex of a space, which is discussed below. For the moment, see below.</p> </blockquote> <p>This cup product operation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>N</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N^\bullet(A)</annotation></semantics></math> is not in general commutative. However, it is a standard fact that it becomes commutative after passing to <a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">cochain cohomology</a>.</p> <p>This suggests that the cup product should be, while not commutative, homotopy commutative in that it makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>N</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N^\bullet(A)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+algebra+in+an+%28%E2%88%9E%2C1%29-category">homotopy commutative monoid object</a>.</p> <p>This in turn should mean that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>N</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N^\bullet(A)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a> for the <a class="existingWikiWord" href="/nlab/show/E-k-operad">E-∞ operad</a>.</p> <p>That this is indeed the case is the main statement in (<a href="#BergerFresse01">Berger-Fresse 01</a>)</p> <h3 id="InSingularCohomology">In singular cohomology</h3> <p>A special case of the cup product on Moore complexes is the complex of <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a>, which is the Moore complex of the cosimplicial algebra of functions on the singular simplicial set of a topological space.</p> <p>Often in the literature by <em>cup product</em> is meant specifically the realization of the cup product on <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>:</mo><mo>=</mo><msup><mi>X</mi> <mrow><msubsup><mi>Δ</mi> <mi>Top</mi> <mo>•</mo></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex">\Pi(X)_\bullet := X^{\Delta_{Top}^\bullet}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/simplex">simplices</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> some <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>R</mi><msup><mo stretchy="false">)</mo> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">Maps(\Pi(X),R)^{\bullet}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/cosimplicial+algebra">cosimplicial ring</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-valued functions on the spaces of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplices. The corresponding <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore cochain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\bullet(X)</annotation></semantics></math> is the cochain complex whose <a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">cochain cohomology</a> is the <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> of the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: a homogeneous element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>p</mi></msub><mo>∈</mo><msup><mi>C</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega_p \in C^p(X)</annotation></semantics></math> is a function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-simplices in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Write, as usual, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">p \in \mathbb{N}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo><</mo><mn>1</mn><mo><</mo><mi>⋯</mi><mo><</mo><mi>p</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[p] = \{0 \lt 1 \lt \cdots \lt p\}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/poset">totally ordered set</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p+1</annotation></semantics></math> elements. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mu : [p] \to [p+q]</annotation></semantics></math> an injective order preserving map and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> some <a class="existingWikiWord" href="/nlab/show/simplicial+object">cosimplicial object</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>μ</mi> <mo>*</mo></msubsup><mi>K</mi><mo>:</mo><msup><mi>K</mi> <mi>p</mi></msup><mo>→</mo><msup><mi>K</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">d_\mu^* K : K^p \to K^{p+q}</annotation></semantics></math> for the image of this map under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <p>Specifically, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">p,q \in \mathbb{N}</annotation></semantics></math> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">L : [p] \to [p+q]</annotation></semantics></math> be the map that sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [p]</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [p+q]</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R : [q] \to [p+q]</annotation></semantics></math> be the map that sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [q]</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>+</mo><mi>q</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i+q \in [p+q]</annotation></semantics></math>.</p> <p>Then the <strong>cup product</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⌣</mo><mo>:</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \smile : C^\bullet(X) \otimes C^\bullet(X) \to C^\bullet(X) </annotation></semantics></math></div> <p>is the cochain map that on homogeneous elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊗</mo><mi>b</mi><mo>∈</mo><msup><mi>C</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \otimes b \in C^p(X) \otimes C^q(X) \subset C^\bullet(X) \otimes C^\bullet(X)</annotation></semantics></math> is defined by the formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⌣</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>d</mi> <mi>L</mi> <mo>*</mo></msubsup><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msubsup><mi>d</mi> <mi>R</mi> <mo>*</mo></msubsup><mi>b</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a \smile b = (d_L^* a) \cdot (d_R^* b) \,. </annotation></semantics></math></div> <blockquote> <p>There is some glue missing here to connect this back to the above general definition, something involving the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a>.</p> </blockquote> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⌣</mo><mi>b</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub></mrow></msub><mo>=</mo><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>p</mi></msub></mrow></msub><mo>⋅</mo><msub><mi>b</mi> <mrow><msub><mi>i</mi> <mi>p</mi></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub></mrow></msub></mrow><annotation encoding="application/x-tex">(a \smile b)_{i_0, \cdots, i_{p+q}} = a_{i_0, \cdots, i_p} \cdot b_{i_p, \cdots, i_{p+q}}</annotation></semantics></math>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>This cup product enjoys the following properties:</p> <ul> <li> <p>it is indeed a cochain complex morphism as claimed, in that it respects the differential: for any homogeneous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊗</mo><mi>b</mi><mo>∈</mo><msup><mi>C</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a\otimes b \in C^p(X) \otimes C^q(X)</annotation></semantics></math> as above we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo>⌣</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo stretchy="false">)</mo><mo>⌣</mo><mi>b</mi><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mi>a</mi><mo>⌣</mo><mo stretchy="false">(</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d(a \smile b) = (d a) \smile b + (-1)^p a \smile (d b) \,. </annotation></semantics></math></div></li> <li> <p>the image of the cup product on <a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">cochain cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⌣</mo><mo>:</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \smile : H^\bullet(C^\bullet(X))\otimes H^\bullet(C^\bullet(X)) \to H^\bullet(C^\bullet(X)) </annotation></semantics></math></div> <p>is associative and distributes over the addition in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(C^\bullet(X))</annotation></semantics></math>.</p> </li> </ul> </div> <p>Accordingly, the cup product makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(C^\bullet(X)) = H^\bullet(X,R)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>: the <strong>cohomology ring</strong> on the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">ordinary cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>See for instance section 3.2 of</p> <ul> <li>Hatcher, <em>Algebraic Topology</em> (<a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html">web</a> <a href="http://www.math.cornell.edu/~hatcher/AT/AT.pdf">pdf</a>)</li> </ul> <h3 id="CupProductInWhiteheadGeneralizedCohomologyTheory">In Whitehead-generalized cohomology</h3> <p>For the cup product in <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative</a> <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology+theories">Whitehead-generalized cohomology theories</a> is defined by the same recipe of “cocycles pre-composed by diagonal and post-composed by product operation”, one just has to observe that <span class="newWikiWord">symmetric smash-<a href="/nlab/new/symmetric+monoidal+smash+product">?</a></span><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal diagonals</a> on <a class="existingWikiWord" href="/nlab/show/suspension+spectra">suspension spectra</a></p> <div class="maruku-equation" id="eq:SmashMonoidalDiagonalOnSuspensionSpectra"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>D</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma^\infty X \overset{ \;\; D_X \;\; }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) </annotation></semantics></math></div> <p>do exists. This is discussed at <em>suspension spectrum – Smash-monoidal diagonalspectrum#SmashMonoidalDiagonals)</em>.</p> <p>With this, given a <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology+theory">Whitehead-generalized cohomology theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde E</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by a <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>E</mi><mo>,</mo><msup><mn>1</mn> <mi>E</mi></msup><mo>,</mo><msup><mi>m</mi> <mi>E</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>SymmetricMonoids</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spectra</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝕊</mi><mo>,</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \big(E, 1^E, m^E \big) \;\; \in \; SymmetricMonoids \big( Ho(Spectra), \mathbb{S}, \wedge \big) </annotation></semantics></math></div> <p>the smash-monoidal diagonal structure <a class="maruku-eqref" href="#eq:SmashMonoidalDiagonalOnSuspensionSpectra">(1)</a> on suspension spectra serves to define the cup product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-)\cup (-)</annotation></semantics></math> in the corresponding <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory structure</a> by:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>c</mi> <mi>i</mi></msub></mrow></mover><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mi>E</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>∪</mo><mo stretchy="false">[</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow></mover><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mi>E</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mi>E</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mrow><msup><mi>m</mi> <mi>E</mi></msup></mrow></mover><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mi>E</mi><mo maxsize="1.8em" minsize="1.8em">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \big[ \Sigma^\infty X \overset{c_i}{\longrightarrow} \Sigma^{n_i} E \big] \,\in\, {\widetilde E}{}^{n_i}(X) \\ & \Rightarrow \;\; [c_1] \cup [c_2] \, \coloneqq \, \Big[ \Sigma^\infty X \overset{ \Sigma^\infty(D_X) }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) \overset{ ( c_1 \wedge c_2 ) }{\longrightarrow} \big( \Sigma^{n_1} E \big) \wedge \big( \Sigma^{n_2} E \big) \overset{ m^E }{\longrightarrow} \Sigma^{n_1 + n_2}E \Big] \;\; \in \, {\widetilde E}{}^{n_1+n_2}(X) \,. \end{aligned} </annotation></semantics></math></div> <h2 id="in_abelian_sheaf_cohomology">In abelian sheaf cohomology</h2> <p>Traditionally the cup product is considered for abelian cohomology, such as <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> and more generally <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>.</p> <p>In that case all coefficient objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A_i</annotation></semantics></math> are complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(A_i)_\bullet</annotation></semantics></math> of sheaves and the pairing that one usually considers is the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>→</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> (A_1)_\bullet \times (A_2)_\bullet \to (A_1 \otimes A_2)_\bullet </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><msub><mo>⊕</mo> <mi>p</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>p</mi></msub><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mi>p</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (A_1 \otimes A_2)_n = \oplus_p (A_1)_p \otimes (A_2)_{n-p} \,. </annotation></semantics></math></div> <p>with differential</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">|</mo></mrow></msup><msub><mi>a</mi> <mn>1</mn></msub><mo>⊗</mo><mi>d</mi><msub><mi>a</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d (a_1 \otimes a_2) = (d a_1) \otimes a_2 + (-1)^{|a_1|} a_1 \otimes d a_2 \,. </annotation></semantics></math></div> <h3 id="in_abelian_ech_cohomology">In abelian Čech cohomology</h3> <p>The cup product has a simple expression in abelian <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_2</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es (of <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s) construct a morphism of Čech complexes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi : C^\bullet(\{U_i\}, A_1) \otimes C^\bullet(\{U_i\}, A_2) \to C^\bullet(\{U_i\}, A_1 \otimes A_2) </annotation></semantics></math></div> <p>by sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>C</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\alpha \in C^p(U,A_1)_\bullet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>∈</mo><msup><mi>C</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\beta \in C^q(U,A_2)_\bullet</annotation></semantics></math> to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo>⊗</mo><mi>β</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub></mrow></msub><mspace width="thickmathspace"></mspace><mo>:</mo><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>α</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>p</mi></msub></mrow></msub><mo>⊗</mo><msub><mi>β</mi> <mrow><msub><mi>i</mi> <mi>p</mi></msub><mo>,</mo><mi>⋯</mi><msub><mi>i</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi(\alpha \otimes \beta)_{i_0, \cdots , i_{p + q}} \;:=\; \alpha_{i_0, \cdots, i_p} \otimes \beta_{i_p, \cdots i_{p+q}} \,. </annotation></semantics></math></div> <p>For instance (<a href="#Brylinski">Brylinski, section (1.3)</a>) spring</p> <h3 id="in_echdeligne_cohomology_ordinary_differential_cohomology">In Čech-Deligne cohomology (ordinary differential cohomology)</h3> <p>For the case that of <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech</a> <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a> with coefficients in <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a>es the above yields the <em><a class="existingWikiWord" href="/nlab/show/Beilinson-Deligne+cup-product">Beilinson-Deligne cup-product</a></em> for <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cap+product">cap product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup-i+product">cup-i product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intersection+pairing">intersection pairing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+cup+product">functional cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer+sum">Baer sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operation">cohomology operation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/power+operation">power operation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Massey+product">Massey product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product+in+differential+cohomology">cup product in differential cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cup+product+in+ordinary+differential+cohomology">cup product in ordinary differential cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler+class">Euler class</a> takes <a class="existingWikiWord" href="/nlab/show/Whitney+sum">Whitney sum</a> to <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a>, see <a href="Euler+class#EulerClassOfWhitneySumIsCupProductOfEulerClasses">this Prop.</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, part III, sections 2 and 3 of <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</p> </li> <li id="May"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, 18.3 and 22.3 of <em>A concise course in algebraic topology</em> (<a href="http://www.maths.ed.ac.uk/~aar/papers/maybook.pdf">pdf</a>)</p> </li> </ul> <p>The cup product in Čech cohomology is discussed for instance in section 1.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean-Luc+Brylinski">Jean-Luc Brylinski</a>, <em>Loop spaces and characteristic classes</em></li> </ul> <p>Recall from the discussion at <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a> that all <a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercomplete</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es are modeled by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>. Accordingly understanding the cup product on simplicial presheaves goes a long way towards the most general description. For a bit of discussion of this see around page 19 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Frederick+Jardine">John Frederick Jardine</a>, <em>Lectures on simplicial presheaves</em> (<a href="http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf">pdf</a>)</li> </ul> <p>An early treatment of cup product can be found in this classic</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hassler+Whitney">Hassler Whitney</a>, <em>On Products in a Complex</em> (<a href="http://www.jstor.org/pss/1968795">JSTOR</a>)</li> </ul> <p>See also</p> <ul> <li id="BergerFresse01"><a class="existingWikiWord" href="/nlab/show/Clemens+Berger">Clemens Berger</a>, <a class="existingWikiWord" href="/nlab/show/Benoit+Fresse">Benoit Fresse</a> <em>Combinatorial operad actions on cochains</em>, Math. Proc. Cambridge Philos. Soc. 137 (2004), 135-174. (<a href="http://arxiv.org/abs/math/0109158">arXiv:math/0109158</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 13, 2025 at 18:14:02. 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