Categorical Model of MALL

• Categorical model of MALL is a framework that uses Seely categories and strong endofunctors to formally represent multiplicative and additive connectives along with fixed points.
• It interprets logical connectives through tensors, internal homs, and exponential comonads, ensuring a rigorous translation from proof theory to categorical structures.
• Dynamic models like geometry of interaction and indexed linear logic extend the framework by capturing operational cut-elimination and guaranteeing denotational normalization.

The categorical model of Multiplicative Additive Linear Logic (MALL) provides a foundational semantics for linear logic through abstract category-theoretic structures. Central to the approach are Seely categories, which formalize the interplay between tensors, additive connectives, and exponentials, and strong endofunctors, which allow the interpretation of fixed points (μ for least, ν for greatest). Dynamic models, such as those leveraging geometry of interaction (GoI) and indexed linear logic, extend these foundations to account for the operational dynamics of cut elimination, reflecting both denotational invariance and the peculiarities of additive connectives.

1. Seely Categories as Categorical Models for MALL

MALL semantics are grounded in Seely categories, which model the logic’s connectives through categorical constructions. A Seely category is a *-autonomous symmetric monoidal closed category $(\mathcal{L}, \otimes, I, \mathbin{%%%%0%%%%}, T, \oplus, 0, \perp)$ equipped with a Seely-monoidal exponential comonad $!$.

Key structural components include:

• Symmetric monoidal closed structure: Objects ($A,B,C,\ldots$), tensor product ($A \otimes B$), unit ($I$), associativity ($\alpha_{A,B,C}$), symmetry ($\sigma_{A,B}$), and internal hom (linear implication) $A \multimap B$ with adjunction $L(A \otimes B, C) \cong L(A, \ell B C)$.
• Dualizing object $\perp$: Ensures *-autonomy via $!$0, giving categorical linear negation.
• Additive structure: Cartesian product ($!$1), projections ($!$2, $!$3), terminal object ($!$4), coproduct ($!$5), injections ($!$6, $!$7), and initial object ($!$8).
• Exponential comonad $!$9: With counit (dereliction) $A,B,C,\ldots$0, comultiplication (digging) $A,B,C,\ldots$1, and Seely isomorphisms $A,B,C,\ldots$2, $A,B,C,\ldots$3.

The Seely category elegantly blends multiplicative and additive fragments and supports the “linear-to-nonlinear” translation required for exponentials. Coherence conditions ensure proper interaction among these structures (Ehrhard et al., 2020).

2. Interpretation of Logical Connectives

The categorical interpretation of MALL connectives is as follows:

• Multiplicatives: $A,B,C,\ldots$4 is categorical tensor product; $A,B,C,\ldots$5 is internal hom.
• Additives: $A,B,C,\ldots$6 is cartesian product; $A,B,C,\ldots$7 is categorical coproduct; $A,B,C,\ldots$8 is terminal object; $A,B,C,\ldots$9 is initial object.
• Exponentials: The “of course” modality $A \otimes B$0 is modeled as a Seely-monoidal comonad with derived contraction ($A \otimes B$1) and weakening ($A \otimes B$2), based on the cartesian-to-tensor structure of $A \otimes B$3 and $A \otimes B$4.

This interpretation facilitates the compositional semantics of MALL formulas within a categorical framework, preserving proof-theoretic properties via categorical constructions (Ehrhard et al., 2020).

3. Fixed Points: Modeling μ and ν in MALL

To extend MALL with least ($A \otimes B$5) and greatest ($A \otimes B$6) fixed points, strong endofunctors are employed. A functor $A \otimes B$7 is strong if equipped with a natural family of strength maps:

$A \otimes B$8

• Greatest fixed point $A \otimes B$9: Interpreted as the carrier of the final coalgebra for $I$0, with

$I$1

• Least fixed point $I$2: Interpreted as the carrier of the initial algebra for $I$3, with

$I$4

Both are strong in all variables and can be freely substituted, enabling robust semantic modeling of inductive and coinductive types within MALL (Ehrhard et al., 2020).

4. Concrete Categorical Instances

Two exemplar models exemplify the categorical semantics:

Model	Objects	Fixed Point Behavior
Set–Rel	Sets, relations	μ = ν
NUTS	Non-uniform totality spaces	μ ≠ ν

• Set–Rel model: In the category of sets and relations, all connectives and exponentials are interpreted in terms of finite sets and binary relations. Fixed points coincide: both $I$5 and $I$6 arise from the same iterative chain, yielding $I$7.
• Non-uniform totality spaces (NUTS): Objects $I$8 comprise a web $I$9 and total sets closed under biorthogonality. Morphisms are total relations. Here, $\alpha_{A,B,C}$0 arises from union over iterations starting at the empty set, while $\alpha_{A,B,C}$1 is the greatest fixpoint of a monotone map on candidate-totalities. In NUTS, $\alpha_{A,B,C}$2 may be strictly smaller than $\alpha_{A,B,C}$3, reflecting distinctions between induction and coinduction (Ehrhard et al., 2020).

5. Denotational Normalization and Proof Dynamics

Semantic invariance under cut-elimination is central to normalization. In any Seely model, cut-elimination preserves denotational interpretation. In the NUTS instance, every closed proof of a purely positive formula denotes a non-empty total set, so a proof of $\alpha_{A,B,C}$4 must yield the “true” or “false” point invariant under reductions.

For any closed proof $\alpha_{A,B,C}$5 of a positive $\alpha_{A,B,C}$6-type, its interpretation in NUTS is nonempty, compelling normalization to a constructor term. This is termed denotational normalization: semantic invariance ensures uniqueness of normal form and forcibly reflects proof-theoretic normalization in the categorical semantics (Ehrhard et al., 2020).

6. Dynamic GoI Models for MALL and Indexed Linear Logic

Further categorical modeling via geometry of interaction (GoI) is realized in a traced symmetric monoidal category. Bucciarelli–Ehrhard’s indexed linear logic MALL(I) introduces indices to track locations of additive superpositions, erasures, and contractions.

Notable structural features:

• Ambient traced category $\alpha_{A,B,C}$7 with trace operation $\alpha_{A,B,C}$8 satisfying Joyal–Street–Verity axioms.
• Reflexive object $\alpha_{A,B,C}$9 with retraction–coretraction $\sigma_{A,B}$0 and distinguished zero morphism $\sigma_{A,B}$1.
• Cut symmetry $\sigma_{A,B}$2 and execution formula $\sigma_{A,B}$3 handle matching/mismatching cuts: mismatches are annihilated by $\sigma_{A,B}$4 action.
• Index-wise execution: For relational points $\sigma_{A,B}$5, execution runs pointwise $\sigma_{A,B}$6; indices diminish upon erasure in cut-elimination.

The invariance theorem asserts that execution formulas are preserved index-wise and converge to the relational interpretation of cut-free proofs. The dynamic semantics thus recover denotational invariance after cut-elimination, while modeling the operational effects of additive interactions (Hamano, 2017).

7. Significance and Research Context

These categorical models delineate an exhaustive semantic framework for propositional MALL, incorporating fixed points. Ehrhard & Jafarrahmani provide rigorous constructions, universal properties, and normalization results, while Bucciarelli–Ehrhard indexed logic and GoI models address dynamic, operational aspects. The distinction of μ and ν in non-uniform totality spaces emphasizes the semantic sensitivity to model choice. The results collectively establish a denotationally complete and dynamically invariant semantics for MALL with fixed points, applicable to normalization, type theory, and program semantics (Ehrhard et al., 2020, Hamano, 2017).