Categorical semantics

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(Monoidal categories)
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{{Definition|title=Monoidal category|
 
{{Definition|title=Monoidal category|
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'',
 
* an object <math>I</math> called ''unit object'',
 
* an object <math>I</math> called ''unit object'',
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commute.
 
commute.
 
}}
 
}}
  +
  +
{{Definition|title=Braided, symmetric monoidal category|
  +
A ''braided'' monoidal category is a category together with a natural isomorphism of components
  +
:<math>\gamma_{A,B}:A\otimes B\to B\otimes A</math>
  +
called ''braiding'', such that the two diagrams
  +
:<math></math>
  +
commute for every objects <math>A</math>, <math>B</math> and <math>C</math>.
  +
  +
A ''symmetric'' monoidal category is a braided monoidal category in which the braiding satisfies
  +
:<math>\gamma_{B,A}\circ\gamma_{A,B}=A\otimes B</math>
  +
for every objects <math>A</math> and <math>B</math>.
   
 
== References ==
 
== References ==

Revision as of 19:08, 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 (\mathcal{C},\otimes,I) is a category \mathcal{C} equipped with

  • a functor \otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C} called tensor product,
  • an object I called unit object,
  • three natural isomorphisms of components

\alpha_{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C)
\qquad
\lambda_A:I\otimes A\to A
\qquad
\rho_A:A\otimes I\to A

called respectively associator, left unitor and right unitor,

such that

  • for every objects A,B,C,D in \mathcal{C}, the diagram

commutes,

  • for every objects A and B in \mathcal{C}, the diagrams

commute.

{{Definition|title=Braided, symmetric monoidal category| A braided monoidal category is a category together with a natural isomorphism of components

\gamma_{A,B}:A\otimes B\to B\otimes A

called braiding, such that the two diagrams

UNIQ48951ec73e8d31b-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

\gamma_{B,A}\circ\gamma_{A,B}=A\otimes B

for every objects A and B.

References

  1. MacLane, Saunders. Categories for the Working Mathematician.
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