Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 10 / i

A monoidal category consists of a category $\mathcal{C}$, a bifunctor $\otimes :\mathcal{C}\times \mathcal{C}\to \mathcal{C}$, a unit object $I$, and natural isomorphisms $a_{X,Y,Z}:(X\otimes Y)\otimes Z\to X\otimes (Y\otimes Z)$, $l_X:I\otimes X\to X$ and $r_X:X\otimes I\to X$. They satisfy the pentagon identity for a monoidal categoryand the triangle identity for a monoidal category $(1_X\otimes l_Y)a_{X,I,Y}=r_X\otimes 1_Y$. These axioms make different coherent rebracketings and unit removals agree.

It defines a bicategory with one object $\ast$. The hom-category $\mathcal{B}(\ast ,\ast )$ is $\mathcal{C}$; objects of $\mathcal{C}$ are its 1-cells and morphisms of $\mathcal{C}$ its 2-cells. Horizontal composition is the tensor bifunctor, the identity 1-cell is $I$, and the associator and unitors are the bicategorical coherence 2-cells. Vertical composition is composition of morphisms in $\mathcal{C}$, and the interchange law follows from bifunctoriality of $\otimes$. Conversely the hom-category of any one-object bicategory has exactly this monoidal structure. Thus a monoidal category is precisely a one-object bicategory, not necessarily a strict one-object 2-category.