Rational parameterized spectra

Question

Consider the homotopical category of rational dg-modules. I suppose this ought to present the rationalization of the homotopy theory of parameterized spectra. Has such "rational parameterized stable homotopy theory" been worked out a little, anywhere?

Here by the homotopical category of dg-modules I mean the obvious thing: the category of pairs $(A,N)$ consisting of a dg-algebra $A$ over $\mathbb{Q}$ and a dg-module $N$ over $A$, with homomorphisms the evident pairs of morphisms, and with weak equivalences those pairs where both components are quasi-isomorphism. This category has been studied a lot by Agustí Roig, who considers minimal dg-modules as tools for computing the rational cohomology of topological fibrations, disregarding rational homotopy groups, see below.

Similarly, parameterized spectra are pairs $(X,E)$ consisting of a space $X$ and a spectrum object $E$ in the slice of the classical homotopy theory over $X$.

In two subcases the above equivalence is well known:

On the one hand, classical rational homotopy theory says that if we restrict to zero spectra inside all parameterized spectra, hence to the classical homotopy theory of spaces, then its rationalization is equivalent to the opposite of rational dg-algebras (on simply connected objects of finite type at least), hence to the sub-category of dg-modules, as above, on the zero modules.

At the other extreme, if we restrict attention to the trivial base space, then the homotopy theory of parameterized spectra reduces to that of plain spectra, rationalization yields $H \mathbb{Q}$-module spectra, and theorem 5.1.6 in Schwede-Shipley 03 says that these are equivalently modeled by the homotopy theory of rational chain complexes, which are the dg-modules over the trivial dg-algebra.

Finally, for any fixed base space we may apply a relative version of this argument, following the general considerations in Schwede 97.

This makes it plausible that the homotopy theory of rational dg-modules is generally equivalent to the rationalization of parameterized spectra, at least for simply-connected finite-type base objects and with homomorphisms of $H \mathbb{Q}$-module spectra understood.

For example on page 2 of Roig-Saralegi 00 the authors give a minimal model, in dg-modules over the minimal Sullivan dg-algebra for the 3-sphere, of the rationalization of the map

$$ S^4 /^h S^1 \to S^3 $$

(where $S^1$ acts canonically on $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$ and where the slash means the homotopy quotient, in contrast to the naive quotient, which is $S^3$). I am after statements such as that this minimal dg-module is (if indeed it is) a rational presentation of the $S^3$-parameterized spectrum $\Sigma^\infty_{S^3}(S^4 /^h S^1)$ (the fiberwise suspension spectrum of this projection).

Answer 1

(This is really a comment, but it got too long.) If $R$ is a ring spectrum and $X$ is a space, I'll write $R\mathrm{-Mod}_X$ for the infinity-category of $R$-modules parametrized over $X$, or in other words $\mathrm{Fun}(X, R\mathrm{-Mod})$. If $R_{X}$ denotes the constant parametrized $R$-module over $X$ with value $R$, then the endomorphisms of $R_X$ can be identified with the function spectrum $D_R(X) := F(\Sigma^{\infty}_+ X, R)$. This induces a functor $D_R(X)\mathrm{-Mod} \to R\mathrm{-Mod}_X$. Using the Barr-Beck theorem for $\infty$-categories, it is not hard to see that if $X$ is a compact object in spaces (i.e. a retract of a finite complex) then this functor is fully faithful. (As far as I know this observation is originally due to Blumberg and Mandell.) So a more precise version of your question is whether this functor is an equivalence if $X$ is simply connected (and rational) and $R = H\mathbb{Q}$. It would be enough to show that the "global sections" functor from $R\mathrm{-Mod}_X$ to $R\mathrm{-Mod}$ detects equivalences (or equivalently the zero object). I don't know whether this is true, unfortunately.

Answer 2

Something similar is proven in section 3 of

• Stefan Schwede, "Spectra in model categories and applications to the algebraic cotangent complex", Journal of Pure and Applied Algebra 120 (1997) 104 (pdf)

Theorem 3.2.3 there says that for $A$ a simplicial algebra over $\mathbb{Q}$, then the homotopy theory of spectrum objects in augmented $A$-algebras is equivalently that of spectrum objects in $A$-modules.

What I am after is effectively this statement, but with simplicial algebras over $\mathbb{Q}$ replaced by cochain dg-algebras in non-negative degree over $\mathbb{Q}$, among them the Sullivan algebras. Because augmented $A$-algebras for $A$ a Sullivan algebra, that's the rational model for the pointed objects in the slice over the rational space corresponding to $A$, and spectrum objects in non-negatively graded dg-modules is equivalently unbounded dg-modules, e.g. by prop. 2.10 (3) in:

• Brooke Shipley, "$H \mathbb{Z}$-algebra spectra are differential graded algebras", Amer. Jour. of Math. 129 (2007) 351-379. (arXiv:math/0209215)

In Schwede's article the model structure on the category of algebras is assumed to be left and right proper. On the one hand this is in order to appeal to the Bousfield-Friedlander theorem. But (as shown in the proof given behind that link, from Goerss-Jardine), only right properness is needed to establish the model structure (and Stanculescu 08 claims that not even right properness is necessary here).

On the other hand a scan through Schwede 97, section 2 shows (I think) that (only) right properness is used to establish the assumptions for the BF-theorem on the spectrification functor $Q$, namely in the proof of lemma 2.1.3 (only) right properness is used on the bottom of p. 92 (16 of 28) in the guise of "the dual of the gluing lemma (Lemma 1.1.9)".

Now the projective model structure on cochain dgc-algebras in non-negative degrees is right proper (being transferred from the projective model structure on chain complexes in non-negative degree) and it readily admits the analogous adjunction$(\mathrm{Sym} \dashv U \circ \mathrm{AugmentationIdeal})$ as in Schwede 97, first lines of section 3.2.

Therefore, it seems to me, the direct analogue of the proof of Schwede 97, lemma 3.2.2 should go through for the projective model structure on dgc-algebras in non-negative degrees (in place of simplicial commutative algebras as considered in the article) and hence the main result theorem 3.2.3 should be obtained, which, in view of classical rational homotopy theory and Schwede-Shipley, theorem 5.1.6, would be the desired statement, at least over a fixed base space.

[edit: Maybe this approach fails for $\mathrm{dgcAlg}^{\geq 0}_{\mathrm{proj}}$ not satisfying enough of the axioms of a simplicial model category to admit $(\Sigma^\infty \dashv \Omega^\infty)$ as a Quillen adjunction. But so we should just pick instead any one of the models of classical rational homotopy theory that is both right proper as well as a simplicially enriched model catgeory. The projective model structure on simplicial Lie algebras should do the trick.]