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What questions do category theorists want to answer?

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There are some good answers already, but I'll try to give a concrete idea of what a category theorist might ask themselves after finding a functorial relation between categories.

The first thing to figure out is if the functor is part of an adjunction. Adjoint functors behave really nicely (they preserve (co)limits) and indicate a particularly strong relationship between the categories.

The next thing would be to figure out if the categories involved have further categorical structure. For instance are they monoidal? Closed monoidal? If so, does the functor preserve the monoidal structure (there are different levels of strictness in which a functor may preserve a monoidal structure). Monoidal structures are particularly nice since they usually let you do a lot of algebra in your category.

Another interesting structure to check is if there is a model structure on your categories. A model structure is what allows you to do homotopy theory im your category. If not a model structure there are other (stronger and weaker) types of structure that lets you do hototopy theory-esque things in your category. There are appropriate notions of functors that preserve these homotopical structures.
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These days most active research in the area is about higher category theory. As such, you won't find many mathematicians who identify as category theorists - they're more likely to say they work in homotopy theory. Since (\infty,1)-categories are pretty well understood at this point, I think people are mostly working on the foundations for (\infty,2)-category theory and so on.

Edit: This may be gobbledygook to the uninitiated. I'm happy to explain further if anyone is interested.
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How will I get a job?
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Oh, nothing specific
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Broadly, we just want to understand how different constructions 'interlock', which can be made rigorous by a wide variety of very controlled methods. In alchemical metaphor: we want to purify solutions to problems, distill and classify structural species (the 'results' of 'performing constructions'), coagulating species into useful combinations, et cetera. I find this colorful language useful because mathematicians habitually appropriate ordinary words.

  
All sciences seek understanding, including category theory. The differences are measured in the scope of their applications. Category theory has been carried forward by the explicit intention to expand its scope, perhaps to include ALL other scopes. My idea of "scope" is similar to the idea of "context" in dependent type theory (which is roughly "cartesian closed category theory" via Lambek-Curry-Howard). Contextuality is important in logic as well as quantum phenomenology. For example, it's worth wondering about the scope of application of a phenomenological physical theory. "Where/when" and "how", exactly, does a description "break down"? When does a logic fail, creating demand for new logics to take its place? Is there a "best" or "right" way to go about doing math, logic, science, and engineering? Many category theorists seek beyond the binaries and Booleans of "classical foundations", based on "material set theory". Rather than settling for negative answers to existence questions, we refine to questions concerning *obstructions* to existence. Can we classify the obstructions? There may be a variety of substantially different obstructions, some being essential and others capable of being "deformed away" with some structural mutation. These lines of inquiry have brought awareness of deeper and richer connections between structures. We discover and invent new species of information, and adapt our worldview to accommodate the expanding scope.  


Exploring the mathematical polycosm feels like hacking through a dense jungle of tangled ideas. The tools of expedition are formal constructions, expressed in formal languages. These languages evolve and adapt to the shifting demands of application, splintering into multitudes and giving birth to new languages capable of differentiating the previous generations (which were born in order to differentiate previous generations of languages, et cetera). Category theory engineers tools for efficient expression and manipulation of linguistic constructions, and that pretty much sums up my motives for studying and applying it.
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Category theory (at least as I undrerstand it) isn't really there to answer specific question, but to provide a "nice", general enough context in which other maths can be done, such as homology, algebraic topology, algebraic geometry, etc

However, some category theorists have specific questions about category theory itself, such as Emily Riehl, whose question is (if I understand correctly) "Does every way of defining ∞-categories lead to the same theorems", but that just seems to be yet another step up in abstraction, to define a nice context to do category theory in, to have a nice context to do more "concrete" mathematics (as concrete as homology can be...)

Still quite new to category theory, but I hope this helps
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They want to answer mathematics without actually doing mathematics.
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Forewarning: I am a bricklayer that has been self teaching maths for a few years. Currently working through category Theory and abstract algebra.

It's the most abstract framework we have to date. At least initially we were discovering that a lot of the disparate things different branches of math (and even science) were doing were actually the same thing if abstracted sufficiently.

The two most practical things that I know of to come out of it this far are functional Programming and Ontology Logs. You may wanna glance over the wiki for OLogs; there's quite a lot there (not on the wiki but the subject of OLogs themselves). We were largely stuck with The Chomsky Hierarchy for linguistics (this applies to computer languages too) for 60 years or so and as cool as they are they tend to break down when attempting to use a Chomsky Hierarchy for a very rigorous application. OLogs are changing that and we have new tools to structure and analyze languages of all forms now. Neat stuff!

For the cutting edge category Theory stuff I think you should listen to a lecture or read up on Emily Rhiel. She's doing some cool stuff with infinite categories that are having direct applications for physics. I suspect you'll find quite a bit of interesting stuffs to report on between Emily and Ontology Logs.

Good luck!!!
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"How can I make my research even more obscure?"
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My masters dissertation recently was looking at generators of certain categories from an algebraic and algebraic geometry perspective and looking at whether the generators are compact. My supervisor is  currently looking at conditions for when bounded derived categories of quasi coherent sheaves of a scheme is compactly generated, i.e. are there any ‘nice’ conditions like compactness, affine, projective, Hausdorff, etc?

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