What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what? Michael
Dear Michael, The question is so impossibly big, and it was asked and answered in various form so many times, and no responsibility for the originality/completeness of the answer can be assumed... So, I am not afraid to begin with a few obvious remarks, looking forward to seeing many other remarks from others: 1. Many people believe that mathematics is about mathematical structures, but what is a mathematical structure in general? According to Bourbaki, one should begin with two finite sequences of sets, say, A, B, C,... and X, Y, Z,...; let us call them constant sets and variable sets respectively. Then build any scale, which is a finite sequence of sets obtained by taking finite products and power sets of the sets above. Then, (briefly and roughly!) call a structure (of a fixed type) an element of one of the sets in the scale satisfying certain conditions. For example: (a) a topological space (defined via open sets) has no constant sets, one variable set X, and its structure is an element t in PP(X) that is closed under finite intersections and arbitrary unions. (b) a vector space has one constant set A ("the set of scalars"), one variable set X ("the set of vectors"), and its structure can be defined as element (a,b,c,d) of P(AxAxA)xP(AxAxA)xP(AxXxX)P(XxXxX), where a, b, c, d are addition of scalars, multiplication of scalars, scalar (-on-vector) multiplication, and addition of vectors respectively; that (a,b,c,d) should satisfy familiar conditions of course. Then, according to Bourbaki again, structures are useless without morphisms - but what is a morphism? It turns out that only isomorphisms can be defined, and the class of morphisms should in each case be CHOSEN depending on the "experience" of working with a given class of structures in such a way, that it is closed under composition and contains all isomorphisms (or, better, also determine isomorphisms as invertible morphisms). Is it possible that the most fundamental concept of mathematics is described as such a monster? Is not it better to study abstract categories? And what is the problem of defining morphisms? To answer this question one should learn about functors, covariant and contravariant ones, and what do they preserve and what not. 2. Set theory is wonderful: it gives precise mathematical definitions to concepts that were only intuitively understood before. But... it often makes definitions complicated. And category theory very often solves this problem by using universal properties. For example the set N - better to say, the structure (N,0,s) - of natural numbers is "designed to count"; therefore N should have the first element 0 (unfortunately 0 is better than 1) and the successor function s from N to N - and be "the best such", i.e. initial such. Moreover, developing basic properties of this structure out of initiality is much easier than out of Peano axioms. In fact all classical number systems have simple elegant definitions via universal properties. Moreover, the axioms of set theory itself are much less elegant than their elementary-topos-theoretic counterparts. 3. We can say that set theory is more fundamental than arithmetics: e.g. children learn addition by counting the number of elements in the disjoint union. But category theory is more fundamental than set theory: e.g. it makes the disjoint union a more natural operation... but, more importantly (a) we all know that, say, a+b=b+a, ab=ba, PvQ<=>QvP,... for all natural numbers a and b and all logical formulas P and Q - but one needs category theory to see these as the same result (and we can add cartesian products, free product, intersection, union, and many other operations to it). (b) or, say, we all know that composites of injections are injections and composites of surjections are surjections - but again, one needs category theory to see these as the same result; (c) and even exponentiation and implication are the same... and categorical logic follows... 4. Linear algebra tells us that instead of working with linear transformations of finite-dimensional vector spaces we can work with matrices, but one cannot formulate this properly without using the concept of equivalence of categories (the category of finite-dimensional K-vector spaces is equivalent to the category of natural numbers with matrices with entries from K as morphisms). And there are so many other equivalences and dualities (that are not isomorphisms) playing fundamental roles in various branches of mathematics. Not to mention that the aforementioned matrices themselves arise from a categorical observation (finite products = finite coproducts). 5. Proper understanding of Eilenberg - Mac Lane work, and the work of their followers, friends, and not-quite-friends in category theory and proper understanding of what the 21st century mathematics would be without it obviously requires far better knowledge of mathematics than the 21st century students have. But may be we should at least say that our Fields Medal (Grothendieck) is not less than any other Fields Medal... George Janelidze ----- Original Message ----- From: "Michael Barr" <barr@math.mcgill.ca> To: "Categories list" <categories@mta.ca> Sent: Friday, October 05, 2007 2:52 PM Subject: categories: Help!
What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what?
Michael
The great thing about categories is that they allow analogies between different mathematical structures: see the paper R. Brown and T. Porter) `Category Theory: an abstract setting for analogy and comparison', In: What is Category Theory? Advanced Studies in Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. An example of the analogy is between the category of sets and the category of directed graphs: ``Higher order symmetry of graphs'', {\em Bull. Irish Math. Soc.} 32 (1994) 46-59. Here one easily sees non Boolean logics, of course. The word `analogy' seems to be underused in teaching undergraduates, but that is what abstraction is about, is it not? A teacher told me after a lecture on knots that was the first time he had heard the word analogy used in relation to mathematics! ( I discussed prime knots.) The other possibility is to advertise categorical structures: I advertised higher dimensional algebra to an international conference of neuroscientists in Delhi in 2003, pointing out the unlikelihood of the brain working entirely serially, and also the concept of colimit with an email analogy. A senior Indian neuroscientist came up to me afterwards and said that was the first time he had heard a seminar by a mathematician which made any sense! This is written up in (R. Brown and T. Porter), `Category theory and higher dimensional algebra: potential descriptive tools in neuroscience', Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 (2003) 80-92. These are all downloadable from http://www.bangor.ac.uk/r.brown/publicfull.htm or my home page. See also http://www.bangor.ac.uk/r.brown/outofline/out-home.html for a general talk. As said before, I see higher dimensional algebra as the study of mathematical structures with operations defined under geometrical conditions, thus allowing a combination of algebra and geometry, in a way which even Atiyah might like (see his paper on `20th century mathematics' Bull LMS 44 (2002) 1-15, in which the words `category' and `groupoid' do not appear). I have found giving general talks makes one think hard about the underlying ideas and motivation. Ronnie ----- Original Message ----- From: "Michael Barr" <barr@math.mcgill.ca> To: "Categories list" <categories@mta.ca> Sent: Friday, October 05, 2007 1:52 PM Subject: categories: Help!
What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what?
Michael
Hi, Michael, I am in the same predicament but, since I am speaking at this math club one week after you (November 6), I do hope to be able to use anything you do in your own talk! I also thought a lot about this problem and discarded one topic after another. Finally, I have decided to speak about the uses of infinitesimals in the synthetic calculus of variations, aiming at giving an algebraic (synthetic) proof of the well known fact that, for a paths functional ("energy"), its critical points agree with the geodesics. This requires that I introduce adjoint functors and cartesian closed categories and the notion of a ring object of line type. If you will do any of these yourself I could use it. Informally, I will argue constructively and acually prove things. Historical considerations may be briefly mentioned at the beggining of the talk, and the conceptual advantages of the synthetic method at the end. This will be an expanded portion of my paper "Synthetic Calculus of Variations" (with M. Heggie) in Contemporary Mathematics 30, 1983. I hope that this helps you as well as me. Best wishes, Marta
From: Michael Barr <barr@math.mcgill.ca> To: Categories list <categories@mta.ca> Subject: categories: Help! Date: Fri, 5 Oct 2007 08:52:29 -0400 (EDT)
What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what?
Michael
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I want to thank all who replied and I will take all your comments seriously. I would love to talk about Stone duality and such but I don't think many of our undergrads have ever heard of a topological space. They have heard of topology of course, but mostly they think it concerns things like toruses and Klein bottles. So they know nothing of the point set underpinnings of algebraic topology. My last term before retirement I taught a course called topology and spent exactly 6 lectures on point-set theory (taking a beeline to the Tychonoff theorem) before introducing pi_1 and covering spaces. The students were last year undergrads and a couple of grad students. Do they know what a boolean algebra is? Probably some do, some don't. Groups and abelian groups they will know about, probably modules, etc. Vector space duality is a familiar example, for finite dimension at least. Hi-tech whiteboards and even video-taping are out. I don't think we have any of the former and the one case that I know of a lecture that was video-taped (a fascinating lecture by Conway in the early '70s in which he showed how the game of Life allowed the simulation of self-reproducing Turing-power automata) seems to have disappeared without a trace. I will probably use a blackboard (or greenboard) and chalk, my favorite medium. One suggestion that does appeal is to start with universal mapping properties to explain products and sums. One thing that always struck me was Bill Lawvere's observation that the dual of the usual definition of function as a subset of a product s.t.... namely as a quotient of a sum s.t.... actually corresponds closely to the usual picture we draw when we introduce functions for the first time. I guess I could do worse than build the whole lecture around universal mapping properties. I could mention the somewhat unmotivated definition of (infinite) product of topological spaces as a perfect example of the universal viewpoint. Especially as topologists had come up with that definition independent of category theory. Michael
George Janelidze wrote:
(a) a topological space (defined via open sets) has no constant sets, one variable set X, and its structure is an element t in PP(X) that is closed under finite intersections and arbitrary unions.
The point of categories being (presumably) to shift the burden of structure from the objects to the morphisms, one would illustrate this point using your example by pointing out that the topological structure imputed to a space by the above definition is at least as well imputed with the definition of a space as the set of continuous functions to the space from the one-point space together with the set of continuous functions from it to the Sierpinski space. It sounded like you were headed in roughly that direction but then moved on to other points before getting there.
(b) a vector space has one constant set A ("the set of scalars"), one variable set X ("the set of vectors"), and its structure can be defined as element (a,b,c,d) of P(AxAxA)xP(AxAxA)xP(AxXxX)P(XxXxX), where a, b, c, d are addition of scalars, multiplication of scalars, scalar (-on-vector) multiplication, and addition of vectors respectively; that (a,b,c,d) should satisfy familiar conditions of course.
Ditto with the one-dimensional space in place of the one-point and Sierpinski space (which itself is a kind of one-dimensional space for topology).
4. Linear algebra tells us that instead of working with linear transformations of finite-dimensional vector spaces we can work with matrices, but one cannot formulate this properly without using the concept of equivalence of categories (the category of finite-dimensional K-vector spaces is equivalent to the category of natural numbers with matrices with entries from K as morphisms).
You may be setting the bar for "proper" higher than necessary to satisfy us engineers. I'm currently involved in a computer project where the question of the proper formulation of matrices came up. One team had formulated them in terms of the Kleisli construction for monads as defined in CTWM, the monad in question being the one that you yourself would surely come up with for the variety Vct_C, C the complex numbers. Unfortunately that formulation was giving the computer conniptions. This could have been construed as bearing out your point were it not for the fact that another team came along with a reformulation of monads that overcame the difficulty. Since I know this list is good at keeping secrets (such as the secret of categories) I'll be happy to share with you all, in my next message, my confidential report on the current status of this reformulation. Our CEO is not convinced of the correctness of the reformulation, the fact that it fixed the buggy behavior notwithstanding, and has asked me for a qualified expert second opinion---where better than this list for a question about monads? Vaughan
Michael Barr wrote:
What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what?
When I was a graduate student (recently), I gave a talk on category theory to other (mostly new) grad students (as part of a series where advanced students discussed their work). I began with my definition of category theory for nonmathematicians ("a general theory of how mathematical structures can fit together"), then gave some basic definitions and an example (duality in finite-dimensional vector spaces). Then I asked the audience a very open-ended question: Tell me what's your favourite branch of mathematics, and I'll tell you what category theory has to say about it (to justify the generality in my beginning statement). What attracted me first to category theory, and what I think remains impressive about it, is that you can you can really make good on this challenge. (It helps to know ahead of time what answers are likely; fortunately there were no pure number theorists at my school.) --Toby Bartels
I would think the best topics would be those that can be described with a minimum of jargon. The problem with category theory is that it is so steeped in its own jargon as to make it quite an effort to strip it out. Here are some topics where I would expect that effort to be minimal, arranged in roughly increasing order of intricacy of definition. This should more than fill a one-hour lecture, especially if there are questions. 1. Thinking of each object T of a category C as both a type and a dual type, characterize a product AxB in C as an object consisting of all pairs of T-elements of A and B over all types T, and A+B as dually consisting of all pairs of T-functionals of A and B over all dual types T in ob(C). Section 6 needs pullbacks, they could be done either here or there, it's neither here nor there. 2. The category FinBip of finite bipointed sets as the theory of cubical sets. The models are arbitrary functors M: Bip --> Set. You could look at \Delta for simplicial sets as well or instead, I'm partial to Bip perhaps because in kindergarten we tended to work more with cubical than simplicial sets (Western Australian kindergartens had strong PTOs reflecting epic entanglements). You could then continue with FinSet^op as the algebraic theory of Boolean algebras, but that would entail giving up one of the other segments. 3. Enriched categories as generalized metric spaces. People who have a hard time with abstract objects mixed in with concrete homsets (I certainly did) will be relieved to know that making the homsets just as abstract as the objects turns the definition of category into a familiar object not normally considered part of the categorical basement. 4. Presheaves on J as colimits of diagrams in J. If you use Yoneda to hide the concept of colimit the idea becomes almost trivial. In the case J = 1, as C starts from 1 and grows towards Set^1 each new set X as a new object of C is installed along with an arbitrary choice of C(1,X). The composites at X are defined by first installing C(X,Y) for all existing Y in C, defining fx: 1 --> X --> Y for each x in X and new f in C(X,Y), and taking C(X,Y) to be maximal subject to Ax[fx=gx] ==> f=g. These composites then uniquely determine the remaining composites gf: X --> Y --> Z and fg: W --> X --> Y for W other than 1. The completion is complete when every new set is necessarily isomorphic to one already present. (Does this have anything to do with Yoneda structures? Trying to read about those I discovered I no longer talked Strine.) For J the ordinals 1 and 2 as respectively the primitive vertex and the primitive edge, namely the two reflexive graphs priming the pump for the rest, there are now two types of element, vertices and edges, with Ax interpreted as quantifying over all elements of both types; otherwise everything is as for J = 1. Point out that whereas all sets are free, the free graphs are just those with trivial incidences. If you do section 6 (triples for matrix multiplication), also point out at some point that whereas Set^1 is tripleable on Set (the identity), Set^J in general is tripleable only on Set^{|J|}, important when talking Czech. 5. Toposes, but *not* the way it is explained on You-tube, which is completely unmotivated and incomprehensible for anyone who hasn't already understood them. The Explanation section http://en.wikipedia.org/wiki/Topos_theory#Explanation in the Wikipedia article on elementary toposes touches on the two points that should be in any explanation of the concept, namely (i) "subobject" predates "topos," witness CTWM which defines it in second-order language, and (ii) monics m: X' --> X are in bijection with pullbacks of the element (hence monic) t: 1 --> \Omega along morphisms f: X --> \Omega, allowing one to speak of *the* characteristic function of a monic, thereby classifying the monics by their characteristic functions, a first-order notion (whence the "elementary" in "elementary topos") that is in full agreement with the second-order notion in (i) when applied to a topos. 6. Matrix multiplication in terms of the Kleisli construction for the triple for Vct_k. I just sent out a crib sheet for that which focused on a difficulty with non-finitary (square summable) linear combinations, but that difficulty is impossible to absorb in the available time, better to stick to the finitary operations where matrix multiplication is tripleable. You could mention the Haskell programming language and how they blended the second component of the triple and Kleisli into a single operator Bind: T(X) --> (X --> T(Y)) --> T(Y), which might get any programmers in the club interested in Haskell; also point out the possibility of replacing (X --> T(Y)) by T(Y*X) and its implications for matrix algebra including Hilbert space. Stick to finite X in T(X) = k^X to save the extra step of defining finitary k^X for infinite X (but if you do decide to do that step it should suffice to point out that 6 of the 16 binary Boolean operations as 2x2 truth tables have constant rows or columns or both and then generalize to infinity). In case You-tube ever has a video on triples you should probably mention any synonyms for "triple" so the students can find the video. Vaughan Michael Barr wrote:
What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what?
Michael
On Fri, 5 Oct 2007, Michael Barr wrote:
What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what?
How about talking about simultaneously existing results in several categories? The Noetherian isomorphism theorems, while not necessarily the easiest to nail down exactly when they hold, have always been a strong motivator at the back of my head for why one might want to look at algebraic entities codifying things like "All Xs and maps between them". -- Mikael Vejdemo Johansson | To see the world in a grain of sand mik@math.su.se | And heaven in a wild flower | To hold infinity in the palm of your hand | And eternity for an hour
On Sun, 7 Oct 2007, Michael Barr wrote:
Hi-tech whiteboards and even video-taping are out. I don't think we have any of the former and the one case that I know of a lecture that was video-taped (a fascinating lecture by Conway in the early '70s in which he showed how the game of Life allowed the simulation of self-reproducing Turing-power automata) seems to have disappeared without a trace. I will probably use a blackboard (or greenboard) and chalk, my favorite medium.
Hi Michael, Video-taping certainly is out, if it ever was in. Taping is non-interactive and just silly. Your attitude towards "Hi-tech whiteboards" sounds too hi-tech as my point indeed was to say what you say about your favourite medium. That is still also my favourite medium, but I accept to write or meet virtually in particular if my audience is a flight distance away. Something is lost when you go virtual, but you also win some. Do you resist virtual whiteboards per se, or would you be interested in trying out a session? Installation is less than 15 minutes, and once we are online, we could spend another 15 discussing idempotent functors extendable to monads where E-M and Kleisli coincide. The mouse is your chalk and your board colour is white. I've used it so much already over the last years so I cannot work without it anymore. I can supervise a student from my home or a hotel room in Tokyo, and nobody knows or even cares who's where. Cheers, Patrik PS And for those who didn't see my mail to Michael, here it is, and apologies to those who view this purely as spam: Date: Sun, 7 Oct 2007 07:19:34 +0200 (MEST) From: Patrik Eklund <peklund@cs.umu.se> To: Michael Barr <barr@math.mcgill.ca> Cc: Patrik Eklund <peklund@cs.umu.se> Subject: Re: categories: Help! Dear Michael, No comment (at this point) on content, but let me refer to a previous mail I sent out on the subject and related to execution. My idea was to suggest a setup of virtual classrooms so that students and teacher indeed all over the world can attend a class. Of course, students and teacher, and in the end content, must be carefully selected. The reason for my suggestion is that the number of students at many sites is usually bery low for these courses and we should join forces. My suggestion is to use "sound-video-whiteboard" techniques as provided e.g. by Adobe and Marratech. I use the latter. "Sound-video" is nothing but Skype, but adding whiteboards, that can be saved and worked with also offline, you have very good possibilities. The whiteboard mainly accepts non-formatted text, drawings and images. You can read doc and ppt file which are "pasted" as bitmaps on the whiteboard. They include desktop sharing if that would be required. Mathematical text I add through LaTeX, compiling, converting to pdf, and using the snapshot tool to paste bitmapped formulas on the whiteboard. Once you get used to it you are actually not (much) slower on the virtual whiteboard as compared to a real whiteboard. Virtual advantages are e.g. - several whiteboards and easy to switch between them - more than one can jointly add to whitebooard content - can save and open (as mentioned) - can prepare whiteboards offline (as mentioned) If this is inline with your thoughts and you would like to try out Marratech, let me know. Best, Patrik
(It helps to know ahead of time what answers are likely; fortunately there were no pure number theorists at my school.)
At the risk of sounding like a cracked record, how about the division category in lieu of the division lattice, namely the coproduct completion of the set P of primes as a discrete category? For a longer story use P* instead of P, P with a final object adjoined. Motivate the division category by pointing out that only the square-free positive integers can be recovered as sups of primes in the division lattice. Vaughan
Vaughan Pratt wrote at last part:
In case You-tube ever has a video on triples you should probably mention any synonyms for "triple" so the students can find the video.
YouTube has a series of 5 video on triples under the name "monads": < http://www.youtube.com/results?search_query=monads&search=Search >, among others by the Catsters < http://www.youtube.com/user/TheCatsters >. --Toby
Speaking of videos, I think that this one could be great for motivating students to learn about categories http://claymath.msri.org/voevodsky2002.mov Besides making some strong statements about the importance of categories in the middle of the talk, it's all related to things that undergraduates know about or are about to learn. - Saul On 10/8/07, Toby Bartels <toby+categories@ugcs.caltech.edu> wrote:
Vaughan Pratt wrote at last part:
In case You-tube ever has a video on triples you should probably mention any synonyms for "triple" so the students can find the video.
YouTube has a series of 5 video on triples under the name "monads": < http://www.youtube.com/results?search_query=monads&search=Search >, among others by the Catsters < http://www.youtube.com/user/TheCatsters >.
--Toby
Dear Mike Some categories that are easily described (even to talented high school students) are: the category fun of functions (where objects are natural numbers and morphisms are functions); the category mat of matrices (again the objects are natural numbers); the category brd of braids; and, the category tang of tangles. There are enough functors amongst these to be interesting. They are all monoidal categories. One can try to discuss other structure the categories have in common so that strong monoidal functors (although I probably wouldn't introduce too much such terminology) preserve it. For example, duals in mat and tang; trace and braid closure; etc. One could try to show how the specialized, seemingly ad hoc Reidemeister moves translate naturally into the braided monoidal setting. A hint about how the "new" (mid 1980s) polynomial link invariants come from a functor tang --> mat might be of interest. Best wishes, Ross On 05/10/2007, at 10:52 PM, Michael Barr wrote:
What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what?
participants (10)
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George Janelidze -
Marta Bunge -
Michael Barr -
Mikael Vejdemo Johansson -
Patrik Eklund -
Ronnie Brown -
Ross Street -
Saul Youssef -
Toby Bartels -
Vaughan Pratt