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class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Stable+Homotopy+Theory'>Introduction</a></em></p> <h1 id='ingredients'>Ingredients</h1> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></li> </ul> <h1 id='contents'>Contents</h1> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/loop+space+object'>loop space object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/suspension+object'>suspension object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/looping'>looping and delooping</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stabilization'>stabilization</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/spectrum+object'>spectrum object</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+derivator'>stable derivator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category+of+spectra'>stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+category'>stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smash+product+of+spectra'>smash product of spectra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Spanier-Whitehead+duality'>Spanier-Whitehead duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/stable+homotopy+theory+-+contents'>Edit this sidebar</a> </p> </div></div> <h4 id='higher_algebra'>Higher algebra</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a></strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+algebra'>universal algebra</a></p> <h2 id='algebraic_theories'>Algebraic theories</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+theory'>algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/2-Lawvere+theory'>2-algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-algebraic+theory'>(∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-monad'>(∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/operad'>operad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%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/diff/algebra+over+a+monad'>algebra over a monad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-monad'>∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+Lawvere+theory'>algebra over an algebraic theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-algebraic+theory'>∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra over an operad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-operad'>∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action'>action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module'>module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a></p> </li> </ul> <h2 id='higher_algebras'>Higher algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+%28infinity%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/diff/smash+product+of+spectra'>smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-algebra'>A-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-space'>A-∞ space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/C_%E2%88%9E-algebra'>C-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-module bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+simplicial+algebras'>model structure on simplicial T-algebras</a> / <a class='existingWikiWord' href='/nlab/show/diff/homotopy+T-algebra'>homotopy T-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+operads'>model structure on operads</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+conjecture'>Deligne conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> </div> <h1 id='contents_2'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#reference'>Reference</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <em>(commutative) ring spectrum</em> refers to a (<a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative</a>) <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a> in the <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+category'>stable homotopy category</a> regarded as <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a> via the <a class='existingWikiWord' href='/nlab/show/diff/smash+product+of+spectra'>smash product of spectra</a>. Hence a ring spectrum is a <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> equipped with a product operation, which is associative (and commutative) up to unspecified <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>.</p> <p>Under the <a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>Brown representability theorem</a>, a ring spectrum represents a <a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a>.</p> <p>Since the concept of <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> is the refinement of the concept of <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a> to <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>/<a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>. The concept of <strong>ring spectrum</strong> is the corresponding generalization of the notion of (<a class='existingWikiWord' href='/nlab/show/diff/ring'>commutative</a>) <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a>.</p> <p>Therefore the use of monoids in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> is limited, what one often needs is further enhancement to the structure of a <a class='existingWikiWord' href='/nlab/show/diff/coherence+law'>homotopy coherent</a> <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ rings</a>/<a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ rings</a>.</p> <p>It turns out that these may still be modeled as ordinary (<a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative</a>) <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoids</a> with respect to a sufficiently good <a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a> as they exist on <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+spectrum'>orthogonal spectra</a>, <a class='existingWikiWord' href='/nlab/show/diff/symmetric+spectrum'>symmetric spectra</a> and <a class='existingWikiWord' href='/nlab/show/diff/S-module'>S-modules</a>. To contrast with the naive monoids in the <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+category'>stable homotopy category</a>, these monoids are called <em><a class='existingWikiWord' href='/nlab/show/diff/highly+structured+spectrum'>highly structured ring spectra</a></em>.</p> <p>But beware that this terminology, in turn, is often abbreviated again to just “ring spectrum”. For more on this see at <em><a class='existingWikiWord' href='/nlab/show/diff/brave+new+algebra'>brave new algebra</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a></em>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/periodic+ring+spectrum'>periodic ring spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+ring+spectra'>model structure for ring spectra</a></p> </li><ins class='diffins'> </ins><ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/Boardman+homomorphism'>Boardman homomorphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/Adams+spectral+sequence'>Adams spectral sequence</a></p> </li></ins> </ul> <h2 id='reference'>Reference</h2> <ul> <li id='Adams74'><a class='existingWikiWord' href='/nlab/show/diff/John+Frank+Adams'>Frank Adams</a>, part III, section 10 of <em><a class='existingWikiWord' href='/nlab/show/diff/Stable+homotopy+and+generalised+homology'>Stable homotopy and generalised homology</a></em>, 1974</li> </ul> <p>Survey includes</p> <ul> <li id='Malkiewich14'><a class='existingWikiWord' href='/nlab/show/diff/Cary+Malkiewich'>Cary Malkiewich</a>, section 1.3 of <em>The stable homotopy category</em>, 2014 (<a href='http://math.stanford.edu/~carym/stable.pdf'>pdf</a>)</li> </ul> <p>Discussion of ring spectra as rings with respect to the <a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric smash product of spectra</a> on <a class='existingWikiWord' href='/nlab/show/diff/S-module'>S-modules</a> includes</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Anthony+Elmendorf'>Anthony Elmendorf</a>, <a class='existingWikiWord' href='/nlab/show/diff/Igor+Kr%CC%8Ciz%CC%8C'>Igor Kriz</a>, <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, sections 2 and 3 of <em><a class='existingWikiWord' href='/nlab/show/diff/Modern+foundations+for+stable+homotopy+theory'>Modern foundations for stable homotopy theory</a></em> (<a href='http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf'>pdf</a>)</li> </ul> <p>A comprehensive account for for <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+spectrum'>orthogonal spectra</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Stefan+Schwede'>Stefan Schwede</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Global+homotopy+theory'>Global homotopy theory</a></em></li> </ul> <p>An account in terms of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a> is in section 7.1 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Algebra'>Higher Algebra</a></em></li> </ul> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+ring'>simplicial</a> ring spectra is in</p> <ul> <li id='GoerssHopkins99'><a class='existingWikiWord' href='/nlab/show/diff/Paul+Goerss'>Paul Goerss</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, <em>Simplicial Structured Ring Spectra</em> (1999) (<a href='http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.9216'>web</a>)</li> </ul> <p>See also the references at <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a>.</p> <ul> <li>Glossary for stable and chromatic honotopy theory (<a class='existingWikiWord' href='/nlab/files/StableChromaticGlossary.pdf' title='pdf'>pdf</a>)</li> </ul> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on February 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