nLab
(infinity,1)-module

Context

Higher algebra

Higher linear algebra

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

Contents

Idea

The notion of (,1)(\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

Properties

Compact modules

Propositon

Let RR be an A-∞ ring. The (∞,1)-category of ∞-modules RModR 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 E_\infty-modules over RR as objects in the stabilization of the over-(∞,1)-category over RR:

Proposition

Let E :=Alg Comm(Grpd)E_\infty := Alg^{Comm}(\infty Grpd) be the (∞,1)-category of E-∞ rings and let RE R \in E_\infty. Then the stabilization of the over-(∞,1)-category over AA

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

is equivalentl to the category of RR-module spectra.

This is (Lurie, cor. 1.5.15).

Stable Dold-Kan correspondence

For RR an ordinary ring, write HRH R for the corresponding Eilenberg-MacLane spectrum.

Theorem

For RR any ring (or ringoid, even) there is a Quillen equivalence

HRModCh (RMod) H R Mod \simeq Ch_\bullet(R Mod)

between model structure on HRH R-module spectra and the model structure on chain complexes (unbounded) of ordinary RR-modules.

This presents a corresponding equivalence of (∞,1)-categories. If RR 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 (,1)(\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 HRH R-algebra spectra and dg-algebras.

Example

For XX a topological space and RR a ring, let C (X,R)C_\bullet(X, R) be the standard chain complex for singular homology H (X,R)H_\bullet(X, R) of XX with coefficients in RR.

Under the stable Dold-Kan correspondence, prop. 1, this ought to be identified with the smash product (Σ + X)HR(\Sigma^\infty_+ X) \wedge H R of the suspension spectrum of XX with the Eilenberg-MacLane spectrum. Notice that by the general theory of generalized homology the homotopy groups of the latter are again singular homology

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

While the correspondence (Σ + X)HRC (X,R)(\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 HRH R-module spectra and Ch (RMod)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)