How to motivate a student of functional analysis
This semester I am teaching rudimentary category theory at graduate level. It is somewhat scary that I should be doing this, but other faculty members do not seem to do much general category theory. I have only few students (and they are very bright) but their areas of research are quite diverse: discrete math/computer science, algebra, algebraic topology, and functional analysis. I can plenty motivate categories for discrete math and computer science, with things like "initial algebras are inductive datatypes, final coalgebras are coinductive (lazy) datatypes". I also know enough general algebra to motivate algebraists with tquestions like "What is an additive category with a single object?". And we will study algebraic theories as well. Algebraic topologists are self-motivated. Nevertheless, we'll do some sheaves towards the end of the course. But how do I show the fun in categories to a student of functional analysis? I would like to give him a class project that he will find close to his interests. The course is covering (roughly) the following material: basic category theory (limits, colimits, adjoints, we mentioned additive and enriched categories), Lawvere's algebraic categories, monads (up to stating Beck's theorem and working out some examples), basics of presheaves and sheaves with a slant toward topology. There must be some functional analysis in there. I would very much appreciate some suggestions. Best regards, Andrej
A student interested in functional analysis presumably knows some about topological vector spaces in general and Mackey spaces in particular. He might be interested in knowing that the full subcategory of Mackey spaces has a *-autonomous structure. This means that if M and N are Mackey there is a topology on the vector space of continuous linear maps M --> N that makes it into a Mackey space, often denoted M -o N, and that if you let M* = M -o C, then the canonical map M --> M** is an isomorphism. There is also a tensor product @ and the usual isomorphism Hom(M@N,P) = Hom(M,N-oP). See M. Barr, On $*$-autonomous categories of topological vector spaces. \cahiers {41} (2000), 243--254.
Dear Adrei, May I suggest you to look at the monograph "Lectures and Exercises on Functional Analysis" by A. Ya. Helemskii published by AMS in 2006, where, I'm sure, you'll find a lot of good motivations for students interested in functional analysis to study category theory. Good Luck and best regards, Yefim _______________________________________________________________________ Prof. Yefim Katsov Department of Mathematics & CS Hanover College Hanover, IN 47243-0890, USA telephones: office (812) 866-6119; home (812) 866-4312; fax (812) 866-7229
Functional Analysis was one of the key origins of categorical concepts and outlook, for example that the functionals themselves should be collected into a single object (Voltera-Hadamard) leads to the Hom functor,etc. This was also one of the roads followed by students in the 1950s, for example from J. L. Kelley's "galactic" treatment of M. H. Stone's functor C. However in North America (as distinct from Europe) more recent functional analysists have accepted categorical methods only grudgingly, and hence piecemeal. On the other hand, students who are not specializing in analysis are often woefully ignorant of the basics of functional analysis that are part of what every mathematician should know. To combat that ignorance in my Advanced Graduate Algebra course I often devoted several weeks to topics from functional analysis. It is a source of examples both interesting and essential. To begin to try to answer Andrej's question, I rapidly recall some examples, and hope others will also comment: The double dual functor on Banch spaces is a protype example of a composite of adjoints becoming a monad. The EM algebras for this monoid were computed by Fred Linton, in an exercise that should be better known. It also illustrates the "descent" principle that C. Houzel cited couple of months ago (what I called semantics of structure of a given functor in my thesis) : Objects constructed by a given functor tend to have, by virtue of that, more structure than originally contemplated in its codomain , hence a lifted version of the functor comes closer to being invertible. As Peter Johnstone just recalled, if we consider commutative monoids with zero and hom them into the particular object of reals, the resulting set is "actually" a compact space whose C-algebra reveals by adjointness that the opposite of the spaces form a full subcategory of the monoids with zero. Again a good exercise, related to Kelley's "square root lemma". Students might wonder why contiuous linear operators are traditionally called "bounded" (when they are not even). For many linear spaces (roughly those where sequentiality suffices) , preserving sequential limits is equivalent to preserving boundedness of sequences (for a linear map). George Mackey started to functorize this crucial observation before categories were fully explicit. Now we can consider the category of all presheaves on the category of all countable sets, define an "underlying" functor from Banach spaces to it, and verify that it actually lands in the subtopos of sheaves for the finite-disjoint-covering topology. Indeed it not only gives abelian group objects in the latter topos, but modules over R, the Dedekind reals of the topos, and a FULL subcategory of those. The above construction has an analogue using instead Johnstone's coherent topos of sheaves on countable compact spaces. ETC Bill On Tue Mar 4 10:20 , Andrej Bauer sent:
This semester I am teaching rudimentary category theory at graduate level. It is somewhat scary that I should be doing this, but other faculty members do not seem to do much general category theory.
...
Dear Andrej, Conventional functional analysis is largely concerned with Banach spaces, and there's certainly alot that can be said about the category of Banach spaces and linear contractions (i.e., continuous linear transformations with norm less-than-or-equal-to 1). For example, it is symmetric monoidal closed (with internal hom, the space of _all_ continuous linear transformations!) and locally countably presentable. In fact, it is a countable-ary quasi-variety, and the full subcategory of its finite-dimensional objects is a good example of a *-autonomous category that is not compact closed. [Although it is not hard to prove the latter directly, it can also be seen as an interesting application of Robin Houston's theorem that products and coproducts can not differ in a compact closed category.] In fact, I think Ban is a fine example which can teach any student of category theory a number of salutary lessons: 1. in category theory, the meaning of isomorphism is fixed---so if you have a pre-existing class of isomorphisms in mind (in this case, the isometric (norm-preserving) isomorphisms), then you must take care in choosing an appropriate class of morphisms; 2a. there's more to defining internal homs than just slapping an extra structure on the external homs; 2b. forgetful functors don't have to be "the obvious thing"; 3. you can't always have your cake and eat it too!---the whole category can not hope to be self-dual, precisely because it is locally presentable (and not a poset). Of course there are also (unital) C*-algebras, and I can make an interesting point about them too---sometimes one needs to consider maps between C*-algebras which are not *-homomorphisms: for example, there are "completely positive maps" and "completely bounded maps". Now, as important as the b.o./f.f. factorisation may be in general, it seems fishy to speak of a category whose objects are C*-algebras but whose morphisms preserve only part of the C*-algebraic structure; and so it was that analysts were led to develop the notions of "operator space" and "operator system" which provide the correct level of structure to define c.b. maps and c.p. maps, respectively. In fact, these are quite interesting categories in their own right: operator spaces are said to model "non-commutative functional analysis" ---but I only have a tenuous grasp of what that is supposed to mean! I meant to discuss quantale theory and Banach sheaves too, but I've run out of time---perhaps someone else will pick up the thread. Cheers, Jeff. --- Andrej Bauer <Andrej.Bauer@fmf.uni-lj.si> wrote:
This semester I am teaching rudimentary category theory at graduate level. It is somewhat scary that I should be doing this, but other faculty members do not seem to do much general category theory.
I have only few students (and they are very bright) but their areas of research are quite diverse: discrete math/computer science, algebra, algebraic topology, and functional analysis.
I can plenty motivate categories for discrete math and computer science, with things like "initial algebras are inductive datatypes, final coalgebras are coinductive (lazy) datatypes".
I also know enough general algebra to motivate algebraists with tquestions like "What is an additive category with a single object?". And we will study algebraic theories as well.
Algebraic topologists are self-motivated. Nevertheless, we'll do some sheaves towards the end of the course.
But how do I show the fun in categories to a student of functional analysis? I would like to give him a class project that he will find close to his interests. The course is covering (roughly) the following material: basic category theory (limits, colimits, adjoints, we mentioned additive and enriched categories), Lawvere's algebraic categories, monads (up to stating Beck's theorem and working out some examples), basics of presheaves and sheaves with a slant toward topology. There must be some functional analysis in there.
I would very much appreciate some suggestions.
Best regards,
Andrej
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Jeff did not mention the excellent very categorical lectures given on operator spaces by Matthias Neufang at the FIELDS INSTITUTE Summer School in Operator Algebras held last summer at the University of Ottawa and that we both attended. I do not know if any version of Matthias' notes is available. The theme of tensor products was important. Not only did his lectures provide good motivation for studying the subject from a categorical viewpoint. He did not do the category theory of operator spaces but rather was explicitly conscious of the categorical content of what he was saying. His notes may be of some interest to others so let us hope he will put some of the material on the web. Tim Quoting Jeff Egger <jeffegger@yahoo.ca>:
Dear Andrej,
Conventional functional analysis is largely concerned with Banach spaces, and there's certainly alot that can be said about the category of Banach spaces and linear contractions (i.e., continuous linear transformations with norm less-than-or-equal-to 1).
Following Jeff Egger, who wrote, in part, "Ban is a fine example which can teach any student of category theory a number of salutary lessons," but asking forgiveness for tooting my own horn, I'd like to point out another one of those lessons -- my old characterization of Banach conjugate spaces as the algebras over the double-dualization monad on {Ban}. Neat mix of Beck Theorem, functional analysis, and more, on pp. 227-240 of: Proc. Conf. Integration, Topology, and Geometry in Linear Spaces, in: Contemporary Mathematics, Volume 2, AMS, Providence, 1980. Might even serve as one student's "individual reading report" project. There's also my even older squib on "Functorial Measure Theory," in pp. 36-49 of: Proc. Conf. Functional Analysis, UC Irvine, 3/28-4/1, 1966, Thompson Book Co., Wash., DC, & Academic Press, London, 1967. This one breathes life into the slogan, "Measures are adjoint to functions." Cheers, -- Fred
Most of the material connecting analysis and category theory seems to be written by specialists in category theory who have observed some of the ways that insights from category theory can be brought to bear (for example look on Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But here's an example which is a book on functional analysis that has a strong use of categories: @book {MR0296671, AUTHOR = {Semadeni, Zbigniew}, TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, NOTE = {Monografie Matematyczne, Tom 55}, PUBLISHER = {PWN---Polish Scientific Publishers}, ADDRESS = {Warsaw}, YEAR = {1971}, PAGES = {584 pp. (errata insert)}, MRCLASS = {46E15 (46M99)}, MRNUMBER = {MR0296671 (45 \#5730)}, MRREVIEWER = {H. E. Lacey}, } -- Bob -- Robert L. Knighten RLK@knighten.org
Every student who learned the basics of operator algebras knows the Gelfand-Naimark representation theorem, usually stated non- categorically as "every commutative unital C*-algebra is isomorphic to the algebra of continuous functions on a compact Hausdorff space". Asking such students to check that this is part of a dual equivalence of categories is probably a good idea, and based on this one can do exercises about particular algebras they know - for instance to compute presentations by generators and relations of C(T^n), the algebra of continuous functions on the n-dimensional torus, which by the duality is instantly reduced to finding a presentation of C(S^1), etc. Also, some students might be willing to work out how the existence of presentations of C*-algebras by generators and relations relates to the fact that the category of C*-algebras is algebraic over Sets. On Mar 4, 2008, at 3:20 PM, Andrej Bauer wrote:
This semester I am teaching rudimentary category theory at graduate level. It is somewhat scary that I should be doing this, but other faculty members do not seem to do much general category theory.
I have only few students (and they are very bright) but their areas of research are quite diverse: discrete math/computer science, algebra, algebraic topology, and functional analysis.
I can plenty motivate categories for discrete math and computer science, with things like "initial algebras are inductive datatypes, final coalgebras are coinductive (lazy) datatypes".
I also know enough general algebra to motivate algebraists with tquestions like "What is an additive category with a single object?". And we will study algebraic theories as well.
Algebraic topologists are self-motivated. Nevertheless, we'll do some sheaves towards the end of the course.
But how do I show the fun in categories to a student of functional analysis? I would like to give him a class project that he will find close to his interests. The course is covering (roughly) the following material: basic category theory (limits, colimits, adjoints, we mentioned additive and enriched categories), Lawvere's algebraic categories, monads (up to stating Beck's theorem and working out some examples), basics of presheaves and sheaves with a slant toward topology. There must be some functional analysis in there.
I would very much appreciate some suggestions.
Best regards,
Andrej
You could also look at MR1471480 (98i:58015) Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis. (English summary) Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3 for which an e-version has been downloadable. However as the review says: "the exposition is based on functional analysis rather than on category theory; this fact will, undoubtedly, allow the subject to reach a wider audience. " Ronnie ----- Original Message ----- From: "Robert L Knighten" <RLK@knighten.org> To: "Categories list" <categories@mta.ca> Sent: Thursday, March 06, 2008 2:37 AM Subject: categories: How to motivate a student of functional analysis
Most of the material connecting analysis and category theory seems to be written by specialists in category theory who have observed some of the ways that insights from category theory can be brought to bear (for example look on Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But here's an example which is a book on functional analysis that has a strong use of categories:
@book {MR0296671, AUTHOR = {Semadeni, Zbigniew}, TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, NOTE = {Monografie Matematyczne, Tom 55}, PUBLISHER = {PWN---Polish Scientific Publishers}, ADDRESS = {Warsaw}, YEAR = {1971}, PAGES = {584 pp. (errata insert)}, MRCLASS = {46E15 (46M99)}, MRNUMBER = {MR0296671 (45 \#5730)}, MRREVIEWER = {H. E. Lacey}, }
-- Bob
-- Robert L. Knighten RLK@knighten.org
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Hi Tim,
Jeff did not mention the excellent very categorical lectures given on operator spaces by Matthias Neufang at the FIELDS INSTITUTE Summer School in Operator Algebras held last summer at the University of Ottawa and that we both attended.
There are, of course, many people I could have credited and cited in my previous posting, but only the newest readers of this list will be unaware of the perils with which such an attempt is fraught. [For instance, I first read of the local presentability of Ban in Adamek and Rosicky's book, but I would not like to hazard a guess as to the origin of this result.] But you are right: I should have made an exception in Matthias' case. I should also credit Vladimir Pestov, a topologist who knows enough category theory to wonder whether operator spaces might be internal Banach spaces in some Grothendieck topos (but perhaps not enough to realise that this might take a student more than one term to prove), for having introduced me to Matthias several years ago. While at Dalhousie, I gave a talk on Pestov's conjecture; but when I started writing up my notes, I was distracted by an unrelated observation about the category of operator spaces which ultimately led to my ill-fated C*-algebra paper. I still haven't gotten back to the original project.
I do not know if any version of Matthias' notes is available.
Nor do I, but I am sure he would rather point people towards the pre-Wikipedia-era "online dictionary" of operator space theory to which he contributed: [German] http://www.math.uni-sb.de/ag/wittstock/projekt99.html [English] http://www.math.uni-sb.de/ag/wittstock/projekt2001.html These notes are quite good in the sense that, to use Tim's words, they are
explicitly conscious of the categorical content
In particular, it is quite gratifying to see a theorem such as "the forgetful functor from operator spaces to Banach space admits both a left and a right adjoint" stated (more or less) ungrudgingly. Cheers, Jeff. P.S. I should say that I don't think that operator spaces would be a suitable topic for an introductory CT course to a general audience; they are rather intricate. But it might be possible to craft an interesting set of exercises for the functional analysis contingent of such a course around operator space theory.
Ronnie points out the very excellent 1997 book on smooth analysis by Kriegl & Michor. In fact, not the reviewer but the authors themselves originally stated the principle of functional analysis "rather than" category theory. It is rather strange since much of the material in the book was arrived at by very categorical means. For example, results published in Kriegl's joint work with Alfred Frolicher are basic. My dismay is reflected in my RCMP paper on Volterra, where I praise the book for its powerful combination of functional analysis "and" category theory. In a related expositional choice the book claims to be about topological vector spaces, but the definition of morphism used betrays the fact that the weaker structures of bounded sequences and of C-infinity paths are the actual underpinning. It would be instructive to know whether this strategy actually widened the audience in the past 10 years. Bill On Thu Mar 6 10:15 , "Ronnie" sent:
You could also look at MR1471480 (98i:58015) Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis. (English summary) Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3
for which an e-version has been downloadable. However as the review says: "the exposition is based on functional analysis rather than on category theory; this fact will, undoubtedly, allow the subject to reach a wider audience. "
Ronnie
----- Original Message ----- From: "Robert L Knighten" RLK@knighten.org> To: "Categories list" categories@mta.ca> Sent: Thursday, March 06, 2008 2:37 AM Subject: categories: How to motivate a student of functional analysis
Most of the material connecting analysis and category theory seems to be written by specialists in category theory who have observed some of the ways that insights from category theory can be brought to bear (for example look on Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But here's an example which is a book on functional analysis that has a strong use of categories:
@book {MR0296671, AUTHOR = {Semadeni, Zbigniew}, TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, NOTE = {Monografie Matematyczne, Tom 55}, PUBLISHER = {PWN---Polish Scientific Publishers}, ADDRESS = {Warsaw}, YEAR = {1971}, PAGES = {584 pp. (errata insert)}, MRCLASS = {46E15 (46M99)}, MRNUMBER = {MR0296671 (45 \#5730)}, MRREVIEWER = {H. E. Lacey}, }
-- Bob
-- Robert L. Knighten RLK@knighten.org
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Dear Andrej, Two examples that have not been mentioned before: * The use of the Giry monad in stochastic processes. This should motivate CS students as well as (functional) analysists. You can also use co-algebras here. * Gelfand's theorem: commutative C*-algebras are precisely the complex numbers in the topos of sheaves over its spectrum. This will also teach them that the axiom of choice is almost never needed in functional analysis and that there are good reasons to avoid it: E.g. continuous fields of C*-algebras. This is the fundamental work by Banaschewski and Mulvey. Bas
Bill is quite right on what the author's say. I'd also be glad of any of Bill's comments on the `Historical remarks on the development of smooth calculus', pp, 79-83, which seem very carefully put. It might interest people to give what seems the origin of the word `convenient category'. In my 1963 paper `Ten topologies for X x Y' (a title frivolously influenced by `Seven brides for seven brothers') I wrote in the Introduction: `It may be that the category of Hausdorff k-spaces is adequate and convenient for all purposes of topology'. `Convenient' here meant cartesian closed. One of the above ten topologies gives monoidal closed on all Hausdorff spaces. Some later writers removed the Hausdorff restrictions (I tried, but I was at that time not too good on final topologies). The current acount in `Topology and Groupoids' was influenced by Eldon Dyer. But for analysis the Kriegl-Michor comments show how the emphasis moved from k-spaces to ideas from Frohlicher, Bill and others. Ronnie ----- Original Message ----- From: <wlawvere@buffalo.edu> To: "Categories list" <categories@mta.ca> Sent: Thursday, March 06, 2008 8:30 PM Subject: categories: Re: How to motivate a student of functional analysis Ronnie points out the very excellent 1997 book on smooth analysis by Kriegl & Michor. In fact, not the reviewer but the authors themselves originally stated the principle of functional analysis "rather than" category theory. It is rather strange since much of the material in the book was arrived at by very categorical means. For example, results published in Kriegl's joint work with Alfred Frolicher are basic. My dismay is reflected in my RCMP paper on Volterra, where I praise the book for its powerful combination of functional analysis "and" category theory. In a related expositional choice the book claims to be about topological vector spaces, but the definition of morphism used betrays the fact that the weaker structures of bounded sequences and of C-infinity paths are the actual underpinning. It would be instructive to know whether this strategy actually widened the audience in the past 10 years. Bill On Thu Mar 6 10:15 , "Ronnie" sent:
You could also look at MR1471480 (98i:58015) Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis. (English summary) Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3
for which an e-version has been downloadable. However as the review says: "the exposition is based on functional analysis rather than on category theory; this fact will, undoubtedly, allow the subject to reach a wider audience. "
Ronnie
----- Original Message ----- From: "Robert L Knighten" RLK@knighten.org> To: "Categories list" categories@mta.ca> Sent: Thursday, March 06, 2008 2:37 AM Subject: categories: How to motivate a student of functional analysis
Most of the material connecting analysis and category theory seems to be written by specialists in category theory who have observed some of the ways that insights from category theory can be brought to bear (for example look on Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But here's an example which is a book on functional analysis that has a strong use of categories:
@book {MR0296671, AUTHOR = {Semadeni, Zbigniew}, TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, NOTE = {Monografie Matematyczne, Tom 55}, PUBLISHER = {PWN---Polish Scientific Publishers}, ADDRESS = {Warsaw}, YEAR = {1971}, PAGES = {584 pp. (errata insert)}, MRCLASS = {46E15 (46M99)}, MRNUMBER = {MR0296671 (45 \#5730)}, MRREVIEWER = {H. E. Lacey}, }
-- Bob
-- Robert L. Knighten RLK@knighten.org
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Bill Lawvere wrote in part:
Students might wonder why contiuous linear operators are traditionally called "bounded" (when they are not even).
Then Jeff Egger wrote in part:
there's certainly alot that can be said about the category of Banach spaces and linear contractions
and:
forgetful functors don't have to be "the obvious thing";
Indeed, the "obvious" forgetful functor from Ban to Set (or Top or Met) takes a Banach space to its space of all points, while the "good" one takes the space to its unit ball. Anyway, if you mix these, then a linear transformation is bounded iff it is bounded as a function from the unit ball to the space of all points. Similarly for compact linear transformations (the image is compact). That may not be the origin of these terms, but it's how I understand them. More related to category theory itself: Jeff also wrote:
in category theory, the meaning of isomorphism is fixed
I'd say that the meaning of a term like "Banach space" necessarily includes the idea of what an isomorphism of such is. If different definitions define equivalent groupoids (or equivalent omega-groupoids in the most general case, as with different definitions of n-category, for example), then we can consider them equivalent defintions. So to define the essence of what Banach spaces are, one must specify (up to equivalence) the groupoid Ban_0 of Banach spaces and linear isometries between them. That said, there is some sense in the category Ban_b of Banach spaces and bounded linear transformations between them, but it is only a secondary notion compared to Ban_0. To be useful at all, it needs some extra structure, such as (at least) the dagger operator (giving duals of morphisms); then the actual isomorphisms of Banach spaces (those in Ban_0) are only the ~unitary~ (dual = inverse) isomorphisms in Ban_b. (In contrast, the category Ban as Jeff defined it needs no extra structure to be a sensible concept, since all of its isomorphisms are in Ban_0 already.) This dagger operator is used, for example, to make Hilb_b (the full subcategory of Ban_b whose objects are Hilbert spaces) into a 2-Hilbert space (from John Baez's HDA4), which is useful if you want examples of 2-Hilbert spaces; but the ~essence~ of what Hilbert spaces are is given by the groupoid Hilb_0 of linear isometries. So here is another lesson of category theory, to be taken together with Jeff's lesson last quoted above: Sometimes different notions of morphism are useful for different purposes. --Toby
participants (12)
-
Andrej Bauer -
Bas Spitters -
Fred E.J. Linton -
Jeff Egger -
Katsov, Yefim -
Michael Barr -
Pedro Resende -
Robert L Knighten -
Ronnie -
Tim Porter -
Toby Bartels -
wlawvere＠buffalo.edu