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I have been wondering for a while about the following question without getting anywhere:

Let $G$ be a group, $N$ a normal subgroup. Can the quotient group $G/N$ be seen as the (category theoretical) colimit of a diagram? If it can, of what diagram?

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up vote 5 down vote accepted

It's the diagram $N \rightrightarrows G$ where one of the arrows is the inclusion and the other arrow is the zero map. (If $N$ is not necessarily normal you will instead get the quotient by the normal closure of $N$.) More generally, see coequalizer and cokernel.

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If you want a detailed discussion of this and other related issues, Aluffi's "Algebra - chapter 0" does a great job. –  Ittay Weiss Mar 22 '13 at 0:12
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