Exploring the Equivalence of Category Theory and Versatility

[hepiaaro]

I am a final year math major thinking of taking Category Theory (CT). The problem is people keep telling me its useless. Why?

I have a biology degree and decided to get a second degree in mathematics. My motivation for studying mathematics is related partly to an idea I stumbled across in high school and to my total love of biology. I recall realizing, at about 16, that scientific models are not always domain specific; that some are versatile. This idea had been swimming around in my head for years until I read a description of CT. And now I'm wandering whether there is a real equivalence between this "versatility" and CT

So, does the equivalence exist? And why do people keep writing it off as "abstract nonsense". Can I use CT as a tool to build versatile "models"?
Thanks!

 

[Allen Beers]

It isn't really useless, it just doesn't have very many application outside of mathematics, one of the major ones being biology.

 

[micromass]

It isn't really useless, it just doesn't have very many application outside of mathematics, one of the major ones being biology.

You're 17 years old and haven't done any abstract mathematics. Maybe you should stay out of this thread.

 

[Mirero]

Although I don't know much category theory, from what other mathematicians say it is more or less something that you learn after you solidify your foundations in the main subjects like algebra, topology, etc., and one wouldn't really be able to capture how useful it is without first becoming really comfortable with such subjects. It really provides motivation more than anything.

For one case, take the concept of a product of two objects in a category. Of course, you could just look at the definitions of such a notion and be on your merry way if you wanted. But it would probably be more useful to notice that, depending on what your objects are, the notion of the product is equivalent to many other concepts in mathematics. For example, if the objects are groups, the product represents their direct product, and if the two objects are topologies, then the product represents their product topology.

As for the "abstract nonsense" thing, I'm pretty sure it's just more a humorous jab than anything else. I don't think anybody actually thinks of category theory as useless.

 

[micromass]

I don't think anybody actually thinks of category theory as useless.

I'm actually sure many people think of category theory as useless. I have seen people who actually adore category theory, and I have seen people who absolutely hate it. I guess I can understand both perspectives. It's nice to like because it gives a unifying language for a lot of mathematics. But it also tends to obscure a lot of geometrical intuition.

 

[mathwonk]

I used to be fascinated by category theory, spending some time reading the fun little book by Peter Freyd, but eventually lost all interest in it as a subject. However I still think there is a moral from category theoretical thinking that is valuable, namely always remember to ask what the mappings are between objects you are studying, not just the static definitions of the objects themselves. E.g. before learning a little category theory one thinks a direct product is a set of ordered pairs, while ever afterwards one thinks of it as an object equipped with projection maps to the factors and having a universal mapping property with respect to such pairs of maps. So the main benefit to me is to think in terms of mappings as more important than objects. This leads one to the right definition of certain constructions. E.g. in algebraic geometry the Zariski tpology on the product of two affine algebraic varieties is not the product topology, because that only has the universal mapping propert for continuous maps, whereas one wants the universal property fo algebraic maps in that setting. Another example is the correct definition of an isomorphism as a structure preserving map with a structure preserving inverse, rather than just a bjiective structure preserving map.
So everyone may benefit from a little categorical perspective, but lots of people do not think it is an intertesting subect in its own right, sort of like many people use grammar when they speak but do not think of grammar as an intersting subject of research.

 

[chrisaldrich]

Category Theory certainly isn't useless, and has some really beautiful and interesting things going on. One of the bigger issues it has is that it is relatively so young as a discipline within math. (Imagine what people said about set theory in 1910.) Your toughest hurdle as an undergraduate is finding a university that is offering it as an accessible class depending on your level of mathematical background. I haven't noticed a lot of universities even offering it at the graduate level yet either.

That being said, David Spivak's new book https://www.amazon.com/gp/product/0262028131/?tag=pfamazon01-20 which came out of MIT Press last year is the closest textbook I've seen to attempt making Category Theory easily accessible to not only an undergraduate audience, but also to a broader range of scientists who are not necessarily steeped in mathematics. For your particular purposes, it actually has a nice handful of biology related examples in the early chapters.

Because it expounds on some of the beauty of category theory and it's application to science, I'll draw your attention to an essay that Ilyas Khan wrote recently: https://www.linkedin.com/pulse/category-theory-bedrock-mathematics-ilyas-khan

I'd recommend category theory more highly, if you plan on pursuing a Ph.D. and research work, but in the meanwhile you might also find some interesting material in John Carlos Baez's Azimuth blog or on the n-Category Cafe.

 

[hepiaaro]

Thanks everyone for your replies!