A Global Model Structure for $\mathbb{K}$-Linear $\infty$-Local Systems

Abstract: Parameterized stable homotopy theory organizes local systems of spectra over homotopy types, governed by a "yoga" of six functors. To provide semantics for the recently developed Linear Homotopy Type Theory (LHoTT), good model categories of these spectra are required, preferably monoidal with respect to the external smash product. We focus on the case of parameterized $H\mathbb{K}$-module spectra ($\infty$-local systems), motivated by recent applications of parameterized homotopy to topological quantum computing. While traditionally treated via dg-categories, we leverage combinatorial model structures on simplicial chain complexes to construct the first dedicated global model structure for $\mathbb{K}$-linear $\infty$-local systems, which offers better control than existing models for general parameterized spectra. In particular, when restricted to base 1-types, our model structure is monoidal with respect to the external tensor product, making it a candidate target semantics for the multiplicative fragment of LHoTT.

• The paper constructs a monoidal model structure for K-linear ∞-local systems, addressing gaps in LHoTT semantics and quantum computing models.
• Model structures on simplicial chain complexes provide tractable frameworks for parameterized topological quantum phenomena.
• The study enhances categorical semantics in topological quantum information by enabling monoidal operations over base groupoids.

A Global Model Structure for $\mathbb{K}$-Linear $\infty$-Local Systems

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Introduction and Motivation

The study develops a robust global model structure for $\mathbb{K}$-linear $\infty$-local systems, motivated by the need for precise categorical semantics in Linear Homotopy Type Theory (LHoTT) and recent demands in topological quantum computing. Parameterized stable homotopy theory integrates the unstable homotopy theory of spaces and the stable theory of spectra into a unified framework of parameterized spectra, which gains additional structure due to monoidality at the "global" (varying base) level, notably supporting both external direct sum and smash tensor product. The canonical need for a monoidal model structure, especially in applications to topological quantum information and the semantics of linear dependent type theories, underpins this investigation.

Existing approaches to parameterized spectra—e.g., those following May-Sigurdsson (MS) and later simplifications—do not offer a monoidal model structure that interacts compatibly with external smash products, with significant limitations for applications such as LHoTT. In many relevant settings, fiberwise stabilization leads to the comparatively tractable setting of $H\mathbb{K}$-module spectra, equivalently described by chain complexes of $\mathbb{K}$-vector spaces via the (stable) Dold-Kan correspondence. However, a direct and global model-categorical treatment for $\mathbb{K}$-linear $\infty$-local systems, tuned for computational control and monoidal operations, has been absent.

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Model Structures: Construction and Properties

The main technical contribution is the construction and analysis of a left proper, combinatorial, monoidal simplicial model structure on simplicial chain complexes $\mathbf{sCh}_\mathbb{K}$, and its amplification to model categories of $\infty$-local systems parameterized by simplicial groupoids.

Key features of the construction:

• For each simplicial groupoid (modeling the base homotopy type), the category of $\infty$0-linear $\infty$1-local systems is given as a category of simplicial functors into $\infty$2, with the projective local model structure (objectwise weak equivalences and fibrations).
• The global category of $\infty$3-linear $\infty$4-local systems is assembled as the Grothendieck integral model structure of these fiberwise model categories as the base varies.

The chain complexes have all objects cofibrant and the model structure is confirmed to be monoidal with respect to the usual tensor product. At the simplicial level, weak equivalences are detected as quasi-isomorphisms on the total complex. The resulting model category is shown to be left proper, combinatorial, and supports strong monoidal functors in base change and extension.

Significant structural outcomes:

• The model structure on $\infty$5 admits enrichment over $\infty$6 and a monoidal closed structure.
• The extension to model categories of $\infty$7-local systems on varying bases satisfies axioms for integral (Grothendieck) model structures.
• The external (or fiberwise) tensor product $\infty$8 on local systems is analyzed in depth, shown to be homotopical and monoidal under stringent circumstances.
• When restricted to base 1-types (i.e., underlying groupoids rather than general higher groupoids), the external product recovers a full-fledged monoidal model structure.

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Technical Results and Claims

Main Theorem Highlights

• Model Structure on Simplicial Chain Complexes: The category $\infty$9 supports a left proper, combinatorial, monoidal, simplicial model category structure with cofibrations, weak equivalences, and fibrations all admitted degreewise (objects are both cofibrant and fibrant).
• Fiberwise Model Structures: For any simplicial groupoid $\mathbb{K}$0, the functor category $\mathbb{K}$1 (simplicial functors) has a projective local model structure, preserved under base change and supporting closed monoidal structure.
• Global Model Category: The Grothendieck construction yields a global combinatorial model category $\mathbb{K}$2 modeling $\mathbb{K}$3-linear $\mathbb{K}$4-local systems over varying bases, with integral weak equivalences and fibrations.
• External Tensor Product: The external tensor $\mathbb{K}$5 is homotopical; it preserves weak equivalences and cofibrations on linear components, matching the derivator expectations as closely as possible given the known obstructions to pushout-product bifunctoriality on general cartesian products of higher groupoids.
• Over 1-Types: Restricting to base groupoids (i.e., 1-types) allows a canonical cartesian monoidal model structure, promoting the global category to a full external monoidal model category, essential for interpreting the multiplicative fragment of LHoTT.

Contrasts and Strong Claims

• The model provides substantially improved control and tractability for parameterized spectra in the $\mathbb{K}$6-linear case versus the classical parameterized spectra setting (modeled by $\mathbb{K}$7-modules) [see analogies: [Malkiewich23], [MaySigurdsson06]], including monoidal model structure for external products over 1-types—a claim not realized for general parameterized spectra in the classical literature.
• The construction unifies and extends approaches based on dg-categories, presenting them within the more flexible and explicit model category framework, equipped for functorial base change and colimit operations.

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Theoretical and Practical Implications

For Categorical Semantics and Homotopy Type Theory

• Semantics for LHoTT: The results lay categorical foundations for LHoTT semantics in the $\mathbb{K}$8-linear setting, particularly the multiplicative (external tensor product) fragment. The monoidality over 1-types enables interpretation of dependent linear/quantum types with variable parameter base up to 1-groupoidal level.
• Structural Control: With all objects cofibrant and explicit generating cofibrations, the model is well-suited for computational manipulations, transfers, and further "enrichment stacking" such as modules over commutative monoids in local systems.

For Topological Quantum Computing

• In topological quantum settings, ground state bundles and local systems naturally appear as flat $\mathbb{K}$9-vector bundles. The present model provides a systematic homotopical framework for these objects when parameter space is a space or 1-groupoid (e.g., braid groupoids, configuration spaces of points).
• The monoidal structure is compatible with "fusion" of parameter spaces, matching physical requirements for modeling topological (braid) gates and quantum information flows.

For Higher Representation Theory

• The model describes linear representations of general $\infty$0-groupoids, with compatibility for extension and base change, synthesizing ideas from homotopical representation theory, dg-category theory, and classical local systems.

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Future Directions

• Extensions Beyond 1-Types: One open direction is to construct tractable external monoidal model structures over higher (non-groupoidal) bases. The current analysis identifies the precise ways in which cartesian monoidal model structures fail on higher groupoids, and future work may adapt or resolve these.
• Refined Geometric Applications: Incorporating additional geometric or analytic structure (e.g., differential graded, Real or equivariant enhancements) is natural in applications to quantum field theory and topological quantum matter.
• Full Homotopical Semantics for LHoTT: The current framework addresses the multiplicative fragment; further progress may support full dependent sum and product types in the linear homotopy context provided LHoTT clarifies its higher semantic expectations.

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Conclusion

This work constructs and analyzes a global, combinatorial, monoidal model category structure for $\infty$1-linear $\infty$2-local systems parameterized by spaces (modeled as simplicial groupoids), and crucially provides monoidality for the external product over base 1-types. The results address outstanding challenges in the formalization of parameterized stable homotopy theory, opening paths for semantics in linear homotopy type theory and concrete modeling of parameterized quantum phenomena in topological and physical contexts. The explicit, functorial presentation enables direct application to representation theory, quantum computation, and beyond.

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References

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