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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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="higher_linear_algebra">Higher linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#basic_idea'>Basic idea</a></li> <li><a href='#RelationToVectorBundlesInIntroduction'>Motivation for and role of modules: generalized vector bundles</a></li> <li><a href='#more_general_perspectives'>More general perspectives</a></li> <ul> <li><a href='#MonadAlgs'>Modules for monoids in 2-categories: modules over monads</a></li> <li><a href='#enriched_presheaves'>Enriched presheaves</a></li> <li><a href='#stabilized_overcategories'>Stabilized overcategories</a></li> </ul> </ul> <li><a href='#Definitions'>Definition</a></li> <ul> <li><a href='#ModuleOverMonoidObject'>Modules over a monoid in a monoidal category</a></li> <li><a href='#InEnrichedCategory'>Presheaves in enriched category theory</a></li> <li><a href='#DefWithOCat'>In terms of stabilized overcategories</a></li> <ul> <li><a href='#ModulesOverARingInTermsOfStabilizedSlices'>Modules over a ring</a></li> <li><a href='#ModulesOverAGroupInTermsOfStabilizedOvercategories'>Modules over a group</a></li> <li><a href='#SimpRings'>Modules over a simplicial ring</a></li> </ul> <li><a href='#OverHigherAndGeneralizedAlgebras'>Modules over an algebra over an operad</a></li> <li><a href='#multiplicatively_cancellative_module_over_a_rig'>Multiplicatively cancellative module over a rig</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#ExamplesOfModulesOverARing'>Of modules over a ring</a></li> <li><a href='#VectorBundlesAndModules'>Vector bundles and modules</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#on_modules_as_enriched_presheaves'>On modules as enriched presheaves</a></li> <li><a href='#on_modules_as_stabilized_overcategories'>On modules as stabilized overcategories</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="basic_idea">Basic idea</h3> <p>The basic idea is that a <em>module</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is an object equipped with an <a class="existingWikiWord" href="/nlab/show/action">action</a> by a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. This is closely related to the concept of a <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a>.</p> <p>A familiar example of a module is a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>: this is a <em>module</em> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in the category <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of abelian groups: every element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> acts on the vector space by multiplication of vectors, and this action respects the addition of vectors, but nothing in the definition of <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> really depends on the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> here is a <a class="existingWikiWord" href="/nlab/show/field">field</a>: more generally it could be any <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> (or even a general <a class="existingWikiWord" href="/nlab/show/rig">rig</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. The analog of a vector space for fields replaced by rings is that of a <em>module</em> over the ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> can be thought of as a generalization of an abelian group, where the operation taking integer multiples of an element (seen as iterated addition) is extended to taking arbitrary multiples with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. In the trivial case a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-module is simply an abelian group.</p> <p>This is the traditional and maybe most common notion of modules. But the basic notion is easily much more general.</p> <h3 id="RelationToVectorBundlesInIntroduction">Motivation for and role of modules: generalized vector bundles</h3> <p>The theory of monoids or rings and their modules, its “meaning” and usage, is naturally understood via the <a class="existingWikiWord" href="/nlab/show/Isbell+duality">duality</a> between <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>:</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is to be thought of as the ring of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> on some <a class="existingWikiWord" href="/nlab/show/space">space</a>,</p> </li> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> is to be thought of as the space of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> on that space.</p> </li> </ol> <p>A classical situation where this correspondence holds precisely is <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, where</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> theorem says that sending a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact 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> to its <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathbb{C})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> with values in the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> constitutes an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> between compact topological spaces and the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Serre-Swan+theorem">Serre-Swan theorem</a> says that sending a <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff</a> topological complex <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> over a compact topological space to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathbb{C})</annotation></semantics></math>-module of its continuous <a class="existingWikiWord" href="/nlab/show/sections">sections</a> establishes an equivalence of categories between that of topological complex vector bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and that of <a class="existingWikiWord" href="/nlab/show/finitely+generated+module">finitely generated</a> <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathbb{C})</annotation></semantics></math>.</p> </li> </ol> <p>In fact, as this example already shows, modules faithfully subsume vector bundles, but are in fact more general. In many contexts one regard modules as the canonical generalization of the notion of vector bundles, with better formal properties.</p> <p>This identification of vector bundles with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules being the spaces of sections of a vector bundle on the space whose ring of functions is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> can be taken then as the very <em>definition</em>: notably in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> Gelfand duality is taken to “hold by definition” in that an <a class="existingWikiWord" href="/nlab/show/algebraic+variety">algebraic variety</a> is essentially by definition the formal dual of a given ring, and the Serre-Swan theorem similarly becomes the statement that the space of sections of a vector bundle over a variety is equivalently given by a module over that ring. (See also at <a class="existingWikiWord" href="/nlab/show/quasicoherent+module">quasicoherent module</a> for more on this.)</p> <p>This <a class="existingWikiWord" href="/nlab/show/Isbell+duality">duality</a> between geometry and algebra allows us to re-interpret many statement about modules in terms of vector bundles. For instance</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of modules corresponds to fiberwise direct sum of vector bundles;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> of a module along a ring homomorphism corresponds to <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of vector bundles along the dual map of spaces;</p> </li> <li> <p>etc.</p> </li> </ul> <p>Using this dictionary for instance the notion of <a class="existingWikiWord" href="/nlab/show/descent">descent</a> of vector bundles can be expressed in terms of <a class="existingWikiWord" href="/nlab/show/monadic+descent">monadic descent</a>, see at <em><a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a></em> for discussion of this point.</p> <h3 id="more_general_perspectives">More general perspectives</h3> <p>The notion of <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> generalizes directly to that of a monoid in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, where it is called a <a class="existingWikiWord" href="/nlab/show/monad">monad</a>. Accordingly the notion of module generalizes to this more general case, where however it is called an <em>algebra over a monad</em> . For more on this see <a href="#MonadAlgs">Modules for monoids in 2-categories: algebras over monads</a> below.</p> <p>Apart from this direct generalization, there are two distinct and separately important perspectives on the notion of module from the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a>:</p> <ul> <li> <p>modules may usefully be thought of in the context of <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> (and the enrichment may be over a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>);</p> </li> <li> <p>modules may usefully be thought of in terms of <a class="existingWikiWord" href="/nlab/show/abelian+category">abelianization</a>/<a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of <a class="existingWikiWord" href="/nlab/show/overcategory">overcategories</a>.</p> </li> </ul> <h4 id="MonadAlgs">Modules for monoids in 2-categories: modules over monads</h4> <p>The notion of <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> generalizes straightforwardly from monoids in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> to monoids in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>: for the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, and more generally for arbitrary 2-categories, these are called <a class="existingWikiWord" href="/nlab/show/monad">monad</a>s.</p> <p>A <a class="existingWikiWord" href="/nlab/show/module+over+a+monad">module over a monad</a> (see there for more details) is defined essentially exactly as that of module over a monoid. For historical reasons, a module over a monad in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> is called an <em>algebra over a monad</em>, because the algebras in the sense of <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a> can be obtained as algebras/modules over a <a class="existingWikiWord" href="/nlab/show/finitary+monad">finitary monad</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>: the modules for a free algebra monad (for certain kind of algebras) on <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, which are the composition of the free algebra functor and its <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> forgetful functor are exactly algebras of that type. Modules over a fixed monad (in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>) are the objects of the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a> of the monad; in an arbitrary bicategory, this category generalized to Eilenberg-Moore objects which may or may not exist.</p> <h4 id="enriched_presheaves">Enriched presheaves</h4> <p>See <a class="existingWikiWord" href="/nlab/show/module+over+an+enriched+category">module over an enriched category</a>.</p> <h4 id="stabilized_overcategories">Stabilized overcategories</h4> <p>A module <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> over a (commutative, unital) ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> may be encoded in another ring: the one that as an abelian group is the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">R \oplus N</annotation></semantics></math> and whose product is defined by the formulas</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mi>r</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>r</mi> <mn>2</mn></msub><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (r_1, n_1) \cdot (r_2,n_2) := (r_1 r_2, r_2 n_1 + r_1 n_2) \,. </annotation></semantics></math></div> <p>This is a <strong>square-0 extension</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. It is canonically equipped with a ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>N</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R \oplus N \to R</annotation></semantics></math> which is the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and sends all elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> to 0. As such, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>N</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R \oplus N \to R</annotation></semantics></math> is an object in the <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CRing</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">CRing/R</annotation></semantics></math>. But a special such object: it is in fact canonically an abelian <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CRing</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">CRing/R</annotation></semantics></math>, where the group operation (over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>!) is given by addition of elements in <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>.</p> <p>From this perspective, it makes sense for general <a class="existingWikiWord" href="/nlab/show/category">categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to think of the <a class="existingWikiWord" href="/nlab/show/abelian+category">abelianization</a> of their <a class="existingWikiWord" href="/nlab/show/overcategory">overcategories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C/A</annotation></semantics></math> as categories of modules over the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Taken all together, this makes the fiberwise <a class="existingWikiWord" href="/nlab/show/abelian+category">abelianization</a> of their <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cod</mi><mo>:</mo><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">cod : [I,C] \to C</annotation></semantics></math> the category of all possible modules over all objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>This general perspective has a nice <a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a> to the context of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-categories</a>: abelianization becomes <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> in this context, and the fiberwise stabilization of the <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a> of any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-category">tangent (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>C</mi></msub><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">T_C \to C</annotation></semantics></math>.</p> <p>For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">sAlg_k</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+ring">simplicial algebras</a> over a ground field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of characteristic 0, we have that the <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>sAlgk</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(sAlgk/A)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is equivalent to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules.</p> <h2 id="Definitions">Definition</h2> <p>We spell out the definition of <em>module</em> for</p> <ul> <li><a href="#ModuleOverMonoidObject">modules over a monoid in a monoidal category</a></li> </ul> <p>Then we give more general definitions</p> <ul> <li> <p><a href="#InEnrichedCategory">modules as presheaves in enriched category theory</a></p> </li> <li> <p><a href="#DefWithOCat">modules as objects in stabilized overcategories</a>.</p> </li> </ul> <h3 id="ModuleOverMonoidObject">Modules over a monoid in a monoidal category</h3> <p>See <a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a>.</p> <h3 id="InEnrichedCategory">Presheaves in enriched category theory</h3> <p>See <a class="existingWikiWord" href="/nlab/show/module+over+an+enriched+category">module over an enriched category</a>.</p> <h3 id="DefWithOCat">In terms of stabilized overcategories</h3> <p>There is a general definition of modules in terms of stabilized <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> of the category of monoids: <em><a class="existingWikiWord" href="/nlab/show/Beck+modules">Beck modules</a></em>, <em><a class="existingWikiWord" href="/nlab/show/tangent+%28infinity%2C1%29-categories">tangent (infinity,1)-categories</a></em>.</p> <h4 id="ModulesOverARingInTermsOfStabilizedSlices">Modules over a ring</h4> <p>The ordinary case of modules over rings is phrased in terms of stabilized overcategories by the following observation, which goes back at least to (<a href="#Beck67">Beck 67</a>), and is found in the important paper of (<a href="#Quillen70">Quillen 70</a>); both listed below. For more see at <em><a class="existingWikiWord" href="/nlab/show/Beck+module">Beck module</a></em>.</p> <div class="num_prop" id="Square0ExtensionsOfRingsAreAbelianSliceObjectsAreModules"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>CRing</mi></mrow><annotation encoding="application/x-tex">R \in CRing</annotation></semantics></math> be a commutative <a class="existingWikiWord" href="/nlab/show/ring">ring</a>. Then there is a canonical <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules and the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi><mo stretchy="false">(</mo><mi>CRing</mi><mo stretchy="false">/</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ab(CRing/R)</annotation></semantics></math> of abelian <a class="existingWikiWord" href="/nlab/show/group+object">group object</a>s in the <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CRing</mi></mrow><annotation encoding="application/x-tex">CRing</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi><mo>≃</mo><mi>Ab</mi><mo stretchy="false">(</mo><mi>CRing</mi><mo stretchy="false">/</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R Mod \simeq Ab(CRing/R) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We first unwind what the structure of an abelian group object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p: K \to R)</annotation></semantics></math> in the overcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CRing</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">CRing/R</annotation></semantics></math> is explicitly</p> <p>The unit of the abelian group object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CRing</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">CRing/R</annotation></semantics></math> is a diagram</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>R</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>K</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>Id</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>p</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>R</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ R &&\to&& K \\ & {}_{\mathllap{Id}}\searrow && \swarrow_{\mathrlap{p}} \\ && R } \,. </annotation></semantics></math></div> <p>This diagram identifies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> with a ring whose underlying <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> is the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R \oplus ker(p)</annotation></semantics></math> of some ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n \in ker(p)</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>⋅</mo><mi>n</mi><mo>∈</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r\cdot n \in ker(p)</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/product">product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>N</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R \oplus N \to R</annotation></semantics></math> with itself in the <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a> is the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> in the original category, hence is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>N</mi><mo>⊕</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">R \oplus N \oplus N</annotation></semantics></math>.</p> <p>The addition operation on the abelian group object is therefore a morphism</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>R</mi><mo>⊕</mo><mi>N</mi><mo>⊕</mo><mi>N</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>R</mi><mo>⊕</mo><mi>N</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>R</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ R \oplus N \oplus N &&\to&& R \oplus N \\ & \searrow && \swarrow \\ && R } \,. </annotation></semantics></math></div> <p>With the above unit, the unit axiom on this operation together with the fact that the top morphism is a ring homomorphism says that this morphism is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>N</mi><mo>⊕</mo><mi>N</mi><mover><mo>→</mo><mrow><mi>Id</mi><mo>⊕</mo><mo stretchy="false">(</mo><mi>Id</mi><mo>+</mo><mi>Id</mi><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo>⊕</mo><mi>N</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R \oplus N \oplus N \stackrel{Id \oplus (Id + Id)}{\to} R \oplus N \,. </annotation></semantics></math></div> <p>Since the ring product in the direct product ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>N</mi><mo>⊕</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">R \oplus N \oplus N</annotation></semantics></math> between two elements in the two copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> vanishes, it therefore has to vanish between two elements in the same copy, too.</p> <p>This says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊕</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">R \oplus N</annotation></semantics></math> is a square-0 extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. Conversely, for every square-0-extension we obtain an abelian group object this way.</p> </div> <p>For instance the square-0-extension of a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> corresponding to the canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> itself is the <a class="existingWikiWord" href="/nlab/show/ring+of+dual+numbers">ring of dual numbers</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> <h4 id="ModulesOverAGroupInTermsOfStabilizedOvercategories">Modules over a group</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/group">group</a>. Taking together the above desriptions</p> <ol> <li> <p>of <a href="#AbelianGroupsWithGAction">Modules over a group as modules over the group ring</a></p> </li> <li> <p>of <a href="#ModulesOverARingInTermsOfStabilizedSlices">Modules over a ring as stabilized overcategories</a></p> </li> </ol> <p>one finds:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-modules is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the category of <a class="existingWikiWord" href="/nlab/show/abelian+group+objects">abelian group objects</a> in the <a class="existingWikiWord" href="/nlab/show/slice">slice</a> of <a class="existingWikiWord" href="/nlab/show/Ring">Ring</a> over the <a class="existingWikiWord" href="/nlab/show/group+ring">group ring</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Mod</mi><mo>≃</mo><mi>Ab</mi><mo stretchy="false">(</mo><msub><mi>Ring</mi> <mrow><mo stretchy="false">/</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G Mod \simeq Ab(Ring_{/\mathbb{Z}[G]}) \,. </annotation></semantics></math></div></div> <p>But there is also a more direct characterization along these lines, not involving the auxiliary construction of group rings.</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-modules is equivalent to the category of <a class="existingWikiWord" href="/nlab/show/abelian+group+objects">abelian group objects</a> in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Mod</mi><mo>≃</mo><mi>Ab</mi><mo stretchy="false">(</mo><msub><mi>Grp</mi> <mrow><mo stretchy="false">/</mo><mi>G</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G Mod \simeq Ab(Grp_{/G}) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The proof is analogous to that of prop. <a class="maruku-ref" href="#Square0ExtensionsOfRingsAreAbelianSliceObjectsAreModules"></a>. One checks that a group homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\hat G \to G</annotation></semantics></math> with the structure of an abelian group object over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> by some <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> which more over is a split extension (the is the neutral element of the abelian group object) and hence is a <a class="existingWikiWord" href="/nlab/show/semidirect+product+group">semidirect product group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>≃</mo><mi>G</mi><mo>⋉</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\hat G \simeq G \ltimes A</annotation></semantics></math>. By the discussion there these are equivalently given by <a class="existingWikiWord" href="/nlab/show/actions">actions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> by group <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a>. This is precisely what it means for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to carry a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-module structure.</p> </div> <p>This construction generalizes to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a>. See at <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></em> the section <em><a href="infinity-action#GModules">∞-action – G-modules</a></em>.</p> <h4 id="SimpRings">Modules over a simplicial ring</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">sAlg_k</annotation></semantics></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sAlg</mi></mrow><annotation encoding="application/x-tex">sAlg</annotation></semantics></math> for short) be the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of commutative <a class="existingWikiWord" href="/nlab/show/simplicial+ring">simplicial algebras</a> over a base field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>sAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">A \in sAlg_k</annotation></semantics></math> there is generally a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi><mo>→</mo><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>sAlg</mi> <mi>k</mi></msub><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A Mod \to Stab(sAlg_k/A) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules to the <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of the <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sAlg</mi></mrow><annotation encoding="application/x-tex">sAlg</annotation></semantics></math>. But in general this functor is neither <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a> nor <a class="existingWikiWord" href="/nlab/show/full+functor">full</a>. If however <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> has characteristic 0, then this is an equivalence.</p> <h3 id="OverHigherAndGeneralizedAlgebras">Modules over an algebra over an operad</h3> <p>There is a notion of <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>. The corresponding notion of modules is described at <em><a class="existingWikiWord" href="/nlab/show/module+over+an+algebra+over+an+operad">module over an algebra over an operad</a></em>.</p> <h3 id="multiplicatively_cancellative_module_over_a_rig">Multiplicatively cancellative module over a rig</h3> <p>A module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/rig">rig</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is called multiplicatively cancellative (in <a href="#NazariGhalandarzadeh19">Nazari & Ghalandarzadeh (2019), Sec. 3</a>) if for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>s</mi><mo>′</mo><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s,s' \in S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≠</mo><mi>m</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">0 \neq m \in M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sm</mi><mo>=</mo><mi>s</mi><mo>′</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">sm = s'm</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>=</mo><mi>s</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">s = s'</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <h3 id="ExamplesOfModulesOverARing">Of modules over a ring</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>.</p> <div class="num_example" id="RingAsModuleOverItself"> <h6 id="example">Example</h6> <p>The ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is naturally a module over itself, by regarding its multiplication map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊗</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R \otimes R \to R</annotation></semantics></math> as a module action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>⊗</mo><mi>N</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">R \otimes N \to N</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>≔</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">N \coloneqq R</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the <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>-fold <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is naturally a module over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>n</mi></msup><mo>≔</mo><msup><mi>R</mi> <mrow><msub><mo>⊕</mo> <mi>n</mi></msub></mrow></msup><mo>≔</mo><msub><munder><mrow><mi>R</mi><mo>⊕</mo><mi>R</mi><mo>⊕</mo><mi>⋯</mi><mo>⊕</mo><mi>R</mi></mrow><mo>⏟</mo></munder> <mrow><mi>n</mi><mspace width="thickmathspace"></mspace><mi>summands</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R^n \coloneqq R^{\oplus_n} \coloneqq \underbrace{R \oplus R \oplus \cdots \oplus R}_{n\;summands} \,. </annotation></semantics></math></div> <p>The module action is componentwise:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>r</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>r</mi><mo>⋅</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>,</mo><mi>r</mi><mo>⋅</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>,</mo><mo>⋅</mo><mi>r</mi><mo>⋅</mo><msub><mi>r</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> r \cdot (r_1, r_2, \cdots, r_n) = (r \cdot r_1, r\cdot r_2, \cdot r \cdot r_n) \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>Even more generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">I \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> any <a class="existingWikiWord" href="/nlab/show/set">set</a>, the direct sum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊕</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mi>R</mi></mrow><annotation encoding="application/x-tex">\oplus_{i \in I} R</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module.</p> <p>This is the <strong><a class="existingWikiWord" href="/nlab/show/free+module">free module</a></strong> (over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>) on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> serves as the <a class="existingWikiWord" href="/nlab/show/basis+of+a+free+module">basis of a free module</a>: a general element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mo>⊕</mo> <mi>i</mi></msub><mi>R</mi></mrow><annotation encoding="application/x-tex">v \in \oplus_i R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/formal+linear+combination">formal linear combination</a> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> </div> <p>For special cases of the ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module is equivalent to other notions:</p> <div class="num_example"> <h6 id="example_4">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">R = \mathbb{Z}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module is equivalently just an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-module, hence an abelian group, is not a free module if it has a non-trivial <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion subgroup</a>.</p> </div> <div class="num_example" id="VectorSpacesArekModules"> <h6 id="example_6">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">R = k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a>, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module is equivalently a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <p>Every <a class="existingWikiWord" href="/nlab/show/finitely+generated+module">finitely generated</a> <a class="existingWikiWord" href="/nlab/show/free+module">free</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-module is a <a class="existingWikiWord" href="/nlab/show/free+module">free module</a>, hence every finite dimensional vector space has a <a class="existingWikiWord" href="/nlab/show/basis+of+a+free+module">basis</a>. For infinite dimensions this is true if the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> holds.</p> </div> <div class="num_example" id="RestrictionAndExtensionOfScalars"> <h6 id="example_7">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">f : S \to R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/rings">rings</a>, <a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a> produces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mi>N</mi></mrow><annotation encoding="application/x-tex">f_* N</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-modules <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> and <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> produces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mi>N</mi></mrow><annotation encoding="application/x-tex">f_! N</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules <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>.</p> </div> <div class="num_example" id="SubmodulesInExamples"> <h6 id="example_8">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> a module and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>n</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{n_i\}_{i \in I}</annotation></semantics></math> a set of elements, the <a class="existingWikiWord" href="/nlab/show/linear+span">linear span</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><msub><mi>n</mi> <mi>i</mi></msub><msub><mo stretchy="false">⟩</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>↪</mo><mi>N</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \langle n_i\rangle_{i \in I} \hookrightarrow N \,, </annotation></semantics></math></div> <p>(hence the completion of this set under addition in <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> and multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>) is a <a class="existingWikiWord" href="/nlab/show/submodule">submodule</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_9">Example</h6> <p>Consider example <a class="maruku-ref" href="#SubmodulesInExamples"></a> for the case that the module is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">N = R</annotation></semantics></math>, the ring itself, as in example <a class="maruku-ref" href="#RingAsModuleOverItself"></a>. Then a <a class="existingWikiWord" href="/nlab/show/submodule">submodule</a> is equivalently (called) an <em><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_10">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>≔</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> R \coloneqq C(X,\mathbb{C}) </annotation></semantics></math></div> <p>be the ring of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with values in the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>.</p> <p>Given a complex <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> 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>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(E)</annotation></semantics></math> for its set of continuous sections. Since for each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">E_x</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>-module (by example <a class="maruku-ref" href="#VectorSpacesArekModules"></a>), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(X)</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathbb{C})</annotation></semantics></math>-module.</p> <p>By the <a class="existingWikiWord" href="/nlab/show/Serre-Swan+theorem">Serre-Swan theorem</a> if <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 class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff</a> and <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/projective+module">projective</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathbb{C})</annotation></semantics></math>-module and indeed there is an equivalence between projective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,\mathbb{C})</annotation></semantics></math>-modules and complex vector bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>More on this below in <em><a href="#VectorBundlesAndModules">Vector bundle and modules</a></em>.</p> </div> <h3 id="VectorBundlesAndModules">Vector bundles and modules</h3> <p>A <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> is a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> over the <a class="existingWikiWord" href="/nlab/show/point">point</a>. For every <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/space">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>, its collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(E)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/section">section</a>s is a module over the monoid/ring of functions 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>. When <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/ringed+space">ringed space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(X)</annotation></semantics></math> is usefully thought of as a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of modules over the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>:</p> <p>For describing vector bundles and their generalization it turns out that this perspective of encoding them in terms of their modules of sections is useful. For instance the category of vector bundles on a space typically fails to be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>. But if instead of looking just as sheaves of modules 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> that arise as sections of vector bundles one generalizes to <a class="existingWikiWord" href="/nlab/show/coherent+sheaf">coherent sheaves of modules</a> then one obtains an abelian category, something like the completion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Vect(X)</annotation></semantics></math> to an abelian category. If one further demands that the category be closed under push-forward operations, such as to obtain a <a class="existingWikiWord" href="/nlab/show/bifibration">bifibration</a> of generalized vector bundles over spaces, one arrives at the notion of <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaf">quasicoherent sheaves of modules</a> over the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a>.</p> <p>But it turns out that the category of <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaf">quasicoherent sheaves</a> over a test space (see there for details) is equivalent simply to the category of <em>all</em> modules over the (functions on) this test space. This means that quasicoherent sheaves of modules have a nice description in terms of the general-abstract-nonsense characterization of modules discussed above:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> our <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of of test <a class="existingWikiWord" href="/nlab/show/space">space</a>s (hence the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> our <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of “functions rings” on test spaces), by the above the assignment of all modules over a test space is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> Mod : C^{op} \to (\infty,1)Cat </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo>:</mo><mi>U</mi><mo>↦</mo><mi>Stab</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Mod : U \mapsto Stab( C/U ) \,. </annotation></semantics></math></div> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/space">space</a> regarded as an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, a “quasicoherent <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>-stack of modules” 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><mi>X</mi><mo>→</mo><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \to Mod \,. </annotation></semantics></math></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><strong>module</strong>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module">(∞,1)-module</a>, <a class="existingWikiWord" href="/nlab/show/2-module">2-module</a>, <a class="existingWikiWord" href="/nlab/show/module+category">module category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module+object">module object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad+for+modules+over+an+algebra">operad for modules over an algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+independence">linear independence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+module">injective module</a>, <a class="existingWikiWord" href="/nlab/show/projective+module">projective module</a>, <a class="existingWikiWord" href="/nlab/show/free+module">free module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+module">Noetherian module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/completion+of+a+module">completion of a module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasicoherent+module">quasicoherent module</a></p> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+derived+stack">module over a derived stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fredholm+module">Fredholm module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Beck+module">Beck module</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+combination">linear combination</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submodule">submodule</a>, <a class="existingWikiWord" href="/nlab/show/quotient+module">quotient module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finitely+generated+module">finitely generated module</a>, <a class="existingWikiWord" href="/nlab/show/presentable+module">presentable module</a>, <a class="existingWikiWord" href="/nlab/show/finitely+presented+module">finitely presented module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+module">projective module</a>, <a class="existingWikiWord" href="/nlab/show/injective+module">injective module</a>, <a class="existingWikiWord" href="/nlab/show/free+module">free module</a>, <a class="existingWikiWord" href="/nlab/show/flat+module">flat module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a>, <a class="existingWikiWord" href="/nlab/show/N-graded+module">N-graded module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Banach+module">Banach module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoidal+functor">module over a monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+an+algebra+over+an+operad">module over an algebra over an operad</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Frank+W.+Anderson">Frank W. Anderson</a>, <a class="existingWikiWord" href="/nlab/show/Kent+R.+Fuller">Kent R. Fuller</a>, <em>Rings and Categories of Modules</em>, Graduate Texts in Mathematics, <strong>13</strong> Springer (1992) [<a href="https://doi.org/10.1007/978-1-4612-4418-9">doi:10.1007/978-1-4612-4418-9</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Edwin+Bland">Paul Edwin Bland</a>, <em>Rings and Their Modules</em>, De Gruyter (2011) [<a href="https://doi.org/10.1515/9783110250237">doi:10.1515/9783110250237</a>, <a href="http://site.iugaza.edu.ps/mashker/files/%D9%85%D8%AD%D8%A7%D8%B6%D8%B1%D8%A7%D8%AA-%D8%AC%D8%A8%D8%B1-%D8%AF%D9%83%D8%AA%D9%88%D8%B1%D8%A7%D8%A9.pdf">pdf</a>]</p> </li> </ul> <p>Lecture notes:</p> <ul> <li id="Lawvere92"> <p><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, pp. 27 of: <em>Introduction to Linear Categories and Applications</em>, course lecture notes (1992) [<a href="https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf">pdf</a>]</p> <blockquote> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules are called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<em>linear spaces</em>)</p> </blockquote> </li> </ul> <p>Lectures notes on <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> of modules / modules over a <a class="existingWikiWord" href="/nlab/show/ringed+space">ringed space</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Murfet">Daniel Murfet</a>, <em>Modules over a ringed space</em> (<a href="http://therisingsea.org/notes/RingedSpaceModules.pdf">pdf</a>)</li> </ul> <p>Exposition of basics of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> and <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a>:</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">geometry of physics – categories and toposes</a></em>, Section 2: <em><a href="geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra">Basic notions of categorical algebra</a></em></li> </ul> <p>Formalization in <a class="existingWikiWord" href="/nlab/show/cubical+type+theory">cubical</a> <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/1lab">1lab</a>: <em><a href="https://1lab.dev/Algebra.Ring.Module.html">Algebra.Ring.Module</a></em></li> </ul> <h3 id="on_modules_as_enriched_presheaves">On modules as enriched presheaves</h3> <p>See also the references at <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> and at <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a>.</p> <h3 id="on_modules_as_stabilized_overcategories">On modules as stabilized overcategories</h3> <p>The observation that the category of modules over a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is equivalent to the category of abelian group objects in the overcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CRing</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">CRing/R</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Beck+module">Beck module</a>) is due to</p> <ul> <li id="Beck67"> <p><a class="existingWikiWord" href="/nlab/show/Jon+Beck">Jon Beck</a>, <em>Triples, algebras and cohomology</em>, Ph.D. thesis, Columbia University, 1967, Reprints in Theory and Applications of Categories, No. 2 (2003) pp 1-59 (<a href="http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html">TAC</a>)</p> </li> <li id="Quillen70"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+G.+Quillen">Daniel G. Quillen</a>, <em>On the (co-)homology of commutative rings</em>, in Proc. Symp. on Categorical Algebra, 65 – 87, American Math. Soc., 1970.</p> </li> </ul> <p>The fully abstract higher categorical concept in terms of <a class="existingWikiWord" href="/nlab/show/stabilization">stabilized</a> <a class="existingWikiWord" href="/nlab/show/overcategory">overcategories</a> and the <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-category">tangent (∞,1)-category</a> appears in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Deformation+Theory">Deformation Theory</a></em></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-modules">(∞,1)-modules</a> over <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebras">A-∞ algebras</a> are discussed in section 4.2 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></li> </ul> <p>Multiplicatively cancellative modules over a rig appear in</p> <ul> <li id="NazariGhalandarzadeh19">Rafieh Razavi Nazari, Shaban Ghalandarzadeh, <em>Multiplication semimodules</em>, 2019 (<a href="https://arxiv.org/abs/1904.11729">arXiv:0704.2106</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 3, 2024 at 13:31:31. 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