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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='algebraic_topology'>Algebraic topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></strong> – application of <a class='existingWikiWord' href='/nlab/show/higher+algebra'>higher algebra</a> and <a class='existingWikiWord' href='/nlab/show/higher+category+theory'>higher category theory</a> to the study of (<a class='existingWikiWord' 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> <h4 id='cohomology'>Cohomology</h4> <div class='hide'> <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+homology'>long exact sequence in cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fiber+infinity-bundle'>fiber ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/principal+infinity-bundle'>principal ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/associated+infinity-bundle'>associated ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/twisted+infinity-bundle'>twisted ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/infinity-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/chain+homology+and+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/topological+automorphic+form'>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+complex'>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/%C3%A9tale+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+homology'>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+infinity-bundle'>principal ∞-bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/universal+principal+infinity-bundle'>universal principal ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/groupal+model+for+universal+principal+infinity-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+infinity-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/%28infinity%2C1%29-module+bundle'>(∞,1)-vector bundle</a> / <a class='existingWikiWord' href='/nlab/show/n-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%E1%B6%9C+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/local+system'>with a local system of coefficients</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/infinity-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+extension'>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'>?ech 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%E1%B6%9C+structure'>twisted spin^c structure</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/twisted+differential+c-structure'>twisted differential c-structures</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/differential+string+structure'>twisted differential string structure</a>, <a class='existingWikiWord' href='/nlab/show/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' title='schreiber'>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/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/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+operation'>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/Poincar%C3%A9+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+structure'>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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definitions'>Definitions</a><ul><li><a href='#general'>General</a></li><li><a href='#sheaftheoretic_case'>Sheaf-theoretic case</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <em>local system</em> – which is short for <em>local system of <a class='existingWikiWord' href='/nlab/show/coefficient'>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+infinity-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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_1' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_2' xmlns='http://www.w3.org/1998/Math/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/%28infinity%2C1%29-functor'>(∞,1)-functor</a>s out of the <a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a>. These in turn are equivalently <a class='existingWikiWord' href='/nlab/show/flat+connection'>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+infinity-connection'>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/%28infinity%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/sheaf'>sheaves</a>.</p> <h3 id='general'>General</h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-sheaf (∞,1)-topos</a>, write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_4' xmlns='http://www.w3.org/1998/Math/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/%28infinity%2C1%29-geometric+morphism'>(∞,1)-geometric morphism</a>, where <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_5' xmlns='http://www.w3.org/1998/Math/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/%28infinity%2C1%29-functor'>(∞,1)-functor</a> and <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LConst</mi></mrow><annotation encoding='application/x-tex'>LConst</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/constant+infinity-stack'>constant ∞-stack</a>-functor.</p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_7' xmlns='http://www.w3.org/1998/Math/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/Infinity-Grpd'>∞Grpd</a> for the <a class='existingWikiWord' href='/nlab/show/core+groupoid'>core</a> <a class='existingWikiWord' href='/nlab/show/infinity-groupoid'>∞-groupoid</a> of the <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a> of finite <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_8' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_9' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_10' xmlns='http://www.w3.org/1998/Math/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+infinity-stack'>locally constant ∞-stack</a></em> on <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_11' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_12' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> or equivalently the object in the <a class='existingWikiWord' href='/nlab/show/over-%28infinity%2C1%29-topos'>over-(∞,1)-topos</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_14' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_15' xmlns='http://www.w3.org/1998/Math/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/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_16' xmlns='http://www.w3.org/1998/Math/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+infinity-stack'>locally constant ∞-stacks</a> or equivalently their classifying <a class='existingWikiWord' href='/nlab/show/cocycle'>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+infinity-bundle'>principal ∞-bundle</a> for discussion of how <a class='existingWikiWord' href='/nlab/show/cocycle'>cocycle</a>s <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_17' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P \to X</annotation></semantics></math>.)</p> <div> <h6 id='remark'>Remark</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> happens to be a <a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n%2B1%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+%28infinity%2C1%29-functor'>adjoint (∞,1)-functor</a> <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Π</mi></mrow><annotation encoding='application/x-tex'>\Pi</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_22' xmlns='http://www.w3.org/1998/Math/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+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%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 class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_23' xmlns='http://www.w3.org/1998/Math/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/representation'>representations</a> (<math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_24' xmlns='http://www.w3.org/1998/Math/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+infinity-groupoid'>fundamental ∞-groupoid</a> <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_26' xmlns='http://www.w3.org/1998/Math/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/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-sheaf (∞,1)-topos</a> over a <a class='existingWikiWord' href='/nlab/show/locally+contractible+space'>locally contractible space</a> is locally <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_27' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_28' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_29' xmlns='http://www.w3.org/1998/Math/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-%28infinity%2C1%29-topos'>over-(∞,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_30' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_31' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_32' xmlns='http://www.w3.org/1998/Math/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/fiber+sequence'>homotopy fiber</a> of <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_33' xmlns='http://www.w3.org/1998/Math/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/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a>, one sees that a <a class='existingWikiWord' href='/nlab/show/cocycle'>cocycle</a> <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_34' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_35' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>. This is precisely a <a class='existingWikiWord' href='/nlab/show/section'>section</a> of the <a class='existingWikiWord' href='/nlab/show/locally+constant+infinity-stack'>locally constant ∞-stack</a> <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_37' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_38' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_39' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> nothing but the intrinsic cohomology of the <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_41' xmlns='http://www.w3.org/1998/Math/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-Mac+Lane+object'>Eilenberg-MacLane object</a> <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_42' xmlns='http://www.w3.org/1998/Math/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/representation'>representations</a> of the <a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a> <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_43' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_44' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_45' xmlns='http://www.w3.org/1998/Math/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/flat+infinity-connection'>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) [[doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1)]</li> </ul> <blockquote> <p><em>A local system on a space <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/functor'>covariant functor</a> from the <a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fundamental groupoid</a> of <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_47' xmlns='http://www.w3.org/1998/Math/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) <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://publications.ias.edu/node/355'>publications.ias:355</a><math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math></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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_50' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_51' xmlns='http://www.w3.org/1998/Math/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 [[pdf](http://homepages.math.uic.edu/~libgober/otherpapers/export/2000sergeytokyo.pdf), <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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_52' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-local systems as sections of constant <math class='maruku-mathml' display='inline' id='mathml_2bc27e4345759aa381bc090945194def30d2eb4a_54' xmlns='http://www.w3.org/1998/Math/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 [[arXiv:0407507](http://arxiv.org/abs/math/0407507)]</li> </ul> <p>for <a class='existingWikiWord' href='/nlab/show/locally+constant+stack'>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/%28infinity%2C1%29-topos'>(∞,1)-topos</a>es.</p> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/Galois+representation'>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> <p> </p> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on June 4, 2023 at 05:15:20 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> See the <a href="/nlab/history/local+system" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/2099/#Item_63">Discuss</a><span class="backintime"><a href="/nlab/show/local+system" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)</span><span class="backintime"><a href="/nlab/revision/local+system/59" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (59 more)</span><a href="/nlab/show/local+system" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/local+system/60" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/local+system" accesskey="S" class="navlink" id="history" rel="nofollow">History (60 revisions)</a><a href="/nlab/rollback/local+system?rev=60" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/local+system/60/cite" style="color: black">Cite</a> <a href="/nlab/source/local+system/60" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>