TITLE: On the philosophy of Category Theory
QUESTION [7 upvotes]: I have been told by my professor that Category Theory is not just a language but is a shift in the way we think. As an example he pointed out that in category theory we do not worry about the objects themselves but he study the morphisms which relate it to other object. But what I do not understand is why is this important.
For example If I want to say a group is commutative then I take a map from $\beta :M \times M \rightarrow M \times M$ which changes $a.b$ to $b.a$. Then I take the usual binary operation map say $\mu $ and then the condition is equivalent to saying that $\mu =\mu \;\mathrm{o}\; \beta$. Can somebody explain why this is advantageous?

REPLY [2 votes]: Writing out definition of a group, for example, in terms of arrows and commutative diagrams rather than sets and elements allows you to easily generalize the concept of a group in the category of sets to other categories. For example, a Lie group is just a group in the category of smooth manifolds, and a topological group is a group in category of topological spaces. 
See this nlab article for the complete definition of a "group object" in terms of arrows and diagrams.