# Intuition behind the definition of a monoidal category

I'm having trouble understanding the intuition behind the definition of a monoidal category (https://en.wikipedia.org/wiki/Monoidal_category). The definition on Wikipedia is a bit long and I'm struggling to see what is the meaning of each point listed in the definition and why should one care about such categories. Can someone give an intuitive explanation of monoidal category?

• I would focus on the examples, not the details of the definition. The definition is just an attempt to formalize what the examples have in common. Commented Mar 29, 2021 at 23:03
• The wiki page seems to go out of its way to avoid the most obvious class of motivating examples, namely concrete categories with binary products. I.e., categories whose objects are sets equipped with some structure (like a group operation or a topology) and a way of transferring the structure on objects $X$ and $Y$ to a structure on the set $X \times Y = \{(x, y) \mid x\in X, y \in Y\}$. For this to give a monoidal category, you also need initial objects (objects with a unique morphism to any object), but that's OK for favourite categories like groups and topological spaces and $R$-modules. Commented Mar 29, 2021 at 23:10
• P.S., I confess the wiki page does mention categories with finite products and correctly says that you need terminal objects (rather than initial objects, as I mistakenly said). I would maintain that this kind of example should be mentioned up front. Commented Mar 29, 2021 at 23:22
• There is a nice section about the idea behind the definition on the nlab page for monoidal category: ncatlab.org/nlab/show/…). Commented Mar 30, 2021 at 0:53
• The approach in Tom Leinster's "Higher Operads, Higher Categories" is via what he calls unbiased monoidal categories. Chapter 3 says that the formalization of momoidal category is not thoroughly understood. The unbiased idea might help put the usual definition in perspective. Commented Mar 31, 2021 at 19:30