nLab (infinity,1)-module

Idea

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.

Definition

See at module over an algebra over an (∞,1)-operad.

Examples

• in the stable (∞,1)-category of spectra: module spectrum over an ring spectrum

Properties

Compact modules

Propositon

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

(HA, prop. 8.2.5.2)

Relation to fiberwise stabilization

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 $R$:

Proposition

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 $A$

$$Stab(E_\infty/R) \simeq A Mod(Spec)$$

is equivalentl to the category of $R$-module spectra.

This is (Lurie, cor. 1.5.15).

Stable Dold-Kan correspondence

For $R$ an ordinary ring, write $H R$ for the corresponding Eilenberg-MacLane spectrum.

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 $R$-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).

Remark

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.

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. , 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 p. 15 of (BCT).

Related concepts

• action, ∞-action

• module, $(\infty,1)$-module

  • (∞,1)-category of (∞,1)-modules

  • tensor product of ∞-modules

  • (∞,1)-module bundle

  • perfect module

• 2-module

• bimodule

  (∞,1)-bimodule

• (∞,n)-module

• representation, ∞-representation

References

Discussion in terms of module objects in symmetric monoidal model categories of spectra includes

• Anthony Elmendorf, Igor Kriz, Peter May, section 5 of Modern foundations for stable homotopy theory (pdf)

Modules over algebras over an arbitrary (∞,1)-operad are discussed in section 3.3 of

• Jacob Lurie, Higher Algebra

Modules specifically over A-∞ algebras are discussed in section 4.2 there.

Further discussion of (infinity,n)-bimodules? is in

• Rune Haugseng, The higher Morita category of $E_n$-algebras (arXiv:1412.8459)

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

• Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules , Topology 42 (2003), 103-153 (pdf)

Applications to string topology are discussed in

• Andrew Blumberg, Ralph Cohen, Constantin Teleman, Open-closed field theories, string topology and Hochschild homology (arXiv:0906.5198)

See the section on string topology at sigma model for more on this.