Categorical semantics
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=== Compact closed categories === |
=== Compact closed categories === |
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− | {{Definition|title=Compact closed category| |
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− | A symmetric monoidal category <math>(\mathcal{C},\tens,I)</math> is ''compact closed'' when every object <math>A</math> has a (left) dual. |
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− | }} |
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{{Definition|title=Dual objects| |
{{Definition|title=Dual objects| |
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{{Lemma| |
{{Lemma| |
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Two left (resp. right) duals of a same object <math>B</math> are necessarily isomorphic. |
Two left (resp. right) duals of a same object <math>B</math> are necessarily isomorphic. |
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+ | }} |
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+ | |||
+ | {{Definition|title=Compact closed category| |
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+ | A symmetric monoidal category <math>(\mathcal{C},\tens,I)</math> is ''compact closed'' when every object <math>A</math> has a right dual <math>A^*</math>. We write |
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+ | :<math>\eta_A:I\to A^*\tens A</math> and <math>\varepsilon:A\tens A^*\to I</math> |
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+ | for the corresponding duality morphisms. |
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}} |
}} |
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Revision as of 15:52, 25 March 2009
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 [1]for a more detailed introduction to category theory. See [2]for a detailed treatment of categorical semantics of linear logic.
Contents |
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 is a category equipped with
- a functor 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
such that
- for every objects A,B,C,D in , the diagram
commutes,
- for every objects A and B in , the diagram
commutes.
Definition (Braided, symmetric monoidal category)
A braided monoidal category is a category together with a natural isomorphism of components
called braiding, such that the two diagrams
and
commute for every objects A, B and C.
A symmetric monoidal category is a braided monoidal category in which the braiding satisfies
for every objects A and B.
Definition (Closed monoidal category)
A monoidal category is left closed when for every object A, the functor
has a right adjoint, written
This means that there exists a bijection
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 morphism , called left evaluation,
for every objects A and B, such that for every morphism there exists a unique morphism making the diagram
commute.
Dually, the monoidal category is right closed when the functor 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 and in a category is an object X of together with two morphisms and such that there exists a unique morphism making the diagram
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 is an object M together with two morphisms
- and
such that the diagrams
and
commute.
Property
Categories with products vs monoidal categories.
Modeling ILL
Introduced in[3].
Definition (Linear-non linear (LNL) adjunction)
A linear-non linear adjunction is a symmetric monoidal adjunction between lax monoidal functors
in which the category has finite products.
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 and consists of
- a functor between the underlying categories,
- a natural transformation f of components ,
- a morphism
such that the diagrams
and
- and
commute for every objects A, B and C of . The morphisms fA,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 and are two monoidal categories and
- and
are two monoidal functors between these categories. A monoidal natural transformation between these monoidal functors is a natural transformation between the underlying functors such that the diagrams
- and
commute for every objects A and B of .
Definition (Monoidal adjunction)
A monoidal adjunction between two monoidal functors
- and
is an adjunction between the underlying functors F and G such that the unit and the counit
- and
induce monoidal natural transformations between the corresponding monoidal functors.
Modeling negation
*-autonomous categories
Definition (*-autonomous category)
Suppose that we are given a symmetric monoidal closed category and an object R of . For every object A, we define a morphism
as follows. By applying the bijection of the adjunction defining (left) closed monoidal categories to the identity morphism , we get a morphism , and thus a morphism by precomposing with the symmetry . The morphism 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 is a bijection for every object A of . A symmetric monoidal closed category is *-autonomous when it admits such a dualizing object.
Compact closed categories
Definition (Dual objects)
A dual object structure in a monoidal category is a pair of objects A and B together with two morphisms
- and
such that the diagrams
and
commute. The object A is called a left dual of B (and conversely B is a right dual of A).
Lemma
Two left (resp. right) duals of a same object B are necessarily isomorphic.
Definition (Compact closed category)
A symmetric monoidal category is compact closed when every object A has a right dual A * . We write
- and
for the corresponding duality morphisms.
Lemma
In a compact closed category the left and right duals of an object A are isomorphic.
Property
A compact closed category is monoidal closed, with closure defined by