Why don't analysts do category theory?

Question

I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects.

Recently, I started taking some functional analysis courses and I discovered that there is almost no category theory done in these courses. But since most of the spaces studied in functional analysis are objects in categories (e.g. the normed spaces form a category), I find it rather strange that the books leave the category theory out.

Is there a reason for this?

Answer 1

Let me ask you a dual question: I am a mathematics student in set theory, why don't category theory students do set theory? I find it strange that most books on category theory have only a naive handling of set theory.

Now let me answer your question. Category theory is an impressive tool for abstraction, but analysis is not always in need for abstraction - it looks for concrete solutions and ideas. In that aspect categories are not too useful. In fact, if you try to insist on concreteness, categories can become a burden when you insist to carry the category around instead of just talking about functions and spaces.

On the other hand, abstract algebra is very fitting for category theory. We focus on "all" groups or "all" modules over a certain ring. Analysis focuses on particular spaces, continuous/differentiable/analytical functions over $\mathbb C$, for example. The situation is similar with set theory, while a very useful tool for formalizing arguments in the better parts of mathematics - it is often neglected and cast aside as a non-issue.

The best tip to remember (as a set theory student) is that while a screwdriver is a very useful tool to carry, you don't really need it if you are making a sandwich.

Answer 2

There are two questions here, in reality, I think.

First, in brief, I am told by many people that I "do functional analysis in the theory of automorphic forms", and I certainly do find a categorical viewpoint very useful. Second, in brief, it is my impression that the personality-types of many people who'd style themselves "(functional) analysts" might be hostile to or disinterested in the worldview of any part of (even "naive") category theory.

In more detail: as a hugely important example, I think the topology on test functions on $\mathbb R^n$ is incomprehensible without realizing that it is a (directed) colimit (direct limit). The archetype of incomprehensible/unmotivated "definition" (rather than categorical characterization) is in Rudin's (quite admirable in many ways, don't misunderstand me!) "Functional Analysis"' definition of that topology.

That is, respectfully disagreeing with some other answers, I do not think the specific-ness of concrete function spaces reduces the utility of a (naive-) categorical viewpoint.

From a sociological or psychological viewpoint, which I suspect is often dominant, it is not hard to understand that many people have a distaste for the structuralism of (even "naive", in the sense of "naive set theory") category theory. And, indeed, enthusiasm does often lead to excess. :)

I might claim that we are in a historical epoch in which the scandals of late 19th and early 20th century set theory prey on our minds (not to mention the mid-19th century scandals in analysis), while some still react to the arguable excesses of (the otherwise good impulses of) Bourbaki, react to certain exuberances of category theory advocates... and haven't yet reached the reasonable equilibrium that prosaically, calmly, recognizes the utilities of all these things.

Edit: since this question has resurfaced... in practical terms, as in L. Schwartz' Kernel Theorem in the simplest case of functions on products of circles, the strong topology on duals of Levi-Sobolev spaces is the colimit of Hilbert space topologies on duals (negatively-indexed Levi-Sobolev spaces) of (positively-indexed) Levi-Sobolev spaces. As I have remarked in quite a few other places, it was and is greatly reassuring to me that a "naive-categorical" viewpoint immediately shows that there is a unique (up to unique isomorphism) reasonable (!) topology there...

Similarly, for pseudo-differential operators, and other "modern" ideas, it is very useful to recast their description in "naive-categorical" terms, thereby seeing that the perhaps-seeming-whimsy in various "definitions" is not at all whimsical, but is inevitable.

A different example is characterization of "weak/Gelfand-Pettis" integrals: only "in my later years" have I appreciated the unicity of characterization, as opposed to "construction" (as in a Riemann/Bochner integral).

Answer 3

I would guess that a lot of what analysis studies is about individual "points" in a way that isn't suitable for category theory. You are often interested in functions that might or might not be in some collection of functions, and it is not always clear that composition of these functions makes any sense. You are often only interested in showing properties at individual points in a space - it is continuous at $x$ or differentiable or bounded. Category theory is less useful for that sort of problem.

Another area where you might even more expect category theory to be involved is linear algebra. In linear algebra, we are entirely talking about composable functions, etc. But the reality is, while category theory gives you a nice overview way of thinking of linear algebra, it doesn't give you any real aid in solving problems about linear algebra.

The most powerful things we can do with category theory are when we start related seemingly different categories. That shows up in analysis in some ways, but at the lower level, the examples are more easily understood as linear algebra (for example, Fourier series.)

It's not that there isn't some useful ways to use category theory in analysis. I'm not well-versed enough to know where it is used or not. However, it seems to me that understanding the fundamentals is not aided enough by category theory to make it worthwhile.

Answer 4

If you want to study functional analysis from categorial point of view see this book. Its first chapter starts from introduction to category theory.

I think that this approach is rare because most functional analyst study very concrete objects and spaces. So category theory become useless for them.

Answer 5

You might also Take a Look at the Book "the geometry of PDE's and mechanics" by agostino Prastaro. You can find a google books version online. For someone looking for category theory in analysis this ought to be fun.. Especially check out the parts about spectral sequences. :)

Answer 6

The level of abstraction inherent in category theory makes it harder to apply it to some of the very concrete questions analysts look at. If you are trying to find Sobolev space properties of solutions to the heat equation just to give an example, the methods currently available to address such questions are pretty hands-on. Fourier transforms, conserved quantities, pseudodifferential operators, etc. The same applies in Fourier analysis in ${\mathbb R}^n$ and various other "concrete" subjects.

There are other areas of analysis where categories are relevant, although somewhat indirectly. In several complex variables for example, they use sheaves and cohomology and so on. I believe most several complex variables people don't use categories that much, but there are analogues of what they do in parts of algebraic geometry where they do use category theory.