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The notion of -module over an monoid object in an (∞,1)-category (for instance an A-∞ ring or E-∞ ring) is the generalization to (stable)homotopy theory of the notion of module over a ring.
See at module over an algebra over an (∞,1)-operad.
(HA, prop. 184.108.40.206)
Relation to fiberwise stabilization
By the discussion an tangent (∞,1)-category we may realize -modules over as objects in the stabilization of the over-(∞,1)-category over :
Let be the (∞,1)-category of E-∞ rings and let . Then the stabilization of the over-(∞,1)-category over
is equivalentl to the category of -module spectra.
This is (Lurie, cor. 1.5.15).
Stable Dold-Kan correspondence
For an ordinary ring, write for the corresponding Eilenberg-MacLane spectrum.
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 220.127.116.11).
See at algebra spectrum_ for the corresponding statement for -algebra spectra and dg-algebras.
For a topological space and a ring, let be the standard chain complex for singular homology of with coefficients in .
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
While the correspondence 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).
Discussion in terms of module objects in symmetric monoidal model categories of spectra includes
Modules over algebras over an arbitrary (∞,1)-operad are discussed in section 3.3 of
Modules specifically over A-∞ algebras are discussed in section 4.2 there.
Further discussion of (infinity,n)-bimodules? is in
The equivalence between the homotopy categories of -module spectra and is due to
Alan Robinson, The extraordinary derived category , Math. Z. 196 (2) (1987) 231–238.
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.