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!
