A monoidal category, also called a tensor category, is a category $\mathcal{C}$ equipped with a bifunctor $\otimes\colon \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ which is associative up to a natural isomorphism, and an object $\mathbb{1}$ which is both a left and right identity for $\otimes$ up to a natural isomorphism.

A *monoidal category*, also called a *tensor category*, is a category $\mathcal{C}$ equipped with a bifunctor $\otimes\colon \mathcal{C}\times\mathcal{C}\to \mathcal{C}$ which is associative up to a natural isomorphism, and an object $\mathbb{1}$ which is both a left and right identity for $\otimes$ up to a natural isomorphism. In symbols, we have natural isomorphisms:

$$A\otimes (B \otimes C) \cong (A \otimes B) \otimes C$$ $$\mathbb{1}\otimes A \cong A \cong A \otimes \mathbb{1}$$

The associated natural isomorphisms are subject to certain coherence conditions which ensure that all the relevant diagrams commute.

# Resources

Tensor Categories by Etingof, Gelaki, Nikshych, and Ostrik