What does it look like to "do math in category theory"? I have just read some introductory stuff about Category theory, and while I understand what a category is, and some examples of it, the purpose of category theory is still vague to me.
It seems there are three frameworks for mathematics:


*

*set theory

*type theory

*category theory
In practice, people don't do math in set theory, because it's unnecessarily technical, and not very helpful. Type theory seems to more closely capture the way mathematicians intuitively think in practice than set theory.
But I'm very unclear about what it means to "do mathematics in category theory". What does this mean in practice? How does a mathematician who "thinks in categorical terms" approach math differently from one who does not, and why?
 A: Simply put, doing "mathematics in category theory" is doing mathematics without (set) elements. Let me be more precise (since my last sentence is factually wrong), objects in categories don't (in general) consist of elements, they build categories and that's it. What you are interested in is how different objects relate to each other and this is captured by notion of morphism between objects.
This can actually be quite satisfying. For example, do you really care that $\mathbb N = \{\emptyset,\{\emptyset\}, \{\emptyset,\{\emptyset\}\},\ldots\}$ (or pick some other set theoretic definition)? Do you really care that $\mathbb Z = (\mathbb N\times \mathbb N)/\sim$, and that in this particular construction $\mathbb N\not\subseteq\mathbb Z$? Do you really care that $(X\times Y)\times Z \neq X\times(Y\times Z)$?
It's likely that the answer is: "No." What you (likely) care about is that there is some object $\mathbb N$ with desired properties, that no matter the construction, $\mathbb N$ embeds in $\mathbb Z$ in a particular way and that $(X\times Y)\times Z \cong X\times(Y\times Z)$ in an obvious manner.
In a way, not working with objects consisting of elements allows you to focus on functionality and not particular implementation.
The obvious question is how would one define a particular object of interest in category theory? You'd do it by invoking some universal property. For example, free group has universal property that any function defined on its generators extends uniquely to group homomorphism. In my opinion, that is way more useful than to actually work with reduced words or something else. I can think of many other similar examples, but this answer will turn in some kind of propaganda material if I continue.
So, it's very likely that in your every day mathematics life you are using a lot of category theory philosophy without thinking about it. Just like set theory.

But, I still didn't quite answer your question. "Mathematics in category theory" is taking all this one step further. You are only working with universal properties, whether they apply to objects, morphisms, functors, natural transformations or categories themselves. You don't check things "element-wise". You are not allowed to compare objects up to equality ($=$) (that's evil, by the way), but up to isomorphism. You care about transformations that are natural, and not dependent on particular choices that you've made (the most famous example is $V\cong V^*$ for finite dimensional vector space is not natural, it depends on choice of basis).
One of the goals is to lay foundations for mathematics that are more like "what mathematicians think like" than ZFC. This should make teaching computers to do formal maths easier (think proof assistants).
The other is simply convenience. Sam Streeter mentioned algebraic geometry. It can also be very helpful in representation theory, algebraic topology etc.
But really, most mathematicians don't do "mathematics in category theory" just as they don't do "mathematics in set theory". They care about solving problems and are satisfied with just the fact that there is some formalism that justifies their work.
So, in a sense "mathematics in category theory" is very like "mathematics in set theory", but with different tools available.
A: This is a very tricky question to interpret and to answer, but don't get me wrong, I like it.
I think that one implication here is that certain mathematicians might view all of mathematics through one of these theories, but I don't think anybody really does that. I think these theories serve as foundations for large subfields of mathematics though.
Philosophically, category theory does seem to be a natural mathematical language, as it reduces everything to the study of objects and relations between them, and that sounds like a good description of what mathematics is at its core.
If you want an example of somebody using category theory for a reason other than aesthetics, I would suggest looking at modern algebraic geometry. Grothendieck built much of the formalism there on category theory due to his so-called "relative point of view" (see here). Adopting a category-theoretic description of algebraic geometry and using universal properties and functoriality to solve problems is way easier than working directly with algebro-geometric objects in the classical setting. Vakil's notes (here) even begin with category theory, setting up the language and style of argument for all else that follows. They give a decent justification of why category theory is so useful.
