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arxiv: 2605.07858 · v1 · submitted 2026-05-08 · 💻 cs.LO

Recognition: no theorem link

A Fibrational Perspective on Differential Linear Logic

Jad Koleilat
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:50 UTC · model grok-4.3

classification 💻 cs.LO
keywords Differential Linear LogicGrothendieck fibrationstangent functorlinear-non-linear adjunctionscategorical semanticsdependent typessequent calculus
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The pith

Categorical models of differential linear logic arise as pairs of Grothendieck fibrations equipped with a tangent functor.

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

The paper establishes that models of differential linear logic can be represented using pairs of Grothendieck fibrations together with a tangent functor. It reaches this by taking the standard fibrational approach to dependent types and adapting it to the linear-non-linear adjunctions that define DiLL. This gives a uniform categorical structure for the differentiation rules and the formula symmetries in the logic. A sympathetic reader would care because the construction is offered as an initial step that could eventually let differential features and dependent types live inside the same framework.

Core claim

The paper claims that categorical models of DiLL can be expressed as a pair of Grothendieck fibrations equipped with a tangent functor. The construction adapts methods from the categorical semantics of type theory directly to linear-non-linear adjunctions, thereby interpreting the differential structure through the tangent functor while keeping the linear and non-linear parts organized by the two fibrations. This yields a new perspective on the sequent calculus of DiLL and is presented as the first step toward unifying it with dependent type theories.

What carries the argument

A pair of Grothendieck fibrations equipped with a tangent functor, adapted from type-theoretic semantics to the linear-non-linear adjunctions of DiLL.

If this is right

  • The tangent functor supplies the categorical counterpart to the differential operator in the sequent calculus.
  • The two fibrations keep the linear and non-linear formulas separate yet related through the adjunction.
  • The sequent rules of DiLL are preserved by the lifting properties of the fibrations.
  • The same structure supplies a natural route for later incorporation of dependent types.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same fibrational setup might be used to interpret concrete programming languages that combine linear types with differentiation.
  • One could test the claim by building the two fibrations explicitly for a well-known concrete model of DiLL.
  • The approach suggests that tangent-category techniques could interact with other fibrational models already used in logic.

Load-bearing premise

The fibrational techniques for dependent types can be transferred to linear-non-linear adjunctions without breaking the differential structure or creating inconsistencies.

What would settle it

An explicit DiLL model in which no pair of Grothendieck fibrations and tangent functor can be defined that respects both the adjunctions and the differentiation rules would refute the claim.

read the original abstract

Differential Linear Logic (DiLL) is a sequent calculus that expresses differentiation via symmetries between linear and non-linear formulas. In this paper, we express categorical models of DiLL as a pair of Grothendieck fibrations equipped with a tangent functor. To do so, we adapt methods from categorical semantics of type theory to linear-non-linear adjunctions. This is a first step towards unifying DILL and dependent types.

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

1 major / 2 minor

Summary. The manuscript claims to express categorical models of Differential Linear Logic (DiLL) as a pair of Grothendieck fibrations equipped with a tangent functor. It does so by adapting standard methods from the categorical semantics of dependent type theory (reindexing, cartesian liftings, and substitution) to the linear-non-linear adjunctions that underlie DiLL models, presenting this as an initial step toward unifying DiLL with dependent types.

Significance. If the adaptation is shown to preserve the differential structure (including dereliction, promotion, and derivative symmetries) while respecting the LNL distinction, the result would supply a fibrational semantics for DiLL that could support dependent contexts and substitution in a differential setting. This would strengthen connections between existing fibrational techniques for type theory and the semantics of linear logic.

major comments (1)
  1. [The fibrational model construction] The central construction (the pair of fibrations with tangent functor) must verify that reindexing functors commute appropriately with the tangent functor so that linear substitution preserves the differential rules of DiLL. Without an explicit naturality or coherence diagram for the derivative with respect to the LNL adjunction, it is unclear whether the model validates the full DiLL sequent calculus or only a fragment.
minor comments (2)
  1. [Introduction and preliminaries] Notation for the two fibrations and the tangent functor should be introduced with explicit diagrams showing the cleavage and the action on objects/morphisms.
  2. [Abstract] The abstract states the result but does not indicate whether the construction is parameter-free or requires additional coherence conditions; a short statement on this point would help readers assess the scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the major comment on the fibrational model construction below and will make revisions to improve clarity as outlined.

read point-by-point responses

  1. Referee: The central construction (the pair of fibrations with tangent functor) must verify that reindexing functors commute appropriately with the tangent functor so that linear substitution preserves the differential rules of DiLL. Without an explicit naturality or coherence diagram for the derivative with respect to the LNL adjunction, it is unclear whether the model validates the full DiLL sequent calculus or only a fragment.

    Authors: We agree that explicit verification of the commutation between reindexing and the tangent functor is essential to ensure the model captures the full DiLL sequent calculus. In the manuscript, this commutation is ensured by the way the tangent functor is defined on the linear fibration while being compatible with the LNL adjunction and the reindexing functors from the non-linear side, adapting the standard cartesian lifting and substitution mechanisms from dependent type theory. However, we recognize that the absence of a dedicated naturality diagram may leave room for ambiguity. In the revised manuscript, we will include an explicit coherence diagram showing the naturality of the derivative with respect to the LNL adjunction, along with a short explanation of how this preserves the differential rules including dereliction and promotion. This addition will make it clear that the construction validates the complete set of DiLL rules. revision: yes

Circularity Check

0 steps flagged

No circularity: fibrational model for DiLL constructed by adapting standard type-theoretic methods to LNL adjunctions

full rationale

The paper's central construction adapts Grothendieck fibration techniques and tangent functors from dependent type semantics to linear-non-linear adjunctions without any quoted equations, definitions, or self-citations that reduce the claimed model back to its inputs by construction. The abstract and described approach treat the adaptation as a novel transfer of existing machinery rather than presupposing the DiLL structure or fitting parameters to the target result. No load-bearing steps collapse into self-definition, renamed known results, or self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract relies on standard categorical constructions without introducing new fitted parameters or invented entities; the central move is an adaptation of existing fibrational semantics.

axioms (2)
  • standard math Grothendieck fibrations can be equipped with tangent functors that preserve the required differential structure
    Invoked when the paper states that models are expressed as fibrations equipped with a tangent functor.
  • domain assumption Methods from the categorical semantics of dependent type theory apply without loss to linear-non-linear adjunctions
    The adaptation step that lets the authors transfer fibrational techniques to DiLL.

pith-pipeline@v0.9.0 · 5345 in / 1538 out tokens · 47903 ms · 2026-05-11T01:50:34.105997+00:00 · methodology

discussion (0)

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Reference graph

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