Categorical semantics
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:<math>\gamma_{B,A}\circ\gamma_{A,B}=A\otimes B</math> |
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for every objects <math>A</math> and <math>B</math>. |
for every objects <math>A</math> and <math>B</math>. |
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+ | == Modeling [[IMALL]] == |
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+ | == Modeling negation == |
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+ | {{Definition|title=*-autonomous category| |
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+ | TODO |
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Revision as of 18:16, 23 March 2009
TODO: why categories? how to extract categorical models? etc.
See [1]for a more detailed introduction to category theory.
Contents |
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 of components
called respectively associator, left unitor and right unitor,
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
- UNIQ608f0e501ac4bb45-math-0000000E-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 IMALL
Modeling negation
Definition (*-autonomous category)
TODO
References
- ↑ MacLane, Saunders. Categories for the Working Mathematician.