From my (limited) understanding, a category is just a class of objects (which form the "vertices") and a class of morphisms between those objects, plus the requirement that associativity and identity hold. Whenever I see them drawn, all I see are multi-digraphs. This being the case, it seems that "category theory" boils down to graph theory. I'm assuming this isn't actually true. What can be expressed by category theory that can't be expressed just by interpreting a multi-digraph in a particular way?
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It depends. Are you fine with graphs with an uncountable number of vertices? What about not even a set's worth? – Randall Aug 09 '18 at 16:28
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This perspective is like saying that rings are just fancy groups. The extra structural restrictions let you do things that you wouldn't be able to do in the "more general" setting. – Ian Aug 09 '18 at 16:28
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Related. – Malice Vidrine Aug 09 '18 at 16:29
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@Ian: I am not sure that quite follows, although I'd be willing to be convinced of the converse. A ring is a group plus another operation. I might be tempted to ask "why do we have group theory when ring theory encompasses it". – Michael Stachowsky Aug 09 '18 at 16:31
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A ring is a group with another operation but the operations are coupled (mostly through the distribution law) such that the freedom of the "addition" is limited. The same thing happens in category theory relative to graph theory. – Ian Aug 09 '18 at 16:32
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Categories are not graphs in the same way in which rings are not groups and metric spaces are not topological spaces. There is a extra structure on the graph which cannot be described by properties alone. – Stefan Perko Aug 09 '18 at 16:35