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A Fibrational Perspective on Differential Linear Logic

This paper proposes a categorical semantics for Differential Linear Logic by modeling it as a pair of Grothendieck fibrations equipped with a tangent functor, thereby adapting type theory methods to linear-non-linear adjunctions as a foundational step toward unifying DiLL with dependent types.

Original authors: Jad Koleilat

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jad Koleilat

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Mixing Two Different Worlds

Imagine you are trying to build a machine that can do two very different things at once:

The "Strict" World (Linear Logic): In this world, resources are precious. If you have one apple, you can use it, but once you use it, it's gone. You can't copy it or throw it away. It's like a strict accountant where every penny must be accounted for exactly once.
The "Flexible" World (Dependent Types): In this world, things can change based on context. Imagine a map where the roads you can take depend on where you are standing. If you are in Paris, the map shows Paris streets; if you are in London, it shows London streets. The rules change depending on the situation.

The Problem:
The paper looks at a logical system called Differential Linear Logic (DiLL). This system tries to do calculus (differentiation) inside logic. Calculus is naturally "flexible" because the slope of a curve depends on exactly where you are on the curve. However, standard DiLL is "strict" (linear). It struggles to express the idea that "the derivative depends on the specific point you are looking at."

The author, Jad Koleilat, asks: Can we build a model where the strict rules of logic can handle the flexible, changing nature of calculus?

The Solution: The "Fibration" (The Bundle of Maps)

To solve this, the author uses a mathematical tool called a Fibration.

The Analogy: The Travel Agency and the Tour Guides
Imagine a giant travel agency (the Base Category). This agency handles the main destinations (like "Paris" or "London").

In a standard model, the agency just lists the destinations.
In this paper's model, for every destination, there is a specific bundle of tour guides (the Fiber) attached to it.
- If the destination is "Paris," the bundle contains guides who speak French and know Paris streets.
- If the destination is "London," the bundle contains guides who speak English and know London streets.

This structure is called a Fibration. It allows the "strict" rules of the travel agency to work with the "flexible" reality that every location has its own specific rules.

The "Linear Simple Category": The Specialized Tour Bus

The author builds a specific type of Fibration called the Linear Simple Category.

The Analogy:
Think of a standard bus (Linear Logic) where passengers (resources) can't get off or duplicate themselves.
Now, imagine a specialized tour bus (The Linear Simple Category).

The bus has a Driver (the non-linear part, like the destination "Paris"). The driver can be copied, ignored, or changed freely.
The bus has Passengers (the linear part, like the "apple"). They must stay on the bus and can't be duplicated.
The magic is that the route the bus takes (the logic) depends on where the driver is going.

This structure allows the author to mix the "strict" passengers with the "flexible" destinations.

The "Linear Tangent Functor": The Differential Engine

The core of the paper is introducing a new engine to this bus system called the Linear Tangent Functor.

The Analogy: The Speedometer and the Map
In calculus, to find the speed (derivative) of a car, you need two things:

The car's current location (the point on the map).
The direction and speed it is moving (the tangent vector).

In the author's model:

The Base Category is the map of all possible locations.
The Linear Simple Category is the collection of all possible "location + speed" pairs.
The Linear Tangent Functor is the machine that takes a "location" and automatically generates the corresponding "location + speed" pair.

The author defines three rules (axioms) for this machine to ensure it behaves like real calculus:

Preservation: If you combine two locations, the machine combines their speed pairs correctly.
Identity: If you have a "pure" vector space (like a straight line), the machine knows exactly how to turn it into a speed pair.
Partial Linearity: This is the most complex rule. It ensures that if you have a function that is "linear" in one part (like speed) but "flexible" in another (like location), the machine can calculate the change correctly. It's like ensuring that if you change the location slightly, the speed calculation updates smoothly without breaking the strict rules of the passengers on the bus.

What Did They Prove?

The paper proves two main things:

Upgrading Old Models: If you take an existing, simpler model of DiLL (called a Differential Seely Category) and put it into this new "Fibration" structure, it still works perfectly. The new structure is a generalization, meaning it covers all the old cases and more.
Creating New Models: Every "slice" (or specific destination) of this new structure acts like a perfect model of Differential Linear Logic. This means the author has successfully created a framework where the strict rules of logic can handle the changing, dependent nature of calculus.

The "So What?" (Without Speculation)

The paper claims this is a first step toward unifying two major fields:

Differential Linear Logic (DiLL): Logic that handles calculus.
Dependent Types: Logic where types depend on values (like "a list of 5 items" vs "a list of 10 items").

By using this "Fibration" (the bundle of maps) approach, the author shows that it is possible to express the derivative of a function as a dependent function. In simple terms, they built a logical "container" that can hold the idea that "the derivative of a function changes depending on which point you are looking at," something previous logical models struggled to do formally.

The paper does not claim to solve real-world physics problems or create new software yet; it is purely a theoretical construction to see if these two complex mathematical worlds can fit together in a single, consistent framework.

Problem Statement
Differential Linear Logic (DiLL) extends Linear Logic (LL) with rules for differentiation (cocontraction, coweakening, codereliction), capturing higher-order differentiation via symmetries between linear and non-linear formulas. While DiLL has established categorical models (Differential Seely Categories) and connections to Cartesian Differential Categories (CDC), it remains a simply typed theory. This presents a conceptual gap: in differential geometry, the differential of a smooth function $f: M \to N$ is a dependent function $D_x(f): TM_x \multimap TN_{f(x)}$ , where the tangent space $TM_x$ depends on the point $x$ . The paper asks whether a dependent version of DiLL exists and how its models can be expressed to accommodate this dependency between linear types (vector spaces) and non-linear types (manifolds).

Methodology
The author employs the framework of Grothendieck fibrations to model dependency, adapting methods from the categorical semantics of dependent type theory to the context of Linear-Non-Linear (LNL) adjunctions.

Linear Simple Fibration: Starting from an LNL adjunction between a cartesian category $\mathcal{C}$ (non-linear) and a symmetric monoidal category $\mathcal{L}$ (linear), the paper constructs a "linear simple category" $LS(\mathcal{C})$ . This category serves as the linear logic equivalent of the simple category for simply typed theories. Objects are pairs $(X, A)$ where $X \in \mathcal{C}$ and $A \in \mathcal{L}$ , representing a context where a linear type $A$ depends on a non-linear context $X$ .
Fibred Adjunctions: The paper demonstrates that the original LNL adjunction lifts to a fibred adjunction between the simple fibration of $\mathcal{C}$ and the linear simple fibration $LS(\mathcal{C})$ .
Linear Tangent Functor: To introduce differentiation, the paper defines a "linear tangent functor" $T: \mathcal{C} \to LS(\mathcal{C})$ . This functor is required to be a product-preserving section of the fibration, satisfying specific axioms regarding the tangent bundle of non-linear objects and the partial linearity of maps.
Generalised Differential Seely Categories (GDSC): By equipping an LNL adjunction with this linear tangent functor, the paper defines a GDSC.

Key Contributions

Linear Simple Fibration: The construction of $LS(\mathcal{C})$ from any LNL adjunction, providing a fibred semantics for linear logic that mirrors the structure of dependent type theories.
Fibred LNL Adjunctions: Proving that any LNL adjunction induces a fibred adjunction between the simple category of the base and the linear simple category, preserving the logical structure fiberwise.
Generalised Differential Seely Categories (GDSC): Defining GDSCs as additive LNL adjunctions equipped with a linear tangent functor satisfying three axioms:
- $T$ is a product-preserving section.
- $T(U(A)) = (U(A), A)$ (the tangent bundle of a vector space object is the object itself).
- A "Partial Linearity Rule" ensuring the tangent functor interacts correctly with weakening and partial linear maps.
Equivalence of Models: Proving that in any GDSC, every fiber of the linear simple fibration is a Differential Seely Category (DSC). Conversely, any DSC induces a GDSC. This establishes that the fibred approach generalizes existing models.
Connection to Generalised CDCs: The paper investigates the relationship between GDSCs and Generalised Cartesian Differential Categories (GCDCs). It shows that the Kleisli category of a DSC is a CDC and that the endofunctor induced by the tangent functor in a GDSC satisfies most axioms of a GCDC, though the interchange rule (GCDC.7) for all morphisms remains an open question.

Results

The paper establishes that the semantics of DiLL can be extended from Seely categories to any additive LNL adjunction with biproducts, provided a linear tangent functor exists.
It is proven that the fibers of the linear simple fibration in a GDSC are themselves DSCs, meaning the fibred structure recovers the standard models of DiLL locally while providing a global dependent structure.
The construction recovers the standard deriving transform $\partial$ in each fiber, satisfying the constant, Leibniz, linear, chain, and interchange rules of DiLL.
The work links the fibred perspective to the broader landscape of differentiation categories, positioning GDSCs as a generalization that sits between DSCs and the emerging theory of fibred tangent categories.

Significance and Claims
The paper claims this work is a "first step" towards unifying Differential Linear Logic and dependent types. By expressing DiLL models as a pair of Grothendieck fibrations equipped with a tangent functor, the author provides a categorical framework capable of handling the dependency of linear types on non-linear contexts, which is intrinsic to differential geometry but absent in standard DiLL.

The significance lies in:

Unification: Offering a path to model dependent DiLL, potentially enabling the expression of types like $\Pi_{x:M} (TM_x \multimap TN_{f(x)})$ .
Generalization: Extending the semantic scope of DiLL beyond Seely categories to any additive LNL adjunction, opening the search for new models arising from tangent categories.
Foundational Bridge: Connecting the categorical semantics of differentiation (DSC, CDC) with the semantics of dependency (fibrations, comprehension categories), suggesting that tangent categories may serve as the natural dependent generalization of GCDCs and models for dependent DiLL.

The author remains modest, noting that while the fibred construction works for the specific case of simple fibrations over LNL adjunctions, the ultimate goal is to find structures in general models of dependent LL (such as those described by Vákár and Lundfall) that satisfy these conditions. The paper does not claim to have fully constructed a dependent DiLL type theory, but rather to have provided the necessary categorical infrastructure (GDSCs) to support such a development.