1
$\begingroup$

I'm beginning to study category theory, and I encountered this result:

In any category with a terminal object 1, any object X is itself a Cartesian product of X and 1. (Halvorson, p. 34)

This certainly clashes with the set-theoretic view of the Cartesian product of two objects (sets in that case) being an entity of a higher rank, and thus non-identical with either of these objects. So, my question is: how to think about Cartesian products in the category theoretic setting?

$\endgroup$
3
  • 5
    $\begingroup$ They mean that $X$ is a product of $X$ and $1$, and this is trivial to check using the definition of a product (en.wikipedia.org/wiki/Product_(category_theory)). Don't confuse it with the cartesian product of sets, which provides a product in $\mathbf{Set}$. $\endgroup$
    – Martin Brandenburg
    Aug 31, 2021 at 10:40
  • $\begingroup$ @MartinBrandenburg Thanks, that's clear now. $\endgroup$
    – fr_
    Aug 31, 2021 at 10:46
  • 2
    $\begingroup$ Sorry I should not post answers as comments, in particular when I always complain about that when others do it :D. I copied the comment to an answer. $\endgroup$
    – Martin Brandenburg
    Aug 31, 2021 at 11:57

1 Answer 1

Reset to default
3
$\begingroup$

They mean that $X$ is a product of $X$ and $1$, and this is trivial to check using the definition of a product. Don't confuse it with the cartesian product of sets, which provides a (possibly different, but isomorphic) product in $\mathbf{Set}$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .