TITLE: Mathematics needed for higher dimensional category theory?
QUESTION [5 upvotes]: I'm a undergrad(third year, Manchester uni and want to do a PhD) that is thinking of doing a PhD in this area or category theory in general.(Sorry for asking it here, Maths exchange stack didn't help as asked twice last week and then today, I spend like an hour a day worrying about what I'm studying and what I should study). Imagine a chess board and you need to move a piece, how you know you are moving the best piece?, now times that by a thousand. 
Just wondering, what branches of Maths should I focus on? As I've been told that topology particular Algebraic topology is the main area for this. A lecturer told me I should focus on getting down Algebraic topology before thinking of doing category theory as most of the examples of category theory are from algebraic topology.
However, another lecturer told me I should be doing logic and particular model theory. I can take a course in it this year, but would mean not doing commutative algebra.
So before you delete this can you please tell me what I should be studying? As it would save me countless hours of worrying if I'm doing the right subjects. 
P.S. I can do most of the stuff in Conceptual Mathematics category book. I don't understand topoi through.

REPLY [20 votes]: It seems to me that category theory (whether higher or not) may be the worst part of mathematics to approach with a narrow viewpoint --- just topology, or just algebraic geometry, or just logic, or just any one area.  To me, much of the value and beauty of category theory lies in how it exhibits connections and similarities among many areas of mathematics.