symmetric monoidal (∞,1)-category of spectra
Stab(E_\infty/R) \simeq A Mod(Spec)
is equivalentl to the category of -module spectra.
This is (Lurie, cor. 1.5.15).
H R Mod \simeq Ch_\bullet(R Mod)
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 -categories appears as (Lurie, theorem 126.96.36.199).
This is a stable version of the Dold-Kan correspondence.
Under the stable Dold-Kan correspondence, prop. 1, this ought to be identified with the smash product of the suspension spectrum of 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) \,.
An ordinary vector bundle is a bundle of -modules for some ring (which should be a field, or otherwise we’d rather say “module bundle”). Generalizing here from a ring to a ring spectrum, we may hence regard -module spectra as (∞,1)-vector spaces, and ∞-bundles of these as (∞,1)-vector bundles. See there for more details.
A comprehensive general discussion is in
The equivalence between the homotopy categories of -mopdule spectra and 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.