Categorical semantics

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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

UNIQ3c4be5ee16794135-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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