Cocteau Category: Modal Categorical Semantics

Updated 6 July 2026

Cocteau category is a categorical semantics framework defined on a symmetric monoidal closed category equipped with exponential, affine, and relevant comonads along with comparison maps.
It models subexponential linear logic by capturing resource-sensitive term formations and proof conversions through structured adjunctions and coherent 2-categorical constructions.
An example using commutative quantales illustrates how conuclei encode discardability and duplicability, demonstrating practical algebraic applications of the framework.

Searching arXiv for the explicit term and primary source. A Cocteau category is a categorical semantics for a specific three-modal fragment of intuitionistic linear logic with subexponentials: one ordinary or exponential modality, one affine-only modality, and one relevant-only modality, together with comparison morphisms from the exponential modality to the other two. In the formulation introduced in "Term Assignment and Categorical Models for Intuitionistic Linear Logic with Subexponentials" (Rogozin, 16 Jul 2025), it serves as the semantic counterpart of the typed calculus ${\bf SILL}(\lambda)_3$ and as the finite prototype later generalized to arbitrary subexponential signatures by the notion of a $\Sigma$ -assemblage.

1. Definition and ambient structure

A Cocteau category is defined on a symmetric monoidal closed category $\mathcal C$ equipped with three symmetric lax monoidal endofunctors

$(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$

subject to three distinct comonadic regimes and two comparison morphisms (Rogozin, 16 Jul 2025). The ambient structure is therefore not merely monoidal; it is explicitly symmetric monoidal closed, so that tensor, internal hom, and resource modalities coexist in the same semantic base.

The definition requires that $\bang_{\bf i}$ be an exponential comonad, $\bang_{\bf a}$ an affine comonad, and $\bang_{\bf r}$ a relevant comonad. In addition, there must be symmetric lax monoidal comonad morphisms

$\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$

These comparison maps express that the intuitionistic modality is stronger than both the relevant and affine modalities. Their compatibility conditions are encoded by the commutativities

$\mathsf c^{\bf r}_A\circ \mu_{\bf ir,A} = (\mu_{\bf ir,A}\otimes \mu_{\bf ir,A})\circ \mathsf c^{\bf i}_A,$

and

$\mathsf d^{\bf a}_A\circ \mu_{\bf ia,A} = \mathsf d^{\bf i}_A.$

The definition is tailored to the three-element subexponential signature with preorder

$\Sigma$ 0

and with structural classes

$\Sigma$ 1

This means that weakening is available at the affine and intuitionistic modes, while contraction is available at the relevant and intuitionistic modes.

2. Exponential, affine, and relevant components

The internal architecture of a Cocteau category is organized by three distinct kinds of symmetric monoidal comonads (Rogozin, 16 Jul 2025). Their difference is exactly the difference between copying, deletion, or both.

Modality	Required structure	Resource behavior
$\Sigma$ 2	Exponential comonad with $\Sigma$ 3 and $\Sigma$ 4	Copying and deletion
$\Sigma$ 5	Relevant comonad with $\Sigma$ 6	Copying without deletion
$\Sigma$ 7	Affine comonad with $\Sigma$ 8	Deletion without copying

An exponential comonad is a symmetric monoidal comonad $\Sigma$ 9 with symmetric monoidal natural transformations

$\mathcal C$ 0

such that each $\mathcal C$ 1 is a cocommutative comonoid, $\mathcal C$ 2 is a comonoid morphism, and both $\mathcal C$ 3 and $\mathcal C$ 4 are coalgebra morphisms. This is the linear-exponential pattern: the modality supports both contraction and weakening.

A relevant comonad is weaker. It is a symmetric monoidal comonad equipped only with

$\mathcal C$ 5

such that each $\mathcal C$ 6 is a cocommutative cosemigroup, $\mathcal C$ 7 is a cosemigroup morphism, and $\mathcal C$ 8 is a coalgebra morphism. It models contraction without weakening.

An affine comonad is dual in emphasis. It is a symmetric monoidal comonad equipped with

$\mathcal C$ 9

satisfying the coherence condition

$(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$0

as given in the source. It models weakening without contraction.

The lax monoidal structures themselves are also part of the semantics. For each modality $(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$1 one has

$(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$2

together with the standard compatibility of $(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$3 and $(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$4 with the monoidal structure.

3. Role in subexponential linear logic

Cocteau categories were introduced to model the three-modal calculus $(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$5, and more broadly to characterize semantics for linear type theories with a single exponential and affine and relevant subexponentials (Rogozin, 16 Jul 2025). Their proof-theoretic role is not only denotational. The paper states that the proof conversions in $(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$6 syntactically represent the axioms of a Cocteau category, so the categorical equations are meant to match the computational equalities of the term calculus.

Type interpretation follows the general pattern later used for $(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$7-assemblages: $(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$8

$(\bang_{\bf i}, \mathsf m^{\bf i}),\qquad (\bang_{\bf a}, \mathsf m^{\bf a}),\qquad (\bang_{\bf r}, \mathsf m^{\bf r}),$9

$\bang_{\bf i}$0

In the Cocteau case, $\bang_{\bf i}$1 ranges over $\bang_{\bf i}$2.

The resource-sensitive term formers are interpreted directly by the modal structure. Dereliction is interpreted by the counit of the corresponding comonad. Weakening or deletion is interpreted by maps of the form

$\bang_{\bf i}$3

for modes in $\bang_{\bf i}$4. Contraction is interpreted by maps of the form

$\bang_{\bf i}$5

for modes in $\bang_{\bf i}$6. Promotion uses comultiplication, comparison morphisms, and monoidal structure maps, with general interpretation

$\bang_{\bf i}$7

In the three-modal case, the only nontrivial comparisons are from $\bang_{\bf i}$8 to $\bang_{\bf i}$9 and from $\bang_{\bf a}$0 to $\bang_{\bf a}$1.

A plausible implication is that Cocteau categories isolate a minimal nontrivial fragment in which the ordinary exponential decomposes into two structurally incomplete modalities while retaining a common stronger source. The paper’s own formulation supports this reading by presenting the Cocteau structure as the special three-mode case of the more general subexponential semantics.

4. Relation to $\bang_{\bf a}$2-assemblages and monoidal adjunctions

A Cocteau category is the special case of a $\bang_{\bf a}$3-assemblage corresponding to the three-element signature above (Rogozin, 16 Jul 2025). In a $\bang_{\bf a}$4-assemblage, each $\bang_{\bf a}$5 is assigned a comonad $\bang_{\bf a}$6, the assignment is contravariantly functorial in the preorder $\bang_{\bf a}$7, and the modes in $\bang_{\bf a}$8, $\bang_{\bf a}$9, and $\bang_{\bf r}$0 are respectively affine, relevant, and exponential. Cocteau categories are therefore the smallest explicitly worked instance of this framework.

The semantic adequacy statement is formulated at the level of assemblages: $\bang_{\bf r}$1 Since every Cocteau category is a $\bang_{\bf r}$2-assemblage for the three-mode signature, the same soundness statement applies to $\bang_{\bf r}$3.

The paper also gives an adjunctional reconstruction. It introduces a Cocteau adjunction consisting of an affine resource modality, a relevant resource modality, an exponential resource modality, and strict monoidal functors from the exponential side to the affine and relevant sides. The two central representation theorems are that every Cocteau adjunction yields a Cocteau category and every Cocteau category induces a Cocteau adjunction. Consequently, every Cocteau category can be viewed as a symmetric monoidal closed category equipped with a family of monoidal adjunctions of a particular kind.

In this reconstruction, each comonad appears as an adjoint composite. For instance,

$\bang_{\bf r}$4

For the relevant modality, the copying structure is reconstructed by

$\bang_{\bf r}$5

The comparison maps are similarly induced by the adjunctional data.

This adjunctional characterization places Cocteau categories in the line of categorical logic that treats modalities as arising from structured adjunctions rather than as primitive endofunctors alone.

5. Stronger 2-categorical characterization

The final section of the defining paper gives a stronger 2-categorical account of Cocteau categories (Rogozin, 16 Jul 2025). The ambient 2-category is

$\bang_{\bf r}$6

where $\bang_{\bf r}$7 is a small category encoding the shape of the three adjunctions and their comparison functors. On this 2-category the paper defines a strict 2-comonad

$\bang_{\bf r}$8

built from $\bang_{\bf r}$9 on the relevant corner, $\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$0 on the exponential corner, and slice-over-$\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$1 on the affine corner. The purpose is to encode relevant, Cartesian, and semicartesian structure simultaneously.

The induced 2-category $\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$2 has as 0-cells Cocteau categories, as 1-cells strict symmetric monoidal functors preserving the three comonads by symmetric lax monoidal comonad morphisms and compatible with $\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$3, and as 2-cells symmetric monoidal natural transformations satisfying the corresponding compatibility conditions.

The principal result is the embedding theorem

$\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$4

This identifies Cocteau categories not just objectwise but functorially and 2-functorially as coalgebra-like structures for a strict 2-comonad on a diagram 2-category. The result is stronger than the adjunctional reconstruction because it also organizes morphisms and 2-cells between Cocteau categories.

A plausible implication is that the notion is designed to be stable under higher-categorical transport: the three-modal semantics is not merely a collection of endofunctors on isolated bases, but a coherent 2-dimensional doctrine.

6. Examples, scope, and terminological ambiguity

The paper gives a concrete example from commutative quantales (Rogozin, 16 Jul 2025). For a commutative quantale $\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$5, define

$\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$6

$\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$7

These determine quantic conuclei $\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$8, and the paper concludes that a commutative quantale $\mu_{\bf ir} : \bang_{\bf i} \Rightarrow \bang_{\bf r}, \qquad \mu_{\bf ia} : \bang_{\bf i} \Rightarrow \bang_{\bf a}.$9 together with these conuclei forms a Cocteau category. In this example, the inequalities $\mathsf c^{\bf r}_A\circ \mu_{\bf ir,A} = (\mu_{\bf ir,A}\otimes \mu_{\bf ir,A})\circ \mathsf c^{\bf i}_A,$ 0 and $\mathsf c^{\bf r}_A\circ \mu_{\bf ir,A} = (\mu_{\bf ir,A}\otimes \mu_{\bf ir,A})\circ \mathsf c^{\bf i}_A,$ 1 are the algebraic counterparts of discardability and duplicability.

The same source states that one can also extract an example from coherence spaces, although the detailed construction is not developed in the supplied text. This suggests that Cocteau categories are intended as genuine semantic structures rather than merely formal packaging.

A recurrent source of confusion is terminological. The categorical notion of Cocteau category is unrelated to the 2022 empathy-based civic-engagement platform COCTEAU, expanded as “Co-Creating The European Union,” which is described instead as a novel empathy-based framework, web-based gamified application, and participatory decision-support platform (Mauri et al., 2022). It is also distinct from several near-sounding notions sometimes invoked when the term is misremembered, such as cluster categories and orbit categories in representation theory (Amiot et al., 2010). In the arXiv record supplied here, the explicit term “Cocteau category” is introduced only in the 2025 paper on subexponential linear logic (Rogozin, 16 Jul 2025).

Within that setting, the term designates a specific semantic structure: a symmetric monoidal closed category equipped with an exponential, an affine, and a relevant comonad, plus comparison morphisms from the exponential modality to the other two. Its significance lies in making the three-mode subexponential fragment mathematically explicit, in connecting proof conversions to categorical equations, in extending to arbitrary signatures through $\mathsf c^{\bf r}_A\circ \mu_{\bf ir,A} = (\mu_{\bf ir,A}\otimes \mu_{\bf ir,A})\circ \mathsf c^{\bf i}_A,$ 2-assemblages, and in admitting both adjunctional and 2-categorical characterizations.