Categorical semantics

Constructing denotational models of linear can be a tedious work. Categorical semantics are useful to identify the fundamental structure of these models, and thus simplify and make more abstract the elaboration of those models.

TODO: why categories? how to extract categorical models? etc.

See for a more detailed introduction to category theory. See for a detailed treatment of categorical semantics of linear logic.

Basic category theory recalled

Definition (Category)

Definition (Functor)

Definition (Natural transformation)

Definition (Adjunction)

Definition (Monad)

Modeling IMLL

A model of IMLL is a closed symmetric monoidal category. We recall the definition of these categories below.

Definition (Monoidal category)

A monoidal category $(\mathcal{C},\otimes,I,\alpha,\lambda,\rho)$ is a category $\mathcal{C}$ equipped with

• a functor $\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ called tensor product,
• an object I called unit object,
• three natural isomorphisms α, λ and ρ, called respectively associator, left unitor and right unitor, whose components are

$\alpha_{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C) \qquad \lambda_A:I\otimes A\to A \qquad \rho_A:A\otimes I\to A$

such that

• for every objects A,B,C,D in $\mathcal{C}$, the diagram

$\xymatrix{ ((A\otimes B)\otimes C)\otimes D\ar[d]_{\alpha_{A\otimes B,C,D}}\ar[r]^{\alpha_{A,B,C}\otimes D}&(A\otimes(B\otimes C))\otimes D\ar[r]^{\alpha_{A,B\otimes C,D}}&A\otimes((B\otimes C)\otimes D)\ar[d]^{A\otimes\alpha_{B,C,D}}\\ (A\otimes B)\otimes(C\otimes D)\ar[rr]_{\alpha_{A,B,C\otimes D}}&&A\otimes(B\otimes (C\otimes D)) }$

commutes,

• for every objects A and B in $\mathcal{C}$, the diagram

$\xymatrix{ (A\otimes I)\otimes B\ar[dr]_{\rho_A\otimes B}\ar[rr]^{\alpha_{A,I,B}}&&\ar[dl]^{A\otimes\lambda_B}A\otimes(I\otimes B)\\ &A\otimes B& }$

commutes.

Definition (Braided, symmetric monoidal category)

A braided monoidal category is a category together with a natural isomorphism of components

$\gamma_{A,B}:A\otimes B\to B\otimes A$

called braiding, such that the two diagrams

$\xymatrix{ &A\otimes(B\otimes C)\ar[r]^{\gamma_{A,B\otimes C}}&(B\otimes C)\otimes A\ar[dr]^{\alpha_{B,C,A}}\\ (A\otimes B)\otimes C\ar[ur]^{\alpha_{A,B,C}}\ar[dr]_{\gamma_{A,B}\otimes C}&&&B\otimes (C\otimes A)\\ &(B\otimes A)\otimes C\ar[r]_{\alpha_{B,A,C}}&B\otimes(A\otimes C)\ar[ur]_{B\otimes\gamma_{A,C}}\\ }$

and

$\xymatrix{ &(A\otimes B)\otimes C\ar[r]^{\gamma_{A\otimes B,C}}&C\otimes (A\otimes B)\ar[dr]^{\alpha^{-1}_{C,A,B}}&\\ A\otimes (B\otimes C)\ar[ur]^{\alpha^{-1}_{A,B,C}}\ar[dr]_{A\otimes\gamma_{B,C}}&&&(C\otimes A)\otimes B\\ &A\otimes(C\otimes B)\ar[r]_{\alpha^{-1}_{A,C,B}}&(A\otimes C)\otimes B\ar[ur]_{\gamma_{A,C}\otimes B}&\\ }$

commute for every objects A, B and C.

A symmetric monoidal category is a braided monoidal category in which the braiding satisfies

$\gamma_{B,A}\circ\gamma_{A,B}=A\otimes B$

for every objects A and B.

Definition (Closed monoidal category)

A monoidal category $(\mathcal{C},\tens,I)$ is left closed when for every object A, the functor

$B\mapsto A\otimes B$

has a right adjoint, written

$B\mapsto(A\limp B)$

This means that there exists a bijection

$\mathcal{C}(A\tens B, C) \cong \mathcal{C}(B,A\limp C)$

which is natural in B and C. Equivalently, a monoidal category is left closed when it is equipped with a left closed structure, which consists of

• an object $A\limp B$,
• a morphism $\mathrm{eval}_{A,B}:A\tens (A\limp B)\to B$, called left evaluation,

for every objects A and B, such that for every morphism $f:A\otimes X\to B$ there exists a unique morphism $h:X\to A\limp B$ making the diagram

$\xymatrix{ A\tens X\ar@{.>}[d]_{A\tens h}\ar[dr]^{f}\\ A\tens(A\limp B)\ar[r]_-{\mathrm{eval}_{A,B}}&B }$

commute.

Dually, the monoidal category $\mathcal{C}$ is right closed when the functor $B\mapsto B\otimes A$ admits a right adjoint. The notion of right closed structure can be defined similarly.

In a symmetric monoidal category, a left closed structure induces a right closed structure and conversely, allowing us to simply speak of a closed symmetric monoidal category.

Modeling the additives

Definition (Product)

A product (X,π₁,π₂) of two coinitial morphisms $f:A\to B$ and $g:A\to C$ in a category $\mathcal{C}$ is an object X of $\mathcal{C}$ together with two morphisms $\pi_1:X\to A$ and $\pi_2:X\to B$ such that there exists a unique morphism $h:A\to X$ making the diagram

$\xymatrix{ &\ar[ddl]_fA\ar@{.>}[d]_h\ar[ddr]^g&\\ &\ar[dl]^{\pi_1}X\ar[dr]_{\pi_2}&\\ B&&C }$

commute.

A category has finite products when it has products and a terminal object.

Definition (Monoid)

A monoid (M,μ,η) in a monoidal category $(\mathcal{C},\tens,I)$ is an object M together with two morphisms

$\mu:M\tens M \to M$ and $\eta:I\to M$

such that the diagrams

$\xymatrix{ &(M\tens M)\tens M\ar[dl]_{\alpha_{M,M,M}}\ar[r]^-{\mu\tens M}&M\tens M\ar[dd]^{\mu}\\ M\tens(M\tens M)\ar[d]_{M\tens\mu}&&\\ M\tens M\ar[rr]_{\mu}&&M\\ }$

and

$\xymatrix{ I\tens M\ar[r]^{\eta\tens M}\ar[dr]_{\lambda_M}&M\tens M\ar[d]_\mu&\ar[l]_{M\tens\eta}\ar[dl]^{\rho_M}M\tens I\\ &M& }$

commute.

Property

Categories with products vs monoidal categories.

Modeling ILL

Introduced in.

Definition (Linear-non linear (LNL) adjunction)

A linear-non linear adjunction is a symmetric monoidal adjunction between lax monoidal functors

$\xymatrix{ (\mathcal{M},\times,\top)\ar[rr]^{(L,l)}&\bot&\ar[ll]^{(M,m)}(\mathcal{L},\otimes,I) }$

in which the category $\mathcal{M}$ has finite products.

$\oc=L\circ M$

This section is devoted to defining the concepts necessary to define these adjunctions.

Definition (Monoidal functor)

A lax monoidal functor (F,f) between two monoidal categories $(\mathcal{C},\tens,I)$ and $(\mathcal{D},\bullet,J)$ consists of

• a functor $F:\mathcal{C}\to\mathcal{D}$ between the underlying categories,
• a natural transformation f of components $f_{A,B}:FA\bullet FB\to F(A\tens B)$,
• a morphism $f:J\to FI$

such that the diagrams

$\xymatrix{ (FA\bullet FB)\bullet FC\ar[d]_{\phi_{A,B}\bullet FC}\ar[r]^{\alpha_{FA,FB,FC}}&FA\bullet(FB\bullet FC)\ar[d]^{FA\bullet\phi_{B,C}}\\ F(A\otimes B)\bullet FC\ar[d]_{\phi_{A\otimes B,C}}&FA\bullet F(B\otimes C)\ar[d]^{\phi_{A,B\otimes C}}\\ F((A\otimes B)\otimes C)\ar[r]_{F\alpha_{A,B,C}}&F(A\otimes(B\otimes C)) }$

and

TODO

commute for every objects A, B and C of $\mathcal{C}$. The morphisms f_A,B and f are called coherence maps.

A lax monoidal functor is strong when the coherence maps are invertible and strict when they are identities.

Definition (Monoidal natural transformation)

Suppose that $(\mathcal{C},\tens,I)$ and $(\mathcal{D},\bullet,J)$ are two monoidal categories and

$(F,f):(\mathcal{C},\tens,I)\Rightarrow(\mathcal{D},\bullet,J) \qquad \text{and} \qquad (G,g):(\mathcal{C},\tens,I)\Rightarrow(\mathcal{D},\bullet,J)$

are two monoidal functors between these categories. A monoidal natural transformation $\theta:(F,f)\to (G,g)$ between these monoidal functors is a natural transformation $\theta:F\Rightarrow G$ between the underlying functors such that the diagrams

TODO

commute for every objects A and B of $\mathcal{D}$.

Definition (Monoidal adjunction)

TODO

Modeling negation

*-autonomous categories

Definition (*-autonomous category)

Suppose that we are given a symmetric monoidal closed category $(\mathcal{C},\tens,I)$ and an object R of $\mathcal{C}$. For every object A, we define a morphism

$\partial_{A}:A\to(A\limp R)\limp R$

as follows. By applying the bijection of the adjunction defining (left) closed monoidal categories to the identity morphism $\mathrm{id}_{A\limp R}:A\limp R \to A\limp R$, we get a morphism $A\tens (A\limp R)\to R$, and thus a morphism $(A\limp R)\tens A\to R$ by precomposing with the symmetry $\gamma_{A\limp R,A}$. The morphism $\partial_A$ is finally obtained by applying the bijection of the adjunction defining (left) closed monoidal categories to this morphism. The object R is called dualizing when the morphism $\partial_A$ is a bijection for every object A of $\mathcal{C}$. A symmetric monoidal closed category is *-autonomous when it admits such a dualizing object.

Compact closed categories

Definition (Compact closed category)

A symmetric monoidal category $(\mathcal{C},\tens,I)$ is compact closed when every object A has a (left) dual.

Definition (Dual objects)

A dual object structure $(A,B,\eta,\varepsilon)$ in a monoidal category $(\mathcal{C},\tens,I)$ is a pair of objects A and B together with two morphisms

$\eta:I\to B\otimes A$ and $\varepsilon:A\otimes B\to I$

such that the diagrams

TODO

commute.

In a compact closed category the left and right duals of an object A are isomorphic.

Property

A compact closed category is monoidal closed.