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href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+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></div> <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> </div> </div> <h1 id="contents">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></li> <ul> <li><a href='#relation_to_multiplicative_generalized_cohomology'>Relation to multiplicative generalized cohomology</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The terminology <em>(commutative) ring spectrum</em> refers either to a (<a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative</a>) <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a> regarded as <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> via the <a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a>, or to the richer structure of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in a <a class="existingWikiWord" href="/nlab/show/model+structure+for+spectra">model structure for spectra</a> equipped with a <a class="existingWikiWord" href="/nlab/show/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/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/homotopy">homotopy</a>, such as in an <a class="existingWikiWord" href="/nlab/show/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/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/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/highly+structured+spectra">highly structured spectra</a>, is equivalent to <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a> structure (<a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+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/highly+structured+ring+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/brave+new+algebra">brave new algebra</a></em> and <em><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></em>.</p> <p>Since the concept of <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> is the refinement of the concept of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> to <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>. The concept of <strong>ring spectrum</strong> is the corresponding generalization of the notion of (<a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative</a>) <a class="existingWikiWord" href="/nlab/show/ring">ring</a>.</p> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring">ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dg-ring">dg-ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/module">module</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dg-module">dg-module</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a></td></tr> </tbody></table> </div> <h2 id="definition">Definition</h2> <p>For details see <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory">Introduction to Stable homotopy theory</a></em>, <em><a class="existingWikiWord" href="/nlab/show/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/Brown+representability+theorem">Brown representability theorem</a>, the <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> represented by a ring spectrum inherits the structure of a <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a>.</p> <p>Conversely via the <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a> a spectrum representing a <a class="existingWikiWord" href="/nlab/show/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/ultracommutative+ring+spectrum">ultracommutative ring spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+ring+spectrum">symmetric ring spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+algebra+spectrum">symmetric algebra spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gorenstein+ring+spectrum">Gorenstein ring spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor+with+smash+products">functor with smash products</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+ring+spectrum">periodic ring spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+ring+spectrum">Hopf ring spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+ring+spectra">model structure for ring spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/commutative+monoid">commutative</a>) <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> in the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a> with respect to the <a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a>:</p> <ul> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, part III, section 10 of <em><a class="existingWikiWord" href="/nlab/show/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/George+Whitehead">George Whitehead</a>)</p> </blockquote> </li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Michael+Boardman">John Michael Boardman</a>, Sections 3,7 of: <em>Stable Operations in Generalized Cohomology</em> [<a href="https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Boardman-StableOperations.pdf" title="pdf">pdf</a>] in: <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/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/Andrew+Baker">Andrew Baker</a>, <a class="existingWikiWord" href="/nlab/show/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/Dai+Tamaki">Dai Tamaki</a>, <a class="existingWikiWord" href="/nlab/show/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/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/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/nlab/show/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/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/Andrew+J.+Blumberg">Andrew J. Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/Teena+Gerhardt">Teena Gerhardt</a>, <a class="existingWikiWord" href="/nlab/show/Michael+A.+Hill">Michael A. Hill</a> (eds.), <em><a class="existingWikiWord" href="/nlab/show/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/H-infinity+ring+spectra">H-infinity ring spectra</a>:</p> <ul> <li id="BrunerMayMcLureSteinberger86">R. Bruner, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/James+McClure">James McClure</a>, M. Steinberger, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Gamma-spaces">Gamma-spaces</a>:</p> <ul> <li id="Schwede99"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <em>Stable homotopical algebra and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-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/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/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a> on <a class="existingWikiWord" href="/nlab/show/S-modules">S-modules</a> includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Anthony+Elmendorf">Anthony Elmendorf</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, sections 2 and 3 of <em><a class="existingWikiWord" href="/nlab/show/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/Anthony+Elmendorf">Anthony Elmendorf</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Mandell">Michael Mandell</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">P. May</a>, <em><a class="existingWikiWord" href="/nlab/show/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/symmetric+spectra">symmetric spectra</a> is in</p> <ul> <li id="Schwede12"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <em><a class="existingWikiWord" href="/nlab/show/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/orthogonal+spectra">orthogonal spectra</a> in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <em><a class="existingWikiWord" href="/nlab/show/Global+homotopy+theory">Global homotopy theory</a></em></li> </ul> <p>An account in terms of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> is in section 7.1 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>Discussion of <a class="existingWikiWord" href="/nlab/show/simplicial+ring">simplicial</a> ring spectra is in</p> <ul> <li id="GoerssHopkins99"><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/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/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> </body></html> </div> <div class="revisedby"> <p> Last revised on September 5, 2023 at 19:34:38. 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