With my studies in category theory trundling along, I thought I’d take moment to share some general resources for typesetting commutative diagrams in . I’ll outline below some of the better resources and recommendations I’ve found, most by much more dedicated and serious users than I. Following that I’ll list a few resources, articles, and writings on some of the more common packages that I’ve seen mentioned.
Naturally, just reading through some of the 20+ page user guides to some of these packages can be quite daunting, much less wading through the sheer number that exist. Hopefully this one-stop-shop meta-overview will help others save some time trying to figure out what they’re looking for.
Gabriel Valiente Feruglio has a nice overview article naming all the primary packages with some compare/contrast information. One will notice it was from 1994, however, and misses a few of the more modern packages including TikZ. His list includes: AMS; Barr (diagxy); Borceux; Gurari; Reynolds; Rose (XY-pic); Smith (Arrow); Spivak; Svensson (kuvio); Taylor (diagrams); and Van Zandt (PSTricks). He lists them alphabetically and gives brief overviews of some of the functionality of each.
J.S. Milne has a fantastic one-page quick overview description of several available packages with some very good practical advise to users depending on the level of their needs. He also provides a nice list of eight of the most commonly used packages including: array (LaTeX); amscd (AMS); DCpic (Quaresma); diagrams (Taylor); kuvio (Svensson); tikz (Tantau); xymatrix (Rose); and diagxy (Barr). It’s far less formal than Feruglio, but is also much more modern. I also found it a bit more helpful for trying to narrow down one or more packages with which to play around.
Based on the recommendations given in several of the resources above, I’ve narrowed the field a bit to some of the better sounding packages. I’ve provided links to the packages with some of the literature supporting them.
Below is a handful of suggestions and thoughts relating to the study group in terms of platforms to assist us in communicating as well as a general outline for the summer. I’m only “leading” this in the sense that I put my foot forward first, but I expect and sincerely hope that others will be active leaders and participants as well, so please take the following only as a suggestion, and feel free to add additional thoughts and commentary you feel might help the group.
Since many within the group are already members of the Google Group “Advanced Physics & Math – Los Angeles.” I suggest we use the email list here as a base of communication. I believe the group is still “private” but am happy to invite the handful of participants who aren’t already members. Those actively participating are encouraged to change their settings so that they receive emails from the group either as they’re posted, or in batches once a day. Those subscribed only once a week or less frequently are likely to miss out on questions, comments, and other matters.
Alternately we might also use the GoodReads.com discussion group within the “Mathematics Students” group. I believe only about three of us so far may already be goodreads members, so this may require more effort for others to join.
If anyone has an alternate platform suggestion for communicating and maintaining resources, I’m happy to entertain it.
I wouldn’t be opposed to setting up a multi-user WordPress site that we could all access and post/cross-post to. Doing this could also allow for use of as well, which may be useful down the line. This would also have the benefit of being open to the public and potentially assisting future students. It also has built-in functionality of notifying everyone of individual posts and updates as they’re entered.
I’ll propose a general weekly meeting online via Google Hangouts on a day and time to be determined. It looks like the majority of respondents are in the Pacific timezone, so perhaps we could shoot for something around 7pm for an hour or so if we do something during weekdays so that East coasters can join without us running too late. If we decide to do something during the weekend, we obviously have a good bit more flexibility.
If we could have everyone start by indicating which days/times absolutely won’t work for them and follow up with their three to four preferred days/times, then we might be able to build a consensus for getting together.
Alternate videoconference options could include Skype, ooVoo, or others, in some part because I know that most participants are already part of the Google ecosystem and know that one or more potential participants is using Google Chromebooks and thus may not be able to use other platforms. Is anyone not able to use Google Hangouts? If we opt for something else, we want something that is ubiquitous for platform, allows screen sharing, and preferably the ability to record the sessions for those who aren’t present.
Ideally the videoconference meetings will be geared toward an inverted classroom style of work in which it would be supposed that everyone has read the week’s material and made an attempt at a number of problems. We can then bring forward any general or specific conceptual problems people may be having and then work as a group toward solving any problems that anyone in the group may be having difficulty with.
I’ll also suggest that even if we can’t all make a specific date and time, that we might get together in smaller groups to help each other out. Perhaps everyone could post one or two regular hours during the week as open “office hours” so that smaller groups can discuss problems and help each other out so that we can continue to all make progress as a group.
Given the diversity of people in the group and their backgrounds, I’ll suggest Spivak’s text which has a gentle beginning and is geared more toward scientists and non-professional mathematicians, though it seems to come up to speed fairly quickly without requiring a large number of prerequisites. It also has the benefit of being free as noted below.
The textbook can be purchased directly through most book retailers. Those looking for cheaper alternatives might find these two versions useful. The HTML version should be exactly in line with the printed one, while the “old version” may not be exactly the same.
Following this, I might suggest we use something like Awody’s text or Leinster’s which are slightly more technical, but still fairly introductory. Those who’d like a more advanced text can certainly supplement by reading portions of those texts as we work our way through the material in Spivak. If all of the group wants a more advanced text, we can certainly do it, but I’d prefer not to scare away any who don’t have a more sophisticated background.
The following schedule takes us from now through the end of the summer and covers the entirety of the book. Hopefully everyone will be able to participate through the end, though some may have additional pressures as the beginning of the Fall sees the start of other courses. Without much prior experience in the field myself, I’ve generally broken things up to cover about 35 pages a week, though some have slightly more or less. Many, like me, may feel like the text really doesn’t begin until week three or four as the early chapters provide an introduction and cover basic concepts like sets and functions which I have a feeling most have at least some experience with. I’ve read through chapter two fairly quickly already myself. This first easy two week stretch will also give everyone the ability to settle in as well as allow others to continue to join the group before we make significant headway.
If anyone has more experience in the subject and wishes to comment on which sections we may all have more conceptual issues with, please let us know so we can adjust the schedule as necessary. I suppose we may modify the schedule as needed going forward, though like many of you, I’d like to try to cover as much as we can before the end of the summer.
Week One: May 24 (24 pages)
Decide on best day/time for meeting
Decide on platform for meetings: Google Hangouts /Skype /ooVoo /Other
1 A brief history of category theory
1.2 Intention of this book
1.3 What is requested from the student
1.4 Category theory references
2 The Category of Sets 9
2.1 Sets and functions
2.2 Commutative diagrams
Week Two: May 31 (50 pages)
3 Fundamental Considerations in Set 41
3.1 Products and coproducts
3.2 Finite limits in Set
Week Three: June 7 (40 pages)
3.3 Finite colimits in Set
3.4 Other notions in Set
Week Four: June 14 (31 pages)
4 Categories and Functors, Without Admitting It 115
Week Five: June 21 (38 pages)
Week Six: June 28 (19 pages)
4.5 Databases: schemas and instances
Week Seven: July 5 (36 pages)
5 Basic Category Theory 203
5.1 Categories and functors
Week Eight: July 12 (28 pages)
5.2 Common categories and functors from pure math
Week Nine: July 19 (48 pages)
5.3 Natural transformations
5.4 Categories and schemas are equivalent, Cat » Sch
I’m putting together a study group for an introduction to category theory. Who wants to join me?
Usually in the Fall and Winter, I’m concentrating on studying some semblance of abstract mathematics with a group of 20-30 kamikaze amateurs under the apt tutelage of Dr. Michael Miller through UCLA Extension. Since he doesn’t offer any classes in the Spring or Summer and we haven’t managed to talk Terence Tao into offering something interesting à laLeonard Susskind, we’re all at a loss for what to do with some of our time.
A small cohort of regulars from Miller’s class has recently taken up plowing through Howard Georgi’s Lie Algebras and Particle Physics. Though this seems very diverting to me given our work on Lie groups and algebras in the Fall and Winter, I don’t see any direct or exciting applications to anything more immediate.
Why Not Try Category Theory?
Since the death of Grothendieck I have seen a growing number of references to the area of category theory from a variety of different fronts.
Most notably, for the past year I’ve been more closely following John Baez’s Azimuth Blog which has frequent posts relating to category theory with applications I can directly use in various areas. Unfortunately I couldn’t attend his recent workshop at NIMBioS on Information and Entropy in Biological Systems, which apparently means I missed meeting Tom Leinster who recently released the textbook Basic Category Theory (Cambridge University Press, 2014). [I was already never going to forgive myself after I missed the workshop, but this fact now seems to be additional salt in the wound.]
The straw that broke the proverbial camel’s back was my serendipitously stumbling across Ilyas Khan‘s excellent post “Category Theory – the bedrock of mathematics?” while doing a Google image search for something entirely unrelated to anything remotely similar to mathematics. His discussion and the breadth of links to interesting and intriguing papers and articles within it and several colleagues thanking me for posting about it have finally forced my hand. (I also find myself wishing that he would write on a more formal basis more frequently.)
So over the past week or so, I’ve done some basic subject area searching, and I’ve picked up David I. Spivak’s book Category Theory for the Sciences (The MIT Press, 2014) to begin plowing through it.
Anyone Care to Join Me?
Since doing abstract math is always more fun with companions, and I know there are several out there who might be interested in some of the areas which category theory touches on, why don’t you join in? Over the coming months of Summer, let’s plot a course through the subject. I’ll suggest Spivak’s book first as it seems to be one of the most basic as well as the broadest out there in terms of applications. (There are also free copies of versions available through arXiv and MIT.) It doesn’t have a huge list of prerequisites either, so a broader category of people might be able to join in as well.
We can have occasional weekly or bi-weekly “meetings” via internet using something like Google Hangouts, Skype, or ooVoo to discuss problems and help each other out as necessary. Ideally those who join will spend at least 3 hours a week, if not more reading the text and working through problems. Following Spivak, we might try dipping into Leinster, Awody, or Mac Lane.
From the author of Category Theory for the Sciences:
Awody, Steve. Category Theory (Oxford Logic Guides, #52). (Oxford University Press, 2nd Edition, 2010)
John Baez, one of the organizers of the workshop, is also going through them and adding some interesting background and links on his Azimuth blog as well for those who are looking for additional details and depth
Category theory looks set to become the dominant foundational basis for all mathematics. It could, in fact, already have achieved that status through stealth.
Beauty, even in Maths, can exist in the eye of the beholder. That might sound a little surprising, when, after all, what could be more objective than mathematics when thinking about truth, and what, therefore, could be more natural than for beauty and goodness, the twin accomplices to truth, to be co-joined ?
In the 70 odd years since Samuel Eilenberg and Saunders Mac Lane published their now infamous paper “A General Theory of Natural Equivalences“, the pursuit of maths by professionals (I use here the reference point definition of Michael Harris – see his recent publication “Mathematics without Apologies“) has become ever more specialised. I, for one, don’t doubt cross disciplinary excellence is alive and sometimes robustly so, but the industrially specialised silos that now create, produce and then sustain academic tenure are formidable within the community of mathematicians.
Beauty, in the purest sense, does not need to be captured in a definition but recognised through intuition. Whether we take our inspiration from Hardy or Dirac, or whether we experience a gorgeous thrill when encountering an austere proof that may have been confronted thousands of times before, the confluence of simplicity and beauty in maths may well be one of the few remaining places where the commonality of the “eye” across a spectrum of different beholders remains at its strongest.
Neither Eilenberg nor Mac Lane could have thought that Category theory, which was their attempt to link topology and algebra, would become so pervasive or so foundational in its influence when they completed and submitted their paper in those dark days of WW 2. But then neither could Cantor, have dreamt about his work on Set theory being adopted as the central pillar of “modern” mathematics so soon after his death. Under attack from establishment figures such as Kronecker during his lifetime, Cantor would not have believed that set theory would become the central edifice around which so much would be constructed.
Of course that is exactly what has happened. Set theory and the ascending magnitude of infinities that were unleashed through the crack in the door that was represented by Cantor’s diagonal conquered all before them.
Until now, that is.
In an article in Science News, Julie Rehmeyer describes Category Theory as “perhaps the most abstract area of all mathematics” and “where math is the abstraction of the real world, category theory is an abstraction of mathematics”.
Slowly, without fanfare, and with an alliance built with the emergent post transistor age discipline of computer science, Category theory looks set to become the dominant foundational basis for all mathematics. It could, in fact, already have achieved that status through stealth. After all, if sets are merely an example of a category, they become suborned without question or query. One might even use the description ‘subsumed’.
There is, in parallel, a wide ranging discussion in mathematics about the so called Univalent Foundation that is most widely associated with Voevodsky which is not the same. The text book produced for the year long univalence programme iniated at the IAS that was completed in 2013 Homotopy type theory – Univalent Foundations Programme states:
“The univalence ax-iom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the “official”doctrines of conventional foundations..”
before going on to present the revelatory exposition that Univalent Foundations are the real unifying binding agent around mathematics.
I prefer to think of Voevodsky’s agenda as being narrower in many crucial respects than Category Theory, although both owe a huge amount to the over-arching reach of computational advances made through the mechanical aid proffered through the development of computers, particularly if one shares Voevodsky’s view that proofs will eventually have to be subject to mechanical confirmation.
In contrast, the journey, post Russell, for type theory based clarificatory approaches to formal logic continues in various ways, but Category theory brings a unifying effort to the whole of mathematics that had to wait almost two decades after Eilenberg and Mac Lane’s paper when a then virtually unknown mathematician, William Lawvere published his now much vaunted “An Elementary Theory of the Category of Sets” in 1964. This paper, and the revolutionary work of Grothendieck (see below) brought about a depth and breadth of work which created the environment from which Category Theory emerged through the subsequent decades until the early 2000’s.
Lawvere’s work has, at times, been seen as an attempt to simply re-work set theory in Category theoretic terms. This limitation is no longer prevalent, indeed the most recent biographical reviews of Grothendieck, following his death, assume that the unificatory expedient that is the essential feature of Category theory (and I should say here not just ETCS) is taken for granted, axiomatic, even. Grothendieck eventually went much further than defining Category theory in set theoretic terms, with both Algebraic Topology and Mathematical Physics being fields that now could not be approached without a foundational setting that is Category theory. The early language and notation of Category Theory where categories ‘C’ are described essentially as sets whose members satisfy the conditions of composition, morphism and identity eventually gave way post Lawvere and then Lambek to a systematic adoption of the approach we now see where any and all deductive systems can be turned into categories. Most standard histories give due credit to Eilenberg and Mac Lane as well as Lawvere (and sometimes Cartan), but it is Grothendieck’s ‘Sur quelques points d’algebre homologique’ in 1957 that is now seen as the real ground breaker.
My own pathway to Category theory has been via my interest in Lie Groups, and more broadly, in Quantum Computing, and it was only by accident (the best things really are those that come about by accident !) that I decided I had better learn the language of Category theory when I found Lawvere’s paper misleadingly familiar but annoyingly distant when, in common with most people, I assumed that my working knowledge of notation in logic and in set theory would map smoothly across to Category theory. That, of course, is not the case, and it was only after I gained some grounding in this new language that I realised just how and why Category theory has an impact far beyond computer science. It is this journey that also brings me face to face with a growing appreciation of the natural intersection between Category theory and a Wittgensteinian approach to the Philosophy of Mathematics. Wittgenstein’s disdain for Cantor is well documented (this short note is not an attempt to justify, using Category theory, a Wittgensteinian criticism of set theory). More specifically however, it was Abramsky and Coecke’s “Categorical Quantum Mechanics” that helped me to discern more carefully the links between Category Theory and Quantum Computing. They describe Category Theory as the ‘language of modern structural mathematics’ and use it as the tool for building a mathematical representation of quantum processes, and their paper is a thought provoking nudge in the ribs for anyone who is trying to make sense of the current noise that surrounds Quantum mechanics.
Awodey and Spivak are the two most impressive contemporary mathematicians currently working on Category Theory in my view, and whilst it is asking for trouble to choose one or two selected works as exemplars of their approach, I would have to say that Spivak’s book on Category Theory for the Sciences is the standout work of recent times (incidentally the section in this book on ‘aspects’ bears close scrutiny with Wittgenstein’s well known work on ‘family resemblances’).
Awodey’s 2003 paper is as good a recent balance between a mathematical and philosophical exposition of the importance of category theory as exists whilst his textbook is often referred to as the standard entry point for working mathematicians.
Going back to beauty, which is how I started this short note. Barry Mazur wrote an article in memory of Saunders Mac Lane titled ‘When is one thing equal to another‘ which is a gem of rare beauty, and the actual catalyst for this short note. If you read only one document in the links from this article, then I hope it is Mazur’s paper.