Should every modern day mathematician care about category theory? As far as I know, category theory is used mainly in topology. I have a dislike towards category theory, similar to my dislike of Bourbakism, and want to avoid it as much as I can. However, the head of our math department (where I have just started my PhD recently) made a speech where he sang praises for category theory and said that in the future every area of mathematics will be affected by it, and every mathematician who ignores category theory will be left in the gutter (his actual words). I was pretty depressed after this meeting. I want to get as objective an answer to this question:

Is it possible to survive in the current mathematics (during the next several decades) as a successful mathematician without caring about category theory? If it is not possible, then what is the minimum required amount of knowledge that every mathematician should get about category theory?

 A: When I was young I didn’t like sheaves or cohomology, so wanted to find something that was algebraic but didn’t involve too much sheaves or cohomology.  I didn’t really need to know much about either to get a tenure track job. But now I’m a more mature person and a more mature mathematician, and I’ve learned to stop worrying and accept cohomology.
All of this is to say that everyone in comments is right, you can certainly be a mathematician without caring about category theory, but strictly avoiding a subject entirely is going to make you an immature mathematician and hold back your development.  You don’t have to love category theory, but it’s a good idea to stop hating it.
A: I say many (most?) mathematicians with thriving research careers completely ignore large parts of mathematics in their work. Probably, they don't even remember what they learned in some of their introductory graduate courses, unless they teach them, and would be unable to pass some comprehensive PhD exams without preparation. What you don't use you forget.
Disliking some parts of mathematics is a way of finding what you really enjoy, a completely natural process. Being broadly educated helps, as long as it does not interfere with research. Learning and doing math are somewhat different activities. One cannot do math without learning some. On the other hand, it is possible to enjoy learning so much that you never actually do anything. There has to be a balance.
In particular, most math research can surely be done without category theory. If you ever need to learn what is, say, a colimit, just read Wikipedia, and follow the references there.
Short term, grad students should focus on finding the kind of math they enjoy doing, and also on passing their exams.
Personally, I revere broadly educated mathematicians, and I strive to become one. Is it a must for a successful career? Not really.
A: You can look at the edit history of this post to see previous versions, which took a different tack whose thread I have honestly lost. I want to take a different tack, though.
What makes this question peculiar is the fact that if you substitute any other area of math for "category theory" in the question, the resultant discussion would look quite different. That is consider the following dialog for various values of $X$:
Professor : Any mathematician who ignores $X$ will be left in the gutter.
Student : I have a distaste for $X$. What's the minimum I should know about $X$ to get by?
I invite the reader to perform the thought experiment of considering the different reactions this exchange would elicit for various values of $X$, such as set theory, group theory, ring theory, combinatorics, functional analysis, topology, category theory.
When I run this thought experiment, I find that in most cases, the professor's pronouncement admits basically two interpretations:

*

*a strong interpretation, where they mean you must be actively be keeping up with current research in $X$.


*a weak interpretation, where they mean that you must have an idea of what $X$ is good for, and that you should be prepared to reach for tools from $X$ when the situation calls for it in your own research.
For most values of $X$, the strong interpretation is a clear stretch, and the onlooker will charitably assume that the weak interpretation is intended. For most values of $X$, that's all there is to it. But when $X$ is category theory, unlike other values of $X$, there's additionally a flame war among the onlookers.
After surviving the latest flame war, I have a theory as to why this is so. My theory is that for most values of $X$, there's a general understanding of how to formulate a weak interpretation of the professor's statement. But when it comes to category theory, people may not be so clear on what kind of weak interpretation should be understood. I propose to remedy this situation with the following pronouncement:
Category theory is good for understanding the naturality vs. choice-dependence of constructions.
This is intended to be parallel to the following pronouncement, which I believe is widely-understood among mathematicans:
Group theory is good for understanding symmetries.
or
Set theory is good for quotienting by equivalence relations.
In each case, the pronouncement doesn't give a complete picture of what $X$ is good for, but gives some kind of launching-off point.
Just as it's reasonable for the professor to say

*

*"questions of symmetry are everywhere in math -- be ready to reach for group-theoretic tools to help understand them"

it's similarly reasonable to say


*"questions of naturality are everywhere in math -- be ready to reach for category-theoretic tools to help understand them".

I hope we can all think of examples illustrating (1). Perhaps the situation is different in the case of (2), and perhaps this points to a shortcoming in general mathematical education. Here's a small example pulled from differential geometry: Let $f : X \to Y$ be a smooth map of manifolds, and let $\omega$ be a differential form on $Y$. Then there is a pullback form $f^\ast(\omega)$ on $X$. You might define $f^\ast(\omega)$ in terms of coordinates, and then wonder whether your definition depends on the choice of coordinates. You can prove that it doesn't, and you can prove things like $g^\ast \circ f^\ast = (f \circ g)^\ast$. The statements of each of these facts are very naturally stated category-theoretically (though the proofs are mostly geometry). There are various routine coordinate-based manipulations you can do on differential forms which are justified by these facts, which again can be nicely summarized in category-theoretic language.
A couple of takeaways from this last example:

*

*The use of category-theoretic language here is not supposed to be earth-shattering or anything. It's pretty banal, really.


*We could continue the flame war by arguing about whether it's necessary to use category-theoretic language here (of course, strictly speaking it isn't). But we don't devolve into such arguments when it comes to examples of using group theoretic-language to understand symmetry. I have a dream that one day we will stop treating category theory differently from group theory in this respect!
