A Cayley Graph encodes group structures as graphs. Is it possible to view such an encoding as a functor between the category of groups and the category of graphs? If so, what would be the upside of such a view, and what would such a functor look like?
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5The Cayley graph is not determined by the group itself, but by the group together with a generating set. – Tobias Kildetoft Sep 20 '17 at 19:38
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2The Cayley graph of a group $G$ is a way of describing $G$ as presented by a given set of generators. So there isn't enough information associated with an object in the category of groups (i.e., a group) to give a Cayley graph.I imagine that, if you wanted to, you could set up a category of groups equipped with a chosen set of generators and give a categorial interpretation of the Cayley graph construction. There are likely some choices to be made about the technical details and how to make the choices will depend on why you want to do all this. – Rob Arthan Sep 20 '17 at 19:40
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1It should work fine, categorically, if you consider the standard Cayley graph to be based on the generating set $G$ for the group $G$. Of course, this is a pretty impractical graph! On the other hand, Rob’s suggestion would require you to restrict to group homomorphisms preserving a chosen generating set, which isn’t really natural. – Kevin Carlson Sep 20 '17 at 21:21