nLab rational parameterized stable homotopy theory

under construction

Idea

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 tat 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

$$\array{ & \text{classical homotopy theory} && \text{stable homotopy theory} \\ & spaces & \underoverset{\underset{\Sigma^\infty}{\longrightarrow}}{\overset{\Omega^\infty}{\longleftarrow}}{} & spectra \\ & & \text{stabilization} \\ \text{rationally} & \text{dg(co)algebras} & \underoverset{\underset{underlying}{\longrightarrow}}{\overset{free}{\longleftarrow}}{} & \text{chain complexes} }$$

Classical homotopy theory and stable homotopy theory are unified and jointly generalized in parameterized stable homotopy theory, whose objects are parameterized spectra, parameterized over a classical homotopy type. The rational parameterized stable homotopy theory to be discussed here is supposed to be the rationalization of this joint generalization, unifiying and jointly generalizing the algebraic model of rational topological spaces by Sullivan algebras and of $H \mathbb{Q}$-module spectra by chain complexes.

$$\array{ & \text{stable homotopy theory} && \text{parameterized stable homotopy theory} && \text{classical homotopy theory} \\ & \text{spectra} &\overset{\phantom{\text{include}}}{\longrightarrow}& \text{parameterized spectra} &\overset{\phantom{\text{project}}}{\longrightarrow}& \text{spaces} \\ & &\text{include}& &\text{project}& \\ \text{rationally} & \text{chain complexes} &\overset{\phantom{\text{include}}}{\longrightarrow}& \text{dg-modules} &\overset{\phantom{\text{project}}}{\longrightarrow}& \text{dg(co)algebras} }$$

Here we (intend to) show that, accordingly, rational parameterized homotopy theory is presented by the the opposite of the homotopical category of dg-modules over cochain differential graded-commutative algebras in non-negative degrees.

under construction

Preliminaries

Chain complexes

Definition

Write

• $Ch_{\bullet,\mathbb{Q}}$ for the category of chain complexes of modules/vector spaces over $\mathbb{Q}$ (i.e. differential of degree -1)

• $Ch^{\bullet}_{\mathbb{Q}}$ for the category of cochain complexes (i.e. differential of degree +1).

For $n \in \mathbb{N}$ write

• $Ch_{\geq n, \mathbb{Q}} \hookrightarrow Ch_{\bullet,\mathbb{Q}}$ for the full subcategory of the chain complexes concentrated in degree $\geq n$;

• $Ch^{\geq n}_{\mathbb{Q}} \hookrightarrow Ch^\bullet_{\mathbb{Q}}$ for the full subcategory of the cochain complexes concentrated in degree $\geq n$.

For $V \in \mathbb{Q} Mod$ a rational vector space, and for $n \in \mathbb{N}$, we write $V[n]$ both for the chain complex as well as for the cochain complex concentrated on $V$ in degree $n$.

dg-Algebras

Definition

Write $dgcAlg^{\geq 0}_{\mathbb{q}}$ for the category of cochain dgc-algebras over the rational numbers concentrated in non-negative degrees.

Say that a morphism in this category is

1. a weak equivalence if it is a quasi-isomorphisms on the underlying chain complexes;

2. a fibration if it is degreewise surjection;

3. a cofibration it it is a relative Sullivan algebra inclusion,

We write

$$(dgcAlg^{\geq 0}_{\mathbb{Q}})_{proj}$$

for the category $dgcAlg^{\geq 0}_{\mathbb{Q}}$ equipped with these three classes of morphisms.

Proposition

The homotopical category $(dgcAlg^{\geq 0}_{\mathbb{Q}})_{proj}$ from def. is a model category, to be called the projective model structure on dgc-algebras in non-negative degrees.

(Bousfield-Gugenheim 76, theorem 4.3)

Example

For $S \in sSet$ a simplicial set, write

$$\Omega^\bullet_{poly}(S) \in dgcAlg^{\geq 0}_{\mathbb{Q}}$$

for the polynomial differential forms with rational coefficients on $S$.

(Bousfield-Gugenheim 76, def. 2.1)

Definition

Write $\mathbb{Q}[0] \coloneqq (\mathbb{Q}[0], d = 0)$ for the dgc-algebra concentrated on the ground field in degree 0, necessarily with vanishing differential. This is the initial object in $dgcAlg^{\geq 0}_{\mathbb{Q}}$.

Write

$$(dgcAlg^{\geq 0}_{\mathbb{Q}})_{/\mathbb{Q}[0]}$$

for the slice category of that of all dgc-algebras (def. ) over $\mathbb{Q}[0]$. Hence an object in this category is a pair consisting of a dgc-algebra $A$ and a dg-algebra homomorphism of the form

$$\epsilon_A \;\colon\; A \longrightarrow \mathbb{Q}[0] \,.$$

This is equivalently called a $\mathbb{Q}[0]$-augmented dgc-algebra. The kernel of the augmentation map $\epsilon$

$$ker_{\epsilon_{A}} \in Ch^{\geq 0}_{\mathbb{Q}}$$

is the augmentation ideal of $(A,\epsilon)$.

Since $\mathbb{Q}[0] \in dgcAlg^{\geq 0}_{\mathbb{Q}}$ carries a unique augmentation $\epsilon = id$, we still write $\mathbb{Q}[0]$ for the ground field regarded as an augmented dgc-algebra. As such this is now a zero object.

Furthermore write

$$\left( (dgcAlg^{\geq 0}_{\mathbb{Q}})_{/\mathbb{Q}[0]} \right)_{proj}$$

for the slice model structure induced on this by the projective model structure on dgc-algebras according to prop. .

See also Bousfield-Gugenheim 76, 4.11

Simplicial Lie algebras

$$(LieAlg_k)^{\Delta^{op}}_{proj} \underoverset {\underset{N}{\longrightarrow}} {\overset{N^\ast}{\longleftarrow}} {\bot} (dgLieAlg_k)_{proj} \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} dgcoAlg_k$$

(…)

Statement

We want to claim the following:

For every $\mathfrak{g} \in (LieAlg_{\mathbb{Q}})^{\Delta^{op}}$ there is a Quillen equivalence

$$SeqSpec\left( \mathfrak{g}Mod \right)_{stable} \underoverset {\underset{SeqSpec((Sym)^{op})}{\longrightarrow}} { \overset{SeqSpec( ker_{\epsilon(-)}^{op} )}{\longleftarrow} } {\simeq_{Qu}} SeqSpec\left( \mathfrak{g} / (LieAlg_{\mathbb{Q}})^{\Delta^{op}} / \mathfrak{g} \right)_{stable}$$

Idea of proof: the analogous statement for simplicial Lie algebras replaced by rational simplicial algebras $cAlg_{\mathbb{A}}^{\Delta^{op}}$ is Schwede 97, theorem 3.2.3. Apart from the connectivity of the $Sym$-construction, all that this proof uses is that simplicial commutative algebras form a right proper simplicial model category. But also the model structure on simplicial Lie algebras is right proper and simplicial.

Related concepts

• rational model for a suspension

• rational homotopy theory

• rational stable homotopy theory

• rational parametrized homotopy theory

• rational equivariant homotopy theory

• rational equivariant stable homotopy theory

References

A classical reference on plain rational homotopy theory is:

• Aldridge Bousfield, Victor Gugenheim, On PL deRham theory and rational homotopy type, Memoirs of the AMS 179 (1976) [ams:memo-8-179]

The equivalence between $H R$-module spectra (unparametrized) and $R$-chain complexes is due to

• Stefan Schwede, section 3 of Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 104 (pdf)

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

Discussion of rational fiberwise suspension spectra:

• Michael C. Crabb, Ioan Mackenzie James, around Prop. 15.8 of: Fiberwise homotopy theory, Springer Monographs in Mathematics, Springer (1998) $[$doi:10.1007/978-1-4471-1265-5, pdf ,pdf$]$

• Yves Félix, Aniceto Murillo Daniel Tanré, Fibrewise stable rational homotopy, Journal of Topology, Volume 3, Issue 4, 2010, Pages 743–758 (doi:10.1112/jtopol/jtq023)

A discussion of full-blown rational parametrized stable homotopy theory is due to

• Vincent Braunack-Mayer, Rational parameterized stable homotopy theory, Zurich, 2018

• Vincent Braunack-Mayer, Strict algebraic models for rational parametrised spectra I, Algebraic & Geometric Topology 21 (2021) 917–1019 [arXiv:1910.14608, doi:10.2140/agt.2021.21.917]

Application to mathematical analysis of duality between M-theory and type IIA string theory:

• Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber, Gauge enhancement of Super M-Branes rational parameterized stable homotopy theory, Communications in Mathematical Physics 371: 197 (2019) (arXiv:1806.01115, doi:10.1007/s00220-019-03441-4)

• Vincent Braunack-Mayer, Parametrised homotopy theory and gauge enhancement, talk at Higher Structures in M-Theory Durham Symposium 2018 , Fortschritte der Physik (2019) (doi:10.1002/prop.201910003m arXiv:1903.02862)