# nLab (infinity,1)-module

### Context

#### Higher algebra

higher algebra

universal algebra

homotopy theory

# Contents

## 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.

## 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

### 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. 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

$\pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R) \,.$

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).

## References

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.

Revised on May 23, 2016 06:52:13 by Urs Schreiber (131.220.184.222)