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.
Last revised on May 23, 2016 at 06:52:13. See the history of this page for a list of all contributions to it.