Categorical semantics
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{{Definition|title=Monoidal category| |
{{Definition|title=Monoidal category| |
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| − | A monoidal category <math>(\mathcal{C},\otimes,I)</math> is a category <math>\mathcal{C}</math> equipped with |
+ | A ''monoidal category'' <math>(\mathcal{C},\otimes,I)</math> is a category <math>\mathcal{C}</math> equipped with |
* a functor <math>\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}</math> called ''tensor product'', |
* a functor <math>\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}</math> called ''tensor product'', |
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* an object <math>I</math> called ''unit object'', |
* an object <math>I</math> called ''unit object'', |
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commute. |
commute. |
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}} |
}} |
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| + | {{Definition|title=Braided, symmetric monoidal category| |
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| + | A ''braided'' monoidal category is a category together with a natural isomorphism of components |
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| + | :<math>\gamma_{A,B}:A\otimes B\to B\otimes A</math> |
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| + | called ''braiding'', such that the two diagrams |
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| + | :<math></math> |
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| + | commute for every objects <math>A</math>, <math>B</math> and <math>C</math>. |
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| + | |||
| + | A ''symmetric'' monoidal category is a braided monoidal category in which the braiding satisfies |
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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>. |
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== References == |
== References == |
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Revision as of 18:08, 23 March 2009
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
- UNIQ1dcd1a8c11f4ff40-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.
