10

In case you have not yet seen it, I thought I would draw your attention to (what is currently) the most recent issue of the American Mathematical Monthly, and, in particular, the article: Leinster, T. Rethinking Set Theory. The American Mathematical Monthly, 121(5), pp. 403-415. An arXiv version can be found here. The abstract says: Mathematicians ...


10

My instinctive reaction is that a "category error" is being made here (in the philosophical sense, not the mathematical sense of category). Namely, category theory is an abstraction of (standard, undergraduate level) abstract algebra, which is itself an abstraction of the sort of very concrete mathematical manipulations most students have seen up to that ...


4

Here is my intuition: First, each object can be used to encode some information which is accessible through morphisms. For example, take the $\mathbf{Set}$ category and two objects $A = \{0\}$, $B = \{1, 2\}$. Now you can encode one bit of information by picking one of the morphisms $f_1(0) = 1$ or $f_2(0) = 2$. Now, suppose that we could assign to each ...


3

I'm going to explain how I think of the category-theoretic definition of products. Unfortunately, this viewpoint won't lead directly to the usual universal property but rather to an equivalent formulation in terms of morphisms into products. I hope that it can still be of some use even if you'd rather go directly to the universal property. In most of the ...


3

Your profile suggests that you are still a teen ager. Therefore my advice is, for now, to explore a lot of things and don't try to specialize too much in any one thing. It is a big world with a lot of possibilities. I'm not suggesting that you give up your study of Category Theory, but there is a lot of other mathematics that is interesting and which ...


1

Dan Ghica has a blog post on discovering category theory for primary school children by way of knot theory. Quite charming, I think, albeit not really an answer to your question. Still, I thought it might be worth noting here. (I came across it when reading Paweł Sobociński’s Graphical Linear Algebra blog.)


1

Great question! Yes, this was Piaget's project after initially working with more classic structuralism (e.g. his text titled Structuralism). He explicitly mentions Mac Lane's work in his book on understanding functions, Epistemology and Psychology of Functions I discuss this a bit in my dissertation, but am not familiar with other work that compares ...


1

Learning how to code and code well is an incredibly useful skill in math research in all fields. (I can code, but not code well, but I've been very lucky to have collaborators who are much better than I am. Around half of my papers involve serious computer calculations) But perhaps more importantly, category theory is one of the closest fields in math ...


1

My advice would be to be open to areas of math that allow for programming skill. You can still take some theoretical classes and such. But there is a lot of good stuff in optimization theory, data processing, cryptography, etc. that uses both math and programming (and even sometimes other fields like OR, business, mechE for finite element analysis, etc.) ...


Only top voted, non community-wiki answers of a minimum length are eligible