Monoidal Categories

Monoidal Category

A monoidal category is a tuple $(\mathbf{C},\otimes ,1,\alpha ,\lambda ,\rho )$ where

• $\mathbf{C}$ is a category;
• functor $\otimes :\mathbf{C}\times \mathbf{C}\to \mathbf{C}$ is called the monoidal product;
• $1\in \text{obj}\mathbf{C}$ is called the monoidal unit;
• $\alpha$ is a natural isomorphism: ${\alpha }_{X,Y,Z}:(X\otimes Y)\otimes Z\cong X\otimes (Y\otimes Z)$for all $\mathbf{C}$-objects $X,Y,Z$, called the associativity isomorphism;
• $\lambda$ and $\rho$ are also natural isomorphisms that $\lambda :1\otimes X\cong X,\rho :X\otimes 1\cong X$for all $\mathbf{C}$-objects $X$, called the left unit isomorphism and the right unit isomorphism respectively.

And the following axioms hold:

• Middle Unity Axiom:
• Pentagon Axiom:

Moreover, a strict monoidal category is a monoidal category in which the components of $\alpha$, $\lambda$ and $\rho$ are all identity morphisms.

Remark

The triangle diagram and the pentagon diagram are somewhat to define the behavior of the tensor product morphisms $1_X\otimes {\lambda }_Y$ etc.