The short answer is no. You just have to think of them as formal sums, in the same way that you can only think of elements of a group algebra as formal sums.
What you can do is think of the path category of a quiver, which is the category whose objects are elements are vertices of the quiver, and whose morphisms are paths in the quiver. A representation of the path algebra as an algebra is essentially just a functor from this path category to vectors spaces.
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