# Equalizers in category of sets/graphs — a couple of examples

i've constructed a couple of equalizers in the category of sets/graphs. If anyone could tell me if my equalizers turned out right, that would be very much appreciated.

example of an equalizer in the category of sets

given:
O1 = {a1, a2, a3, a4} --set 1
O2 = {b1, b2, b3, b4} --set 2
f12 = {(a1->b1), (a2->b2), (a3->b3), (a4->b4)} --set morphism 1
g12 = {(a1->b1), (a2->b1), (a3->b2), (a4->b4)} --set morphism 2

solution (correct?)
equalizer:
eq_kernel = {a1, a4}
eq_injection = {(a1->a1), (a4->a4)}

example of an equalizer in the category of graphs

given:
graph object 1
GO1 = (E1,V1,SRC1,TGT1) -- graph object
E1 = {e11, e12} -- edges
V1 = {v11, v12} -- vertices
SRC1 = {(e12->v12), (e11->v11)} -- source
TGT1 = {(e12->v12), (e11->v11)} -- target

graph object 2
GO2 = (E2,V2,SRC2,TGT2) -- graph object
E2 = {e21, e22} -- edges
V2 = {v21, v22} -- vertices
SRC2 = {(e22->v22), (e21->v21)} -- source
TGT2 = {(e22->v22), (e21->v21)} -- target

gmorph1 = (Dom1, EdgeArrow1, NodeArrow1, Codom1)
Dom1 = GO1
EdgeArrow1 = {(e12->e22), (e11->e21)}
NodeArrow1 = {(v12->v22), (v11->v21)}
Codom1 = GO2

gmorph2 = (Dom2, EdgeArrow2, NodeArrow2, Codom2)
Dom2 = GO1
EdgeArrow2 = {(e12->e21), (e11->e21)}
NodeArrow2 = {(v12->v21), (v11->v21)}
Codom2 = GO2

solution (correct?)
equalizer:

EQ_KERNEL = (EEQ,VEQ,SRCEQ,TGTEQ)
EEQ = {e11} -- edges
VEQ = {v11} -- vertices
SRCEQ = {(e11->v11)} -- source
TGTEQ = {(e11->v11)} -- target

EQ_INJECTION = (DomEQ, EdgeArrowEQ, NodeArrowEQ, CodomEQ)
DomEQ = EQ_KERNEL
EdgeArrowEQ = {(e11->e11)}
NodeArrowEQ = {(v11->v11)}
CodomEQ = EQ_KERNEL

-spirit

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So both your examples are correct. The way you wrote all that is really dense and more complicated than need be though.

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Thanks a lot, David! How would I go about writing it less complicated? –  spirit Mar 29 '11 at 19:26
Using latex and a more standard mathematical presentation style. In particular, don't be afraid to write sentences. –  BBischof Jul 23 '11 at 14:09