I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because it seems like a quite a lot of topological and group theoretic concepts can be defined most succinctly using categorical concepts, and the categorical definitions are more revealing. So my question is: (1) Is it possible/beneficial to teach analysis using category theory? and (2) Are there any good textbooks that use this method?

7$\begingroup$ Defining is a lot different than motivating or using. $\endgroup$ – Steve Huntsman Sep 15 '10 at 0:56

9$\begingroup$ Which parts of analysis did you have in mind, and what level of course or treatment are you thinking of? I tend to think that one should look for the nail before using the hammer, especially if the nail turns out to be a screw... $\endgroup$ – Yemon Choi Sep 15 '10 at 0:58

$\begingroup$ What I think might be interesting is to see if one could profitably take a (more) structuralist perspective when teaching analysis. But I don't know if that is the same as "using categories". $\endgroup$ – Yemon Choi Sep 15 '10 at 1:05

$\begingroup$ I was mainly wondering if basic concepts like the derivative and integral could be defined axiomatically, in the sense of the Eilenberg–Steenrod axioms. $\endgroup$ – Daniel Miller Sep 15 '10 at 1:30

$\begingroup$ The Lebesgue integral can be defined by a universal property: golem.ph.utexas.edu/category/2014/07/… $\endgroup$ – David Roberts May 28 '16 at 11:24
I hesitate to let this out, but there's always this cute little note that I learned from another MO answer (I don't know which one): http://www.maths.gla.ac.uk/~tl/glasgowpssl/banach.pdf. Maybe this will satisfy your curiosity, but I maintain that it takes a warped mind to identify such a categorical formulation of integration as the "right" way to think about integrals.
The advantage of categorical thinking in my view is that it helps to organize computations and arguments involving several different kinds of structures at the same time. For instance, (co)homology is all about capturing useful invariants associated to a complicated structure (e.g. a geometric object) in a much simpler structure (e.g. an abelian group). When we want to determine how the invariants behave under certain operations on the complicated structure (e.g. products, (co)limits) it helps to have a theory already set up to tell us what will happen to the simpler structure. That's where category theory comes into its own, and instances of this paradigm are so ubiquitous in algebra and topology that category theory has taken on a life of its own. It seems that people working in those areas have found it convenient to build categorical constructions into the foundations of their work in order to emphasize generality (one can treat algebraic varieties and solutions to diophantine equations on virtually the same footing), keep track of different notions of equivalence (e.g. homotopy versus homeomorphism), build new kinds of spaces (e.g. groupoids), and to achieve many other aims.
In many kinds of analysis, this kind of abstraction isn't necessary because there's often only one structure to keep track of: $\mathbb{R}$. When you think about it, analysis is only possible because we are willing to seriously overburden $\mathbb{R}$. Take, for example, the expression "$\frac{d}{dt}\int_X f_t(x) d\mu(x)$" and consider all of the different ways real numbers are being used. It is used as a geometric object (odds are X is built out of some construction involving the real numbers or a subspace thereof), a way to give $X$ additional structure (it wouldn't hurt to guess that $\mu$ is a real valued measure), a parameter ($t$), and a reference system ($f$ probably takes values in $\mathbb{R}$ or something related to it). In algebraic geometry, one would probably take each of these roles seriously and understand what kind of structure they are meant to bring to the problem. But part of the power and flexibility of analysis is that we can sweep these considerations under the rug and ultimately reduce most complications to considerations involving the real numbers.
All that being said, the tools of category theory and homological algebra actually have started to make their way into analysis. Because of the fact that analysts generally consider problems tied to certain very specific kinds of structure, they have historically focused on providing the sharpest and most detailed solutions to their problems rather than extracting the crude, qualitative invariants for which cohomological thinking is most appropriate. However, as analysts have become more and more attuned to the deep relationships between functional analysis and geometry, they have turned to ideas from category theory to help keep things organized. Ktheory and Khomology have become indispensable tools in operator theory; there is even a bivariant functor $KK(,) $ from the category of C*algebras to the category of abelian groups relating the two constructions, and many deep theorems can be subsumed in the assertion that there is a category whose objects are C*algebras and whose morphism spaces are given by $KK(A,B)$. Cyclic homology and cohomology has also become extremely relevant to the interface between analysis and topology.
So ultimately I think it all comes down to what kinds of subtleties are most relevant in a given problem. There is just something fundamentally different about the kind of thinking required to estimate the propagation speed of the solution operator for a nonlinear PDE compared to the kind of thinking required to relate the fixed point theory in characteristic 0 of a linear group acting on a variety to that in characteristic p.

$\begingroup$ uhm, the original link at the top seems now broken, does anyone have a working one? $\endgroup$ – bananastack May 17 '15 at 22:00

4$\begingroup$ @Paul Siegel Indeed the link is broken. I believe this is the new address: maths.ed.ac.uk/~tl/glasgowpssl/banach.pdf. Can anyone confirm this? $\endgroup$ – Alp Uzman Jun 9 '15 at 18:46

$\begingroup$ Maybe there is so much category theory in functional analysis because the latter is essentially topological algebra (over $\mathbb{R}$ or $\mathbb{C}$)... $\endgroup$ – Qfwfq May 28 '16 at 13:44

1$\begingroup$ @Qfwfq: I don't think many functional analysts would agree that there is "so much category theory in functional analysis". $\endgroup$ – Nik Weaver May 28 '16 at 14:10

$\begingroup$ @NikWeaver: yeah, you're right, I meant relative to other branches of analysis :) $\endgroup$ – Qfwfq May 29 '16 at 15:17
Others can definitely give better opinions, but I currently have "Lectures and exercises on functional analysis" checked out from the library, and I have been enjoying the few parts that I've read so far.
I can not comment on the use of category theory in analysis, but for people who aren't very comfortable with more abstract fields where category theory plays a major role a book like the one above is great since it goes over a lot of basic category theory while keeping the main characters from analysis. At the very least it's a great way to get accustomed to the language.

1$\begingroup$ I briefly skimmed over that book in the library a year or two ago... a nice read, though in places I felt in a strange way that AYaH wasn't going far enough. (Certainly more people at this level should say that $\ell^1$ is the free Banach space functor, left adjoint to the forgetful unit ball functor, and similar stuff. How much this helps people with Rolle's theorem, I don't know... ) $\endgroup$ – Yemon Choi Sep 15 '10 at 1:32

$\begingroup$ I made the answer CW, since I just realized the reason why I posted it as an answer instead of a comment was so that I see what people think of this book. :) $\endgroup$ – Gjergji Zaimi Sep 15 '10 at 1:42

$\begingroup$ I need to reread that book. Some of it perhaps didn't seem as new or interesting to me as it should have, just because I've read parts of the author's more specialist works on homological algebra (CartanEilenberg vintage) in the Banach and LCTVS worlds. $\endgroup$ – Yemon Choi Sep 15 '10 at 2:41
This community wiki answer is addressed to the OP's comment that he is looking for an "axiomatic" approach to the integral.
I don't (yet) understand what axioms have to do with category theory. In particular, with respect to the example you give, I don't see what is particularly categorical about the EilenbergSteenrod axioms (unless you mean to count the functorial nature of co/homology as one of the axioms).
As an example of an axiomatic treatment of the (Riemann) integral, see Section 2 of
http://math.uga.edu/~pete/243integrals1.pdf
(Note: this is nothing very original. For instance, shortly after I wrote this I saw that Lang had almost the same treatment in his undergraduate analysis text.)
Here I see no category theory whatsoever. Is this what you had in mind? Why or why not?
Perhaps you were talking about the Lebesgue integral rather than the Riemann integral. In that respect, I would say that the Daniell approach to the Lebesgue integral (i.e., characterizing it in terms of the completion of a certain normed linear space) feels "axiomatic" to me but still not categorical.

3$\begingroup$ Since I may sometimes give the impression of being insufficiently categorically enlightened, shall I be the one to plug the perspective in Tom Leinster's note here: maths.gla.ac.uk/~tl/glasgowpssl (I should emphasise that I don't know if one'd want to base a course on integration around this, or even use it, but I do find it an interesting point of view.) $\endgroup$ – Yemon Choi Sep 15 '10 at 2:37

$\begingroup$ @Yemon: the link seems interesting. Moreover, it seems at least as relevant to the question as a whole than my throwaway answer (I gave an example of an axiomatic characterization of an integral which is not categorical in nature; you gave one which is). Maybe you should post it as a separate answer? $\endgroup$ – Pete L. Clark Sep 15 '10 at 3:42

$\begingroup$ Oops, sorry to duplicate  I spent awhile typing up my answer. $\endgroup$ – Paul Siegel Sep 15 '10 at 4:35

$\begingroup$ That's fine, Paul; your answer is much more thorough than anything I'd have got round to writing. (I am rather fond of TL's note as I was in the audience at that talk and failed, to my chagrin, to guess the right answer.) $\endgroup$ – Yemon Choi Sep 15 '10 at 7:21

1$\begingroup$ For an algebraic characterisation of analysis see Peter Freyd's Real Algebraic Analysis: tac.mta.ca/tac/volumes/20/10/2010abs.html. As its publication in Theory and Applications of Categories suggests, there's category theoretic thinking going on it. $\endgroup$ – David Corfield Sep 15 '10 at 8:09