Categorical semantics
Constructing denotational models of linear can be a tedious work. Categorical model 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.
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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 diagrams
commute.
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
- UNIQ463dfd5f1855b55b-math-00000011-QINU
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.
Modeling the additives
Definition (Product)
Definition (Monoid)
Property
Categories with products vs monoidal categories.
Modeling IMALL
Modeling negation
Definition (*-autonomous category)
TODO
References
- ↑ MacLane, Saunders. Categories for the Working Mathematician.