I'm studying quiver representation on "Introduction to representation theory" by Etingof and others. Often I read on this book about a representation being injective or surjective at some vertex (of a quiver) i.
What does precisely mean, for a quiver representation, to be injective (surjective) at some vertex?
Is this relate to the direct sum of the functions corresponding to the arrows pointing in and out of i?
Thanks to everyone.
From the context, I'm pretty sure that what is meant is:
If $i$ is a source vertex of the quiver $Q$, then a representation $V$ of $Q$ "is injective at $i$" if the obvious map $$V(i)\to\bigoplus_{i\to j}V(j)$$ is injective, where the direct sum is indexed by all arrows starting at $i$.
And "surjective at $i$" is the dual notion for a sink vertex.