# Categories

• Jul 29th 2010, 10:38 AM
Math Major
Categories
Let $\displaystyle C$ be a category with $\displaystyle Obj(C) = \mathbb{Z}^+$ and for two positive integers $\displaystyle a$ and $\displaystyle b$, there is exactly one morphism $\displaystyle a \rightarrow b$ iff $\displaystyle a$ divides $\displaystyle b$ without remainder, and $\displaystyle Hom(a,b) = \emptyset$ 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.
• Jul 30th 2010, 12:01 AM
NonCommAlg
Quote:

Originally Posted by Math Major
Let $\displaystyle C$ be a category with $\displaystyle Obj(C) = \mathbb{Z}^+$ and for two positive integers $\displaystyle a$ and $\displaystyle b$, there is exactly one morphism $\displaystyle a \rightarrow b$ iff $\displaystyle a$ divides $\displaystyle b$ without remainder, and $\displaystyle Hom(a,b) = \emptyset$ 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.