Category-theoretic cross product and set-theoretic cross product

Question

I recently proved as an exercise the associativity of cross product as defined in category theory. But in set theory, cross product is not associative. It seems intuitive to me that cross should be associative, but it also seems that the set theoretic definition doesn't cause set theorists any problems. Are there examples in which the associativity of cross does matter, so that the category theoretic def. is preferable for more than just intuitive reasons?

Answer 1

The Cartesian product of sets is what is usually referred to as the product of two sets, and it is denoted with $\times$. It's certainly associative. The intuition is that when you form $(X\times Y)\times Z$ and $X\times (Y\times Z)$, they are both basically the same as the set of ordered triples $X\times Y\times Z$. While the first two are not equal as sets, they are "rearrangable" with a canonical isomorphism of sets so that they look alike, so there is not much reason to consider them different.

---

No longer believe the user has this problem: You might possibly be confusing it with the 3-d vector algebra cross product (denoted by the same $\times$), which is not associative. This is a product between vectors (usually from $\Bbb R^3$), not sets.