(also nonabelian homological algebra)
Rational stable homotopy theory is the simple special case of stable homotopy theory under rationalization.
It is a classical fact that the rationalization of classical homotopy theory (of topological spaces or simplicial sets) – called rational homotopy theory – is considerably more tractable than general homotopy theory, as exhibited by the existence of small concrete dg-algebraic models for rational homotopy types: minimal Sullivan algebras or equivalently their dual dg-coalgebras. A similar statement holds for the rationalization of stable homotopy theory i.e. the homotopy theory of spectra (of topological spaces or simplicial sets): rational spectra are equivalent to rational chain complexes, i.e. to dg-modules over $\mathbb{Q}$. This is a dg-model for rational stable homotopy theory compatible with that of classical rational homotopy theory in that the stabilization adjunction that connects classical homotopy theory to stable homotopy theory is, under these identifications, modeled by the forgetful functor from dg-(co-)algebras to chain complexes
By the stable Dold-Kan correspondence, the (stable) homotopy theory of rationalized spectra, namely of $H \mathbb{Q}$-module spectra is equivalent to that of chain complexes of modules/vector space over the rational numbers
(Schwede-Shipley 03, theorem 5.1.6, see also at stable Dold-Kan correspondence).
Observe that $H \mathbb{Q}$-module spectra are just the rational spectra. Since rationalization of spectra is the smashing localization $(-)\wedge H \mathbb{Q}$ (here) every rational spectrum $X \simeq H \mathbb{Q} \wedge X$ carries an $H \mathbb{Q}$-module structure. Conversely, just as a $\mathbb{Q}$-module structure on an abelian group is unique if it exists, so an $H \mathbb{Q}$-module spectrum structure on a spectrum is essentially unique if it exists, due to the fact that the multiplication map $H \mathbb{Q} \wedge H \mathbb{Q} \to H \mathbb{Q}$ is an equivalence.
Theorem parallels that of classical rational homotopy theory:
There is an equivalence of homotopy theories between the homotopy theory of nilpotent rational topological spaces of finite type with that of cochain dgc-algebras over $\mathbb{Q}$ in non-negative degree
The following composite total derived functors
(where the key part in the middle right, involving $Sym$, is from prop. , the left middle part involving connectivity is from prop. , and the equivalences on the far left and far right are those from theorem and theorem , respectively)
agree with the restriction of the stabilization infinity-adjunction
to simply connected rational homotopy types of finite type.
By the nature of the classical model structure on topological spaces, every simply connected object of $\mathrm{Ho}(\mathrm{Top}^{\ast/})$ is represented by one that is a retract of a transfinite composition of pushouts of the pointed connected generating cofibrations $S^n \to D^n$, for $n \geq 1$. More abstractly, the (∞,1)-category of simply connected homotopy types is generated under (∞,1)-colimits from the n-spheres of dimension $n \geq 2$. (See the analogous argument used in the Brown representability theorem here).
Since $\Sigma^\infty$ is a left adjoint functor it is sufficient to check that the two functors agree on $S^n$ for $n \geq 1$. That they agree on $D^n$ is immediate. We hence need to consider their value on the positive dimensional $n$-spheres:
Let $n$ be odd. Then the minimal Sullivan model for $S^n$ is $\mathrm{Sym}( \mathbb{R}[n] )$. Since every dgc-algebra is fibrant in the projective model structure, the value of $\mathbb{R}(U \circ \mathrm{ker}(\epsilon_{(-)}))$ on this model is represented by the value of $U \circ \mathrm{ker}(\epsilon_{(-)})$ on that model, which is the cochain complex $\mathbb{Q}[n]$, hence equivalently the corresponding chain complex. This is indeed the rationalization of $\Sigma^\infty S^n$.
Next let $n$ be even with $n \geq 1$. Then the minimal Sullivan model for $S^n$ is $\mathrm{Sym}( \mathbb{R}[n] \oplus \mathbb{R}[2n-1], d c_{2n-1} = c_{n} \wedge c_n )$. The underlying chain complex of the augmentation ideal is spanned by the elements $c_{n}^{\wedge^k}$ in degree $n k$ and by the elements $c_{n}^{\wedge^k} \wedge c_{2n-1}$ in degree $nk+2n-1$. The former elements are all closed, but except for $k = 1$ are all in the image of the latter elements, none of which is closed. Hence this chain complex is quasi-isomorphic to $\mathbb{R}[n]$. Again, this is indeed the rationalization of $\Sigma^\infty S^{n}$
Here we collect some background definitions, notations and facts for ease of reference in the main text above.
Throughout, let $k$ be a field of characteristic zero.
Write
$\mathrm{Ch}_{k,\bullet}$ for the category of chain complexes of $k$-vector spaces, i.e. the category of $\mathbb{Z}$-graded $k$-vector spaces $V$ equipped with a linear map $\partial_V : V \to V$ of degree -1 such that $\partial_V \circ \partial_V = 0$ (the differential);
$\mathrm{Ch}_k^\bullet$ for the category of cochain complexes of $k$-vector spaces, i.e. as before, but with differential $d_V$ of degree +1.
For $V$ a $k$-vector space, and $n \in \mathbb{N}$ we write $V[n]$ for the (co-)chain complex concentrated on $V$ in degree $k$.
For $n \in \mathbb{Z}$ write
for the full subcategories on the (co-)chain complexes that are concentrated in degrees $\geq n$.
The inclusions from def. have right resp. left adjoints, which we denote by
and
These are Quillen adjunctions with respect to the projective model structure on chain complexes.
We write $\mathrm{dgcAlg}^\bullet_k$ for the category of differential graded-commutative algebras over $k$, i.e. commutative monoids in $\mathrm{Ch}^\bullet_k$. We write $(\mathrm{dgcAlg}^\bullet_k)_{/k[0]}$ for its slice category over $k = (k[0], d = 0) \in \mathrm{dgcAlg}^{\geq 0}$, hence for the category of augmented dgc-algebras, hence for dgc-algebras $A$ equipped with a homomorphism $\epsilon_A : A \to (k[0], d = 0)$.
Finally we write
for the full subcategory of the dgc-algebras in non-negative degree, hence for the commutative monoids in $\mathrm{Ch}^{\geq 0}$, and similarly for the augmented case
There are adjunctions between the categories from def. of the form
and
where
$U$ forms the underlying chain complex;
$\mathrm{ker}(\epsilon_{(-)})$ forms the augmentation ideal;
$\mathrm{Sym}$ forms the free graded-commutative dg-algebra;
$\mathrm{Sym}_{/k[0]}$ forms the free graded-commutative dg-algebra and regards it with its canonical augmentation over $k$.
Moreover, these are Quillen adjunctions: the first with respect to the projective model structure on chain complexes and the projective model structure on dgc-algebras and the second with respect to the corresponding slice model structure.