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From my study of both these theories, they are defined differently but seem to look the same. What exactly is the main points of relation between these two fields?

Clemens Bartholdy
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  • Category Theory is studying relations between mathematical structures where arrows are used between the structures. Graph Theory is studying objects with relations between them given as edges, these edges can be directed or undirected, can even have multiple edges between objects. They are definitely not the same. – oliverjones Jun 30 '22 at 23:03
  • The arrows in category theory should be thought of as analogous to functions or relations on sets, perhaps respecting additional structure like a group homomorphism. The edges in graphs can indicate relationships like adjacency, which don't have a canonical "composition" operation. – Dustan Levenstein Jun 30 '22 at 23:06
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    It is true that every category gives rise to a corresponding graph, but almost all the important information is lost in this process. Specifically, when we just look at the "graph version" of a category, we forget the composition structure. Consider, for example, two non-isomorphic groups $G,H$ with the same cardinality. These can be thought of as categories $\mathcal{C}_G,\mathcal{C}_H$ with one object and the same number of morphisms. Meanwhile, the "graph versions" of $\mathcal{C}_G$ and $\mathcal{C}_H$ are isomorphic since $G$ and $H$ have the same cardinality. – Noah Schweber Jun 30 '22 at 23:23
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    So the similarity is in fact extremely superficial: a graph doesn't carry the same sort of algebraic structure as a category (we can't "compose edges" in a graph), and that's A Big Deal. – Noah Schweber Jun 30 '22 at 23:24
  • Thanks @NoahSchweber Could you please post that as answer, I'll accept it – Clemens Bartholdy Jun 30 '22 at 23:25

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It is true that every category gives rise to a corresponding graph, but almost all the important information is lost in this process. Specifically, when we just look at the "graph version" of a category, we forget the composition structure. Consider, for example, two non-isomorphic groups $G,H$ with the same cardinality. These can be thought of as categories $\mathcal{C}_G,\mathcal{C}_H$ with one object and the same number of morphisms. Meanwhile, the "graph versions" of $\mathcal{C}_G$ and $\mathcal{C}_H$ are isomorphic since $G$ and $H$ have the same cardinality.

So the similarity is in fact extremely superficial: a graph doesn't carry the same sort of algebraic structure as a category (we can't "compose edges" in a graph), and that's A Big Deal.

(In jargon, the forgetful functor from Cats to Graphs is very non-full.)

Noah Schweber
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    That said, we can freely build a category by composing arrows given in a graph. In jargon, the forgetful functor from $\mathbf{Cats}$ to $\mathbf{Graphs}$ has a left adjoint. Moreover, this makes your "suprificial" similarity precise: This adjunction is monadic, which means categories look like "algebras" over their underlying graphs. See here, for instance. – Chris Grossack Jul 01 '22 at 01:20