Is a background in Category Theory enough for starting a PhD in Category Theory? Hello! 
I have just finished my master's course on which I focused on category theory, because I found it a very interesting subject, but my background, as an undergraduate student, was on functional analysis and analytic topology. So I did not arrive to category theory from a "normal" path, i.e logic, set theory, then algebraic topology etc. But I am now very interested on category theory and I would even like to do a PhD on this subject. 
My question is, how easy would it be in your opinion, to start learning about things like cohomology theories or abstract homological algebra, by looking at them from a category theoretic point of view, in which I have a stronger background now, than starting, say in homology, from studying singular homology or other simpler cases. 
Or for another example, if you consider sheaves, would it be useful - in a technical sense- to know and use the definition of a sheaf which uses restrictions of intersections, or does it suffice, for the applications of this idea, to understand the definition as a presheaf satisfying the unique amalgamation on matching families condition? So, I guess what I am asking is, is your ability to generalize concepts, when you are considering research level questions, dependent on your good understanding of the underlying theory in its simple forms? I look, for example at the definition of a product of topological spaces, and see how beautiful it looks to describe using categorical language, but does it feel easier to handle now just because I knew already what this object is?
Thanks!
 A: Tim Porter is absolutely right about the difference between studying and using category theory. It seems nowadays that there are not so many people studying category theory for its own sake, but many times people have to develop new things in category theory in order to do something of interest in their primary field (this happens all the time in homotopy theory). If you're set on category theory for its own sake I recommend checking out logic programming, kleisli categories, allegories, etc. Or perhaps homotopy type theory.
My advisor introduces category theory to his classes as 'a way for your brain to remember more things.' This is true on many levels (I can replace homomorphism, continuous, holomorphic, etc in my vocabulary by 'morphism') but it also highlights the unifying perspective. Having a solid foundation in category theory as well as knowledge of how things go in many examples can give you a great intuition when you come to a new field and let you pick it up much faster. But just knowing category theory and no examples will not be good. So start gobbling!
Not everyone has drunk the koolaid yet on category theoretic language, especially in analysis so be sure you can speak the classical language too. I like to practice my this skill by translating seminar talks I go to into my own language. This is a great way to distill the essence of the talk and also practice recognizing and using category theory in a classical story
You asked about learning from a categorical point of view as opposed to how one without category theory would. I think the right answer is to learn both, but try to distill the essence of the one which seems less natural. As you progress you'll find you think of the same object in radically different ways and this can be quite useful (I think about simplicial presheaves in like 8 different ways). It's also good to go back to basics when you sit down to prepare a talk or write a paper because that will show you what was essential in the classical story. From there you'll see what's truly essential in the new thing you are presenting, what the natural questions should be (maybe this leads to your next project), and how to best present what you did. The good idea in many good papers is to view something classical in a new light and then use that viewpoint to make a hard question easy
A: You seem to be wanting to USE category theory rather than just to study it as such, although the two are linked. The detailed answer to you problem also depends on the background of the people who are in your chosen PhD school.  Category theory is great as an area (as are some others) because if you take it seriously you will be forced to learn a lot of other areas of mathematics in searching for the enlightenment about what certain (categorical) constructions.  There are lots of ideas out there and sometimes you should just launch in and work back, finding out what you don't know, and remedying the lack! 
There are several blogs, sets of lecture notes, the n-Lab, etc. on which you can train yourself. (I gave an MSc course in Ottawa some years ago and have put an extended version of it on my homepage on the n-Lab, try it, it is freely available, (but is a bit long as I went on adding to it!) It tried to build intuition on cohomology and what it means, starting from a very classical basis and then looking at group cohomology, simplicial stuff, and so on. It wanders around looking at some, to me, fun stuff, so dip in and see if some topic grabs you then, as Andrej say, gobble it up. The course (The Menagerie) takes a categorical viewpoint but is not that abstractly categorical.)
(Edit: Another good example of the sort of thing you might browse through is the Stacks project.  The n-Lab has a page with that title and a link, in case you want to look.)
A: If you're like Bill Lawvere you will have no trouble doing everything with categories. Also, it may seem like you've gone some way in learning math, but trust me, there's much more to come. So don't worry about your background, what course you might not have taken, or whether one view of a subject is better than another. Just keep gobbling up everything that comes in your path and when in doubt as to which way to do something, do it both ways.
If however you're a problem solver, you should just learn combinatorics or some other subject rife with insanely hard yet easily understood problems, and live happily ever after.
