Categorical semantics
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Revision as of 18:08, 23 March 2009 by Samuel Mimram (Talk | contribs)
TODO: why categories? how to extract categorical models? etc.
Categories recalled
See [1]for a more detailed introduction to category theory.
Monoidal categories
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|title=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
- UNIQ5f0f210865d24ac2-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.
References
- ↑ MacLane, Saunders. Categories for the Working Mathematician.