Rational Stable Parametrized Homotopy Theory

• Rational stable parametrized homotopy theory is the study of stable homotopy invariants for families of spectra, translated into algebraic models using complete dg Lie algebras.
• A sequence of Quillen equivalences converts parametrized topological data into complete differential graded modules over the universal enveloping algebra.
• The algebraic model facilitates direct computation of fiberwise stable homotopy groups, multiplicative structures, and base-change effects on arbitrary base spaces.

Rational stable parametrized homotopy theory is the paper of stable homotopy-theoretic phenomena for families of spaces (or spectra) parametrized over arbitrary bases, after passage to rational coefficients and stabilized in the sense of spectra. The field aims to encode parametrized topological invariants entirely within algebraic frameworks, enabling direct computation and classification of stable invariants using tools from homological and Lie algebra methods. Recent advances culminate in the construction of explicit algebraic models for the rational stable homotopy theory of parametrized spectra over general base spaces, connecting geometric constructions with representations of complete differential graded Lie algebras and modules over their universal enveloping algebras (Félix et al., 14 Sep 2025).

1. Algebraic Modeling Framework

The foundational algebraic model developed for rational stable parametrized homotopy theory is based on a sequence of Quillen equivalences arising from the interplay between simplicial, Lie algebraic, and module-theoretic categories. Given a base simplicial set $B$, one chooses a complete differential graded Lie algebra (cdgl) Lie model $L$ for $B$ (via Quillen or Sullivan methods). A retractive cdgl over $L$ is a cdgl $M$ equipped with a "structure map" $L \to M$ and a retraction $M \to L$ such that $M \cong L \oplus K$ for a suitable "fiber" cdgl $K$.

The stable model category of spectra of retractive simplicial sets over $B$, denoted $\mathbf{Sp}_B$, is shown to be Quillen equivalent to the stable model category of spectra of retractive cdgls over $L$ ($\mathbf{Sp}_L$), then to spectra of connected $\widehat{UL}$-modules ($\mathbf{Sp}_{\widehat{UL}}^0$), and finally to the abelian category of complete differential graded modules over the completed universal enveloping algebra $\widehat{UL}$ of $L$, denoted $\mathbf{cdgm}_{\widehat{UL}}$. These equivalences induce a strong monoidal equivalence at the homotopy category level:

$$\operatorname{Ho}(\mathbf{Sp}_B^{\mathbb{Q}}) \simeq \operatorname{Ho}(\mathbf{cdgm}_{\widehat{UL}})$$

where all stable and monoidal structures are preserved. This correspondence translates the entire rational stable parametrized homotopy theory into a computationally tractable algebraic setting (Félix et al., 14 Sep 2025).

2. Sequence of Quillen Equivalences

The sequence of Quillen equivalences is critical for the passage from parametrized topological data to strict algebraic models:

• Step 1: $\mathbf{Sp}_B \leftrightarrows \mathbf{Sp}_L$. The functor is an extension of the classical Quillen adjunction between simplicial sets and cdgls, reformulated for spectra of retractive objects and stabilized using Hovey's machinery.
• Step 2: $$\mathbf{Sp}_L \leftrightarrows \mathbf{Sp}_{\widehat{UL}}^0$$. A fiber functor extracts the non-base part (i.e., the $K$ component in $M \simeq L \oplus K$) and regards it as a module over $UL$. The stabilization here is akin to linearization, as only the fiberwise data determines the invariant in the rational context.
• Step 3: $$\mathbf{Sp}_{\widehat{UL}}^0 \leftrightarrows \mathbf{cdgm}_{\widehat{UL}}$$. Here, stabilization again passes the information to the derived category of (complete, graded) $\widehat{UL}$-modules.

At the homotopy category level, the composition of these functors defines a strong monoidal equivalence, ensuring that fiberwise smash products correspond to derived (completed) tensor products of modules, and base change operations are modeled by module-theoretic extension or restriction of scalars.

3. Explicit Algebraic Invariants

This model provides precise computational access to stable invariants. For a rational $B$-spectrum $X$,

• The fiberwise stable homotopy groups $\pi^{st}_n(X)$ correspond to the $(n-1)$-st homology groups $H_{n-1}(M_X)$ of the associated complete $\widehat{UL}$-module $M_X$ under the equivalence.
• The parametrized smash product over the base $B$, $X \wedge_B Y$, is sent to the completed tensor product $M_X \widehat{\otimes}_{\widehat{UL}} M_Y$ (with diagonal $\widehat{UL}$-action).
• The unit for the smash product, i.e., the fiberwise sphere spectrum $S_B$, corresponds to the algebra $\widehat{UL}$ itself.

This translation enables fiberwise computations (such as Ext and Tor invariants, and base-change effects) to be carried out completely within the homological algebra of the ring $\widehat{UL}$.

4. Generality and Base Space Dependence

A salient feature of this approach is its applicability to arbitrary base spaces, including spaces with nontrivial fundamental or higher homotopy groups. The algebraic model for a given base $B$ is determined by a choice of complete dg Lie model $L$ of $B$ (which can be constructed using the Quillen or Sullivan methodologies by applying minimal model theory and sheafifying over simplicial covers if needed). The universal enveloping algebra $\widehat{UL}$ is completed with respect to the lower central filtration, ensuring convergence and well-definedness in the non-simply connected context.

This approach extends prior results, such as the strictly simply-connected case treated by Braunack-Mayer, to the generality of non-simply connected and even highly nontrivial spaces, and is compatible with various forms of parametrized and equivariant stable homotopy theory.

5. Monoidal and Homotopical Structures

The developed equivalence is strong monoidal: the monoidal structure (smash product over $B$) on parametrized spectra is identified with (completed) tensor products over $\widehat{UL}$. For any pair $$X, Y \in \operatorname{Ho}(\mathbf{Sp}_B^{\mathbb{Q}})$$,

$$\Psi(X \wedge_B Y) \cong \Psi(X) \otimes_{\widehat{UL}} \Psi(Y)$$

where $\Psi$ is the derived composite realization-to-algebra functor, and $\Psi(S_B) \cong \widehat{UL}$.

Furthermore, base-change under a map $f: B \to B'$ at the level of spectra is modeled algebraically by extension or restriction of scalars along the induced morphism $\widehat{UL} \to \widehat{UL'}$.

6. Computational Applications and Significance

The algebraic model provides direct tools for computing rational fiberwise stable homotopy groups, constructing model-based fibrewise generalized cohomology, and understanding multiplicative and base-change structures in parametrized settings. It enables the parametrized version of calculations for such invariants as:

• Rational stable homology and cohomology of section spaces,
• Computation and classification of rational ring spectra and module spectra parametrized by nontrivial $B$,
• Explicit computation of fiberwise traces, transfers, and duality phenomena.

This approach thus supports practical computations in fields such as parametrized index theory, stable gauge theory, and categorified representation theory, significantly broadening the reach of rational stable parametrized homotopy techniques by leveraging the detailed homological algebra of $\widehat{UL}$-modules.

7. Schematic Summary Table

Category	Notation	Description
Parametrized spectra	$\mathbf{Sp}_B$	Spectra of retractive simplicial sets over $B$
Ret. Lie-algebra spectra	$\mathbf{Sp}_L$	Spectra of retractive cdgls over $L$
Ret. $\widehat{UL}$-mods	$\mathbf{Sp}_{\widehat{UL}}^0$	Spectra of connected $\widehat{UL}$-modules
Dg-module model	$\mathbf{cdgm}_{\widehat{UL}}$	Complete d.g. $\widehat{UL}$-modules (final target)

The category equivalences are mediated by explicit Quillen adjunctions, preserving stable and monoidal structures throughout.

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This algebraic translation—preserving all stable and monoidal data—constitutes a comprehensive and computationally robust approach to rational stable parametrized homotopy theory over arbitrary base spaces (Félix et al., 14 Sep 2025).