# Category theory for graph theory research

I am doing research in algebraic graph theory, focusing on the relation between graphs and groups (especially the representing groups as graphs) for my Ph.D. In particular, one of the ideas is to study groups using their endomorphisms and define related graphs.

1. Are ideas from category theory likely to be helpful in this field?
2. If so, which book would be a best introduction to category theory from the point of view of someone interested in algebra and graph theory?

I looked at other questions about book recommendations for category theory, but they ask for books related to set theory and foundations, or programming, or ask for a general introduction. Also, the first question is more important than the second, and I would really like to hear some views from category theory experts about possible applications in graph theory and algebra research. Please let me know if I should add more details.

• If you can't find one, write one :) Jun 29, 2014 at 14:30
• @Shaun Well, if they've found connections between category theory and integration (I saw your questions), then I think there is hope for my question :) Jun 29, 2014 at 14:33
• There is always the book *category theory for the working mathematician". It comes well-recommended, and although I have never quite got around to reading it, it has such a promising title! Jun 30, 2014 at 20:34
• Once you've got the basics under your belt, this might be a good reference. Jul 1, 2014 at 3:50
• "Are ideas from category theory likely to be helpful in this field?" Yes, I think so. Define lots of different categories of graphs connected by different forgetful functors, then ask which of those functors have left-adjoints. This should be enough to get you started. Jul 1, 2014 at 3:52

Fix a class of groups $\mathcal{C}$. Does every (countable) group occur as the outer automorphism group of a group from the class $\mathcal{C}$?
I fixed a class $\mathcal{C}$, and I managed to prove that the above result held for this class so long as I could prove that a certain group (in reality, a class of very similar groups) had a malnormal subgroup (with certain additional properties). So, I then took the fibre product of a "subgroup" in the ambient free group and (with a bit of effort) proved that its malnormality fell down to my group.