+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Stable Homotopy theory +--{: .hide} [[!include stable homotopy theory - contents]] =-- #### Higher algebra +--{: .hide} [[!include higher algebra - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea A _module spectrum_ is a [[module]] over a [[ring spectrum]]. As for [[ring spectra]], this has different interpretations according to how much [[coherence]] is considered: A _homotopy module spectrum_ over a [[homotopy commutative ring spectrum]] is a [[module object]] in the [[stable homotopy category]] equipped with its derived [[smash product of spectra]]. On the other hand if "ring spectrum" means [[E-∞ ring]] (or [[A-∞ ring]]) then a module over a ring spectrum is a [[module over an algebra over an (∞,1)-operad]] for the [[commutative operad]]: for $R$ an [[E-∞ ring]] (an [[algebra over an (∞,1)-operad|∞-algebra]] over [[Comm]]), an $R$-module spectrum is a [[spectrum]] equipped with an $R$-[[action]]. ## Properties ### General By the discussion an [[tangent (∞,1)-category]] we may realize $E_\infty$-modules over $R$ as objects in the [[stabilization]] of the [[over-(∞,1)-category]] over $R$: +-- {: .num_prop} ###### Proposition Let $E_\infty := Alg^{Comm}(\infty Grpd)$ be the [[(∞,1)-category]] of [[E-∞ ring]]s and let $R \in E_\infty$. Then the [[stabilization]] of the [[over-(∞,1)-category]] over $R$ $$ Stab(E_\infty/R) \simeq R Mod(Spec) $$ is equivalent to the category of $R$-module spectra. =-- This is ([Lurie, cor. 1.5.15](#Lurie)). ### Stable Dold-Kan correspondence {#StableDoldKanCorrespondence} For $R$ an ordinary [[ring]], write $H R$ for the corresponding [[Eilenberg-MacLane spectrum]]. +-- {: .num_theorem #StableDoldKan} ###### Theorem For $R$ any [[ring]] (or [[ringoid]], even) there is a [[Quillen equivalence]] $$ H R Mod \simeq Ch_\bullet(R Mod) $$ between model structure on $H R$-module spectra and the [[model structure on chain complexes]] (unbounded) of ordinary $R$-[[module]]s. This presents a corresponding [[equivalence of (∞,1)-categories]]. If $R$ is a commutative ring, then this is an equivalence of [[symmetric monoidal (∞,1)-categories]]. =-- This equivalence on the level of [[homotopy categories]] is due to ([Robinson](#Robinson)). The refinement to a Quillen equivalence is ([SchwedeShipley, theorem 5.1.6](#SchwedeShipley)). See also the discussion at _[[stable model categories]]_. A direct description as an equivalence of $(\infty,1)$-categories appears as ([Lurie, remark 7.1.1.16](#Lurie)). If $R$ is a commutative ring, then this equivalence underlies an equivalence of symmetric monoidal $(\infty,1)$-categories by ([Lurie, theorem 7.1.2.13](#Lurie)). +-- {: .num_remark} ###### Remark This is a stable version of the [[Dold-Kan correspondence]]. See at _[[stable Dold-Kan correspondence]]_ for more. =-- See at __[[algebra spectrum]]_ for the corresponding statement for $H R$-algebra spectra and [[dg-algebras]]. +-- {: .num_example} ###### Example For $X$ a [[topological space]] and $R$ a ring, let $C_\bullet(X, R)$ be the standard [[chain complex]] for [[singular homology]] $H_\bullet(X, R)$ of $X$ with coefficients in $R$. Under the stable Dold-Kan correspondence, prop. \ref{StableDoldKan}, this ought to be identified with the [[smash product]] $(\Sigma^\infty_+ X) \wedge H R$ of the [[suspension spectrum]] of $X$ with the [[Eilenberg-MacLane spectrum]]. Notice that by the general theory of [[generalized homology]] the [[homotopy groups]] of the latter are again singular homology $$ \pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R) \,. $$ While the correspondence $(\Sigma^\infty_+ X) \wedge H R \sim C_\bullet(X,R)$ under the above equivalence is suggestive, maybe nobody has really checked it in detail. It is sort of stated as true for instance on <a href="http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.5198v1.pdf#page=15">p. 15 </a> of ([BCT](#BCT)). =-- ### As fibers for $(\infty,1)$-vector bundles An ordinary [[vector bundle]] is a [[bundle]] of $k$-[[module]]s for $k$ some [[ring]] (which should be a [[field]], or otherwise we'd rather say "module bundle"). Generalizing $k$ here from a ring to a [[ring spectrum]], we may hence regard $K$-module spectra as [[(∞,1)-vector space]]s, and [[associated ∞-bundle|∞-bundles]] of these as [[(∞,1)-vector bundle]]s. See there for more details. ## Related concepts [[!include homological and higher algebra -- table]] * [[comodule spectrum]] * [[ring spectrum]], [[algebra spectrum]] * [[(∞,1)-category of (∞,1)-modules]] ## References The notion of homotopy module spectra ([[module objects]] in the [[stable homotopy category]]): * {#Adams74} [[Frank Adams]], part III, section 10 of _[[Stable homotopy and generalised homology]]_, 1974 Review: * [[Urs Schreiber]], _[[Introduction to Stable homotopy theory]] -- [Homotopy ring spectra](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra)_, 2016 Discussion in terms of [[symmetric spectra]]: * {#Schwede12} [[Stefan Schwede]], chapter IV _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf)) and in terms of [[(infinity,1)-category theory]] in * {#Lurie} [[Jacob Lurie]], chapter 7 of _[[Higher Algebra]]_. The equivalence between the [[homotopy categories]] of $H R$-module spectra and $Ch_\bullet(R Mod)$ is due to * {#Robinson} Alan Robinson, _The extraordinary derived category_ , Math. Z. 196 (2) (1987) 231&#8211;238. The refinement of this statement to a [[Quillen equivalence]] is due to * {#SchwedeShipley} [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_ , Topology 42 (2003), 103-153 ([pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf)) Applications to [[string topology]] are discussed in * {#BCT} [[Andrew Blumberg]], [[Ralph Cohen]], [[Constantin Teleman]], _Open-closed field theories, string topology and Hochschild homology_ ([arXiv:0906.5198](http://arxiv.org/abs/0906.5198)) See the section on string topology at [[sigma model]] for more on this. [[!redirects module spectra]] [[!redirects module-spectra]] [[!redirects module-spectrum]] [[!redirects homotopy module spectrum]] [[!redirects homotopy module spectra]]