For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", and the like), where $P$ consists of paths in $Q$, and multiplication in $P$ is composition (or zero when the domain doesn't match the codomain).
However, as I have been writing something in more detail, I find that it's not always possible to find a corresponding quiver for some given category. For example, when $C$ is the thin category representing the partial order of $\mathbb R$ (objects are real numbers, and morphisms are pairs $(x, y)$ with $x \le y$), I cannot find the corresponding quiver.
My questions are
- Am I just not aware of the quiver that will give rise to the category in question?
- If it is really the case that there are no corresponding quivers for some categories, then this "category ring" is a more general object that the path algebra. Should it still be called the path algebra?
- Why do people usually start with a quiver, then make it into a category to define the path algebra? Why not start with a category?
Edit: Thank you for the answers below by Aaron and Julian. So the answer to my question number 1 is that I was not aware of the quiver algebra with relations. Now that I am, I have a follow-up question. Is the path algebra with relations the same as the category ring when $R$ is a field? (Why would one want to consider the quotient over a two-sided ideal that does not come from identifying paths anyway?)