Category theory for graph theory research

Question

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.

Answer 1

The book categories for the working mathematician, by Saunder's MacLane, comes well-recommended. Although I have never quite got around to reading it, it has such a promising title! Note that Saunder's MacLane was one of the original category theorists.

For an explicit example of graph theory and category theory working together to prove results on groups, there is a well-known link between the category of graphs and subgroups of free groups. The classical reference is Topology of Finite Graphs by Stallings. For example, if the non-diagonal component of the fibre product of a graph with itself is simply connected then the corresponding subgroup is malnormal. Dani Wise lifted this idea to the more general category of "cubical complexes", which allowed him to prove some famous open problems in group theory and G&T (for example, he proved the virtually Haken conjecture, and that every one-relator group with torsion is residually finite). I found Wise's paper The residual finiteness of positive one-relator groups to be especially helpful.

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You say you want to use category theory and graphs to look at endomorphisms of groups. Well, one of the questions I was attacking in my PhD thesis was the following.

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.

This doesn't quite answer your question, but I thought it relevant enough to mention. If you want, I can send you a copy of my thesis.