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Let $G$ and $G'$ be quivers. If their path categories $Path[G]$ and $Path[G']$ are isomorphic, does is follow that $G$ is isomorphic to $G'$?

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  • $\begingroup$ Vertices are isomorphism classes of objects. How would you characterize edges among all morphisms? $\endgroup$
    – Dmitry Vaintrob
    Commented Dec 15, 2021 at 19:20
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    $\begingroup$ This is true even if you just ask for Morita equivalence of their path algebras over $\mathbb C$ which is of course implied by isomorphism of path category. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 15, 2021 at 19:48

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Yes. Given a Quiver $G$, you can identify $G$ as the subquiver of $Path[G]$ of arrows that are not identity and cannot be written as composite of non-identity arrows. So any isomorphism between $Path[G]$ and $Path[G']$ send elements of $G$ to elements of $G'$ and restrict to an isomorphism between $G$ and $G'$.

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