# Thread: Categories

1. ## Categories

Let $C$ be a category with $Obj(C) = \mathbb{Z}^+$ and for two positive integers $a$ and $b$, there is exactly one morphism $a \rightarrow b$ iff $a$ divides $b$ without remainder, and $Hom(a,b) = \emptyset$ otherwise. In this category, verify that there are products and coproducts. What are they?

My question - I've worked through it, and I came up with the products being the greatest common divisor of two integers and the coproducts the least common multiple. I just wanted to make sure I had come up with the correct interpretation before moving on.

2. Originally Posted by Math Major
Let $C$ be a category with $Obj(C) = \mathbb{Z}^+$ and for two positive integers $a$ and $b$, there is exactly one morphism $a \rightarrow b$ iff $a$ divides $b$ without remainder, and $Hom(a,b) = \emptyset$ otherwise. In this category, verify that there are products and coproducts. What are they?

My question - I've worked through it, and I came up with the products being the greatest common divisor of two integers and the coproducts the least common multiple. I just wanted to make sure I had come up with the correct interpretation before moving on.
yes, it is correct.