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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#InAnInfinityTopos'>In an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos – twisted nonabelian (sheaf) cohomology</a></li> <li><a href='#InAStabilizedInfinityTopos'>In a stabilized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos – twisted ES-type (sheaf) cohomology</a></li> <li><a href='#InASymmetricMonoidalInfinityCategory'>In a general symmetric monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#twisted_cohomology_with_trivial_twisting_cocycle'>Twisted cohomology with trivial twisting cocycle</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#SectionsAsTiwstedFunctions'>Sections as twisted functions…</a></li> <ul> <li><a href='#InfSections'>… and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-sections as twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-functions</a></li> </ul> <li><a href='#twisted_ktheory'>twisted K-theory</a></li> <li><a href='#actions_on_spectra'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Actions on spectra</a></li> <li><a href='#group_cohomology_with_coefficients_in_a_module'>Group cohomology with coefficients in a module</a></li> <li><a href='#twisted_bundles'>Twisted bundles</a></li> <li><a href='#cohomology_with_local_coefficients'>Cohomology with local coefficients</a></li> <ul> <li><a href='#effective_computation_of_cohomology_with_local_coefficients'>Effective computation of cohomology with local coefficients</a></li> </ul> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#chronology_of_literature_on_twisted_cohomology'>Chronology of literature on twisted cohomology</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>By the discussion at <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em>, plain cohomology is the study of</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-spaces">(∞,1)-categorical hom-spaces</a> in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a> (for “<a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a>”)</p> </li> <li> <p>or their <a class="existingWikiWord" href="/nlab/show/stabilizations">stabilizations</a> to <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-categories">stable (∞,1)-categories</a> of <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> (for “<a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a>”)</p> </li> <li> <p>or generally in <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-categories">symmetric monoidal (∞,1)-categories</a></p> </li> </ul> <p>and maybe fully generally in any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> whatsoever.</p> <p>So for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/object">object</a> the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of any other object <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> 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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(X,A)</annotation></semantics></math>. Notice that this is equivalently the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo>×</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}(X, X \times A)</annotation></semantics></math> of the trivial <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>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> over <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 idea of <em>twisted cohomology</em> then is to consider general <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>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundles">fiber ∞-bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and take the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted cohomology 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> to the type of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of this.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></th><th>twisted cohomology</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> of <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> of spaces of <a class="existingWikiWord" href="/nlab/show/sections">sections</a></td></tr> </tbody></table> <p>Given an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos <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>, then also its arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^I</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos, over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><msup><mi>Grpd</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">\infty Grpd^I</annotation></semantics></math> and it also sits over <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> by the <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a>, constituting an “<a class="existingWikiWord" href="/nlab/show/extension">extension</a>” of <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> by itself:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>incl</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>cod</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{H} \\ \downarrow^{\mathrlap{incl}} \\ \mathbf{H}^I \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,. </annotation></semantics></math></div> <p>The intrinsic cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^I</annotation></semantics></math> under this fibration is <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian</a> twisted cohomology as discussed in some detail in <em><a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">Principal ∞-bundles – theory, presentations and applications</a></em>.</p> <p>Notice that “stable cohomology”, which is traditionally called <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> may be thought of as the lowest order <a class="existingWikiWord" href="/nlab/show/Goodwillie+calculus">Goodwillie</a> approximation to <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a>: where a cocycle in nonabelian cohomology is a map to any <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>, a cocycle in <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> is a map into a <a class="existingWikiWord" href="/nlab/show/stable+homotopy+type">stable homotopy type</a>.</p> <p>In this sense the <a class="existingWikiWord" href="/nlab/show/tangent+%28infinity%2C1%29-topos">tangent (infinity,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T \mathbf{H}</annotation></semantics></math> is the lowest order linear approximation to the <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo lspace="0em" rspace="thinmathspace">Stab</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>incl</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>cod</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Stab(\mathbf{H}) \\ \downarrow^{\mathrlap{incl}} \\ T\mathbf{H} \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,. </annotation></semantics></math></div> <p>Higher-order approximations should involve a notion of higher-order forms of the tangent (∞,1)-topos, in parallel with the relationship between the <a class="existingWikiWord" href="/nlab/show/jet+bundles">jet bundles</a> and <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of a manifold. It is clear that whatever we may say in detail about the <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>th-<a class="existingWikiWord" href="/nlab/show/jet+%28%E2%88%9E%2C1%29-category">jet (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>k</mi></msup><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">J^k \mathbf{H}</annotation></semantics></math>, its intrinsic cohomology is a version of twisted cohomology which is in between <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> and stable i.e. <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a>.</p> <p>It seems that a layered analysis of nonabelian cohomology this way in higher homotopy theory should eventually be rather important, even if it hasn’t received any attention at all yet. It seems plausible that a generalization of <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> which approximates classes of <a class="existingWikiWord" href="/nlab/show/principal+infinity-bundles">principal infinity-bundles</a> not just by <a class="existingWikiWord" href="/nlab/show/universal+characteristic+classes">universal characteristic classes</a> in <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> and hence in stable cohomology, but that one wants to consider the whole Goodwillie Taylor tower of approximations to it.</p> <h2 id="Definition">Definition</h2> <p>We discuss concrete realizations of the above <a href="#Idea">general idea</a> in some cases of interest:</p> <ul> <li> <p><em><a href="#InAnInfinityTopos">In an ∞-topos</a></em></p> </li> <li> <p><em><a href="#InAStabilizedInfinityTopos">In a stabilized ∞-topos</a></em></p> </li> <li> <p><em><a href="#InASymmetricMonoidalInfinityCategory">In a general symmetric monoidal ∞-category</a></em></p> </li> </ul> <h3 id="InAnInfinityTopos">In an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos – twisted nonabelian (sheaf) cohomology</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathcal{C} = \mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>. Let <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> be any <a class="existingWikiWord" href="/nlab/show/object">object</a>, to be called the <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> object.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Aut}(A) \in Grp(\mathbf{H})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a> of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbf{Aut}(A) \in \mathbf{H}</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>. There is a canonical <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Aut}(A)</annotation></semantics></math> on <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> exhibited by the corresponding universal <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ρ</mi> <mi>A</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\to& A//\mathbf{Aut}(A) \\ && \downarrow^{\mathrlap{\rho_A}} \\ && \mathbf{B}\mathbf{Aut}(A) } \,. </annotation></semantics></math></div> <p>Let <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> be any object.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>twist</strong> for <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>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> on <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> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi \colon X \to \mathbf{B}\mathbf{Aut}(A)</annotation></semantics></math> 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>. The corresponding <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated</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>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> over <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> which is the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>χ</mi> <mo>*</mo></msup><msub><mi>ρ</mi> <mi>A</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ρ</mi> <mi>A</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>χ</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \chi^\ast \rho_A &\to& A//\mathbf{Aut}(A) \\ \downarrow && \downarrow^{\mathrlap{\rho_A}} \\ X &\stackrel{\chi}{\to}& \mathbf{B}\mathbf{Aut}(A) } </annotation></semantics></math></div> <p>we call the <a class="existingWikiWord" href="/nlab/show/local+coefficient+%E2%88%9E-bundle">local coefficient ∞-bundle</a> for twisted <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>-cohomology classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted <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>-cohomology</em> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msup><mi>χ</mi> <mo>*</mo></msup><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(\chi^\ast \rho_A) \in \infty Grpd \,. </annotation></semantics></math></div> <p>The <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted cohomology set</strong> 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> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msup><mi>χ</mi> <mo>*</mo></msup><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \pi_0 \Gamma_X(\chi^\ast \rho_A) \in Set </annotation></semantics></math></div></div> <p>Special cases of this definition are implicit in traditional literature. The above statement appears in this form in (<a href="#NikolausSchreiberStevenson12">Nikolaus-Schreiber-Stevenson 12</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted cohomology is equivalently the ordinary cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> 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><msub><mi>ρ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\rho_A</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> of <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> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbf{Aut}(A)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msup><mi>χ</mi> <mo>*</mo></msup><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>χ</mi><mo>,</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(\chi^\ast \rho_A) \simeq \mathbf{H}_{/\mathbf{B}\mathbf{Aut}(A)}(\chi, \rho_A) \,. </annotation></semantics></math></div></div> <h3 id="InAStabilizedInfinityTopos">In a stabilized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-topos – twisted ES-type (sheaf) cohomology</h3> <p>Let now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = Stab(\mathbf{H})</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> in an ambient <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>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msub><mi>CRing</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \in CRing_\infty(\mathbf{H})</annotation></semantics></math> be a corresponding <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a> object. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>↪</mo><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> GL_1(E) \hookrightarrow \mathbf{Aut}(E) \in Grp(\mathbf{H}) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+units">∞-group of units</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>Now a twist <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi \;\colon\; X \to \mathbf{B}GL_1(E)</annotation></semantics></math> classifies an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-lines. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-cohomology is again the (stable) homotopy type of sections of this.</p> <p>For the case of <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> (see the references there) this description goes back to <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>. The above general abstract description is developed in (<a href="#ABG10">Ando-Blumberg-Gepner 10</a>).</p> <p>For more details see</p> <ul> <li> <p>at <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></em> the section <em><a href="%28infinity%2C1%29-module+bundle#SectionsAndTwistedCohomology">Sections and twisted cohomology</a></em></p> </li> <li> <p>at <em><a class="existingWikiWord" href="/nlab/show/twisted+bivariant+cohomology">twisted bivariant cohomology</a></em> the section <em><a href="bivariant+cohomology+theory#AxiomatizazionInHomotopyTheory">Axiomatization in homotopy theory</a></em> .</p> </li> </ul> <div class="num_remark" id="PicardGeneralizingGL1"> <h6 id="remark_2">Remark</h6> <p>There are canonical maps</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>B</mi></mstyle><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>E</mi><mi>Line</mi><mo>↪</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>E</mi><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}GL_1(E) \simeq E Line \hookrightarrow Pic(E Mod) \hookrightarrow E Mod \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pic(E Mod)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/Picard+%E2%88%9E-groupoid">Picard ∞-groupoid</a>. This suggest to speak not just of twists of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>E</mi><mi>Line</mi><mo>↪</mo><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\chi \colon X \to \mathbf{B}GL_1(E) \simeq E Line \hookrightarrow E Mod</annotation></semantics></math> but more generally of twists of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo lspace="verythinmathspace">:</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\chi \colon Pic(E Mod) \hookrightarrow E Mod</annotation></semantics></math>. While these in general no longer define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundles">fiber ∞-bundles</a> (so that sections of them are strictly speaking in general no longer locally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-cohomology cocycles), this more general notion has the advantage that it makes sense also in <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-categories">symmetric monoidal (∞,1)-categories</a> different from those of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(\mathbf{H})</annotation></semantics></math>.</p> <p>This we turn to <a href="#InASymmetricMonoidalInfinityCategory">below</a>.</p> </div> <h3 id="InASymmetricMonoidalInfinityCategory">In a general symmetric monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category</h3> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>If in the above situation we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, E Mod]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundles">(∞,1)-module bundles</a> on <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>, then given an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\chi \in [X,E Mod]</annotation></semantics></math> its homotopy type of sections, hence the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>-twisted cohomology 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> is equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><mi>Mod</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>𝕀</mi> <mi>X</mi></msub><mo>,</mo><mi>χ</mi><mo stretchy="false">)</mo><mo>∈</mo><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Hom_{[X, E Mod]}(\mathbb{I}_X, \chi) \in \infty Grpd \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕀</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{I}_X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> object, the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a> over <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> </div> <p>In view of this and remark <a class="maruku-ref" href="#PicardGeneralizingGL1"></a> one considers the following.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a>.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>∈</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi \in Pic(\mathcal{C})</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/Picard+%E2%88%9E-groupoid">Picard ∞-groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> we call a <em>twist</em> for <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X, A \in \mathcal{C}</annotation></semantics></math> any two <a class="existingWikiWord" href="/nlab/show/objects">objects</a>, we say that the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>−</mo><mi>twisted</mi></mrow><annotation encoding="application/x-tex">\chi-twisted</annotation></semantics></math> cohomology 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> 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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></em> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>χ</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}(X, \chi \otimes A) \in \infty Grpd \,. </annotation></semantics></math></div></div> <h2 id="properties">Properties</h2> <h3 id="twisted_cohomology_with_trivial_twisting_cocycle">Twisted cohomology with trivial twisting cocycle</h3> <blockquote> <p>old material, to be harmonized…</p> </blockquote> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">{*} \to B</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a>. Then</p> <ul> <li> <p>we say that the <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>→</mo><mo>*</mo><mo>→</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X \to * \to B) \in \mathbf{H}(X,B)</annotation></semantics></math></p> <p>is the <strong>trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cocycle</strong> on <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> </li> <li> <p>the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:\hat{B}\to B</annotation></semantics></math> induces a <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to \hat B \stackrel{f}{\to} B</annotation></semantics></math></p> <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>.</p> </li> </ul> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mo>*</mo><mo stretchy="false">]</mo><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">([*],f)</annotation></semantics></math>-twisted cohomology with trivial twisting cocycle is equivalent to the ordinary <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> with coefficients in the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">[</mo><mo>*</mo><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≃</mo><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"> \mathbf{H}_{[*]}(X,f) \simeq \mathbf{H}(X,A) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By definition, the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <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> is the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\to& * \\ \downarrow && \downarrow \\ \hat B &\stackrel{f}{\to}& B } </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>. Since the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid valued <a class="existingWikiWord" href="/nlab/show/derived+hom-space">hom</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> is <a class="existingWikiWord" href="/nlab/show/exact+functor">exact</a> with respect ot <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a> (by definition of homotopy limits), it follows that for every object <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>, there is fibration sequence of <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>const</mi> <mo>*</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{H}(X,A) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{const_{*}}} \\ \mathbf{H}(X,\hat B) &\to& \mathbf{H}(X,B) } \,. </annotation></semantics></math></div> <p>By definition of twisted cohomology, this identifies</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>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">[</mo><mo>*</mo><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X,A) \simeq \mathbf{H}_{[*]}(X,f) \,. </annotation></semantics></math></div></div> <p>For this reason, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is pointed, it is customary to call the set of equivalence classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">[</mo><mi>c</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0\mathbf{H}_{[c]}(X;f)</annotation></semantics></math> the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>-twisted <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>-cohomology 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></strong>, and to denote it by the symbol</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mo stretchy="false">[</mo><mi>c</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_{[c]}(X,A) </annotation></semantics></math></div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>The cohomology fibration sequence <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><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,A) \to \mathbf{H}(X,\hat B) {\to} \mathbf{H}(X,B)</annotation></semantics></math> can be seen as an <strong>obstruction problem</strong> in cohomology:</p> <ul> <li>the <strong>obstruction</strong> to lifting a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat B</annotation></semantics></math>-cocycle to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-cocycle is its image in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cohomology (all with respect to the given <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a>)</li> </ul> <p>But it also says:</p> <ul> <li><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>-cocycles are, up to equivalence, precisely those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat B</annotation></semantics></math>-cocycles whose class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cohomology is the class of the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-cocycle.</li> </ul> </div> <h2 id="examples">Examples</h2> <h3 id="SectionsAsTiwstedFunctions">Sections as twisted functions…</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> and <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 manifold, both regarded a 0-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> objects in the <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 on the site <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> (that of <a class="existingWikiWord" href="/nlab/show/Lie+infinity-groupoid">Lie infinity-groupoid</a>s), a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">X \to V</annotation></semantics></math> is simply smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued function on <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>Now let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a Lie group with smooth <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\rho : \mathbf{B}G \to Vect</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mo>•</mo><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\rho(\bullet) = V</annotation></semantics></math>. Then the corresponding <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">V//G</annotation></semantics></math> sits in the <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mover><mo>→</mo><mi>p</mi></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V \to V//G \stackrel{p}{\to} \mathbf{B}G \,. </annotation></semantics></math></div> <p>Hence we can ask for the <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>-twisted cohomology 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> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. Now, a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g : X \to \mathbf{B}G</annotation></semantics></math> is one classifying a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> on <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>. By looking at this in <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a> it is immediate to convince onself that cocycles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to V//G</annotation></semantics></math> such that the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mover><mo>→</mo><mi>p</mi></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to V//G \stackrel{p}{\to} \mathbf{B}G</annotation></semantics></math> is equivalent to the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> are precisely the <a class="existingWikiWord" href="/nlab/show/section">section</a>s of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+bundle">associated vector bundle</a>:</p> <p>on a patch <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> of a good cover over wich <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> has been trivialized, the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to V//G</annotation></semantics></math> is simply a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\sigma_i : U_i \to V</annotation></semantics></math>. Then on double overlaps it is a smooth <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></msub><mo>→</mo><msub><mi>σ</mi> <mi>j</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\sigma_i|_{U_{i j}} \to \sigma_j|_{U_i j}</annotation></semantics></math> whose components in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> are required to be those of the given cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>. That means exactly that the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>σ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sigma_i)</annotation></semantics></math> are glued on double overlaps precisely as the local trivializations of a global section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi></mrow><annotation encoding="application/x-tex">\sigma : X \to P \times_G V</annotation></semantics></math> would.</p> <p>Hence we find the <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>-twisted cohomology is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>sections</mi><mspace width="thickmathspace"></mspace><mi>of</mi><mspace width="thickmathspace"></mspace><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_{[g]}(X,V) = \{sections\; of\; P \times_G V\} \,. </annotation></semantics></math></div> <p>In this sense <strong>a section is a twisted function</strong>.</p> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mover><mo>→</mo><mi>p</mi></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">V//G \stackrel{p}{\to} \mathbf{B}G</annotation></semantics></math> is not itself a homotopy fiber, but is a <em>lax</em> fiber, in that we have a <a class="existingWikiWord" href="/nlab/show/2-limit">lax pullback</a> (really a <em>comma object</em> )</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Vect</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ V//G &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& Vect } \,, </annotation></semantics></math></div> <p>where in the bottom right corner we have <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> (regarded as a <a class="existingWikiWord" href="/nlab/show/stack">stack</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> in the evident way) and where the right vertical morphism sends the point to the ground field vector space <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> (or rather sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">U \in CartSp</annotation></semantics></math> to the trivial vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">U \times k</annotation></semantics></math> ).</p> <p>We may paste to this the homotopy pullback along the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g : X \to \mathbf{B}G</annotation></semantics></math> to obtain</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Vect</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P\times_G V &\to& V//G &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G &\to& Vect } \,. </annotation></semantics></math></div> <p>This makes is manifest that a section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi></mrow><annotation encoding="application/x-tex">\sigma : X \to P \times_G V</annotation></semantics></math> is also the same as a natural transformation from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>k</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">const_k : X \to Vect</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>g</mi></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">X \stackrel{g}{\to} \mathbf{B}G \to Vect</annotation></semantics></math>.</p> <p>Notice moreover that in the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = U(1)</annotation></semantics></math> and for ground field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{C}</annotation></semantics></math> we may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math> as the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mi>Line</mi><mo>↪</mo><mi>ℂ</mi><mi>Mod</mi><mo>=</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\mathbb{C} Line \hookrightarrow \mathbb{C} Mod = Vect</annotation></semantics></math> and think of the twisting cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>g</mi></mover><mi>ℂ</mi><mi>Line</mi><mo>↪</mo><mi>ℂ</mi><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \stackrel{g}{\to} \mathbb{C}Line \hookrightarrow \mathbb{C}Mod \,. </annotation></semantics></math></div> <h4 id="InfSections">… and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-sections as twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-functions</h4> <p>Regarded this way, the above discussion has a generalization to the case where the <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> is replaced with any <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a> <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> and we consider</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>τ</mi></mover><mi>R</mi><mi>Line</mi><mo>↪</mo><mi>R</mi><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \stackrel{\tau}{\to} R Line \hookrightarrow R Mod \,. </annotation></semantics></math></div> <p>Twisted cohomology in terms of such morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> is effectively considered in</p> <ul> <li id="AndoBlumbergGepner10"><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em>Twists of K-theory and TMF</em>, in <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a> et al. (eds.), <em>Superstrings, Geometry, Topology, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras</em>, volume 81 of <em>Proceedings of Symposia in Pure Mathematics</em>, 2009 (<a href="http://arxiv.org/abs/1002.3004">arXiv:1002.3004</a>)</li> </ul> <p>and in unpublished work of <a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a> et al. For more on this see the discussion at <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></em>.</p> <p>More generally one can hence twist with maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> X \to Pic(R) \hookrightarrow R Mod </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/Picard+%E2%88%9E-group">Picard ∞-group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math>.</p> <p>See also at <em><a href="infinity-group+of+units#AugmentedDefinition">∞-group of units – augmented definition</a></em>.</p> <h3 id="twisted_ktheory">twisted K-theory</h3> <p>In the context of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> a coefficient object for <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> is a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</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>: the <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>-cohomology of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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> with coefficients in <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> is the set of homotopy classes of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \to A</annotation></semantics></math>. For instance, as a model of the degree-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> space in the <a class="existingWikiWord" href="/nlab/show/K-theory+spectrum">K-theory spectrum</a> one can take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>Fred</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = Fred(H)</annotation></semantics></math>, the space of Fredholm operators on a separable Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>. There is a canonical action on this space of the projective unitary group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>P</mi><mi>U</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = P U(H)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>U</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P U(H)</annotation></semantics></math> has the homotopy type of an <a class="existingWikiWord" href="/nlab/show/Eilenberg-Mac+Lane+space">Eilenberg-Mac Lane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathbb{Z},2)</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>U</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P U(H)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> defines a class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in H^3(X,\mathbb{Z})</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">ordinary integral cohomology</a> (this may also be thought of as the class of a twisting <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>). The twisted K-theory (in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>) 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> with that class as its twist is the set of homotopy classes of sections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>P</mi><msub><mo>×</mo> <mrow><mi>P</mi><mi>U</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow></msub><mi>Fred</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to P \times_{P U(H)} Fred(H)</annotation></semantics></math> of the associated bundle.</p> <h3 id="actions_on_spectra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Actions on spectra</h3> <p>The above example generalizes straightforwardly to the case that</p> <ul> <li> <p><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> is a <a class="existingWikiWord" href="/nlab/show/spectrum">connective spectrum</a>, i.e. <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> that is an infinite <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>. (We need to assume a connective spectrum given by an infinite loop space so that <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> can be regarded as living in the category of topologicall spaces along with the other objects, such as classifying spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of nonabelian groups);</p> </li> <li> <p>with a (topological) <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> acting on <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> by automorphisms and</p> </li> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">P \to X.</annotation></semantics></math></p> </li> </ul> <p>In this case there is an established definition of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> with coefficients <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> <em>twisted</em> by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> as follows.</p> <p>From the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> we obtain the associated <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>-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \times_G A \to X</annotation></semantics></math>. Then a twisted <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>-cocycle on <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> is defined to be a <a class="existingWikiWord" href="/nlab/show/section">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">X \to P \times_G A</annotation></semantics></math> of this associated bundle. The collection of homotopy classes of these sections is the <em>twisted <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>-cohomology 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></em> with the twist specified by the class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>.</p> <p>This is the definition of twisted cohomology as it appears for instance essentially as definition 22.1.1 of the May–Sigursson reference below (when comparing with their definition take their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to be the trivial group and identify their <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> with our <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>).</p> <p>It is clearly a particular case of the general definition of twisted cohomology given above:</p> <ul> <li> <p>the <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 <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> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of topological spaces</p> </li> <li> <p>the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </li> <li> <p>the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{B}</annotation></semantics></math> is the homotopy quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>≃</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">A//G\simeq \mathbf{E}G\times_G A</annotation></semantics></math>.</p> </li> <li> <p>the twisting cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> is the element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Top</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Top}(X,\mathbf{B}G)</annotation></semantics></math> defining the principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P\to X</annotation></semantics></math>.</p> </li> </ul> <p>Indeed, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is pointed, we have a fibration sequence</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>A</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> A \to A//G \to \mathbf{B}G </annotation></semantics></math></div> <p>and the homotopy pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>P</mi> <mi>A</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mrow><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mi>f</mi></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>c</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \array{ P_A &\to& {A//G} \\ \downarrow && \downarrow{f} \\ X &\stackrel{c}{\to}& \mathbf{B}G }\, </annotation></semantics></math></div> <p>is the <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>-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P\times_G A\to X</annotation></semantics></math>.</p> <p>The obstruction problem described by this example reads as folllows:</p> <ul> <li>the obstruction to lifting a (“nonabelian” or “twisted”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">A//G</annotation></semantics></math>-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g : X \to A//G</annotation></semantics></math> to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">^</mo></mover><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\hat g : X \to A</annotation></semantics></math> is its image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">\delta g </annotation></semantics></math> in first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mi>g</mi><mo>∈</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>Top</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\delta g \in H^1(X,G) :=\pi_0 \mathbf{Top}(X, \mathbf{B} G)</annotation></semantics></math>.</li> </ul> <p>Read the other way round it says:</p> <ul> <li><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>-cocycles are precisely those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-twisted <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>-cocycles whose twist vanishes.</li> </ul> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>Since the associated bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">P \times_G A</annotation></semantics></math> is in general no longer itself a spectrum, twisted cohomology is not an example of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized Eilenberg-Steenrod cohomology</a>.</p> <p>To stay within the spectrum point of view, May–Sigurdsson suggested that twisted cohomology should instead be formalized in terms of <em><a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></em>, where one thinks of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">P \times_G A</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/parameterized+spectrum">parameterized family of spectra</a>.</p> </div> <h3 id="group_cohomology_with_coefficients_in_a_module">Group cohomology with coefficients in a module</h3> <p>Some somewhat trivial examples of this appear in various context. For instance <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> on a group with coefficients in a nontrivial module can be regarded as an example of twisted cohomology. See there for more details.</p> <p>Compare this to the example below of cohomology “with local coefficients”. It is the same principle in both cases.</p> <h3 id="twisted_bundles">Twisted bundles</h3> <p>To get a feeling for how the definition does, it is instructive to see how for the fibration sequence coming from an ordinary central extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">K \to \hat G \to G</annotation></semantics></math> of ordinary groups as</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>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mover><mo>→</mo><mi>ω</mi></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>K</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}\hat G \to \mathbf{B}G \stackrel{\omega}{\to} \mathbf{B}^2 K </annotation></semantics></math></div> <p>classified by a <a class="existingWikiWord" href="/nlab/show/group+cohomology">group 2-cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math>-cohomology produces precisely the familiar notion of <a class="existingWikiWord" href="/nlab/show/twisted+bundles">twisted bundles</a>, with the twist being the lifting <a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a> that obstructs the lift of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-bundle to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math>-bundle.</p> <p>This is also the first example in the list in the last section of</p> <ul> <li><a class="existingWikiWord" href="/schreiber/show/Background+fields+in+twisted+differential+nonabelian+cohomology">Background fields in twisted differential nonabelian cohomology</a></li> </ul> <p>and contains examples that are of interest in the wider context of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>.</p> <p>See also <a href="http://golem.ph.utexas.edu/category/2009/03/twisted_differential_string_an.html">Twisted Differential String- and Fivebrane-Structures</a>.</p> <h3 id="cohomology_with_local_coefficients">Cohomology with local coefficients</h3> <p>What is called <strong>cohomology with local coefficients</strong> is twisted cohomology with the twist given by the class represented by the universal cover space of the base space, which is to say: by the action of the fundamental group of the base space.</p> <p>In the classical case of ordinary cohomology, C. A. Robinson in 1972 constructed a <em>twisted</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>π</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\pi,n)</annotation></semantics></math>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>π</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K(\pi,n)</annotation></semantics></math>, so that, for nice spaces, the cohomology with local coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>H</mi><mo stretchy="false">˜</mo></mover> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde H^n(X,\pi)</annotation></semantics></math> with respect to a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mo>:</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\varepsilon:\pi_1(X)\to Aut(\pi)</annotation></semantics></math> is given by homotopy classes of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>π</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\to \tilde K(\pi,n)</annotation></semantics></math> compatible with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">\varepsilon.</annotation></semantics></math></p> <p>More generally, for any 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>, let <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> be a coefficient object that is equipped with an action of the first <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X)</annotation></semantics></math> 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>. (Such an action is also called an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-valued <a class="existingWikiWord" href="/nlab/show/local+system">local system</a> on <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>Then there is the <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a></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>A</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \to A//\pi_1(X) \to \mathbf{B} \pi_1(X) </annotation></semantics></math></div> <p>of this action.</p> <p>Notice that there is a canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c : X \to \mathbf{B} \pi_1(X)</annotation></semantics></math>, the one that classifies the universal cover 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>Then <strong><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>-cohomology with local coefficients</strong> on <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> is nothing but the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>-twisted <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>-cohomology 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> in the above sense.</p> <h4 id="effective_computation_of_cohomology_with_local_coefficients">Effective computation of cohomology with local coefficients</h4> <p>By <em>effective</em>, we mean involving as much as possible only calculations within finite dimensional linear algebra. For definiteness, we work in the smooth context and require the locally constant sheaf <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">\mathcal{A}</annotation></semantics></math> to have stalks of finite dimensional vector spaces over a field <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> (<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">\mathbb{R}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>). Let <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> be a connected <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>-dimensional manifold. The sheaf <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">\mathcal{A}</annotation></semantics></math> can then be seen as the sheaf of germs of locally constant sections of a vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\to X</annotation></semantics></math> endowed with a flat connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>. The fibers of <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> are isomorphic with the stalks of <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">\mathcal{A}</annotation></semantics></math>, all of which are isomorphic to some finite dimensional vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{A}</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\tilde{X} \to X</annotation></semantics></math> denote the universal cover 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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>=</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi = \pi_1(X)</annotation></semantics></math> its fundamental group. It is well known that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> acts by <a class="existingWikiWord" href="/nlab/show/deck+transformation">deck transformation</a> diffeomorphisms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{X}</annotation></semantics></math> and also induces a <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> representation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>π</mi><mo>→</mo><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho\colon \pi \to \mathbf{Aut}(\bar{A})</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{A}</annotation></semantics></math>.</p> <p>Consider the vector bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>p</mi></msup><mi>X</mi><msub><mo>⊗</mo> <mi>X</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\Lambda^p X \otimes_X A</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>p</mi></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">\Lambda^p X</annotation></semantics></math> is the bundle of differential <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>-forms, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>A</mi> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^p_A(-)</annotation></semantics></math> denoting the sheaf of its sections (differential forms <em>twisted</em> by <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>). Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mo>∇</mo></msub><mo lspace="verythinmathspace">:</mo><msubsup><mi>Ω</mi> <mi>A</mi> <mi>p</mi></msubsup><mo>→</mo><msubsup><mi>Ω</mi> <mi>A</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">d_\nabla \colon \Omega^p_A \to \Omega^{p+1}_A</annotation></semantics></math> denote the correspondingly twisted de Rham differential, defined by the property that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mo>∇</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo>⊗</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>ω</mi><mo>⊗</mo><mi>a</mi><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><mi>ω</mi><mo stretchy="false">|</mo></mrow></msup><mi>ω</mi><mo>∧</mo><mo>∇</mo><mi>a</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> d_\nabla(\omega \otimes a) = d\omega \otimes a + (-1)^{|\omega|} \omega \wedge \nabla a , </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> is the ordinary de Rham differential, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\nabla a</annotation></semantics></math> is seen as a section of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mn>1</mn></msup><mi>X</mi><msub><mo>⊗</mo> <mi>X</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\Lambda^1 X \otimes_X A</annotation></semantics></math>, and with the wedge operation acting in the obvious way. The complex of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>A</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mo>∇</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega^\bullet_A(-),d_\nabla)</annotation></semantics></math> is then a <span class="newWikiWord">soft sheaf<a href="/nlab/new/soft">?</a></span> resolution of the sheaf <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">\mathcal{A}</annotation></semantics></math> of locally constant sections of <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>. Its cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>A</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mo>∇</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(X;A,\nabla) = H^p(\Omega^\bullet_A(X),d_\nabla)</annotation></semantics></math> is then isomorphic to the sheaf cohomology 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> with coefficients in the locally constant sheaf <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">\mathcal{A}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(X,\mathcal{A}) \simeq H^p(X;A,\nabla)</annotation></semantics></math>.</p> <p>Now, the bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\to X</annotation></semantics></math> and the connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> both pull back to the universal covering space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{X}</annotation></semantics></math>, that is to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{A} \to \tilde{X}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{\nabla}</annotation></semantics></math>. Since now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{X}</annotation></semantics></math> is simply connected, we can globally trivialize this bundle as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>≃</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo>×</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{A} \simeq \tilde{X} \times \bar{A}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{\nabla}</annotation></semantics></math> to the trivial connection thereon. Similarly, the structure of the sheaf of twisted differential forms can be simplified to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>Ω</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\Omega^p_{\tilde{A}}(-) \simeq \Omega^p(-) \otimes \bar{A}</annotation></semantics></math>, with the action of the twisted de Rham differential given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></msub><mo stretchy="false">(</mo><mi>ω</mi><mo>⊗</mo><mover><mi>a</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>ω</mi><mo>⊗</mo><mover><mi>a</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">d_{\tilde{\nabla}} (\omega \otimes \bar{a}) = d\omega \otimes \bar{a}</annotation></semantics></math>. This observation allows us to conclude that, on the universal covering space, we have the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo>;</mo><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">H^p(\tilde{X}; \tilde{A},\tilde{\nabla}) \simeq H^p(\tilde{X}) \otimes \bar{A}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(\tilde{X})</annotation></semantics></math> denotes the ordinary de Rham cohomology.</p> <p>The pull back along a deck transformation diffeomorphism, induces a linear action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> on forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^p(\tilde{X})</annotation></semantics></math>. Combined with the holonomy representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{A}</annotation></semantics></math>, this defines a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^p_{\tilde{A}}(X)</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mi>π</mi></msup><mo>⊆</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^p_{\tilde{A}}(\tilde{X})^\pi \subseteq \Omega^p_{\tilde{A}}</annotation></semantics></math> denote the subspace of twisted forms that is invariant under the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>. It is not hard to notice the isomorphism of complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mi>π</mi></msup><mo>,</mo><msub><mi>d</mi> <mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></msub><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>A</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mo>∇</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega^\bullet_\tilde{A}(\tilde{X})^\pi, d_{\tilde{\nabla}}) \simeq (\Omega^\bullet_A(X), d_\nabla)</annotation></semantics></math> and hence of their cohomologies, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mi>π</mi></msup><mo>,</mo><msub><mi>d</mi> <mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(X; A,\nabla) \simeq H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^\pi, d_{\tilde{\nabla}})</annotation></semantics></math>. Furthermore, the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^p_{\tilde{A}}(\tilde{X})</annotation></semantics></math> commutes with the differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></msub></mrow><annotation encoding="application/x-tex">d_{\tilde{\nabla}}</annotation></semantics></math> and hence induces an action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo>;</mo><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">H^p(\tilde{X}; \tilde{A},\tilde{\nabla}) \simeq H^p(\tilde{X}) \otimes \bar{A}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> also acts in the obvious and compatible way on each tensor factor. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><msup><mo stretchy="false">)</mo> <mi>π</mi></msup></mrow><annotation encoding="application/x-tex">(H^p(\tilde{X})\otimes \bar{A})^\pi</annotation></semantics></math> denote the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-invariant subspace.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Suppose that there exists a decomposition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mi>π</mi></msup><mo>⊕</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mover><mi>π</mi><mo stretchy="false">^</mo></mover></msup></mrow><annotation encoding="application/x-tex">\Omega^p_{\tilde{A}}(\tilde{X}) \simeq \Omega^p_{\tilde{A}}(\tilde{X})^\pi \oplus \Omega^p_{\tilde{A}}(\tilde{X})^{\hat{\pi}}</annotation></semantics></math> as representations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mi>p</mi></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mover><mi>π</mi><mo stretchy="false">^</mo></mover></msup></mrow><annotation encoding="application/x-tex">\Omega^p_{\tilde{A}}(\tilde{X})^{\hat{\pi}}</annotation></semantics></math> having no <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-invariant subspace. Then we have the following isomorphism for each <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>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><msup><mo stretchy="false">)</mo> <mi>π</mi></msup></mrow><annotation encoding="application/x-tex">H^p(X,\mathcal{A}) \simeq (H^p(\tilde{X})\otimes \bar{A})^\pi</annotation></semantics></math>.</p> </div> <p>Whether the decomposition hypothesis actually holds may depend on the properties of the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>. For instance, it does hold if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> is compact (finite, in particular). Other cases, have to be examined individually.</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Start with the short exact sequence of complexes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mi>π</mi></msup><mo>→</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mover><mi>π</mi><mo stretchy="false">^</mo></mover></msup><mo>→</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to \Omega^\bullet_{\tilde{A}}(\tilde{X})^\pi \to \Omega^\bullet_{\tilde{A}}(\tilde{X}) \to \Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}} \to 0 . </annotation></semantics></math></div> <p>The corresponding long exact sequence in cohomology is equivalent to the short exact sequences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo>→</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mover><mi>π</mi><mo stretchy="false">^</mo></mover></msup><mo>,</mo><msub><mi>d</mi> <mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></msub><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to H^p(X,\mathcal{A}) \to H^p(\tilde{X}) \otimes \bar{A} \to H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}}, d_{\tilde{\nabla}}) \to 0 </annotation></semantics></math></div> <p>for each value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>. The reason that all the connecting maps in the long exact sequence are zero is representation theoretic, since all the relevant maps are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-equivariant. Since, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{p+1}(X,\mathcal{A})</annotation></semantics></math> carries a trivial representation by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>, while the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><msup><mo stretchy="false">)</mo> <mover><mi>π</mi><mo stretchy="false">^</mo></mover></msup><mo>,</mo><msub><mi>d</mi> <mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}}, d_{\tilde{\nabla}})</annotation></semantics></math> representation has no <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-invariant subspace, by <a class="existingWikiWord" href="/nlab/show/Schur%27s+lemma">Schur's lemma</a>, the only equivariant map from the latter to the former is zero. From the same observation, we easily see that the inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(X,\mathcal{A})</annotation></semantics></math> must coincide with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-invariant subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><msup><mo stretchy="false">)</mo> <mi>π</mi></msup></mrow><annotation encoding="application/x-tex">(H^p(\tilde{X}) \otimes \bar{A})^\pi</annotation></semantics></math>.</p> </div> <p>The presentation of cohomology 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> with local coefficients <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">\mathcal{A}</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-invariant de Rham cohomology of the universal covering space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{X}</annotation></semantics></math> twisted by the holonomy representation on the stalk <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{A}</annotation></semantics></math> is originally due to (<a href="#Eil47">Eilenberg 47</a>). It is also discussed in Chapter VI of (<a href="#Whit78">Whitehead 78</a>). The idea to look at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-invariant subspace of the twisted de Rham cohomology of the universal covering space scan be found in an answer by <a href="#MichorMO">Peter Michor</a> on <a href="http://mathoverflow.net/">MathOverlflow</a>.</p> <p>The above result can be seen as an effective way to compute the sheaf cohomology groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(X,\mathcal{A})</annotation></semantics></math> since all it requires is the knowledge of the following finite dimensional representations of the fundamental group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>=</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi = \pi_1(X)</annotation></semantics></math>: the deck transformations on the de Rham cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^k(\tilde{X})</annotation></semantics></math> of the covering space, and the holonomy representation on a typical stalk <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{A}</annotation></semantics></math> of the locally constant sheaf <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">\mathcal{A}</annotation></semantics></math>. Obtaining the invariant subspace of their tensor product can then be done using usual representation theory methods, which involve only finite dimensional linear algebra. Unfortunately, it appears that the requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> is finite is rather important for the argument. It is not entirely clearly how to proceed if, for instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\pi = \mathbb{Z}</annotation></semantics></math> or is non-abelian and infinite.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+Umkehr+map">twisted Umkehr map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+cohomology">twisted differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+equivariant+cohomology">twisted equivariant cohomology</a></p> </li> <li> <p><a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos+--+structures#TwistedCohomology">twisted cohomology</a> in <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos+--+structures">cohesive (∞,1)-topos – structures</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+ordinary+cohomology">twisted ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <span class="newWikiWord">twisted tmf<a href="/nlab/new/twisted+tmf">?</a></span></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+ordinary+cohomology">twisted ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Bredon+cohomology">twisted Bredon cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+equivariant+K-theory">twisted equivariant K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+equivariant+cohomology">twisted equivariant cohomology</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Original articles on <a class="existingWikiWord" href="/nlab/show/twisted+ordinary+homology">twisted ordinary homology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kurt+Reidemeister">Kurt Reidemeister</a>, <em>Überdeckungen von Komplexen</em>, Crelle <strong>173</strong> (1935), 164-173 [<a href="http://dx.doi.org/10.1515/crll.1935.173.164">doi:10.1515/crll.1935.173.164</a>]</li> </ul> <p>and independently:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Norman+E.+Steenrod">Norman E. Steenrod</a>, <em>Homology with local coefficients</em>, Annals of Mathematics (Second Series) <strong>44</strong> 4 (1943) 610-627 [<a href="https://doi.org/10.2307/1969099">doi:10.2307/1969099</a>]</li> </ul> <p>On representing twisted <a class="existingWikiWord" href="/nlab/show/Eilenberg%E2%80%93MacLane+spaces">Eilenberg–MacLane spaces</a> and <a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a> for twisted ordinary cohomology:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, <em>Cohomology operations with local coefficients</em>, American Journal of Mathematics <strong>85</strong> 2 (1963) 156–188 [<a href="http://dx.doi.org/10.2307/2373208">doi:10.2307/2373208</a>]</p> </li> <li> <p>M. Bullejos, E. Faro, M. A. García-Muñoz, <em>Homotopy colimits and cohomology with local coefficients</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 1 (2003), p. 63-80 (<a href="http://www.numdam.org/item?id=CTGDC_2003__44_1_63_0">numdam:CTGDC_2003__44_1_63_0</a>)</p> </li> </ul> <p>The case of twisted <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> twisted by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Johann+Sigurdsson">Johann Sigurdsson</a>, Section 22.1 of: <em><a class="existingWikiWord" href="/nlab/show/Parametrized+Homotopy+Theory">Parametrized Homotopy Theory</a></em>, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (<a href="https://bookstore.ams.org/surv-132">ISBN:978-0-8218-3922-5</a>, <a href="https://arxiv.org/abs/math/0411656">arXiv:math/0411656</a>, <a href="http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <em>Twisted stable homotopy theory</em> PhD thesis 2005 (<a href="http://dspace.mit.edu/handle/1721.1/7582">dspace:1721.1/7582</a>)</p> </li> </ul> <p>This in turn is based on the definition of <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> given in</p> <ul> <li id="AtiyahSegal04"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Twisted K-theory</em>, Ukrainian Math. Bull. <strong>1</strong>, 3 (2004) (<a href="http://arxiv.org/abs/math/0407054">arXiv:math/0407054</a>, <a href="http://iamm.su/en/journals/j879/?VID=10">journal page</a>, <a href="http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf">published pdf</a>)</li> </ul> <p>Review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <em>Twisted cohomology</em>, in <em><a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [<a href="https://arxiv.org/abs/2401.03966">arXiv:2401.03966</a>]</li> </ul> <p>Details on Larmore’s work on twisted cohomology are at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Larmore+twisted+cohomology">Larmore twisted cohomology</a>.</li> </ul> <p>The abstract discussion of twisted nonabelian cohomology in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-toposes is in</p> <ul> <li id="NikolausSchreiberStevenson12"><a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em><a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">Principal ∞-bundles – theory, presentations and applications</a></em></li> </ul> <p>The abstract discussion of twisted ES-type cohomology in the <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category+of+spectra">stable (infinity,1)-category of spectra</a> is in</p> <ul> <li id="ABG10"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em>Twists of K-theory and TMF</em>, in Robert S. Doran, Greg Friedman, <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <em>Superstrings, Geometry, Topology, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras</em>, Proceedings of Symposia in Pure Mathematics 81, American Mathematical Society 2010 (<a href="http://arxiv.org/abs/1002.3004">arXiv:1002.3004</a>, <a href="https://bookstore.ams.org/pspum-81">ISBN:978-0-8218-4887-6</a>)</p> </li> <li id="ABGHR14"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categorical approach to <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>-line bundles, <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>-module Thom spectra, and twisted <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>-homology</em>, Journal of Topology <strong>7</strong> 3 (2014) 869893 [<a href="http://arxiv.org/abs/1403.4325">arXiv:1403.4325</a>, <a href="https://doi.org/10.1112/jtopol/jtt035">doi:10.1112/jtopol/jtt035</a>]</p> </li> </ul> <p>The presentation of cohomology with local coefficients in terms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math>-equivariant de Rham cohomology on the universal covering space is discussed in</p> <ul id="MichorMO"> <li id="Eil47"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <em>Homology of spaces with operators I</em>, Trans. Amer. Math. Soc. 6 (1947), 378-417. (<a href="http://dx.doi.org/10.1090/S0002-9947-1947-0021313-4">doi</a>)</p> </li> <li id="Whit78"> <p><a class="existingWikiWord" href="/nlab/show/George+Whitehead">George Whitehead</a>, <em>Elements of homotopy theory</em>, Springer-Veriag, 1978.</p> </li> <li> <p><a href="http://mathoverflow.net/users/26935/peter-michor">Peter Michor</a>, <a href="http://mathoverflow.net/a/129246/2622">answer</a> to MathOverflow question <a href="http://mathoverflow.net/q/129246"><em>de Rham cohomology and flat vector bundles</em></a>, (version: 2013-04-30).</p> </li> </ul> <h3 id="chronology_of_literature_on_twisted_cohomology">Chronology of literature on twisted cohomology</h3> <p>The oldest meaning of twisted cohomology is that of <strong>cohomology with local coefficients</strong> (see above).</p> <p>For more on the history of that notion see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/History+of+cohomology+with+local+coefficients">History of cohomology with local coefficients</a></li> </ul> <p>In the following we shall abbreviate</p> <ul> <li>tc = twisted cohomology</li> </ul> <p>Searching MathSciNet for <em>twisted cohomology</em> led to the following chronology: It missed older references in which the phrase was not used but the concept was in the sense of local coefficient systems – ancient and honorable.</p> <p>Most notably missing are</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kurt+Reidemeister">Kurt Reidemeister</a> (1938) Topologie der Polyeder und kombinatorische</li> </ul> <p>Topologie der Komplexe_, Mathematik und ihre Anwendungen in Physik und Technik,_(But note that reprints appear, sans reviews. There is a short English and longer German review on Zentralblatt)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a> (1942,1943)</p> </li> <li> <p>Olum (thesis 1947, published 1950)</p> </li> </ul> <p>Next come several that involve twisted differentials more generally.</p> <p>Few are in terms of homotopy of spaces</p> <p>tc ops should be treated as a single phrase – it may be that the ops are twisted, not the cohomology</p> <ul> <li> <p>1966 McClendon thesis – summarized in</p> </li> <li> <p>1967 Emery Thomas tc ops</p> </li> <li> <p>1967 Larmore tc ops</p> </li> <li> <p>1969 McClendon tc ops</p> </li> <li> <p>1969 Larmore tc</p> </li> <li> <p>1970 Peterson tc ops</p> </li> <li> <p>1971 McClendon tc ops</p> </li> <li> <p>1972 Deligne Weil conjecture for K3 tc – meaning?</p> </li> <li> <p>1972 Larmore tc</p> </li> <li> <p>1973 Larmore and Thomas tc</p> </li> <li> <p>1973 Larmore tc</p> </li> <li> <p>gap</p> </li> <li> <p>1980 Coelho & Pesennec tc</p> </li> <li> <p>1980 Tsukiyama sequel to McClendon</p> </li> <li> <p>1983 Coelho & Pesennec tc</p> </li> <li> <p>1985 Morava but getsted at 1975 ??</p> </li> <li> <p>1986 Fried tc</p> </li> <li> <p>1988 Baum & Connes ??</p> </li> <li> <p>1989 Lott torsion</p> </li> <li> <p>1990 Dwork ??</p> </li> <li> <p>1993 Gomez–Tato tc minimal models</p> </li> <li> <p>1993 Duflo & Vergne diff tc</p> </li> <li> <p>1993 Vaisman tc and connections</p> </li> <li> <p>1993 Mimachi tc and holomorphic</p> </li> <li> <p>1994 Kita tc and intersection</p> </li> <li> <p>1995 Cho, Mimachi and Yoshida tc and configs</p> </li> <li> <p>1995 Cho, Mimachi tc and intersection</p> </li> <li> <p>1996 Iwaski and Kita tc de rham</p> </li> <li> <p>1996 Asada nc geom and strings</p> </li> <li> <p>1997 H Kimura tc de Rham and hypergeom</p> </li> <li> <p>1998 <a class="existingWikiWord" href="/nlab/show/Michael+Farber">Michael Farber</a>, <span class="newWikiWord">Gabriel Katz<a href="/nlab/new/Gabriel+Katz">?</a></span>, <span class="newWikiWord">Jerome Levine<a href="/nlab/new/Jerome+Levine">?</a></span>, <a href="http://www.sciencedirect.com/science/article/pii/S0040938397827309">Morse theory of harmonic forms</a>, Topology, (Volume 37, Issue 3, May 1998, Pages 469–483)</p> </li> <li> <p>1998 Knudson tc SL_n</p> </li> <li> <p>1998 Morita tc de Rham</p> </li> <li> <p>1999 Kachi, Mtsumoto, Mihara tc and intersection</p> </li> <li> <p>1999 Hanamura & Yoshida Hodge tc</p> </li> <li> <p>1999 Felshtyn & Sanchez–Morgado Reidemeister torsion</p> </li> <li> <p>1999 Haraoka hypergeom</p> </li> <li> <p>2000 Tsou & Zois tc de rham</p> </li> <li> <p>2000 Manea tc Czech</p> </li> <li> <p>2001 Royo Prieto tc Euler</p> </li> <li> <p>2001 Takeyama q-twisted</p> </li> <li> <p>2001 Gaberdiel &Schaefr–Nameki tc of Klein bottle</p> </li> <li> <p>2001 Iwaskai tc deRham</p> </li> <li> <p>2001 Proc Rims tc and DEs and several papers in this book</p> </li> <li> <p>2001 Knudson tc SL_n</p> </li> <li> <p>2001 Royo Prieto tc as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>+</mo><mi>k</mi><mo>∧</mo></mrow><annotation encoding="application/x-tex">d+k\wedge</annotation></semantics></math></p> </li> <li> <p>2001 Barlewtta & Dragomir tc and integrability</p> </li> <li> <p>2002 Lueck <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">L^2</annotation></semantics></math></p> </li> <li> <p>2002 Verbitsky HyperKahler, torsion, etc</p> </li> <li> <p>2003 Etingof & Grana tc of Carter, Elhamdadi and Saito</p> </li> <li> <p>2003 Cruikshank tc of Eilenberg</p> </li> <li> <p>2003 various in Proc NATO workshop</p> </li> <li> <p>2003 Dimca tc of hyperplanes</p> </li> <li> <p>2004 Kirk & Lesch tc and index</p> </li> <li> <p>2004 Bouwknegt, Evslin, Mathai tc and tK</p> </li> <li> <p>2004 Bouwknegt, Hannbuss, Mathai tc in re: T-duality</p> </li> <li> <p>2005 Bouwknegt, Hannbuss, Mathai tc in re: T-duality</p> </li> <li> <p>2005 Bunke & Schick tc in re: T-duality</p> </li> <li> <p>2006 Dubois tc and Reidemeister (elsewhere he considers twisted Reidemesiter)</p> </li> <li> <p>2006 Bunke & Schick tc in re: T-duality</p> </li> <li> <p>2006 Sati</p> </li> <li> <p>2006 Atiyah & Segal tc and tK</p> </li> <li> <p>2007 Mickelsson & Pellonpaa tc and tK</p> </li> <li> <p>2007 Sugiyama in re: Galois and Reidemeister</p> </li> <li> <p>2007 Bunke, Schick, Spitzweck tc in re: gerbes</p> </li> <li> <p>2008 Kawahara hypersurfaces</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 10, 2024 at 00:19:17. 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