What's there to do in category theory?

Question

Disclaimer: I posted this question on MSE only a few days ago; and received very few comments. I know that the etiquette is to wait a bit more than that before moving a post from MSE to MO, but I figured that posting it on MO would be an actual improvement because there would be some actual researchers in category theory on this site, willing to give details about what it is they do, what's interesting about it, etc, whereas there may be less of them on MSE. If this isn't appropriate, I'll remove this post, and if it's the case I'm sorry for the disturbance.

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I'm sure anyone who's heard of categories has also heard the classical "Well obviously there aren't any real theorems in category theory, it's much too general", or something in the likes of it.

Now obviously this argument is invalid (although its conclusion may be correct) because the same could be said of set theory, but there are clearly many really important theorems and results in set theory (I guess I don't have to justify that's it a huge field of research).

Now these theorems come from the fact that, when we do set theory, we don't just look at $\in$, we look at "derived stuff", like transitive sets, well-ordered sets, models of certain things, filters, etc. (I'm just giving a few examples to explain what I mean, I perfectly know that there's much much more to set theory than just those).

So the same thing should apply to category theory : of course we're not going to prove of we just stand there with our arrows and objects; you have to consider interesting ones, with more properties etc.

My question is about these (sorry for the lengthy intrduction). I know that a big part of category theory (although I don't really know in what proportion) is devoted to studying topoi(/ses ?) and for instance cartesian closed categories.

But I'm also guessing that there's much more than that to category theory; and my problem is that I don't know much about what is currently studied, what the major subfields of category theory are, or for that matter what subfields there are; so that when I want to refute the argument given at the very beginning I'm a bit stuck because I feel like I'm reducing category theory to topos theory and abelian categories.

Here's the actual question (after the too wordy introduction) : could you give some examples of subfields of research (if possible, currently, or previously very active fields) in category theory, paradigmatic questions or theorems in those subfields; how they're interesting in themselves and for some, how they can be interesting for other areas in maths (more than just giving a common language) ?

Answer 1

There is a majestic paper by Mac Lane

MacLane, Saunders. "Possible programs for categorists." Category Theory, Homology Theory and their Applications I. Springer, Berlin, Heidelberg, 1969. 123-131.

whose opening line is one of the most beautiful I've ever read:

Communication among Mathematicians is governed by a number of unspoken rules. One of these specifies that a Mathematician should talk about explicit theorems or concrete examples, and not about speculative programs. I propose to violate this excellent rule.

I've often wondered what does remains of those suggestions, and I strongly recommend you (if you haven't already) to have a look at this inspiring note: it is a masterpiece of neat exposition and it is replete with the hope that category theory becomes deeper and stronger, with the passing of time.

It is organized in brief, lapidary short sections, and proposes several directions in which category theory can, should or will go: in a few words

• we shall find new general concepts,
• we shall polish and adapt old ones through (hard work and) time,
• we shall reach a deeper understanding of structured and low-dimensional higher categories (monoidal categories, bi- and tri-categories and their multiple applications),
• we shall link category theory to differential geometry, mathematical analysis and mathematical physics,
• we shall ground category theory on a real foundation (or even better we shall use it as a foundation).

I'll leave you the pleasure of reading the note for yourself. My opinion (which is only the humble feeling of a young craftman) is that there are few items we can feel we have completely solved, even 50 years later.

Of course, today we have more higher category theory than we could ever hope for. Of course, we have a few people working in axiomatic cohesion. Of course, we have people in type theory and in HoTT. And also, we are lucky because today few people question the "importance of being abstract". But there is so much still to do!

And the best way I can explain what I'm saying is by adding an item to the otherwise complete Mac Lane's list.

• We shall work together to let more and more mathematicians see how profound, and beautiful, and inspiring, and elegant category theory is.

Category theory is huge, but few people outside pure mathematics apply it. Many people know that it exists, but few people appreciate its elegant, tautological statements and try to apply it to different things (those who do it are outstanding mathematicians, way better than I will ever be). This is what makes pure mathematics vital: a bunch of flippant engineers and physicists and biologists shaking it, breaking it, deforming it. We shall give other people tools to package immensely deep ideas in an extremely low volume ("rings are spaces"; "homotopy theory is localization"; "the Yoneda lemma"...).

Last, but not least, I feel we shall communicate why we feel lucky: category theory is an island of beauty in the already beautiful land of mathematics, and we are in love with it. We shall communicate the bliss we feel when we do it.

Answer 2

Through extended TQFT and the cobordism hypothesis, many questions in topological quantum field theory have been turned into explicit questions about higher category theory.

A TQFT is formalized as a symmetric monoidal functor $Z\colon\mathsf{Bord}_n\to\mathsf C$, where $\mathsf C$ is some symmetric monoidal $(\infty, n)$-category and $\mathsf{Bord}_n$ is the symmetric monoidal cobordism $(\infty, n)$-category.

Given a group homomorphism $\rho\colon H_n\to \mathrm O_n$, it's possible to consider manifolds with $H_n$-structure, e.g. $\mathrm{SO}_n$ gives an orientation, $\mathrm{Spin}_n$ gives a spin structure, and define a cobordism category of manifolds with $H_n$-structure and hence TQFTs with $H_n$-structure.

The cobordism hypothesis says that a TQFT is determined by the object of $\mathsf C$ it assigns to a point, and further characterizes which objects can occur: a TQFT with $H_n$-structure is the same data as an $H_n$-homotopy fixed point in the subcategory of fully dualizable objects in $\mathsf C$. (For more on the cobordism hypothesis, check out Dan Freed's exposition.)

For example, an $\mathrm{SO}_2$-homotopy fixed point in the Morita 2-category of algebras is the same thing as a semisimple Frobenius algebra, and this implies that a semisimple Frobenius algebra determines and is determined by a 2D oriented TQFT.

More generally, given a dimension and a symmetry type $H_n\to\mathrm O_n$, we would like a similar algebraic characterization, so that we can study TQFTs with algebra and then use algebra to construct TQFTs. In dimension 3, there are examples of algebraic constructions of TQFTs, most notably the Turaev-Viro-Barratt-Westbury (TVBW) construction, whose input data is a spherical fusion category. So these should relate to $\mathrm{SO}_3$-homotopy fixed point structures in some symmetric monoidal 3-category, and characterizing these is as far as I know an open question. This and several related questions are examined by Douglas, Schommer-Pries, and Snyder, among others.

Answer 3

I apologize in advance for this very long answer. I am pretty sure that many people could write a better version of it. Unfortunately, they are not doing it. So, here we are.

The very beginning of the question asks for some even classical results in category theory that are relevant outside category theory. My knowledge is very limited thus I will list just few examples. A good research on google could produce the same, even better, result. Few example are better exposed because they are closer to my understanding.

1. The theory of algebraic theories, known also with the name of Lawevere theories.
2. The theory of monads.

3. Abelian categories.

4. Between 70's and 80's there was a very interesting fashion, that eventually died and I do not understand why. Consider a category, say Hausdorff spaces, can we find a full embedding into a category of algebras? This is a very natural question, that we can rephrase in the following way: given a category, how effective can algebra be when studying it? Pultr and Kucera gave a partial answer to this question in The category of compact Hausdorff spaces is not algebraic if there are too many measurable cardinals. In the same fashion Freyd proved that the homotopy category of topological spaces is not concrete. Thus there is no way of building a beautiful fundamental group which classifies homotopical algebra, somehow higher algebra is required by the complexity of the category. One could also relate this article to this subject.

5. The theory of locally presentable categories. Given a category $\mathcal{K}$ there is a natural notion of size for its objects called presentability rank. This was introduced by Makkai and Paré. Presentability rank coincides with cardinality (up to some subtleties) in category of models of a theory. It coincides with density of a space in metric spaces, with cardinality of an Hilbert basis in the case of Hilbert spaces. Do you think that such a unifying gadget should be studied? This is just a feature of this theory, the easiest to formulate.

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As people told you the community seems to be working on HoTT and higher categories but there are also other projects around.

1. There is an open connection between accessible categories and abstract elementary classes that lately attracted the interest of Grossberg, Boney and Vasey.
2. Still some people are working on categorical logic à la Makkai, extending some completeness results to infinitary logic. This is the case of C. Espindola.
3. Still some people are working on topos theory, this is the case of Nate Ackerman. O. Caramello is trying to apply this framework to algebraic geometry. The same is trying to do Ingo Blechschmidt.
4. W. Kubis gave a shorter proof of the uniqueness of the Gurarii space with a strongly category theoretic approach. This is linked to its presentation of Fraissé theory.

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As a final remark, I will list three question that I asked on this platform. Obviously this is not interest of any research but they witness how naturally questions in category theory are related to other fields of mathematics and are not just a rewriting of known results.

1. How to compute cocontinuity of a functor.

2. Free objects in first order theories.

3. Model existence theorem in elementary topoi.

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Now that I have produced some mathematical argumentation in defense of my position, I would like to say that MO is an online community that everybody reads, especially young students.

I do believe that category theory has proven to be useful in many fields and it is time to stop answering to this kind of questions. One could say that this is just a genuine attempt to ask for the activity of a field but this is just rhetoric. It is impossible for people outside a subject to get into its research just asking "what are you guys doing?" Imagine to do it with algebraic geometry. You simply wouldn't understand the answer.

This kind of questions cannot be well answered because of the ignorance of the asker. Unfortunately this produces a misperception in the average reader, that will pretend there is no technical research in the subject.

Answer 4

Category Theory is deeply rooted in Computer Science.

A by now Classic book is:

Barr, Wells, "Category Theory for Computing Science", 1998.

I also know that software companies (in the Netherlands) apply it.