A Fibrational Perspective on Differential Linear Logic

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.

• The paper constructs a linear simple fibration from any linear-non-linear adjunction, enabling dependent-style semantics for maps that are nonlinear in context but linear in their argument.
• Generalised Differential Seely Categories arise by equipping additive LNL adjunctions with linear tangent functors satisfying three axioms, and every fibre then forms a Differential Seely Category.
• The framework recovers ordinary Differential Seely Categories, connects to Cartesian differential categories, and identifies open problems including full dependence, non-synthetic examples, and the unresolved GCDC.7 condition.

Motivation and overview

Differential Linear Logic (DiLL) extends Linear Logic with cocontraction, coweakening, and codereliction, capturing higher-order differentiation logically; its semantics is given by differential Seely categories (DSC), whose deriving transform satisfies the constant, Leibniz, linear, chain, and interchange rules (2605.07858). The paper's starting observation is that differentiation is intrinsically dependent: the differential of a smooth map $f : M \to N$ has type $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$, a family of linear types indexed by non-linear ones. DiLL itself is simply typed, so the question arises whether a dependent version exists. The author takes a first step by recasting the categorical models of DiLL in fibrational terms, adapting machinery from the categorical semantics of dependent type theory to linear-non-linear adjunctions (LNL adjunctions) (2605.07858).

The central contributions are threefold. First, from any LNL adjunction the paper constructs a linear simple fibration $ls : LS(C) \to C$, the linear-logic analogue of Jacobs' simple fibration for simply typed theories. Second, it shows that any LNL lifts to a fibred LNL adjunction between the simple fibration $s : S(C) \to C$ and this new fibration. Third, requiring a section of $ls$ satisfying three axioms — a linear tangent functor $T : C \to LS(C)$ — yields what the paper calls a Generalised Differential Seely Category (GDSC), and proves that every fibre of $LS(C)$ is then a DSC. This strictly generalises the semantics of DiLL from Seely categories to any additive LNL adjunction with biproducts.

The linear simple fibration

Given an LNL adjunction $F \dashv U$ between a cartesian category $(C,\times,I)$ and a symmetric monoidal category $(L,\otimes,1)$, the linear simple category $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$0 has objects pairs $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$1 with $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$2, $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$3, and morphisms $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$4 where $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$5 in $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$6 and $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$7 in $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$8. Composition uses the contraction on $\Pi_{x:M}(TM_x \multimap TN_{f(x)})$9 induced by the comonoid structure of cartesian objects transported along $ls : LS(C) \to C$0 via the lax–colax duality of Melliès. The projection $ls : LS(C) \to C$1 is a split fibration, and its fibres carry both a symmetric monoidal product $ls : LS(C) \to C$2 and, when $ls : LS(C) \to C$3 has products, fibrewise cartesian products $ls : LS(C) \to C$4, making $ls : LS(C) \to C$5 a monoidal fibration in Shulman's sense (2605.07858).

Two functors lift the original adjunction: $ls : LS(C) \to C$6 applies $ls : LS(C) \to C$7 pointwise using the laxator $ls : LS(C) \to C$8, and $ls : LS(C) \to C$9 applies $s : S(C) \to C$0 pointwise using the strong monoidal structure $s : S(C) \to C$1. Both are split fibred functors, and they form a fibred adjunction $s : S(C) \to C$2 whose unit and counit project to identities; moreover, each fibre is itself an LNL adjunction. The paper notes that over the terminal object the fibres $s : S(C) \to C$3 and $s : S(C) \to C$4 recover the original data exactly, so no information is lost. The induced comonad $s : S(C) \to C$5 acts fibrewise as $s : S(C) \to C$6, with fibred contraction and weakening obtained from the colax structure of $s : S(C) \to C$7.

A useful reading is that $s : S(C) \to C$8 collects maps that are non-linear in $s : S(C) \to C$9 but linear in their second argument: a morphism $ls$0 is "linear in its second component" precisely when it factors through some $ls$1 followed by a projection. The construction also supports linear sigma types: $ls$2 is left adjoint to reindexing along $ls$3, and, following Vákár, the comonad $ls$4 can be recovered entirely from this coproduct structure, with contraction and weakening arising as its comultiplication and counit (2605.07858). The construction specialises to Cruttwell–Gallagher–Lemay–Pronk's $ls$5 in the Seely setting, and to the ordinary simple fibration when $ls$6 trivially.

Generalised differential Seely categories

Assuming $ls$7 additive (CMon-enriched, so the product is a biproduct), the paper defines a GDSC as an additive LNL equipped with a functor $ls$8, the linear tangent functor, subject to three axioms:

• t.1: $ls$9 is a product-preserving section of $T : C \to LS(C)$0;
• t.2: $T : C \to LS(C)$1;
• t.3: the family $T : C \to LS(C)$2 (built from biproduct injections and the canonical comparison $T : C \to LS(C)$3) is natural, i.e., defines a transformation from the weakening functor $T : C \to LS(C)$4 to $T : C \to LS(C)$5.

Geometrically, t.1 says every object carries a trivial bundle of tangent vectors, t.2 says the tangent bundle of a vector space is its self-product, and t.3 encodes the identity $T : C \to LS(C)$6 for maps linear in their second argument. From $T : C \to LS(C)$7, the paper extracts a differential $T : C \to LS(C)$8 as the vertical component of $T : C \to LS(C)$9, partial differentials $LS(C)$0, and proves the expected calculus: $LS(C)$1 (chain rule for composition), compatibility of $LS(C)$2 with pairing, and explicit formulas for the differential of bilinear maps such as the laxator $LS(C)$3 (2605.07858).

The main theorem states that in any GDSC, every slice $LS(C)$4 is a DSC. The proof constructs a fibrewise deriving transform $LS(C)$5, where $LS(C)$6 is built from the unit of the fibred adjunction. Verifying naturality, the linear rule, the chain rule, and the Leibniz rule requires substantial computation — notably the differential of the Seely isomorphism and of the diagonal — after which Blute–Cockett–Seely–Trimble's criterion confirms $LS(C)$7 is a deriving transform. Two consequences follow immediately: the base category $LS(C)$8's partner $LS(C)$9 is itself a DSC (via the fibre over $F \dashv U$0), and conversely, every DSC yields a GDSC by taking the Kleisli adjunction $F \dashv U$1 and defining $F \dashv U$2, which the linear and chain rules make functorial (2605.07858). Hence GDSCs are a genuine common generalisation: they coincide with DSCs on Seely categories while extending the semantics to arbitrary additive LNL adjunctions with biproducts.

This also situates the work relative to Kerjean–Rogers–Maestracci, who derive much of the DSC structure from a differentiation functor on co-slices but cannot recover the monoidal rule for want of axioms governing partial differentiation; axiom t.3 here supplies precisely that missing control over differentiation in context (2605.07858).

Relation to Cartesian differential categories

The paper connects GDSCs to the Cartesian side of the correspondence: the Kleisli of a DSC is a CDC, and every CDC embeds fully faithfully into the Kleisli of a DSC (Garner). In a GDSC one can define $F \dashv U$3, and the composite $F \dashv U$4 guarantees axioms GCDC.1 through GCDC.5 hold, making $F \dashv U$5 a candidate Generalised Cartesian Differential Category. GCDC.6 follows essentially from t.3, and GCDC.7 holds for morphisms of shape $F \dashv U$6, since these are representable in $F \dashv U$7 and their second differentials can be computed via the generalised deriving transform. However, the paper concedes that GCDC.7 for all morphisms remains open, and suggests that an additional axiom on GDSCs may be required to force $F \dashv U$8 to be a GCDC in full generality (2605.07858).

Limitations and open questions

Several restrictions are acknowledged explicitly. All monoidal categories are assumed strict, purely to simplify calculations. The fibrational framework used is deliberately minimal — the pair $F \dashv U$9 over a fixed base — whereas the author argues the natural habitat is a fully dependent LL model in the sense of Vákár or Lundfall, allowing general pairs of fibred LNL adjunctions rather than only these two specific fibrations. The correspondence with GCDCs is incomplete pending GCDC.7, and no non-synthetic examples of GDSCs beyond DSCs themselves are exhibited; whether tangent categories (e.g., convenient manifolds, or Eilenberg–Moore categories of DSCs) furnish genuine GDSCs is left as the guiding open problem. Finally, the existence of a dependent DiLL sequent calculus matching these fibred models is not addressed here and remains future work (2605.07858).

Conclusion

The paper demonstrates that models of DiLL admit a clean fibrational presentation: an additive LNL adjunction with biproducts becomes a model of DiLL exactly when its linear simple fibration admits a linear tangent functor satisfying three geometrically motivated axioms. Every fibre of the resulting fibration is a differential Seely category, and every DSC arises this way, so the framework strictly subsumes prior semantics while exposing the dependence structure implicit in differentiation. The result reframes the search for dependent DiLL as the search for tangent-category-like structures whose differential bundle fibrations model dependent linear logic.