symmetric monoidal (∞,1)-category of spectra
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
The notion of $(\infty,1)$-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.
Let $R$ be an A-∞ ring. The (∞,1)-category of ∞-modules $R Mod$ is a compactly generated (∞,1)-category and the compact objects coincide with the perfect modules
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 $A$
is equivalentl 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 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).
module, $(\infty,1)$-module
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 $H R$-module spectra and $Ch_\bullet(R Mod)$ 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.