Graph theory MOC

Quiver homomorphism

A homomorphism of quivers is just a natural transformation of the corresponding functors, cat i.e. a pair of functions and mapping vertices and edges respectively such that

for all , or equivalently

for all .

Proof these are equivalent

Suppose ^H1 holds. Let , i.e. for some such that and , in which case

and thus . Therefore .

Now suppose ^H2 holds. Let , , and . Then , so , whence

as required.

These form the morphisms in Category of quivers.

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