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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='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#relation_to_multiplicative_generalized_cohomology'>Relation to multiplicative generalized cohomology</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The terminology <em>(commutative) ring spectrum</em> refers either 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>, or to the richer structure of a <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a> in a <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+spectra'>model structure for spectra</a> equipped with a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric smash product of spectra</a>.</p> <p>In the first case a ring spectrum is a <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> equipped with a unit and product operation, which is associative, unital (and commutative) just up to unspecified <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>, such as in an <a class='existingWikiWord' href='/nlab/show/diff/H-space'>H-space</a> structure. Accordingly, these might be called “H-ring spectra”, but it is traditional to just call them “ring spectra.” (An <em><a class='existingWikiWord' href='/nlab/show/diff/H-infinity+ring+spectrum'>H-infinity ring spectrum</a></em> (<a href='#BrunerMayMcLureSteinberger86'>Bruner-May-McClure-Steinberger 86</a>) is such an H-ring spectrum equipped with some extra structure modeling <a class='existingWikiWord' href='/nlab/show/diff/extended+power'>extended power</a> operations.)</p> <p>In the second case the structure is much richer; and in good cases, such as for <a class='existingWikiWord' href='/nlab/show/diff/highly+structured+spectrum'>highly structured spectra</a>, is equivalent to <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a> structure (<a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a> structure).</p> <p>To distinguish the two situations further qualification is being used. Sometimes one says <strong>homotopy ring spectrum</strong> to explicitly refer to the first case (e.g. <a href='#Schwede12'>Schwede 12, chapter II 4.1</a>) or one says “<a class='existingWikiWord' href='/nlab/show/diff/highly+structured+spectrum'>highly structured ring spectrum</a>” to refer explicitly to the second case. 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> <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> <table><thead><tr><th><a class='existingWikiWord' href='/nlab/show/diff/algebra'>algebra</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/homological+algebra'>homological algebra</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a></th></tr></thead><tbody><tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/differential+graded+ring'>dg-ring</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/module'>module</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/dg-module'>dg-module</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectrum</a></td></tr> </tbody></table> <h2 id='definition'>Definition</h2> <p>For details see <em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Stable+Homotopy+Theory'>Introduction to Stable homotopy theory</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Stable+homotopy+theory+--+1-2'>Part 1-2 -- Structured spectra</a></em>.</p> <h2 id='properties'>Properties</h2> <h3 id='relation_to_multiplicative_generalized_cohomology'>Relation to multiplicative generalized cohomology</h3> <p>Under the <a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>Brown representability theorem</a>, the <a class='existingWikiWord' href='/nlab/show/diff/generalized+cohomology'>generalized cohomology theory</a> represented by a ring spectrum inherits the structure of a <a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a>.</p> <p>Conversely via the <a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>Brown representability theorem</a> a spectrum representing a <a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a> inherits the structure of (at least) an H-ring spectrum. See <a href='multiplicative+cohomology+theory#BrownRepresentabilityByRingSpectra'>there</a>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ultracommutative+ring+spectrum'>ultracommutative ring spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+ring+spectrum'>symmetric ring spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectral+symmetric+algebra'>symmetric algebra spectrum</a></p> </li> <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/Gorenstein+ring+spectrum'>Gorenstein ring spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/functor+with+smash+products'>functor with smash products</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/Hopf+E-%E2%88%9E+algebra'>Hopf 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> <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> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+spectral+sequence'>multiplicative spectral sequence</a></p> </li> </ul> <h2 id='references'>References</h2> <p>The notion of (commutative) homotopy ring spectra, i.e. (<a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative</a>) <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoids</a> in the <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+category'>stable homotopy category</a> with respect to the <a class='existingWikiWord' href='/nlab/show/diff/smash+product+of+spectra'>smash product of spectra</a>:</p> <ul> <li id='Adams74'> <p><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 (<a href='https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/adams-shgh.pdf'>pdf</a>)</p> <blockquote> <p>(attributed there to <a class='existingWikiWord' href='/nlab/show/diff/George+Whitehead'>George Whitehead</a>)</p> </blockquote> </li> </ul> <p>Review:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Boardman'>John Michael Boardman</a>, Sections 3,7 of: <em>Stable Operations in Generalized Cohomology</em><span> <del class='diffmod'> (</del><ins class='diffmod'> [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf),</ins></span><a href='https://hopf.math.purdue.edu/Boardman/stabop.pdf'>pdf</a><span><del class='diffmod'> )</del><ins class='diffmod'> ]</ins> in:</span><a class='existingWikiWord' href='/nlab/show/diff/Ioan+James'>Ioan Mackenzie James</a> (ed.) <em><a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Algebraic+Topology'>Handbook of Algebraic Topology</a></em> Oxford 1995 (<a href='https://doi.org/10.1016/B978-0-444-81779-2.X5000-7'>doi:10.1016/B978-0-444-81779-2.X5000-7</a>)</p> </li> <li id='BakerRichter04'> <p><a class='existingWikiWord' href='/nlab/show/diff/Andrew+Baker'>Andrew Baker</a>, <a class='existingWikiWord' href='/nlab/show/diff/Birgit+Richter'>Birgit Richter</a> <em>Structured ring spectra</em>, London Mathematical Society Lecture Notes Series 315, Springer 2004 (<a href='https://www.cambridge.org/de/academic/subjects/mathematics/algebra/structured-ring-spectra?format=PB'>ISBN:9780521603058</a>)</p> </li> <li id='TamakiKono06'> <p><a class='existingWikiWord' href='/nlab/show/diff/Dai+Tamaki'>Dai Tamaki</a>, <a class='existingWikiWord' href='/nlab/show/diff/Akira+Kono'>Akira Kono</a>, Appendix C.2 of: <em>Generalized Cohomology</em>, Translations of Mathematical Monographs, American Mathematical Society, 2006 (<a class='existingWikiWord' href='/nlab/files/GeneralizedCohomology.pdf' title='pdf'>pdf</a>, <a href='https://bookstore.ams.org/mmono-230'>ISBN: 978-0-8218-3514-2</a>)</p> </li> <li id='Malkiewich14'> <p><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>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Urs+Schreiber'>Urs Schreiber</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Stable+Homotopy+Theory'>Introduction to Stable homotopy theory</a> – <a href='Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra'>Homotopy ring spectra</a></em>, 2016</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Birgit+Richter'>Birgit Richter</a>, <em>Commutative ring spectra</em> (<a href='https://arxiv.org/abs/1710.02328'>arXiv:1710.02328</a>, <a href='https://www.math.uni-hamburg.de/home/richter/rev-commringspectra.pdf'>pdf</a>), in: <a class='existingWikiWord' href='/nlab/show/diff/Andrew+Blumberg'>Andrew J. Blumberg</a>, <a class='existingWikiWord' href='/nlab/show/diff/Teena+Gerhardt'>Teena Gerhardt</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hill'>Michael A. Hill</a> (eds.), <em><a class='existingWikiWord' href='/nlab/show/diff/Stable+categories+and+structured+ring+spectra'>Stable categories and structured ring spectra</a></em>, MSRI Book Series, Cambridge University Press.</p> </li> </ul> <p>On <a class='existingWikiWord' href='/nlab/show/diff/H-infinity+ring+spectrum'>H-infinity ring spectra</a>:</p> <ul> <li id='BrunerMayMcLureSteinberger86'>R. Bruner, <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <a class='existingWikiWord' href='/nlab/show/diff/James+McClure'>James McClure</a>, M. Steinberger, <em><math class='maruku-mathml' display='inline' id='mathml_e0702d5ab52373cf38be23b1a832ef30e9a29dca_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>H_\infty</annotation></semantics></math>-Ring Spectra and their Applications</em>, Lecture Notes in Mathematics 1176, Springer 1986 (<a href='https://link.springer.com/book/10.1007/BFb0075405'>doi:10.1007/BFb0075405</a>, <a href='http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf'>pdf</a>)</li> </ul> <p>Discussion of connective ring spectra as monoids with respect to a smash product on <a class='existingWikiWord' href='/nlab/show/diff/Gamma-space'>Gamma-spaces</a>:</p> <ul> <li id='Schwede99'> <p><a class='existingWikiWord' href='/nlab/show/diff/Stefan+Schwede'>Stefan Schwede</a>, <em>Stable homotopical algebra and <math class='maruku-mathml' display='inline' id='mathml_e0702d5ab52373cf38be23b1a832ef30e9a29dca_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>-spaces</em>, Math. Proc. Camb. Phil. Soc. (1999), 126, 329 (<a href='http://www.math.uni-bonn.de/people/schwede/Gamma.pdf'>pdf</a>)</p> </li> <li id='Lawson09'> <p><a class='existingWikiWord' href='/nlab/show/diff/Tyler+Lawson'>Tyler Lawson</a>, <em>Commutative Γ-rings do not model all commutative ring spectra</em>, Homology Homotopy Appl. Volume 11, Number 2 (2009), 189-194. (<a href='http://projecteuclid.org/euclid.hha/1296138517'>Euclid</a>)</p> </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> <p><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>)</p> </li> <li id='EKMM'> <p><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/Michael+Mandell'>Michael Mandell</a>, <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>P. May</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Rings%2C+modules+and+algebras+in+stable+homotopy+theory'>Rings, modules and algebras in stable homotopy theory</a></em>, AMS Mathematical Surveys and Monographs Volume 47 (1997) (<a href='http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf'>pdf</a>)</p> </li> </ul> <p>A comprehensive account for <a class='existingWikiWord' href='/nlab/show/diff/symmetric+spectrum'>symmetric spectra</a> is in</p> <ul> <li id='Schwede12'><a class='existingWikiWord' href='/nlab/show/diff/Stefan+Schwede'>Stefan Schwede</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Symmetric+spectra'>Symmetric spectra</a></em>, 2012 (<a href='http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf'>pdf</a>)</li> </ul> <p>and for <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+spectrum'>orthogonal spectra</a> 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 homotopy theory (<a class='existingWikiWord' href='/nlab/files/StableChromaticGlossary.pdf' title='pdf'>pdf</a>)</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on September 5, 2023 at 19:34:38. 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