From Drew Armstrong’s notes on adjoint functors:
Once upon a time, my opinion of category theory was the same as my opinion of Facebook: if I ignore it for long enough, hopefully it will go away. It is now my educated opinion that category theory will not go away, and in fact the language of category theory will continue to spread until it becomes the default foundation of mathematics.
More posts on category theory:
Speaking as a recent graduate student, it already _is_ the foundation of all pure mathematical fields.
Check out Lee Lady’s post “how to do mathematical research”. CT was invented to tie together and simplify some things (related to AT). Foundations was boring/tedious/metaphysical when it was talked about in set-theoretic terms and it continues to be boring in CT.
A good person on boring vs exciting is Terry Gannon, _Moonshine Beyond the Monster_. I posted some quotes on my blog isomorphism.es/tagged/terry+gannon — but if you pony up for the book you will definitely see the difference between boring logician metaphysics and finding something unexpected (I always use the example of homotopy types of n-spheres from OEIS. You can hear the audience laugh in Milnor’s 1954 (?) differential topology lecture. Hopf fibration is another example.