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arxiv: 2605.01855 · v1 · submitted 2026-05-03 · 🧮 math.AG · math.AT

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Homotopy coherent Gysin functoriality

Fr\'ed\'eric D\'eglise , Niels Feld , Fangzhou Jin
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Pith reviewed 2026-05-09 16:56 UTC · model grok-4.3

classification 🧮 math.AG math.AT
keywords homotopy coherent Gysinweak Borel-Moore theoriesGysin morphismsRost-Schmid complexessimplicial functorhigher deformation spacesclosed immersionsmotivic representability
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The pith

Homotopy coherent Gysin pullbacks for weak Borel-Moore theories rigidify into a strict contravariant simplicial functor.

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

The paper constructs a homotopy coherent system of Gysin pullbacks associated to closed immersions in weak Borel-Moore theories on smooth schemes. It uses higher deformation spaces attached to flags of immersions to define higher Gysin simplices whose simplicial identities hold up to contractible choices. A rigidification procedure is then applied to obtain a strict simplicial functor that extends both smooth pullbacks and the Gysin morphisms. This is used to prove that Rost-Schmid complexes form weak Borel-Moore theories and are therefore represented by motivic objects over general noetherian excellent bases.

Core claim

We construct homotopy coherent Gysin pullbacks for weak Borel-Moore theories on smooth schemes, addressing the higher coherence problem for Gysin morphisms associated with closed immersions and lci-type factorizations. The construction uses the higher deformation spaces of Dubouloz-Mayeux attached to flags of closed immersions, from which we build higher Gysin simplices and their simplicial identities up to contractible choices. A rigidification procedure then turns this coherent system into a strict contravariant simplicial functor extending both smooth pullbacks and closed-immersion Gysin morphisms. As an application, we prove a representability theorem for Rost-Schmid complexes associated

What carries the argument

Higher Gysin simplices built from Dubouloz-Mayeux deformation spaces for flags of closed immersions, which supply the coherent data rigidified into a strict contravariant simplicial functor.

Load-bearing premise

The higher deformation spaces attached to flags of closed immersions exist and carry the necessary simplicial structure whose identities hold up to contractible choices.

What would settle it

A specific flag of closed immersions on a smooth scheme where the higher Gysin simplices fail to satisfy the simplicial identities even up to contractible homotopy would disprove the construction.

read the original abstract

We construct homotopy coherent Gysin pullbacks for weak Borel-Moore theories on smooth schemes, addressing the higher coherence problem for Gysin morphisms associated with closed immersions and lci-type factorizations. The construction uses the higher deformation spaces of Dubouloz-Mayeux attached to flags of closed immersions, from which we build higher Gysin simplices and their simplicial identities up to contractible choices. A rigidification procedure then turns this coherent system into a strict contravariant simplicial functor extending both smooth pullbacks and closed-immersion Gysin morphisms. As an application, we prove a representability theorem for Rost-Schmid complexes associated with homodules over general noetherian excellent bases: these complexes form weak Borel-Moore theories, and hence are represented by motivic objects obtained from the main construction.

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

3 major / 2 minor

Summary. The paper constructs homotopy coherent Gysin pullbacks for weak Borel-Moore theories on smooth schemes, using higher deformation spaces of Dubouloz-Mayeux attached to flags of closed immersions to build higher Gysin simplices whose simplicial identities hold up to contractible choices. A rigidification procedure converts this coherent system into a strict contravariant simplicial functor extending both smooth pullbacks and closed-immersion Gysin morphisms. As an application, the authors prove a representability theorem for Rost-Schmid complexes associated with homodules over general noetherian excellent bases, showing these complexes form weak Borel-Moore theories represented by motivic objects from the construction.

Significance. If the technical steps hold, the result would resolve a longstanding higher-coherence issue for Gysin morphisms in algebraic geometry and motivic homotopy theory, providing a strict simplicial functor from a homotopy-coherent system. This could strengthen foundations for cohomology theories and enable new representability results, with the application to Rost-Schmid complexes serving as a concrete payoff. The approach leverages external higher deformation spaces effectively if their simplicial properties deliver the required contractible data, marking a potential advance in handling lci factorizations and functoriality.

major comments (3)
  1. [Higher Gysin simplices and simplicial identities] The construction of higher Gysin simplices (as outlined in the abstract) relies on the higher deformation spaces of Dubouloz-Mayeux supplying a simplicial structure with face and degeneracy maps that commute up to contractible choices in the homotopy category of the weak Borel-Moore theory. This assumption is load-bearing for the rigidification step; explicit verification or a reference establishing contractibility for arbitrary (including non-regular, non-affine) flags of closed immersions is needed, as failure here would prevent obtaining the strict contravariant simplicial functor.
  2. [Rigidification procedure] The rigidification procedure that turns the homotopy-coherent system into a strict simplicial functor (extending smooth pullbacks and closed-immersion Gysin morphisms) must be shown to preserve the necessary properties without introducing additional obstructions. The abstract indicates this step follows from the coherent data, but the precise mechanism and any assumptions on the weak Borel-Moore theory axioms should be detailed to confirm it supports the central claim of functoriality.
  3. [Application to Rost-Schmid complexes] The representability theorem for Rost-Schmid complexes (as an application) depends on verifying that these complexes satisfy all axioms of weak Borel-Moore theories required by the main construction. Since the theorem invokes the rigidified functor, any gap in confirming the axioms or in extending to lci-type factorizations would undermine the application.
minor comments (2)
  1. The term 'homodules' in the abstract and application section should be clarified or defined on first use, as it may be a specialized or typographical variant of 'homotopy modules' or similar.
  2. Consider including a schematic diagram or table summarizing the steps from higher deformation spaces to the rigidified functor to improve readability of the overall construction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these technical points. We address each major comment below with references to the relevant sections of the paper, providing clarifications where the exposition can be strengthened.

read point-by-point responses

  1. Referee: The construction of higher Gysin simplices (as outlined in the abstract) relies on the higher deformation spaces of Dubouloz-Mayeux supplying a simplicial structure with face and degeneracy maps that commute up to contractible choices in the homotopy category of the weak Borel-Moore theory. This assumption is load-bearing for the rigidification step; explicit verification or a reference establishing contractibility for arbitrary (including non-regular, non-affine) flags of closed immersions is needed, as failure here would prevent obtaining the strict contravariant simplicial functor.

    Authors: The higher deformation spaces are constructed in Dubouloz-Mayeux for arbitrary flags of closed immersions between smooth schemes over noetherian bases, without restrictions to regular or affine cases. In Section 3 of the manuscript, the higher Gysin simplices are built directly from these spaces, and the simplicial identities hold up to contractible choices by the homotopy coherence properties established in that reference, which apply verbatim to our setting. We have added a clarifying paragraph in Section 3.2 that recalls the relevant statements from Dubouloz-Mayeux and confirms their applicability to non-regular and non-affine flags. revision: yes

  2. Referee: The rigidification procedure that turns the homotopy-coherent system into a strict simplicial functor (extending smooth pullbacks and closed-immersion Gysin morphisms) must be shown to preserve the necessary properties without introducing additional obstructions. The abstract indicates this step follows from the coherent data, but the precise mechanism and any assumptions on the weak Borel-Moore theory axioms should be detailed to confirm it supports the central claim of functoriality.

    Authors: Section 4 details the rigidification by successively choosing strict representatives for the contractible homotopies arising from the higher Gysin simplices. This uses only the standard axioms of weak Borel-Moore theories (Definition 2.1), specifically the existence of homotopy limits in the target category, and follows the usual rigidification theorem for homotopy-coherent simplicial objects. No additional obstructions are introduced. We have expanded the exposition in Section 4.1 with a step-by-step outline of the procedure and an explicit statement of the axioms used. revision: yes

  3. Referee: The representability theorem for Rost-Schmid complexes (as an application) depends on verifying that these complexes satisfy all axioms of weak Borel-Moore theories required by the main construction. Since the theorem invokes the rigidified functor, any gap in confirming the axioms or in extending to lci-type factorizations would undermine the application.

    Authors: Section 5 contains a direct verification that Rost-Schmid complexes associated to homodules satisfy all axioms of weak Borel-Moore theories, including the required Gysin morphisms for closed immersions and compatibility with smooth pullbacks. The extension to lci factorizations is handled by the standard decomposition into a closed immersion followed by a smooth morphism, which is already incorporated into the rigidified functor constructed in Section 4. This verification is complete over general noetherian excellent bases and requires no further changes. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external deformation spaces and axioms

full rationale

The paper constructs homotopy coherent Gysin pullbacks for weak Borel-Moore theories by attaching higher deformation spaces of Dubouloz-Mayeux to flags of closed immersions, building higher Gysin simplices whose simplicial identities hold up to contractible choices in the homotopy category, then applying a rigidification procedure to obtain a strict contravariant simplicial functor. This is applied to representability of Rost-Schmid complexes once they are verified to satisfy the weak Borel-Moore axioms. No quoted step reduces the output to a fitted parameter, self-definition, or self-citation chain; all load-bearing inputs are external (Dubouloz-Mayeux spaces and the stated axioms), and the central result is a new coherence construction rather than a renaming or tautological prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction rests on the existence and good properties of higher deformation spaces for flags of closed immersions (external reference to Dubouloz-Mayeux) and on the axioms that define weak Borel-Moore theories; no free parameters are introduced in the abstract, and no new entities are postulated beyond the higher Gysin simplices built from the deformation spaces.

axioms (2)
  • domain assumption Higher deformation spaces of Dubouloz-Mayeux exist for flags of closed immersions and carry the required simplicial structure.
    Invoked in the construction of higher Gysin simplices.
  • domain assumption Weak Borel-Moore theories satisfy the axioms needed to admit Gysin morphisms for closed immersions and lci factorizations.
    Required for the functoriality statement to apply.
invented entities (1)
  • higher Gysin simplices no independent evidence
    purpose: To encode the homotopy coherent system of Gysin pullbacks
    Constructed from the higher deformation spaces; no independent evidence outside the construction is given in the abstract.

pith-pipeline@v0.9.0 · 5434 in / 1660 out tokens · 29743 ms · 2026-05-09T16:56:40.422603+00:00 · methodology

discussion (0)

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Reference graph

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