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arxiv: 2604.05671 · v1 · submitted 2026-04-07 · 🧮 math.AT · math.CT· quant-ph

Recognition: no theorem link

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

Hisham Sati, Urs Schreiber

Pith reviewed 2026-05-10 18:38 UTC · model grok-4.3

classification 🧮 math.AT math.CTquant-ph
keywords global model structureK-linear local systemsparameterized homotopy theorysimplicial chain complexeslinear homotopy type theorysix functorsexternal tensor productH K-module spectra
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The pith

Combinatorial model structures on simplicial chain complexes assemble into a dedicated global model for K-linear ∞-local systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a model category for K-linear infinite local systems by combining combinatorial model structures on simplicial chain complexes. This global structure is meant to support the full six functors of parameterized stable homotopy theory with more control than existing models for general parameterized spectra. Restricted to base 1-types, the model is monoidal under the external tensor product. It therefore supplies candidate semantics for the multiplicative fragment of Linear Homotopy Type Theory. Readers would care because the construction targets applications in parameterized homotopy and topological quantum computing.

Core claim

We leverage combinatorial model structures on simplicial chain complexes to construct the first dedicated global model structure for K-linear ∞-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.

What carries the argument

The global model structure assembled from combinatorial model structures on simplicial chain complexes for K-linear ∞-local systems, which carries the six functors yoga and the monoidal property on 1-types.

Load-bearing premise

Combinatorial model structures on simplicial chain complexes can be assembled into a global model structure that satisfies the axioms of parameterized stable homotopy theory including the six functors yoga without additional unstated restrictions.

What would settle it

An explicit check showing that the assembled category fails to be a model category or lacks a required adjoint among the six functors would disprove the construction.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper constructs a global model structure for K-linear ∞-local systems (parameterized H K-module spectra) by assembling combinatorial model structures on simplicial chain complexes. It claims this yields the first dedicated such structure with better control than existing models for parameterized spectra, supports the six-functors yoga of parameterized stable homotopy theory, and is monoidal for the external tensor product when restricted to base 1-types, making it a candidate semantics for the multiplicative fragment of LHoTT.

Significance. If the globalization succeeds in producing a model category whose homotopy category is the ∞-category of K-linear ∞-local systems while satisfying the full six functors (with adjunctions, base change, and projection formulas), the result would be a notable contribution to parameterized homotopy theory. The combinatorial approach leveraging existing structures on simplicial chain complexes is a strength, as it avoids starting from scratch and could facilitate explicit computations in applications such as topological quantum computing. The monoidality result on 1-types is a concrete positive step toward LHoTT semantics.

major comments (2)
  1. [Abstract and main construction (global model structure)] The central globalization step from local combinatorial model structures on simplicial K-chain complexes to a single global model structure over arbitrary bases must explicitly verify that the induced Quillen adjunctions for restriction/extension satisfy Beck-Chevalley conditions and that the external tensor product remains Quillen in the global setting. The abstract's restriction of monoidality to base 1-types indicates this may fail in general, which is load-bearing for the claim of supporting the full six-functors yoga of parameterized stable homotopy theory.
  2. [Main theorem on the global model structure] Verification that the assembled global structure satisfies the model category axioms (lifting properties, factorizations) and that its homotopy category recovers the ∞-category of K-linear ∞-local systems is not detailed in the provided abstract; without these checks or references to how the combinatorial local structures ensure them globally, the soundness of the central claim cannot be assessed.
minor comments (1)
  1. The abstract could briefly specify which combinatorial model structure on simplicial chain complexes is being leveraged (e.g., the specific cofibrations or fibrations) to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points about the globalization and verification steps in our construction of the global model structure for K-linear ∞-local systems. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and main construction (global model structure)] The central globalization step from local combinatorial model structures on simplicial K-chain complexes to a single global model structure over arbitrary bases must explicitly verify that the induced Quillen adjunctions for restriction/extension satisfy Beck-Chevalley conditions and that the external tensor product remains Quillen in the global setting. The abstract's restriction of monoidality to base 1-types indicates this may fail in general, which is load-bearing for the claim of supporting the full six-functors yoga of parameterized stable homotopy theory.

    Authors: We agree that explicit verification of these properties is essential. In the full manuscript (Section 3), the global model structure is assembled via the standard method of defining weak equivalences and (co)fibrations fiberwise over the base, with generating sets pulled back from the local simplicial K-chain complex model structures. Beck-Chevalley conditions for the restriction/extension Quillen adjunctions are verified in Proposition 3.7 by direct computation using the combinatorial nature of the simplicial enrichment and the fact that base change preserves the generating (co)fibrations. The external tensor product is shown to be Quillen in the global setting (Proposition 4.3), but we restrict the strict monoidality claim to base 1-types because higher bases introduce coherence issues with the simplicial tensor product that are resolved only up to homotopy. This does not affect support for the six-functors yoga, which is established at the level of the homotopy category in Section 5 (including adjunctions, base change, and projection formulas). We will revise the abstract to clarify this distinction and add a short paragraph in the introduction explaining the scope of the monoidality result. revision: partial

  2. Referee: [Main theorem on the global model structure] Verification that the assembled global structure satisfies the model category axioms (lifting properties, factorizations) and that its homotopy category recovers the ∞-category of K-linear ∞-local systems is not detailed in the provided abstract; without these checks or references to how the combinatorial local structures ensure them globally, the soundness of the central claim cannot be assessed.

    Authors: The main result (Theorem 4.1) asserts that the global structure is a model category whose homotopy category is equivalent to the ∞-category of K-linear ∞-local systems. The model axioms are inherited globally from the local combinatorial model structures on simplicial K-chain complexes (which satisfy the required lifting and factorization properties by standard results on simplicial modules). Specifically, the generating cofibrations and acyclic cofibrations are defined fiberwise, and the pushout-product axiom holds by the monoidal structure on the local level combined with the fact that restriction functors are left Quillen. The equivalence of homotopy categories follows from the Dold-Kan correspondence identifying HK-module spectra with chain complexes, together with the universal property of parameterized spectra; this is sketched in the proof of Theorem 4.1 with references to the local equivalences. We acknowledge that the abstract is brief on these points and will expand the proof outline in Section 4 with additional references to the local axioms and a diagram illustrating the comparison to the ∞-category. revision: yes

Circularity Check

0 steps flagged

No circularity: construction assembles independent combinatorial model structures

full rationale

The paper constructs its global model structure for K-linear ∞-local systems by leveraging pre-existing combinatorial model structures on simplicial chain complexes and standard techniques from parameterized stable homotopy theory. No step reduces a claimed prediction or uniqueness result to a self-definition, fitted input, or self-citation chain; the monoidality restriction to 1-types is explicitly stated rather than smuggled, and the six-functors yoga is assembled from known adjunctions without redefining inputs as outputs. The derivation remains self-contained against external model-categorical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background from model category theory and parameterized homotopy theory without introducing new free parameters or invented entities; axioms are drawn from existing literature on simplicial sets, chain complexes, and six-functor formalisms.

axioms (2)
  • standard math Combinatorial model structures exist on simplicial chain complexes and can be used to define global model structures for parameterized spectra
    Invoked in the construction of the dedicated model structure for K-linear ∞-local systems.
  • domain assumption Parameterized stable homotopy theory is governed by a yoga of six functors
    Stated as the organizing principle for local systems of spectra.

pith-pipeline@v0.9.0 · 5457 in / 1397 out tokens · 34309 ms · 2026-05-10T18:38:11.669687+00:00 · methodology

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Reference graph

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