Referred to as "Abstract Nonsense" even by practicing Pure Mathematicians, Category Theory is a bit like Graph Theory or Group Theory in the sense that it defines a "Thing" - in this case a Category - and then looks at the implications of the definition. The idea is that many other things in mathematics tend to be examples of this thing.

So certain types of Categories turn out to be groups, certain types of Categories turn out to be Graphs, and so on. In this way, any results in Category Theory will apply to Groups and Graphs.

Another observation is that we often perform the same types of operations on things: products, sums, sub-thing, and so on. The idea is that perhaps the Category captures the commonality between these things.

Here's an introduction:

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CategoryTheory
GraphTheory
GroupTheory
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EulerCycle
EvaristeGalois
FourColourTheorem
GaloisTheory
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