# Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]

I've read looked at the examples in most category theory books and it normally has little Analysis. Which, is strange as I've even seen lattice theory be used to motivate a whole book on category theory.

I was wondering is there a nice application of category theory to functional analysis?

It's weird as read that higher category theory is used in Quantum mechanics as it foundation, yet QM has heavy use of Hilbert spaces.

Sorry for cross posting to MSE. However, somebody there suggested I posted it here to get better response (well, more than two replies). I'm surprised that most books on category theory have very little mention of Analysis. Especially since Grothendieck originally studied functional analysis.

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Personally, I think this question is a bit vague - functional analysis is by now a very broad church - but in any case you should make this question "Community Wiki" by clicking the appropriate box. This is the preferred option for questions which seek a big list of rankable items, rather than a single "right" or definitive answer –  Yemon Choi Dec 13 '11 at 19:42
Also, please add a link to the cross-post (just so that people here don't unwittingly duplicate what has been said there). Thanks –  Yemon Choi Dec 13 '11 at 19:44
@Yemon Choi I was more looking for books that try to put a foundation on Mathematical Analysis via Category theory sort of like the Mathematics made difficult book where he tries to put elementary Maths in terms of Category theory. –  Dan Dec 13 '11 at 19:52
I think Dmitri Pavlov has a lot to say on this topic, see for eaxample <a href="mathoverflow.net/questions/20740/…; –  Michael Greinecker Dec 13 '11 at 22:01
@Dan: could you perhaps reword your question to make it clearer what kind of "uses" you have in mind. Your question suggests that you are looking for applications in analysis, but then your comment says you are looking for things which give analysis a category-theoretic foundation. If you make the question more precise then people may be able to give more focused/relevant answers –  Yemon Choi Dec 14 '11 at 0:23

## 15 Answers

In relatively mundane, but intensely useful and practical, ways, the naive-category-theory attitude to characterize things by their interactions with other things, rather than to construct (without letting on what the goal is until after a sequence of mysterious lemmas), is enormously useful to me.

E.g., it was a revelation, by now many years ago, to see that the topology on the space of test functions was a colimit (of Frechet spaces). Of course, L. Schwartz already worked in those terms, but, even nowadays, few "introductory functional analysis" books mention such a thing. I was baffled for some time by Rudin's "definition" of the topology on test functions, until it gradually dawned on me that he was constructing a thing which he would gradually prove was the colimit, but, sadly, without every quite admitting this. It is easy to imagine that it was his, and many others', opinion that "categorical notions" were the special purview of algebraic topologists or algebraic geometers, rather than being broadly helpful.

Similarly, in situations where a topological vector space is, in truth, a colimit of finite-dimensional ones, it is distressingly-often said that this colimit "has no topology", or "has the discrete topology", ... and thus that we'll ignore the topology. What is true is that it has a unique topology (since finite-dimensional vector spaces over complete non-discrete division rings such as $\mathbb R$ or $\mathbb C$ do, and the colimit is unique, at least if we stay in a category of locally convex tvs's). Also, every linear functional on it is continuous (!). But it certainly is not discrete, because then scalar multiplication wouldn't be continuous, for one thing. But, despite the prevalence of needlessly inaccurate comments on the topology, the fact that all linear maps from it to any other tvs are continuous mostly lets people "get by" regardless.

Spaces with topologies given by collections of semi-norms are (projective/filtered) limits of Banach spaces. Doctrinaire functional analysts seem not to say this, but it very nicely organizes several aspects of that situation. An important tangible example is smooth functions on an interval $[a,b]$, which is the limit of the Banach spaces $C^k[a,b]$. Sobolev imbedding shows that the (positively-indexed) $L^2$ Sobolev spaces $H^s[a,b]$ are {\it cofinal} with the $C^k$'s, so have the same limit: $H^\infty[a,b]\approx C^\infty[a,b]$, and such.

All very mundane, but clarifying.

[Edit:] Partly in response to @Yemon Choi's comments... perhaps nowadays "functional analysts" no longer neglect practical categorical notions, but certainly Rudin and Dunford-Schwartz's "classics" did so. I realize in hindsight that this might have been some "anti-Bourbachiste" reaction. Peter Lax's otherwise very useful relatively recent book does not use any categorical notions. Certainly Riesz-Nagy did not. Eli Stein and co-authors's various books on harmonic analysis didn't speak in any such terms. All this despite L. Schwartz and Grothendieck's publications using such language in the early 1950s. Yosida? Hormander?

I do have a copy of Helemskii's book, and it is striking, by comparison, in its use of categorical notions. Perhaps a little too formally-categorical for my taste, but this isn't a book review. :)

I've tried to incorporate a characterize-rather-than-construct attitude in my functional analysis notes, and modular forms notes, Lie theory notes, and in my algebra notes, too. Oddly, though, even in the latter case (with "category theory" somehow traditionally pigeon-holed as "algebra") describing an "indeterminate" $x$ in a polynomial ring $k[x]$ as being just a part of the description of a "free algebra in one generator" is typically viewed (by students) as a needless extravagance. This despite my attempt to debunk fuzzier notions of "indeterminate" or "variable". The purported partitioning-up of mathematics into "algebra" and "analysis" and "geometry" and "foundations" seems to have an unfortunate appeal to beginners, perhaps as balm to feelings of inadequacy, by offering an excuse for ignorance or limitations?

To be fair (!?!), we might suppose that some tastes genuinely prefer what "we" would perceive as clunky, irrelevant-detail-laden descriptions, and, reciprocally, might describe "our" viewpoint as having lost contact with concrete details (even though I'd disagree).

Maybe it's not all completely rational. :)

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Which functional analysts are these, then, who don't like to speak of Frechet spaces as (projective/inverse) limits of Banach spaces in an appropiate category of TVSes? –  Yemon Choi Dec 14 '11 at 0:35
Helemskii has a book "Lectures on Functional Analysis" (or some similar title) where he covers some of the results one would normally associate with Rudin Chapters 1 to 4, but with forgeful functors and the such-like. Unfortunately I haven't got access to a copy to check. –  Yemon Choi Dec 14 '11 at 0:37
Oh, and +1 (this is more or less my own use of category theory) –  Yemon Choi Dec 14 '11 at 0:44
I find it particularly helpful about the limits of vector spaces, that the dual of a limit is the colimit of the duals.... –  Marc Palm Dec 14 '11 at 8:28
@pm Not in the category of Banach spaces and linear contractions it isn't... the dual of a coproduct is the product of the duals –  Yemon Choi Dec 14 '11 at 9:33

I've never completely understood what counts as "an application of category theory". With other areas of mathematics an "application" of area A to area B is generally a result which translates a problem in B into the language of A, solves the problem using the main theorems of A, and translates the solution back into the language of B. The problem is that I don't know what the main theorems of category theory are (or even if there are any "main theorems").

What I can say is that many interesting and nontrivial categories do arise in certain parts of functional analysis and it is useful to understand the structure of these categories. The specific part of functional analysis that I have in mind is the theory of operator algebras. For instance, in C*-algebra theory one considers a category whose objects are C*-algebras and whose morphisms are given by groups $KK(A,B)$ which simultaneously generalize K-theory and K-homology. Many of the deepest theorems in the subject are organized around the "Kasparov product" which is nothing more than the composition law $KK(A,B) \times KK(B,C) \to KK(A,C)$ in this category. KK-theory and its close cousin E-theory can be characterized according to homotopy invariance and various functorial properties.

On a related note, Connes' noncommutative geometry program (arguably part of functional analysis) relies heavily on the tools of category theory and homological algebra. Even at its inception the program was based on an analogy between de Rham cohomology and the periodic cyclic homology of a "smooth subalgebra" of a C*-algebra. In the process of investigating the relationship between cyclic homology and K-theory people have realized that it is useful to take seriously the category of projective modules over such smooth subalgebras rather than passing to K-theory. The work of Jonathan Block is particularly relevant; you might have a look here, for example.

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Concerning the first paragraph: There are some theorems in "pure" category theory which are nontrivial (General Adjoint Functor Theorem and the related Freyd's Representability Criterion, Beck's Monadicity Theorem, Recoqnition Theorems of locally presentable categories, Brown's Representability Theorem for triangulated categories,...). Unfortunately many people outside of category theory still believe that these general theorems are not useful at all, although the last decades have brought so many applications ... –  Martin Brandenburg Dec 14 '11 at 13:04
Thanks for the examples - based on this question I was actually contemplating posting a question of the form "What are the main theorems in category theory?" I confess that I don't know the statements of any of those results, and I have heard of only two of them. Thus I can make no intelligent statement about whether or not they have applications to analysis, except that it wouldn't surprise me if they come up in noncommutative geometry (given the flavor of some papers in that area). –  Paul Siegel Dec 14 '11 at 16:01
Well, I decided to post that question after all: mathoverflow.net/questions/83437/…. I'm very curious now, and I hope you'll contribute! –  Paul Siegel Dec 14 '11 at 16:53
To add just a little bit to this answer I would say that the whole issue of Morita equivalence in C*-algebras (and, if you wish, in its geometrical counterpart i.e. Poisson geometry) certainly springs out from categorical ideas. –  Nicola Ciccoli Dec 15 '11 at 7:38
For a simpler example of a "main theorem in category theory" there's the Yoneda lemma. –  Noah Snyder Dec 15 '11 at 16:37

The chain rule for differentiation

$D(f \circ g ) = Df \circ Dg$

is the first example of functoriality one meets and counts as analyis I guess! Of course to properly interpret this it is best to think of f and g as maps between manifolds and Df and Dg as their tangent maps defined on the tangentbundles of the manifolds. The functor of differentiation is thus the functor of taking the tangent bundle (on objects) and tangent maps (on functions).

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Probably this is already known to many readers here, but I'll add it because we are in CW mode:

It is possible to construct and characterize $L^1[0,1]$ categorically, namely as the "smallest" pointed Banach space $X$ satisfying the fixed point $X \cong X \times X$. See this note by Tom Leinster and this MO discussion. The integral $\int_{0}^{1} : L^1[0,1] \to \mathbb{R}$ comes from the universal property.

Although I have do admit that this does not compute $\int_{0}^{1} \sin(\pi x) ~ dx$ or alike, I think this is a quite enlightening and strinkingly simple characterization of such a complicated (well at least it is not just a toy example) Banach space. All you have to know is that you can identify a function with two functions, the rest follows. So this example goes in the direction of Paul Garret's answer. I don't know if it simplifies computations or even enables us to do new ones, but it follows one of the goals of category theory: Unification.

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Nice example, but: "complicated" - really? I would say that Tom's result (spinning off from Freyd's) is a lovely characterization of a natural Banach space. Moreover, to me its force is that one gets the integral for free. –  Yemon Choi Dec 14 '11 at 19:15

The substantial book

Kriegl, A. and Michor, P.W., The convenient setting of global analysis, Mathematical Surveys and Monographs, Volume 53. American Mathematical Society, Providence, RI (1997).

does use categorical notions and methods; a pdf is available.

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First, a disclaimer: I am not even close to being an analyst. Second, I don't know of any applications of category theory to the areas of analysis that you mention. I don't think we have got to that point yet, for the reason given below. But here is an answer to a more general question that I hope you'll find illuminating.

I think the thing to remember here is that category theory is 'structural mathematics'. That is, it seeks to understand mathematical objects and constructions purely in terms of abstract external structure, as opposed to internal details about how an object is put together. In areas like algebra and computer science, this sort of structure is already there and visible, so for example it's easy to define the notion of 'group object' or 'monoid action', and to discuss constructions like quotients and semidirect products and so on in purely structural terms.

My impression of analysis is that the structures involved are less unequivocal and less clearly visible, and consequently it's harder to give a broad unified structural picture of the kind that we're used to in algebra. So the category-theoretic or structural understanding of analysis is a good deal less well-developed. But there are some interesting facts about certain structures found in (very elementary) analysis and topology:

• Metric spaces are a particular kind of enriched category, and the notion of Cauchy completeness has a very neat definition in that context.
• Compact Hausdorff spaces are the algebras for a monad on Set (the ultrafilter monad).
• Topological spaces are lax algebras for the same monad on the bicategory Rel of relations.
• Normed vector spaces can be viewed as enriched categories with duals (Lawvere), or as compact-closed ordinary categories equipped with a certain kind of functor (see Geoff Cruttwell's thesis).
• C-star algebras are monadic over Set (although I gather not much more is known about this).
• Non-standard analysis has a nice interpretation in terms of the filter-quotient construction on toposes.
• There is a notion of Fourier or z-transform for Joyal's 'structure types'.

There are probably many more (and this answer is CW so passers-by are invited to add them).

Personally, I think that the structural and material viewpoints on mathematics complement each other very well, so I'd be delighted if someone could point out (or write!) a structural account of -- a sort of 'Mac Lane and Birkhoff' for -- even elementary analysis.

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Where can I read about C^* algebras being monadic? –  David Carchedi Dec 13 '11 at 21:38
@David: there is a paper of Pelletier and Rosicky which at least mentions this result, see my answer mathoverflow.net/questions/8550/… –  Yemon Choi Dec 13 '11 at 22:02
@Yemon: yes, that's the one I was thinking of. –  Finn Lawler Dec 13 '11 at 22:50
Finn, I honestly don't know how you are using the word 'material' here, but I'm pretty most sure analysts don't care how the real numbers are defined, as long as you get a complete ordered field. That's structural, not material, in the sense I am familiar with (set theoretic constructions). Do you have a good example of what you mean by "material" in analysis? –  Todd Trimble Dec 14 '11 at 2:55
@Todd: I didn't mean to suggest that analysis takes a material approach where, say, algebra takes a structural approach -- all I meant by the last paragraph was that, as Paul Garrett points out in his answer, texts on analysis typically eschew the structural point of view, and that it would be nice if there were more structurally-oriented accounts of analysis to complement the traditional ones. –  Finn Lawler Dec 14 '11 at 19:27

The following book treats parts of Banach space theory and harmonic analysis from the point of view of category theory: Johann Cigler, Viktor Losert, Peter W. Michor: Banach modules and functors on categories of Banach spaces. Lecture Notes in Pure and Applied Mathematics 46, Marcel Dekker Inc., New York, Basel, (1979), MR 80j:46112, Zbl 411.46044. Review in Bull. AMS 3,2 (1980) xv+282 pp., Scanned pdf here.

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I had been meaning to invest in a copy one day, so the generosity of making it available is appreciated by this semi-categorical analyst at least! –  Yemon Choi Oct 2 '12 at 9:39

At the suggestion of Yemon, I have moved my comment to an answer. The Gelfand representation gives an equivalence between the category of commutative, unital $C^*$-algebras and the opposite category of compact Hausdorff spaces.

Breifly, let $A$ be a $C^\*$-algebra, and let $\Sigma$ be the collection of nonzero homomorphisms $A \rightarrow \mathbb{C}$. Then $\Sigma$ sits inside $A^\*$, the dual of $A$. Thus we can endow it with the weak-$\*$ topology. With this topology, if we consider $C(\Sigma)$, the algebra of continuous functions $\Sigma \rightarrow \mathbb{C}$, it turns out that we obtain a canonical isometric $\*$-isomorphism $A \rightarrow C(\Sigma)$.

The functor $CommC^\*Alg \rightarrow CptHdTop^{op}$ given by $A \mapsto \Sigma$ defined above is an equivalence of categories. A great reference for the details is Conway's book on functional analysis (but he doesn't mention categories or functors).

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As far as I understand, this is one of the jumping off points in certain non-commutative geometry/pseudo-geometry programs. When studying non-commutative C*-algebras, the thought is that they should correspond to some non-commutative space analogous to the commutative case. –  Eric A. Bunch Dec 15 '11 at 2:13
Without reference to the morphisms, of course, this theorem is in many if not most books on functional analysis. (The force of the categorical viewpoint is the relationship between various $C^\ast$-algebras, not just the fact that the Gelfand transform maps onto C(max ideal space))) –  Yemon Choi Dec 15 '11 at 4:21
+1 because you mentioned it as an application to analysis. Knowing that a commutative C-algebra is the same as functions on a compact space is helpul, e.g. in spectral theory. The other direction is much less useful (knowing that a compact space is the same as the spectrum of a C-algebra does not tell you anything. –  Johannes Ebert Dec 15 '11 at 12:21
One can prove the universal property of the Stone-Čech compactification by provings its associated C^*-algebra has the appropriate property. To do this one should really look at bounded continuous functions and the spectrum as giving an adjunction which restricts to an equivalence on the usual suspects. –  Benjamin Steinberg Dec 15 '11 at 20:19
@Yemon: Definitely, reference to the morphisms is necessary to fully realize the categorical viewpoint. My category theorist friend would be horrified to see me make no mention of what the functor does on morphisms! I hope he can forgive me if he ever reads this. –  Eric A. Bunch Dec 16 '11 at 0:16

Let me expand on the second example of Lian. Category theory proves trivially that the category of profinite abelian groups is dual to the category of discrete torsion abelian groups. Indeed the former is the pro-completion of the category of finite abelian groups and the latter is the ind-completion. So one just needs the trivial fact that the category of finite abelian groups is self-dual.

Similarly, assuming the Peter-Weyl theorem (you can't get around this) one has that the category of compact abelan groups is the pro-completion of the category of compact abelian Lie groups. The category of discrete abelian groups is the ind-completion of the category of finitely generated abelian groups. So the duality between compact abelian groups and discrete abelian groups boils down to the structure theorem for finitely generated abelian groups, the finite case and that Z is dual to the circle.

So category theory organizes these proofs (the proof still boils down to the same nuts and bolts).

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I would suggest that the following three applications of category theory to functional analysis can be useful (they have points of contact with some of the earlier answers): they concern three topics (which are related) and at the least provide a unifying thread---in my opinion, they do more---to many themes in abstract analysis (spaces of measures, distributions, analytic functionals, the Riesz representation theorem and Gelfand-Naimark duality). These are: extensions of categories, free topological vector spaces and extended duality.

Extensions of categories. The basic example is the extension of the category of Banach spaces (with continuous linear mapings as morphisms) to the class of locally convex spaces (Wiener) and convex bornological spaces (Buchwalter and Hogbe-Nlend). The first has, of course, long occupied a place in the mainstream of functional analysis, the latter less so. They can be regarded as the categories obtained by "adding inductive, respectively, projective limits. This informal notion has been formalised in the first edition of the book "Saks spaces and applications to functional analysis". Exactly the same process can be applied to the category of metric spaces (where we obtain that of uniform spaces), that of Banach algebras (convex bornological algebras and locally multiplicatively convex algebras). The examples can be multiplied indefinitely. We mention two further extensions which we shall refer to below---compactological spaces (Buchwalter) (inductive limits of compact spaces) and Saks spaces (adding projective limits to the category of Banach spaces with linear contractions as morphisms---paradoxically, despite the fact that this category {\it has} limits). These example shows that this process often leads to us rediscovering America. However, it does have the advantage over Columbus that we are rediscovering a plethora of continents with a single expedition, often ones which are known but have hardly been investigated so that a rescrutinising may be worthwhile (after all, Columbus himself had been anticipated by native Americans).

Free locally convex spaces. The basic example here is the following. If we start with the unit interval $I$, then we can consider the free vector space over this set. If we then supply it with the finest locally convex topology which agrees with the original one on $I$ and complete it, we obtain a locally convex space which has the universal property that every continuous function from $I$ into a Banach spaces lifts to a unique continuous linear operator thereon (and is characterised by this property). Not surprisingly, this is just the space of Radon measures on $I$. Our point is that this construction can be carried out in an infinity of analogous situations and leads to a unified aproach to a large array of spaces which are of interest in analysis. We mention a small sample---compact spaces (universal property for continuous functions), compactologies and completely regular spaces (bounded, continuous functions), metric spaces (bounded Lipschitz-continuous functions), uniform spaces (bounded uniformly continuous functions), compact rectangles in $n$-space ($C^\infty$-functions), compact manifolds (again $C^\infty$-functions), open subsets of $n$-dimensional complex space (holomorphic functions). These lead to a long list of interesting spaces (of bounded Radon measures, of uniform measures, of distributions, of analytic functionals) and many of their basic properties can be deduced from general principles which arise from this method of construction (most important example, duality theorems). Again, many of these spaces are known but the historical path to their discovery was long and stony. It is of advantage to have a natural unified approach to their construction. Again, there aare further important cases which are indeed known but seem to have passed out of the mainstream despite an obvious demand for them. A particularly important and, in my opinion, unfortunate example (and a perennial favourite for queries in this forum) is the topic of extensions of the Riesz representation, which we shall now discuss.

Extensions of duality. As we have just mentioned, the classical example is the Riesz representation theorem for compacta. It was initially shown by Buck (for locally compact space and later, by other researchers, to the class of completely regular spaces) that this can be extended to the non-compact case by using the so-called strict topology which can be most succinctly described here as the finest locally convex topology on the space $C^b(S)$ of bounded, continuous fuctions on $S$ which agrees with that of compact convergence on the unit ball for the supremum norm. Again a number of queries on this forum suggest that this topic which barely entered into the mainstream despite the prominence of its proponents (Buck, Beurling and Herz---mainly motivated by questions in harmonic analysis) and, sadly, seems to have vanished without a trace. There is a simple and natural scheme at work here. If we have a duality between two of the central catogories of analysis, then we can extend it in an obvious way to one between say the category obtiained by adding inductive limits to the first one and projective limits to the second one. This leads almost automatically to the above extension of the Riesz representation to the case of completely regular spaces (or, better, compactological spaces). If we take as our starting point one of the central dualities of abstract analysis (that between a Banach space and its dual as a Banach space, or, if one wants a symmetric duality, as a Waelbroeck space, between a compact space and the Banach space of continuous functions thereon, resp. the same space regarded as a $C^*$-algebra, between a metric space and the space of bounded, Lipschitz functioons, then we can apply this process to obtain a large classes of extended dualities which are useful in abstract analysis.

Why is this useful? Lack of space prevents an elaborate justification of these three methods but I would like to mention the following version of Occam's rasor. One can speculate that one of the reasons for the fact that many of these extended dualities failed to enter into the mainstream of abstract analysis lies in the fact that to analysists accustomed to the Banach space settings, the structures employed here seemed unattractively elaborate and artifical, not to say baroque. (The strict topology is not only not normable, but is not a member of the accepted "nice" classes of locally convex spaces---Frechet, $DF$-, even barrelled or bornological. Of course, any such extension must necessarily be more elaborate than the original duality and the above considerations show that the ones presented here are the simplest that can succeed in the given situations and so are inevitable.

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Good stuff - much food for thought. Do you know if I could find further details in, say, the North-Holland book of Cooper? ;-) –  Yemon Choi Nov 18 '12 at 19:09
very much so. as noted in the text, the stuff on category theory is only in the first edition. –  jbc Nov 19 '12 at 8:56

The theory of interpolation spaces is one such example. The classical interpolation theorems of M. Riesz, Thorin and Marcinkiewicz and their generalizations by A. Zygmund, E. M. Stein, G. Weiss, and C. Fefferman are basic tools in harmonic analysis. The Riesz-Thorin theorem yields a quick proof of the Hausdorff-Young inequality on the $L^p \to L^{p'}$ boundedness of the Fourier transform, as well as the classic proof of the $L^p$-boundedness of the Hilbert transform by M. Riesz. The Marcinkiewicz interpolation theorem is effectively a generalization of the proof of the Hardy-Littlewood maximal inequality and is a crucial tool in proving the $L^p$-boundedness of a wide range of singular integrals---say, those of convolution type or the Calder\'{o}n-Zygmund operators.

The interpolation theorem of Riesz-Thorin and Marcinkiewicz have been generalized substantially by A. Calder\'{o}n and J. Peetre, respectively, and modern treatments of the subject---Bergh-L\"{o}fstr\"{o}m, Bennett-Sharpley, etc.---often utilize category theory to make the notion of interpolation precise. To this end, we need the notion of a Banach couple, which is a pair $(A_0,A_1)$ of Banach spaces such that there exists a Hausdorff topological vector space $X$ such that we have continuous linear embeddings $A_0 \hookrightarrow X$ and $A_1 \hookrightarrow X$. The collection of Banach couples forms a category, whose typical morphism $T:(A_0,A_1) \to (B_0,B_1)$ is a bounded linear operator $T:A_0 + A_1 \to B_0 + B_1$ such that the restrictions $T|_{A_0}$ and $T|_{A_1}$ are bounded linear operators mapping into $B_0$ and $B_1$, respectively. An interpolation space for a Banach couple $(A_0,A_1)$ is a Banach space $A$ such that

1. $A_0 \cap A_1 \hookrightarrow A \hookrightarrow A_0 + A_1$;
2. if $T:A_0 + A_1 \to A_0 + A_1$ is a bounded linear operator such that $T|_{A_0}:A_0 \to A_1$ and $T|_{A_1}: A_1 \to A_1$ are bounded linear operators, then the restriction of $T$ on $A$ is a bounded linear operator into $A$;

and an interpolation functor a functor from the category of Banach couples to the category of Banach spaces that sends Banach couples to corresponding interpolation spaces and transforms morphisms between Banach couples into the corresponding bounded linear operators between sums of Banach spaces. With this terminology, we can view an interpolation theorem as a construction of such a functor.

Since this definition is quite general, it is conceivable to have interpolation theorems that are not quite as nice as the classical ones. To characterize the nicer ones, we define exact interpolation spaces $A$ and $B$ to be interpolation spaces of Banach couples $(A_0,A_1)$ and $(B_0,B_1)$ such that

$$\|T\|_{A \to B} \leq \max \left( \|T\|_{A_0 \to B_0} , \|T\|_{A_1 \to B_1} \right)$$

for all morphisms $T: (A_0,A_1) \to (B_0,B_1)$ and an exact interpolation functor an interpolation functor that produces exact interpolation spaces. Now, the theorem of Aronszajin and Gagliardo states that every interpolation space admits an exact interpolation functor that sends the corresponding Banach couple to the interpolation space. With the aid of category-theoretic methods, we thus have a guarantee that our efforts in finding a particular interpolation theorem will not go wasted.

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Nice example. On a related note, there was some work by Kaijser and Wick-Pelletier (like B-S-D, only two actual people there) on formulating interpolation couples in an even more overtly category-theoretic way. Unfortunately I've forgotten the details and don't have references to hand, but I've always thought it deserved to be better known. –  Yemon Choi Dec 22 '11 at 8:13

Of course the equivalence of categories mentioned by Eric A. Bunch is true only for commutative $C^\ast$-algebras. There however is a quite similar result for a wider category of non-commutative $C^\ast$-algebras: a continuous-trace $C^\ast$-algebra $A$ with Hausdorff spectrum $X$ is isomorphic to the $C^\ast$-algebra $\Gamma_0(X,\mathcal{A})$ of continuous sections vanishing at $\infty$ of some continuous $C^\ast$-bundle $\mathcal{A}\to X$; the latter is actually a Dixmier-Douady bundle, in the sense that it has typical fiber the algebra $\mathbb{K}(H)$ of compact operators on some separable Hilbert space. This construction yields an equivalence between the category of continuous-trace $C^\ast$-algebras with Hausdorff spectrum and pairs $(X,\mathcal{A})$.

When it comes to study $K$-theory of such $C^\ast$-algebras, the above equivalence may be very useful, for if $A$ corresponds to the pair $(X,\mathcal{A})$, then $K_\ast(A)$ is isomorphic to the twisted $K$-theory ${}^{\mathcal{A}}K^\ast(X)$, which in the finite-dimensional case may be interpreted in terms of geometric objects (twisted vector bundles). For instance, let $G$ be a compact Hausdorff group. Then the dual space of $G$ is homeomorphic to the the spectrum $X$ of the $C^\ast$-algebra $C^\ast(G)$ (which is actually continuous-trace). Now, for $k\in \mathbb{N}$, let $X_k$ be the (open) subspace of $X$ consisting of (equivalence classes of) irreducible representations of $G$ of rank $k$. Then, for each $k$, there is an Azumaya bundle $\mathcal{A}_k\to X_k$ (i.e. a Dixmier-Douady bundle of finite dimension), and there is an isomorphism $K_\ast(C^\ast(G))\cong \bigoplus_k {}^{\mathcal{A}_k} K^\ast(X_k)$ (cf. http://front.math.ucdavis.edu/0201.5207 for a generalization of this example).

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Perhaps this example is too naive, but one can view the Riesz Representation Theorem categorically as saying that integration with respect to a measure is a natural equivalence of the functor which takes a compact Hausdorff space $X$ and produces the Banach space of finite signed measures on $X$, and the functor which takes and $X$ and produces the dual of $C(X)$. There is a lovely article on this by Hartig: http://www.jstor.org/pss/2975760

Another example that comes to mind is Pontryagin Duality as presented here: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102911979

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Isn't the Banach-Mazur metric space on norms over ${\mathbb R}^n$ such an application?

I think also to the Atiyah-Singer index Theorem.

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Here are four articles on category theory and analysis (see section IV of that collection). I liked this one: R. Boerger, Fubini's Theorem from a Categorical Viewpoint, manages to conclude Fubini's theorem from the fact that left adjoints preserve coproducts...

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I've only had a superficial look at the paper, but it seems to me that what people usually call Fubini's theorem is an analytical fact that under certain boundedness conditions one can exchange the order of two integrals. Is the paper cited just saying, at the end of the day, that one can exchange the order of two finite sums? Near the bottom of the first page the author says things "since we are not worried about convergence" (as he's in some finite world) -- but surely what people usually call Fubini's theorem is precisely an issue of convergence? –  Kevin Buzzard Dec 13 '11 at 21:25
@Kevin: shush, everyone knows that these kind of technicalities are of no concern to Proper Mathematicians... ;) –  Yemon Choi Dec 13 '11 at 22:04
Further to my previous comment: "unlike Modo, Sergeant Colon did understand the meaning of the word 'irony'. He thought it meant 'something like iron'." –  Yemon Choi Dec 14 '11 at 0:25
"Is the paper cited just saying, at the end of the day, that one can exchange the order of two finite sums?" I don't think, the conclusion is just that. If you look at 6.2, they say the integral of the product measure is the double integral over each measure separately (for simple functions). So my take on the category theory, is that it is concluding the product measure acts functorially, as expected by the suspected diagram chase?! So I guess $\int_{A\times B}f(x,y)(\mu_A\times\mu_B)(x,y)=\int_A\int_B f(x,y)\mu_A(x)\mu_B(y)$ is the Fubini theorem. I agree there is no analysis. –  Junkie Dec 14 '11 at 5:10
A "simple function" is a very restrictive class of function -- for example, it has finite image. This is why I'm wondering whether we're really doing sums, not integrals. –  Kevin Buzzard Dec 14 '11 at 7:44