TITLE: Can we have the category of integers?
QUESTION [6 upvotes]: In beginning to learn category theory, I've tried to come up with my own examples of categories.
Could we have a category Int, where the objects are integers, and arrows, all mappings between integers?
If you can't (which I suspect might be the case), what can't be an object/arrow?

REPLY [8 votes]: Sure you can. You just have to decide what the mappings between the integers are, and what composition is. For example, one can define the poset category on $\Bbb{Z}$ whose objects are the integers, and has a unique arrow $n\to m$ whenever $n\le m$. You can check that there is a composition defined by transitivity of $\le$ that makes this setup into a category.