The Category Theory Site Is Now Live

Administrative notes and a new website for the Category Theory Summer Study Group 2015

Platform Choice

I’ve made a few posts here [1] [2] about a summer study group for category theory. In an effort to facilitate the growing number of people from various timezones and differing platforms (many have come to us from Google+, Tumblr, Twitter, GoodReads, and friends from Dr. Miller’s class in a private Google Group), I’ve decided it may be easiest to set up something completely separate from all of these so our notes, resources, and any other group contributions can live on to benefit others in the future. Thus I’ve built Category Theory: Summer Study Group 2015 on WordPress.  It will live as a sub-domain of my personal site until I get around to buying a permanent home for it (any suggestions for permanent domain names are welcome).


We’ve actually had a few people already find the new site and register before I’ve announced it, but for those who haven’t done so yet, please go to our participant registration page and enter your preferred username and email address.  We’ll email you a temporary password which you can change when you login for the first time. Those who want to use their pre-existing WordPress credentials are welcome to do so.

Once you’ve registered, be sure to update your profile (at least include your name) so that others will know a little bit more about you. If you’d like you can also link your account [or sign up for one and then link it] so that you can add a photo and additional details.  To login later, there’s a link hidden in the main menu under “Participants.”

You can also add your details to the form at the bottom of the Participants page to let others know a bit more about you and where you can be reached. Naturally this is optional as I know some have privacy issues. In the notes, please leave your location/timezone so that we can better coordinate schedules/meetings.

Category Theory Blog

Your username/password will allow you to post content directly to the study group’s blog. This can be contributed notes, questions, resources, code, photos, thoughts, etc. related to category theory and related areas of mathematics we’ll be looking at. Initially your posts will be moderated (primarily only to prevent spam), and over time your status will be elevated to allow immediate posting and editing. If you have any questions or need administrative help, I’m easy to find and happy to help if you get into trouble. Most of the interface will hopefully be easy to understand.

For those with questions, please try to read posts as you’re able and feel free to comment with hints and/or solutions.  I’ve created “categories” with the chapter titles from the text we’re using to facilitate sorting/searching. Depending on the need, we can granularize this further as we proceed. There is also the ability to tag posts with additional metadata or upload photos as well.

As appropriate, I’ll take material out of the blog/posts stream and place it into freestanding pages for easier reference in the future. As an example, I’ve already found some material on YouTube and MIT’s Open Course Ware site (Spivak posted his 2013 class using our same text, though it unfortunately doesn’t include video or audio) that may be relevant to many.

For those interested, WordPress supports most basic LaTeX, though I doubt it supports any of the bigger category theory diagramming packages, so feel free to draw out pictures/diagrams, photograph them, and upload them for others to see if necessary.

As an advocate of the open web and owning one’s own data, I highly recommend everyone publish/post their content here as well as to their favorite site/platform of choice as they see fit.


In emails and chatter around the web, I haven’t heard any major objections to the proposed textbook so far, so unless there are, I’m assuming that it should serve most of us well. Hopefully everyone has a copy by now (remember there are free versions available) and has begun reading the introductory material.  Those requiring a bit more mathematical rigor and challenge can supplement with additional texts as I’m sure I and many others will. If you’re posting questions to the site about problems/questions from other texts, please either state them explicitly or tag them with the author’s last name as well as the problem/exercise number. (I’ll try to make them all canonical on the back end as we progress, so don’t worry too much if you’re not sure how or what to tag them with.)

Conference Call Tool

At the moment, most people have been fairly open to the three big platforms, though a few on either Linux or Chromebooks don’t have access to be able to install/operate anything but Google Hangouts, so I’m presently proposing that we adopt it for our group. Nearly everyone in the group already has a gmail account, so I don’t expect it to be an undue burden. If you haven’t used it before, please download/install any plugins you may require for your platform in advance of our first “call.”

Meeting Times

I’ve only heard back from a small handful of people on availability so far, but it doesn’t look like it will be difficult to find an appropriate time.  If you haven’t already done so, please fill out the “survey,” so we can determine a good call time for next week. If necessary, we can do additional times to help serve everyone’s needs. We don’t want to leave out any who sincerely want to participate.

Office Hours

As most of the participants are spread over the United States, Europe, and Asia, I’m suggesting that everyone carve out a standing block of time (we can call them “office hours”) that they can use to be available (via Google Hangouts or otherwise) to help out others having difficulty or who have questions. Since there isn’t a “professor” I’m hoping that we can all serve each other as unofficial teaching assistants to get through the process, and having standing office hours may be the easiest way to catch others for help in addition to the web site itself.

Questions? Comments? Snide Remarks?

If you have any questions, or I’ve managed to miss something, please don’t hesitate to make a comment below.  I’m hoping to get enough responses by Friday/Saturday this week to post our first meeting time for next week.


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Commutative Diagrams in LaTeX

A meta-review of resources for typesetting commutative diagrams in TeX & LaTeX. Save time in trying to find the right commutative diagram package on CTAN.


With my studies in category theory trundling along, I thought I’d take  moment to share some general resources for typesetting commutative diagrams in \LaTeX. 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.

Feruglio Summary

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.

Feruglio, Gabriel Valiente. Typesetting Commutative Diagrams.  TUGboat, Volume 15 (1994), No. 4

Milne Summary

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.

Milne, J.S. Guide to Commutative Diagram Packages.

Spivak Pseudo-recommendations

David Spivak, the author of Category Theory for the Sciences, seems to prefer XY-pic, diagXY, and TikZ based on his website from which he links to guides to each of these.

Resources for some of the “Bigger” Packages

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.

Diagxy: Michael Barr

XY-pic: Kristoffer Rose & Ross Moore

Diagrams: Paul Taylor

TikZ-CD: Florêncio Neves

Is there a particular package you recommend? Feel free to add your thoughts, comments, and additional resources in the comments below.

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Category Theory Summer Study Group 2015

A suggested syllabus for a summer study group on category theory.


Initial details for putting  the group together can be found at

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.

Primary resources:

General Communication

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 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 \LaTeX 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.

Primary Textbook

Spivak, David I. Category Theory for the Sciences. (The MIT Press, 2014)

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.

Additional References

Proposed Schedule

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)

Administrative tasks

  • Purchase Textbook
  • 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)

  • 2.3 Ologs
  • 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
  • 4.1 Monoids
  • 4.2 Groups

Week Five: June 21 (38 pages)

  • 4.3 Graphs
  • 4.4 Orders

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

Week Ten: July 26 (45 pages)

  • 6 Fundamental Considerations of Categories
  • 6.1 Limits and colimits

Week Eleven: August 2 (15 pages)

  • 6.2 Other notions in Cat

Week Twelve: August 9 (26 pages)

  • 7 Categories at Work 375
  • 7.1 Adjoint functors

Week Thirteen: August 16 (32 pages)

  • 7.2 Categories of functors

Week Fourteen: August 23 (19 pages)

  • 7.3 Monads

Week Fifteen: August 30 (23 pages)

  • 7.4 Operads

Additional resources

Requested/Required Responses from participants:

Preferred platform(s) for communications:

Email and/or online discussions

Platform Can use Can’t use Prefer Not to Use
Google Group
WordPress Site
GoodReads Group


Platform Can use Can’t use Prefer Not to Use
Google Hangouts


Dates and times you absolutely CAN’T make for meetings (please include your local time zone):




Dates and times you prefer (please include your local time zone):




One or two time periods during the week you could generally/reliably be available for “office hours”:


Any other thoughts on the material above:

  • Textbooks
  • Schedule
  • Additional resources for the group
  • Other

If you’d like to join us, please fill out the contact information and details below based on the material above:

Please indicate which videoconference platforms you are able to use by placing a checkmark in the corresponding boxes below. If you’re technically unable to use one or more, please indicate which in the “general comments” box above, and provide the reason why.

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8th Annual North American School of Information Theory (NASIT)

8th Annual North American School of Information Theory (NASIT) (

August 10-13, 2015 – UC San Diego, La Jolla, California
Application deadline: June 7, 2015

The School of Information Theory will bring together over 100 graduate students, postdoctoral scholars, and leading researchers for four action-packed days of learning, stimulating discussions, professional networking and fun activities, all on the beautiful campus of the University of California, San Diego (UCSD) and in the nearby beach town of La Jolla.

  • Tutorials by some of the best known researchers in information theory and related fields
  • Poster presentations by student participants with feedback and discussion
  • Panel discussion on “IT: Academia vs. Industry Perspectives”
  • Social events and fun activities
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BIRS Workshop: Advances and Challenges in Protein-RNA: Recognition, Regulation and Prediction (15w5063)

15w5063: Advances and Challenges in Protein-RNA: Recognition, Regulation and Prediction (Banff International Research Station |
BIRS 5 day worksop, arriving in Banff, Alberta Sunday, June 7 and departing Friday, June 12, 2015

In the years since the first assembly of the human genome, the complex and vital role of RNA and RNA binding proteins in regulation of the genome expression has expanded through the discovery of RNA-binding proteins and large classes of non-coding RNA that control many developmental decisions as part of protein- RNA complexes. Our molecular level understanding of RNA regulation has dramatically improved as many new structures of RNA–protein complexes have been determined and new sophisticated experimental technologies and dedicated computational modeling have emerged to investigate these interactions at the whole-genome level. Further deep insight on the molecular mechanisms that underline genome expression regulation is critical for understanding fundamental biology and disease progression towards the discovery of new approaches to interfere with disease progression.

The proposed workshop will bring together experts in RNA biology with structural biologists that focus on RNA-protein complexes, as well as computational biologists who seek to model and develop predictive tools based on the confluence of these experimental advances. The workshop intends to foster new collaborations between experimental and computational biologists and catalyze the development of new and improved technologies (such as single cell binding methods) that merge experimental analysis with novel mathematical and computational techniques to better understand the rules of protein-RNA recognition and RNA-based biological regulation.

The organizers of the workshop are both leaders in the field of protein-RNA recognition and interactions: Yael Mandel-Gutfreund has been working in the field of protein-Nucleic Acids interactions since 1994. Her main research interest is protein-RNA recognition and regulation. She has developed several tools and web servers for predicting RNA binding proteins and RNA binding motifs. Yael is the head to the computational molecular laboratory at the Technion and the president of the Israeli society of Bioinformatics and Computational Biology. Gabriele Varani has been working in the field of RNA structure and protein-RNA interactions since 1987. His main research interest is the structural basis for protein-RNA recognition and the prediction and design of RNA-binding proteins. He determined some of the first few structures of protein-RNA complexes and developed computational tools to analyze and predict the specificity of RNA -binding proteins. His group applies these tools to design RNA-binding proteins with new specificity to control gene expression. Our invitation to participate in the workshop has been met with great enthusiasm by the researchers. More than 20 principle investigators have already confirmed their interest in attending. Six of the confirmed participants are female scientists including the organizer Yael Mandel-Gutfreund as well as Traci Hall, Lynne Maquat, Elena Conti, Susan Jones, Drena Dobbs. We also have invited and confirmed the participation of young and promising researchers including Markus Landthaler, Gene Yeo, Jernej Ule, Uwe Ohler and others. Our confirmed participants come from many different countries: US, Canada, UK, Scotland, Germany, Spain, Switzerland, Poland and Israel. Two confirmed participants as well as the organizer have attended the BIRS workshops in the past.

A key objective of the workshop is to bring together researchers with experimental, mathematical and computational background to share results and discuss the main advances and challenges in the prediction, analysis and control of RNA-protein recognition and RNA-based regulation of gene expression. Towards this aim, we plan to adopt the format of previous BIRS meetings in which invited participants (including selected students) will present relatively short presentations of 20 minutes plus 10 minutes of active discussions. This format will leave aside ample time for informal discussions to foster exchanges between participants. To stress the collaborative, multidisciplinary nature of the workshop, we plan to dedicate each of the workshop sessions to a specific topic that will comprise presentations of structural, experimental and computational approaches, rather than create session focused on a particular approach. Each session we will include at least one lecture from a young scientist/postdoctoral fellow/student to be chosen among attendees by the organizers.

Suggested preliminary schedule:

  • Day 1: Modeling and high throughput approaches to genome-wide analysis of protein-RNA interactions
  • Day 2: Predicting and designing new RNA binding proteins
  • Day 3: Generating and modeling RNA-based regulatory networks
  • Day 4: Principles of RNA regulation by RNA binding proteins
  • Day 5: Conclusion round table discussion on the present and future challenges of the field
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The Information Universe Conference

"The Information Universe" Conference in The Netherlands in October hits several of the sweet spots for areas involving information theory, physics, the origin of life, complexity, computer science, and microbiology.

Yesterday, via a notification from Lanyard, I came across a notice for the upcoming conference “The Information Universe” which hits several of the sweet spots for areas involving information theory, physics, the origin of life, complexity, computer science, and microbiology. It is scheduled to occur from October 7-9, 2015 at the Infoversum Theater in Groningen, The Netherlands.

I’ll let their site speak for itself below, but they already have an interesting line up of speakers including:

Keynote speakers

  • Erik Verlinde, Professor Theoretical Physics, University of Amsterdam, Netherlands
  • Alex Szalay, Alumni Centennial Professor of Astronomy, The Johns Hopkins University, USA
  • Gerard ‘t Hooft, Professor Theoretical Physics, University of Utrecht, Netherlands
  • Gregory Chaitin, Professor Mathematics and Computer Science, Federal University of Rio de Janeiro, Brasil
  • Charley Lineweaver, Professor Astronomy and Astrophysics, Australian National University, Australia
  • Lude Franke, Professor System Genetics, University Medical Center Groningen, Netherlands
Infoversum Theater, The Netherlands
Infoversum Theater, The Netherlands

Conference synopsis from their homepage:

The main ambition of this conference is to explore the question “What is the role of information in the physics of our Universe?”. This intellectual pursuit may have a key role in improving our understanding of the Universe at a time when we “build technology to acquire and manage Big Data”, “discover highly organized information systems in nature” and “attempt to solve outstanding issues on the role of information in physics”. The conference intends to address the “in vivo” (role of information in nature) and “in vitro” (theory and models) aspects of the Information Universe.

The discussions about the role of information will include the views and thoughts of several disciplines: astronomy, physics, computer science, mathematics, life sciences, quantum computing, and neuroscience. Different scientific communities hold various and sometimes distinct formulations of the role of information in the Universe indicating we still lack understanding of its intrinsic nature. During this conference we will try to identify the right questions, which may lead us towards an answer.

  • Is the universe one big information processing machine?
  • Is there a deeper layer in quantum mechanics?
  • Is the universe a hologram?
  • Is there a deeper physical description of the world based on information?
  • How close/far are we from solving the black hole information paradox?
  • What is the role of information in highly organized complex life systems?
  • The Big Data Universe and the Universe : are our numerical simulations and Big Data repositories (in vitro) different from real natural system (in vivo)?
  • Is this the road to understanding dark matter, dark energy?

The conference will be held in the new 260 seats planetarium theatre in Groningen, which provides an inspiring immersive 3D full dome display, e.g. numerical simulations of the formation of our Universe, and anything else our presenters wish to bring in. The digital planetarium setting will be used to visualize the theme with modern media.

The Information Universe Website

Additional details about the conference including the participants, program, venue, and registration can also be found at their website.

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Category Theory Anyone?

I'm putting together a study group for an introduction to category theory. Who wants to join me?

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 la Leonard 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?

If you’re going to get lost and confused in the high weeds, you may as well have company, right?

Chris Aldrich


Category Theory, Anyone?
Category Theory, Anyone?

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:

This book is designed to be read by scientists and other people. It has very few mathematical prerequisites; for example, it doesn’t require calculus, linear algebra, or statistics. It starts by reintroducing the basics: What is a set? What is a function between sets?

That said, having a teacher or resident expert will be very helpful. Category theory is a “paradigm shift”—it’s a new way of looking at things. If you progress past the first few chapters, you’ll see that it’s a language for having very big thoughts and making unusually deep analogies.

To make real progress in this book (unless you’re used to reading university-level math books on your own) it will be useful to periodically check your understanding with someone who has some training in the subject. Seek out a math grad student or even a Haskell expert to help you. A growing number of people are learning basic category theory.

In order to really learn this material, a formal teacher or a professor would be best. Encourage your local university math department to offer a course in Category Theory for the Sciences. I can recommend this in good faith, because I went to special efforts to make this book available for free online. An old version of the book exists on the math arXiv, and a new MIT Press-edited version exists in HTML form on their website (see URLs below). That said, the print version, available here on Amazon and elsewhere, is much easier to read, if you want to get serious and you can afford it.

This book contains about 300 exercises and solutions. For those who wish to teach a course in the subject, there are 193 additional exercises and solutions behind a professors-only wall on the MIT Press website (see URL below). You simply have to request access.

To everyone: I hope you enjoy the book, and get a lot out of it!

Old version:
HTML version:

David Spivak, mathematician
in Description of Category Theory for the Sciences on



Awody, Steve. Category Theory (Oxford Logic Guides, #52). (Oxford University Press, 2nd Edition, 2010)

Lawvere, F. William & Schanuel, Stephen H. Conceptual Mathematics: A First Introduction to Categories. (Cambridge University Press, 2nd Edition, 2009)

Leinster, Tom. Basic Category Theory (Cambridge Studies in Advanced Mathematics, #143). (Cambridge University Press, 2014)

Mac Lane, Saunders. Categories for the Working Mathematician (Graduate Texts in Mathematics, #5). (Springer, 2nd Edition, 1998)

Spivak, David I. Category Theory for the Sciences. (The MIT Press, 2014)


Why write a new textbook on Category Theory, when we already have Mac Lane’s ‘Categories for the Working Mathematician’? Simply put, because Mac Lane’s book is for the working (and aspiring) mathematician. What is needed now, after 30 years of spreading into various other disciplines and places in the curriculum, is a book for everyone else.

Steve Awody, mathematician
on page iv of Category Theory (Oxford Logic Guides, #52). (Oxford University Press, 2nd Edition, 2010)

If you’d like to join us, please leave a comment below and be sure to include your email address in the comment form so we can touch base regarding details.

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Videos from the NIMBioS Workshop on Information and Entropy in Biological Systems

Videos from the NIMBioS workshop on Information and Entropy in Biological Systems from April 8-10, 2015 are slowly starting to appear on YouTube.

Videos from the April 8-10, 2015, NIMBioS workshop on Information and Entropy in Biological Systems are slowly starting to appear on YouTube.

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

Additonal resources from the Workshop:


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Popular Science Books on Information Theory, Biology, and Complexity

The beginning of a four part series in which I provide a gradation of books and texts that lie in the intersection of the application of information theory, physics, and engineering practice to the area of biology.

Previously, I had made a large and somewhat random list of books which lie in the intersection of the application of information theory, physics, and engineering practice to the area of biology.  Below I’ll begin to do a somewhat better job of providing a finer gradation of technical level for both the hobbyist or the aspiring student who wishes to bring themselves to a higher level of understanding of these areas.  In future posts, I’ll try to begin classifying other texts into graduated strata as well.  The final list will be maintained here: Books at the Intersection of Information Theory and Biology.

Introductory / General Readership / Popular Science Books

These books are written on a generally non-technical level and give a broad overview of their topics with occasional forays into interesting or intriguing subtopics. They include little, if any, mathematical equations or conceptualization. Typically, any high school student should be able to read, follow, and understand the broad concepts behind these books.  Though often non-technical, these texts can give some useful insight into the topics at hand, even for the most advanced researchers.

Complexity: A Guided Tour by Melanie Mitchell (review)

Possibly one of the best places to start, this text gives a great overview of most of the major areas of study related to these fields.

Entropy Demystified: The Second Law Reduced to Plain Common Sense by Arieh Ben-Naim

One of the best books on the concept of entropy out there.  It can be read even by middle school students with no exposure to algebra and does a fantastic job of laying out the conceptualization of how entropy underlies large areas of the broader subject. Even those with Ph.D.’s in statistical thermodynamics can gain something useful from this lovely volume.

The Information: A History, a Theory, a Flood by James Gleick (review)

A relatively recent popular science volume covering various conceptualizations of what information is and how it’s been dealt with in science and engineering.  Though it has its flaws, its certainly a good introduction to the beginner, particularly with regard to history.

The Origin of Species by Charles Darwin

One of the most influential pieces of writing known to man, this classical text is the basis from which major strides in biology have been made as a result. A must read for everyone on the planet.

Information, Entropy, Life and the Universe: What We Know and What We Do Not Know by Arieh Ben-Naim

Information Theory and Evolution by John Avery

The Touchstone of Life: Molecular Information, Cell Communication, and the Foundations of Life by Werner R. Loewenstein (review)

Information Theory, Evolution, and the Origin of Life by Hubert P. Yockey

The four books above have a significant amount of overlap. Though one could read all of them, I recommend that those pressed for time choose Ben-Naim first. As I write this I’ll note that Ben-Naim’s book is scheduled for release on May 30, 2015, but he’s been kind enough to allow me to read an advance copy while it was in process; it gets my highest recommendation in its class. Loewenstein covers a bit more than Avery who also has a more basic presentation. Most who continue with the subject will later come across Yockey’s Information Theory and Molecular Biology which is similar to his text here but written at a slightly higher level of sophistication. Those who finish at this level of sophistication might want to try Yockey third instead.

The Red Queen: Sex and the Evolution of Human Nature by Matt Ridley

Grammatical Man: Information, Entropy, Language, and Life  by Jeremy Campbell

Life’s Ratchet: How Molecular Machines Extract Order from Chaos by Peter M. Hoffmann

Complexity: The Emerging Science at the Edge of Order and Chaos by M. Mitchell Waldrop

The Big Picture: On the Origins of Life, Meaning, and the Universe Itself (Dutton, May 10, 2016) 

In the coming weeks/months, I’ll try to continue putting recommended books on the remainder of the rest of the spectrum, the balance of which follows in outline form below. As always, I welcome suggestions and recommendations based on others’ experiences as well. If you’d like to suggest additional resources in any of the sections below, please do so via our suggestion box. For those interested in additional resources, please take a look at the ITBio Resources page which includes information about related research groups; references and journal articles; academic, research institutes, societies, groups, and organizations; and conferences, workshops, and symposia.

Lower Level Undergraduate

These books are written at a level that can be grasped and understood by most with a freshmen or sophomore university level. Coursework in math, science, and engineering will usually presume knowledge of calculus, basic probability theory, introductory physics, chemistry, and basic biology.

Upper Level Undergraduate

These books are written at a level that can be grasped and understood by those at a junior or senor university level. Coursework in math, science, and engineering may presume knowledge of probability theory, differential equations, linear algebra, complex analysis, abstract algebra, signal processing, organic chemistry, molecular biology, evolutionary theory, thermodynamics, advanced physics, and basic information theory.

Graduate Level

These books are written at a level that can be grasped and understood by most working at the level of a master’s level at most universities.  Coursework presumes all the previously mentioned classes, though may require a higher level of sub-specialization in one or more areas of mathematics, physics, biology, or engineering practice.  Because of the depth and breadth of disciplines covered here, many may feel the need to delve into areas outside of their particular specialization.

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Nicolas Perony: Puppies! Now that I’ve got your attention, complexity theory | TED

Animal behavior isn't complicated, but it is complex. Nicolas Perony studies how individual animals — be they Scottish Terriers, bats or meerkats — follow simple rules that, collectively, create larger patterns of behavior. And how this complexity born of simplicity can help them adapt to new circumstances, as they arise.

For those who are looking for a good, simple, and entertaining explanation of the concept of emergent properties and behavior within complexity theory (or Big History), I just came across a nice TED talk that simplifies complexity using a few animal examples including a cute puppy video as well as a bat and a meerkat example. The latter two also have implications for evolution and survival which are lovely examples as well.

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Schools of Thought in the Hard and Soft Sciences

A framework for determining the difference between the hard and soft sciences.

A recent post in one of the blogs at Discover Magazine the other day had me thinking about the shape of science over time.

Neuroscientists don’t seem to disagree on the big issues. Why are there no big ideas in neuroscience?

Neuroskeptic, Where Are The Big Ideas in Neuroscience? (Part 1)

The article made me wonder about the divide between the ‘soft’ and ‘hard’ sciences, and how we might better define and delineate them. Perhaps in a particular field, the greater the proliferation of “schools of though,” the more likely something is to be a soft science? (Or mathematically speaking, there’s an inverse relationship in a field between how well supported it is and the number of schools of thought it has.) I consider a school of thought to be a hypothetical/theoretical proposed structure meant to potentially help advance the state of the art and adherents join one of many varying camps while evidence is built up (or not) until one side carries the day.

Firmness of Science vs. # of Schools of Thought
Simple linear approximation of the relationship, though honestly something more similar to y=1/x which is asymptotic to the x and y axes is far more realistic.

Theorem: The greater the proliferation of “schools of though,” the more likely something is to be a soft science.

Generally in most of the hard sciences like physics, biology, or microbiology, there don’t seem to be any opposing or differing schools of thought. While in areas like psychology or philosophy they abound, and often have long-running debates between schools without any hard data or evidence to truly allow one school to win out over another. Perhaps as the structure of a particular science becomes more sound, the concept of schools of thought become more difficult to establish?

For some of the hard sciences, it would seem that schools of thought only exist at the bleeding edge of the state-of-the-art where there isn’t yet enough evidence to swing the field one way or another to firmer ground.

Example: Evolutionary Biology

We might consider the area of evolutionary biology in which definitive evidence in the fossil record is difficult to come by, so there’s room for the opposing thoughts for gradualism versus punctuated equilibrium to be individual schools. Outside of this, most of evolutionary theory is so firmly grounded that there aren’t other schools.

Example: Theoretical Physics

The relatively new field of string theory might be considered a school of thought, though there don’t seem to be a lot of other opposing schools at the moment. If it does, such a school surely exists, in part, because there isn’t the ability to validate it with predictions and current data. However, because of the strong mathematical supporting structure, I’ve yet to hear anyone use the concept of school of thought to describe string theory, which sits in a community which seems to believe its a foregone conclusion that it or something very close to it represents reality. (Though for counterpoint, see Lee Smolin’s The Trouble with Physics.)

Example: Mathematics

To my knowledge, I can’t recall the concept of school of thought ever being applied to mathematics except in the case of the Pythagorean School which historically is considered to have been almost as much a religion as a science. Because of its theoretical footings, I suppose there may never be competing schools, for even in the case of problems like P vs. NP, individuals may have some gut reaction to which way things are leaning, everyone ultimately knows it’s going to be one or the other (P=NP or P \neq NP). Many mathematicians also know that it’s useful to try to prove a theorem during the day and then try to disprove it (or find a counterexample) by night, so even internally and individually they’re self-segregating against creating schools of thought right from the start.

Example: Religion

Looking at the furthest end of the other side of the spectrum, because there is no verifiable way to prove that God exists, there has been an efflorescence of religions of nearly every size and shape since the beginning of humankind. Might we then presume that this is the softest of the ‘sciences’?

What examples or counter examples can you think of?

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String Theory, Black Holes, and Information

Amanda Peet presented the a lecture entitled "String Theory Legos for Black Holes" at the Perimeter Institute for Theoretical Physics.

Four decades ago, Stephen Hawking posed the black hole information paradox about black holes and quantum theory. It still challenges the imaginations of theoretical physicists today. Yesterday, Amanda Peet (University of Toronto) presented the a lecture entitled “String Theory Legos for Black Holes” yesterday at the Perimeter Institute for Theoretical Physics. A quick overview/teaser trailer for the lecture follows along with some additional information and the video of the lecture itself.

The “Information Paradox” with Amanda Peet (teaser trailer)

“Black holes are the ‘thought experiment’ par excellence, where the big three of physics – quantum mechanics, general relativity and thermodynamics – meet and fight it out, dragging in brash newcomers such as information theory and strings for support. Though a unification of gravity and quantum field theory still evades string theorists, many of the mathematical tools and ideas they have developed find applications elsewhere.

One of the most promising approaches to resolving the “information paradox” (the notion that nothing, not even information itself, survives beyond a black hole’s point-of-no-return event horizon) is string theory, a part of modern physics that has wiggled its way into the popular consciousness.

On May 6, 2015, Dr. Amanda Peet, a physicist at the University of Toronto, will describe how the string toolbox allows study of the extreme physics of black holes in new and fruitful ways. Dr. Peet will unpack that toolbox to reveal the versatility of strings and (mem)branes, and will explore the intriguing notion that the world may be a hologram.

Amanda Peet Amanda Peet is an Associate Professor of Physics at the University of Toronto. She grew up in the South Pacific island nation of Aotearoa/New Zealand, and earned a B.Sc.(Hons) from the University of Canterbury in NZ and a Ph.D. from Stanford University in the USA. Her awards include a Radcliffe Fellowship from Harvard and an Alfred P. Sloan Foundation Research Fellowship. She was one of the string theorists interviewed in the three-part NOVA PBS TV documentary “Elegant Universe”.

Web site:

Dr. Amanda Peet’s Lecture “String Theory Legos for Black Holes”

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Richard Dawkins Interview: This Is My Vision Of “Life” |

This Is My Vision Of "Life" by John Brockman (
The's interview with Richard Dawkins.

Richard Dawkins [4.30.15]

“My vision of life is that everything extends from replicators, which are in practice DNA molecules on this planet. The replicators reach out into the world to influence their own probability of being passed on. Mostly they don’t reach further than the individual body in which they sit, but that’s a matter of practice, not a matter of principle. The individual organism can be defined as that set of phenotypic products which have a single route of exit of the genes into the future. That’s not true of the cuckoo/reed warbler case, but it is true of ordinary animal bodies. So the organism, the individual organism, is a deeply salient unit. It’s a unit of selection in the sense that I call a “vehicle”.  There are two kinds of unit of selection. The difference is a semantic one. They’re both units of selection, but one is the replicator, and what it does is get itself copied. So more and more copies of itself go into the world. The other kind of unit is the vehicle. It doesn’t get itself copied. What it does is work to copy the replicators which have come down to it through the generations, and which it’s going to pass on to future generations. So we have this individual replicator dichotomy. They’re both units of selection, but in different senses. It’s important to understand that they are different senses.”

Richard Dawkins
Richard Dawkins

RICHARD DAWKINS is an evolutionary biologist; Emeritus Charles Simonyi Professor of the Public Understanding of Science, Oxford; Author, The Selfish Gene; The Extended Phenotype; Climbing Mount Improbable; The God Delusion; An Appetite For Wonder; and (forthcoming) A Brief Candle In The Dark.

Watch the entire video interview and read the transcript at

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