Categorical semantics

TODO: why categories? how to extract categorical models? etc.

Categories recalled

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.