The concept of monoidal category is in such widespread use that one might expect – or hope, at least – that its formalization would be thoroughly understood. Nevertheless, it is not. Here we look at five different possible definitions, plus one infinite family of definitions, of monoidal category. We prove equivalence results between almost all of them.

Apart from the wish to understand a common mathematical structure, there is a reason for doing this motivated by higher-dimensional category theory. A monoidal category in the traditional sense is, as observed in Section 1.5, the same thing as a bicategory with only one object. Similarly, any proposed definition of weak n-category gives rise to a notion of monoidal category, defined as a one object weak 2-category. So if we want to be able to compare the (many) proposed definitions of weak n-category then we will certainly need a firm grip on the various notions of monoidal category and how they are related. (This is something like a physicist's toy model: a manageably low-dimensional version of a higher-dimensional system.) Of course, if two definitions of weak n-category happen to induce equivalent definitions of monoidal category then this does not imply their equivalence in the general case, but the surprising variety of different notions of monoidal category means that it is a surprisingly good test.

In the classical definition of monoidal category, any pair (X1, X2) of objects has a specified tensor product X1 ⊗ X2, and there is also a specified unit object. In Section 3.1 we consider a notion of monoidal category in which any sequence X1,…,Xn of objects has a specified product; these are called ‘unbiased monoidal categories’.