symmetric monoidal (∞,1)-category of spectra
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 Comm), an $R$-module spectrum is a spectrum equipped with an $R$-action.
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 $R$:
Let $E_\infty := Alg^{Comm}(\infty Grpd)$ be the (∞,1)-category of E-∞ rings and let $R \in E_\infty$. Then the stabilization of the over-(∞,1)-category over $R$
is equivalent to the category of $R$-module spectra.
This is (Lurie, cor. 1.5.15).
For $R$ an ordinary ring, write $H R$ for the corresponding Eilenberg-MacLane spectrum.
For $R$ any ring (or ringoid, even) there is a Quillen equivalence
between model structure on $H R$-module spectra and the model structure on chain complexes (unbounded) of ordinary $R$-modules.
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). The refinement to a Quillen equivalence is (SchwedeShipley, theorem 5.1.6). See also the discussion at stable model categories. A direct description as an equivalence of $(\infty,1)$-categories appears as (Lurie, theorem 7.1.2.13).
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.
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. 1, 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
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 p. 15 of (BCT).
An ordinary vector bundle is a bundle of $k$-modules 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 spaces, and ∞-bundles of these as (∞,1)-vector bundles. See there for more details.
algebra | homological algebra | higher algebra |
---|---|---|
abelian group | chain complex | spectrum |
ring | dg-ring | ring spectrum |
module | dg-module | module spectrum |
An early account is
A comprehensive general discussion in terms of symmetric spectra is in
and in terms of (infinity,1)-category theory in
The equivalence between the homotopy categories of $H R$-module spectra and $Ch_\bullet(R Mod)$ is due to
The refinement of this statement to a Quillen equivalence is due to
Applications to string topology are discussed in
See the section on string topology at sigma model for more on this.