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href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy</a></p> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> </li> </ul> </div></div> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" 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='#definitions'>Definitions</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#sheaftheoretic_case'>Sheaf-theoretic case</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>local system</em> – which is short for <em>local system of <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> for <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em> – is a system of coefficients for <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>.</p> <p>Often this is presented or taken to be presented by a <a class="existingWikiWord" href="/nlab/show/locally+constant+sheaf">locally constant sheaf</a>. Then cohomology with coefficients in a local system is the corresponding <a class="existingWikiWord" href="/nlab/show/sheaf+cohomology">sheaf cohomology</a>.</p> <p>More generally, we say a <em>local system</em> is a <a class="existingWikiWord" href="/nlab/show/locally+constant+stack">locally constant stack</a>, … and eventually a <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>.</p> <p>Under suitable conditions (if we have <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a>) local systems 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> correspond to <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s out of the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, or more generally to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s out of the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a>. These in turn are equivalently <a class="existingWikiWord" href="/nlab/show/flat+connections">flat connections</a> (this relation is known as the <em><a class="existingWikiWord" href="/nlab/show/Riemann-Hilbert+correspondence">Riemann-Hilbert correspondence</a></em>) or generally <a class="existingWikiWord" href="/nlab/show/flat+%E2%88%9E-connections">flat ∞-connections</a>.</p> <h2 id="definitions">Definitions</h2> <p>A notion of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> exists intrinsically within any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>. We discuss local systems first in this generality and then look at special cases, such as local systems as ordinary <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a>.</p> <h3 id="general">General</h3> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-sheaf (∞,1)-topos</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>LConst</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a>-functor.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>:</mo><mo>=</mo><mi>core</mi><mo stretchy="false">(</mo><mi>Fin</mi><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">\mathcal{S} := core(Fin \infty Grpd) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> for the <a class="existingWikiWord" href="/nlab/show/core">core</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of finite <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>-groupoids. (We can drop the finiteness condition by making use of a higher <a class="existingWikiWord" href="/nlab/show/universe">universe</a>.) This is canonically a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>𝒮</mi></mrow><annotation encoding="application/x-tex">* \to \mathcal{S}</annotation></semantics></math>, with points the terminal groupoid.</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/object">object</a>, a <strong>local system</strong> or <em><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a></em> 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</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>LConst</mi><mi>𝒮</mi></mrow><annotation encoding="application/x-tex"> \tilde \nabla \colon X \longrightarrow LConst \mathcal{S} </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> or equivalently the object in the <a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> (P \to X) \in \mathbf{H}/X </annotation></semantics></math></div> <p>that is classified by <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> under the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</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>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>LConst</mi><mi>𝒵</mi></mtd></mtr> <mtr><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><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mover></mtd> <mtd><mi>LConst</mi><mi>𝒮</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\to& LConst \mathcal{Z} \\ \downarrow && \downarrow \\ X &\stackrel{\tilde \nabla}{\to}& LConst \mathcal{S} } </annotation></semantics></math></div> <p>In other words, local systems are <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stacks">locally constant ∞-stacks</a> or equivalently their classifying <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> for <a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a>.</p> </div> <p>(See <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> for discussion of how <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>X</mi><mo>→</mo><mi>LConst</mi><mi>𝒮</mi></mrow><annotation encoding="application/x-tex">\tilde \nabla : X \to LConst \mathcal{S}</annotation></semantics></math> classify morphisms <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> <div> <h6 id="remark">Remark</h6> <p>If <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> happens to be a <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a> in that there is the further <a class="existingWikiWord" href="/nlab/show/left+adjoint">left</a> <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a> <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd </annotation></semantics></math></div> <p>we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>. In this case, by the adjunction hom-equivalence we have</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>LConst</mi><mi>𝒮</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, LConst \mathcal{S}) \simeq Func(\Pi(X), \mathcal{S}) \,. </annotation></semantics></math></div> <p>This means that local systems are naturally identified with <a class="existingWikiWord" href="/nlab/show/representations">representations</a> (<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>-<a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a>s, as it were) of the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>LConst</mi><mi>𝒮</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝒮</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Maps(X, LConst \mathcal{S}) \simeq Maps(\Pi(X), \mathcal{S}) \,. </annotation></semantics></math></div> <p>This is essentially the basic statement around which <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> revolves.</p> <p>The <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-sheaf (∞,1)-topos</a> over a <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a> is locally <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>-connected, and many authors identify local systems on such a topological space with representations of its <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a>.</p> </div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Given a local system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>X</mi><mo>→</mo><mi>LConst</mi><mi>𝒮</mi></mrow><annotation encoding="application/x-tex">\tilde \nabla : X \to LConst \mathcal{S}</annotation></semantics></math>, the 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 this <strong>local system of coefficients</strong> is the intrinsic <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,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><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbf{H}/X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>P</mi> <mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> H(X,\tilde \nabla) := \mathbf{H}_{/X}(X, P_{\tilde \nabla}) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></msub></mrow><annotation encoding="application/x-tex">P_{\tilde\nabla}</annotation></semantics></math> is 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><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \nabla</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Unwinding the definitions and using the universality of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>, one sees that 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>c</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in \mathbf{H}(X,\tilde \nabla)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</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>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>c</mi></mover></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>LConst</mi><mi>𝒮</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{c}{\to}&& * \\ & \searrow &\swArrow& \swarrow \\ && LConst \mathcal{S} } </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>. This is precisely a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a> <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>.</p> </div> <h3 id="sheaftheoretic_case">Sheaf-theoretic case</h3> <p>Local systems can also be considered in abelian contexts. One finds the following version of a local system</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <strong>linear local system</strong> is a <a class="existingWikiWord" href="/nlab/show/locally+constant+sheaf">locally constant sheaf</a> on 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> (or manifold, analytic manifold, or algebraic variety) whose stalk is a finite-dimensional <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>.</p> </div> <p>Regarded as a sheaf <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> with values in <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s, such a linear local system serves as the coefficient for <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>. As discussed there, this is in degree <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> nothing but the intrinsic cohomology of 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>-topos with coefficients in the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+object">Eilenberg-MacLane object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>F</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n F</annotation></semantics></math>.</p> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>On a connected topological space this is the same as a sheaf of sections of a finite-dimensional <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> equipped with flat <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>; and it also corresponds to the <a class="existingWikiWord" href="/nlab/show/representations">representations</a> of the <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>,</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x_0)</annotation></semantics></math> in the typical stalk. On an analytic manifold or a variety, there is an equivalence between the category of non-singular coherent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/D-module">modules</a> and local systems 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> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+vector+bundle">flat vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Gauss-Manin+connection">Gauss-Manin connection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinity-local+system">infinity-local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+local+system">inner local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+local+system">simplicial local system</a>: within Sullivan’s (1977) theory of <em>Infinitesimal computations in topology</em>, he refers to ‘local systems’ several times. This seems to be simplicial in nature. <a class="existingWikiWord" href="/nlab/show/simplicial+local+system">This</a> entry explores some of the uses of that notion based on Halperin’s lecture notes on minimal models</p> <ul> <li>D. Sullivan, <em>Infinitesimal computations in topology</em> (<a href="http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1977__47_/PMIHES_1977__47__269_0/PMIHES_1977__47__269_0.pdf">pdf</a>)</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>, <a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a>, <a class="existingWikiWord" href="/nlab/show/twisted+infinity-bundle">twisted infinity-bundle</a></p> </li> </ul> <h2 id="references">References</h2> <p>An early version of the definition of local system appears in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>: <em>Homology with local coefficients</em>, Annals 44 (1943) pp. 610 - 627,</li> </ul> <p>This is before the formal notion of <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> was published by <a class="existingWikiWord" href="/nlab/show/Jean+Leray">Jean Leray</a>. (Wikipedia’s entry on <a href="http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29#History">Sheaf theory</a> is interesting for its historical perspective on this.)</p> <p>A definition appears as an exercise in</p> <ul> <li id="Spanier66"><a class="existingWikiWord" href="/nlab/show/Edwin+Spanier">Edwin Spanier</a>, <em>Algebraic topology</em>, McGraw Hill (1966), Springer (1982) [<a href="https://link.springer.com/book/10.1007/978-1-4684-9322-1">doi:10.1007/978-1-4684-9322-1</a>]</li> </ul> <blockquote> <p><em>A local system on a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant functor</a> from the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to some <a class="existingWikiWord" href="/nlab/show/category">category</a>.</em> [p. 58]</p> </blockquote> <p>Then the first major account with discussion of the relation to <a class="existingWikiWord" href="/nlab/show/twisted+de+Rham+cohomology">twisted de Rham cohomology</a>:</p> <ul> <li id="Deligne70"><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em>Equations différentielles à points singuliers réguliers</em>, Lecture Notes Math. <strong>163</strong>, Springer (1970) [<a href="https://publications.ias.edu/node/355">publications.ias:355</a>]</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Voisin02"> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a> (translated by <a class="existingWikiWord" href="/nlab/show/Leila+Schneps">Leila Schneps</a>), Section I 9.2.1 of: <em><a class="existingWikiWord" href="/nlab/show/Hodge+theory+and+Complex+algebraic+geometry">Hodge theory and Complex algebraic geometry</a> I</em>, Cambridge Stud. in Adv. Math. <strong>76, 77</strong>, 2002/3 (<a href="https://doi.org/10.1017/CBO9780511615344">doi:10.1017/CBO9780511615344</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexandru+Dimca">Alexandru Dimca</a>, Section 2.5 of: <em>Sheaves in Topology</em>, Universitext, Springer (2004) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-3-642-18868-8">doi:10.1007/978-3-642-18868-8</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anatoly+Libgober">Anatoly Libgober</a>, <a class="existingWikiWord" href="/nlab/show/Sergey+Yuzvinsky">Sergey Yuzvinsky</a>, <em>Cohomology of local systems</em>, Advanced Studies in Pure Mathematics <strong>27</strong>, Mathematics Society of Japan (2000) 169-184 [<a href="http://homepages.math.uic.edu/~libgober/otherpapers/export/2000sergeytokyo.pdf">pdf</a>, <a href="https://doi.org/10.2969/aspm/02710169">doi:10.2969/aspm/02710169</a>]</li> </ul> <p>A blog exposition of some aspects of linear local system is developed here:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Speyer">David Speyer</a>, <em>Three ways of looking at a local system</em></p> <ul> <li> <p><a href="http://sbseminar.wordpress.com/2009/04/20/three-ways-of-looking-at-a-local-system-introduction-and-connection-to-cohomology-theories/">Introduction and connection to cohomology theories</a></p> </li> <li> <p><a href="http://sbseminar.wordpress.com/2009/04/21/local-systems-the-path-groupoid-approach/">the path groupoid approach</a></p> </li> <li> <p><a href="http://sbseminar.wordpress.com/2009/04/30/three-ways-of-looking-at-a-local-system-the-infinitesimal-perspective/">the infinitesimal perspective</a></p> </li> <li> <p><a href="http://sbseminar.wordpress.com/2009/05/06/the-infinitesimal-site/">the infinitesimal site</a></p> </li> </ul> </li> </ul> <p>A clear-sighted description of locally constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-stacks / <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>-local systems as sections of constant <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>-stacks is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pietro+Polesello">Pietro Polesello</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Waschkies">Ingo Waschkies</a>, <em>Higher monodromy</em>, Homology, Homotopy and Applications <strong>7</strong> 1 (2005) 109-150 [<a href="http://arxiv.org/abs/math/0407507">arXiv:0407507</a>]</li> </ul> <p>for <a class="existingWikiWord" href="/nlab/show/locally+constant+stacks">locally constant stacks</a> on <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s. The above formulation is pretty much the evident generalization of this to general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es.</p> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Galois+representations">Galois representations</a> as encoding local systems in <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a> includes</p> <ul> <li id="Lovering">Tom Lovering, <em>Étale cohomology and Galois Representations</em>, 2012 (<a href="http://tlovering.files.wordpress.com/2012/06/essay-body1.pdf">pdf</a>)</li> </ul> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on August 22, 2024 at 10:03:26. 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