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From Wikipedia: "a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph". Any talk about quivers I've heard starts by explaining that they are just another way of viewing (a general type of) graphs, but that the focus of interest is different from that of graph theorists, whence the different terminology. However I've never heard of a metagraph before, and from the link you provided (which is not very clear) you cannot use a metagraph in place of a quiver (or of a graph), because you apparently are not even allowed to consider its set of vertices (called objects). This is not so much because they might be too numerous to be a set (like when they would form a set-theoretic proper class), but because the theory in question is not founded in set theory in the first place.

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I understood the distinguishing factor of both a meta-graph and a quiver to be that it is formally defined by axioms (functional rules) whereas a graph/directed-graph is formally defined in terms of sets (structural rules). It seems like both a meta graph and a Quiver can have concrete instances that are defined in terms of sets, but it is not necessary. Is there any proposition you can think of applies to quivers but not the to metagraphs (or vice versa)? – smartcaveman Aug 20 '12 at 1:10
I don't understand what you mean by the functional/structural opposition; the axioms of groups (or whatever) are they functional or structural? Can you list the axioms of metagraphs, or quivers? The metagraph link given is particularly unhelpful, it only makes negative claims; as far as I can see it set up a language, with no axioms. But for one thing, according to your quiver link, you can talk about the set of vertices of a quiver $G:X\to\mathrm{Set}$ (slick definition), it is the object $G(X_0)$ of the category Set, manifestly a set. The metagraph link explicitly says the contrary. – Marc van Leeuwen Aug 20 '12 at 6:23
I think our disconnect about meta graph can be clarified from proofwiki's graph article. Particularly, it says that "A graph is an interpretation of a metagraph within set theory". While the formal definition of the quiver uses set theory, I was under the impression that a quiver could exist independently of set theory, because it is used to define a category, which is not itself dependent upon set theory. – smartcaveman Aug 20 '12 at 17:52
Furthermore, you can talk about the sets of vertices of a metagraph. It's just that if you do, you are referring to a specialization of a metagraph which is a graph. It seems like the same can be said about a quiver. – smartcaveman Aug 20 '12 at 17:53
As far as I can tell re: axioms of metagraphs, there are 2: (1) Iff G is a meta graph, then G has [0..*] objects. (2) Iff G is a meta graph then G has [0..*] morphisms that connect objects. – smartcaveman Aug 20 '12 at 17:55

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