Category Archives: Category Theory


Could someone explain Exercise

Find an operation on the set M = \{1, 2, 3, 4\}, i.e., a legitimate function f : M \times M \rightarrow M, such that f cannot be the multiplication formula for a monoid on M. That is, either it is not associative or no element of M can serve as a unit.


Week Four Meeting: §4.1 – §4.2

This week’s Google Hangout (RSVP here) will cover problems/questions from week four of the syllabus:

  • 4.1 Monoids
  • 4.2 Groups

If you’re joining us in progress, please feel free to add in any questions you might have about previous material as well – it’s never too late to join us all.

Those who aren’t able to jump into the hangout (due to hardware issues or the 10 person limit) are encouraged to chat within the hangout IM and follow along with the live stream. If you’re stuck and can’t make it, it will be archived on our YouTube Channel for later consumption.

Participant count

As of this week there are now 32 participants in the group! Thanks to everyone who is participating, as I expected we’d have only about 4 when we started this whole thing.

Dr. Martin Codrington’s “Category Theory: The Beginner’s Introduction”

Dr. Martin Codrington just uploaded an excellent-looking set of videos to YouTube:

Week Three Meeting: §3.3 – §3.4

This week’s Google Hangout (RSVP here) will cover problems/questions from week three of the syllabus:

  • 3.3 Finite colimits in Set
  • 3.4 Other notions in Set

If you’re joining us in progress, please feel free to add in any questions you might have about previous material as well – it’s never too late to join us all.

Those who aren’t able to jump into the hangout (due to hardware issues or the 10 person limit) are encouraged to chat within the hangout IM and follow along with the live stream. If you’re stuck and can’t make it, it will be archived on our YouTube Channel for later consumption.

Gitter Instant Messaging Client

Group participant Steven Shaw has kindly set up an open chat/IM space for us using Gitter through GitHub (apparently one of the benefits of having some hard-core coders in the group).  Gitter is one of the few IM clients out there that allows the use of LaTeX.  You can log in with your GitHub account and feel free to post questions/thoughts there as well. Since it’s always up, feel free to use it during our online meetings or throughout the week.

Thanks Steven!

Coming Up: Week Four §4.1-§4.2

We’ll have finished some of the introductory material and be getting into some new material, so those who were waiting for the serious material to start, get ready. We’ll be covering:

  • 4.1 Monoids
  • 4.2 Groups

Online video

For those who aren’t aware, or haven’t checked recently, we’ve been adding a lot of material in the resources section of the site here.  In particular, I’ll draw your attention to the video section which includes The Catsters’ Category Theory Videos.

Week One Meeting: §1.1 – §2.2

Week One’s Archived Video

For those who may have missed it, the video recording of our first meeting follows below.

I would usually intimate that some may also use it for review, but it primarily contains a brief overview of some of the resources available along with some administrative material overview. It’s much more scant on actual mathematics than I hope/expect they will generally be in the future.

Administrative Note

It appears that we had over 20 people for the first session, though we’re limited to 10 active participants who have access to streaming their audio/video into the session. Apologies to others who weren’t able to more actively participate.

Keep in mind that one should hopefully still be able to add additional material via the hangouts IM functionality or by the Q&A functionality (see notes below). For those who are in the audio/video portion of the hangout, you’ll be expected to participate and contribute to the discussion.s (Perhaps you might present a problem/solution to the group?)

If you don’t have much to say (or don’t have the proper equipment (webcam or microphone)), kindly “step” out of the broadcast and watch the live stream for a while and allow others to have a shot as well. Perhaps we might arrange some method for people to rotate in/out on a regular basis? Suggestions for this are welcome.

Those not actively participating in the session can always watch the live stream through the group’s YouTube channel.

I did notice one or two interesting side-conversations taking place within the hangout’s chat (though I’m at a loss to know if/where it was archived). At present, we’ve got more than enough time in these sessions that instead of typing respondents are more than welcome to bring up their commentary to supply everyone with a more fleshed out conversation. (There does seem to be a difference between the IM/chat within the main window of the stream and that from the separate hangouts window, which is archived and accessible after the fact, so perhaps using the latter is preferable for archive purposes, as well as being more accessible to the balance of the group.


Google Hangouts has a functionality known as Q&A to which one can write in questions that the group can work on answering during the session.  To access it at any time, go to the page for the hangout, click on the “play button” in the video portion of the screen, then in the top right hand corner of the “video” (which obviously won’t be playing until the set meeting time) click on the 3×3 square grid (just to the right of the question mark icon), and choose the Q&A pop up option. This will open up a bar on the right hand side of the screen where one can click on ask a new question at the bottom to post their question.

You can also always register at the group’s main site and post your questions there for everyone to work on/answer via the comments section during the week

Coming Up: Week Two §2.3-§3.2

This week’s Google Hangout (RSVP here) will cover problems/questions from week two of the syllabus:

  •  2.3 Ologs
  • 3 Fundamental Considerations in Set
  • 3.1 Products and coproducts
  • 3.2 Finite limits in Set

Everyone will generally be expected to have read the appropriate sections and bring their questions/issues so the group can attempt to cover and clarify any issues anyone may be having.

If it helps one or more people to ensure that they’ve got the material down well, I’m sure the group would welcome anyone who might like to present/walk their way through one or more of the problems in the relevant sections – particularly problems whose answers left out some reasonable level of detail. If you’d like to offer to do this, please put a comment in below, so we can schedule some time during the session to accommodate this.

Though we’re off to a “slow” start, things will pick up rapidly as we progress, so please don’t hesitate to ask questions here on the blog, through hangouts, or via anyone’s office hours.

Online Video Conference Meetings

Now that we’re a few days into the group and most everyone seems to be registered and has had some time to respond to the initial questionnaire, I thought I’d set a time for our online meeting(s)/sessions.

Most people have indicated to me that they’ve either already bought/received the text and have started reading, or are about to begin. I have a feeling  that most will find the first several chapters very basic, (but we’re all here to help those that don’t).

There are currently 25 people registered!

As a reminder, as mathematically “sophisticated” readers, we’ll be using a “flipped” format for our meetings, so everyone will generally be expected to have read the appropriate sections and worked on some problems/examples ahead of the meetings so that they can bring any problems/issues they may have to get some help from the rest of the group.

Upcoming meetings

As an initial meeting, let’s aim for:

Friday, June 5, 2015 at 6:45 pm Pacific / 9:45 pm Eastern on Google Hangouts.

In this session, we can get any  basic administrative things out of the way and discuss any problems/issues anyone may have with the first two chapters which primarily cover some initial basics including set theory and functions.

For our regular, weekly standing meetings, let’s shoot for

Monday evenings at 6:45pm Pacific / 9:45 pm Eastern.

Hopefully this weekly time will work for those on both coasts of the Americas without any undue burden.  We can attempt to record sessions for those who aren’t able to make it due to time zone or other conflicts, but no one seemed to have any issues with Mondays and we seem to be roughly split with participants on both coasts.

The second meeting will be on:

Monday, June 8, 2015 at 6:45 pm Pacific / 9:45 pm Eastern on Google Hangouts.

We’ll cover everyone’s questions from the following sections (which everyone will have been expected to have read beforehand):

2.3 Ologs; 3.1 Products and coproducts; and 3.2 Finite limits in Set

If necessary, outside of this, we can try to hold an alternate time on Saturday, which was the other day no one seemed to have issues with. An earlier time may help those who live outside the Americas as well. Anyone who’d like an alternate time is invited to mention it in the comments below.

Due to platform requirements and the diversity of the participants, Google Hangouts seems to work for everyone and allows video, audio, screensharing, and most of the other useful features we might want.  As I recall, one doesn’t necessarily need a Google+ account, but can login through their gmail interface (typically with a browser plugin), or via the hangout app on the bigger cell phone platforms.

You can click on the individual links for the appropriate date to find/join the particular hangout on this page (above), or on the individual links listed within the syllabus.

Office Hours

As a reminder, most participants have indicated office hours during which they are available to chat with others to offer assistance or help. I’d hope that everyone would try to login to Google Hangouts and make themselves relatively available to others to offer assistance, if they’re able during their stated office hours. Remember that helping others can assist you in reviewing/clarifying the material for yourself as well.

As always, additional assistance can also be easily had by making a post here to the “blog” with a specific question or problem and everyone can take a stab at helping out through the comments on that particular post.

If you haven’t already done so, feel free to do the following:

We all look forward to seeing you soon!

How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics

For those who are intimidated by the thought of higher mathematics, but are still considering joining our Category Theory Summer Study Group, I’ve just come across a lovely new book by Eugenia Cheng entitled How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics.

It just came out in the U.S. market on May 5, 2015, so it’s very new in the market. My guess is that even those who aren’t intimidated will get a lot out of it as well. A brief description of the book follows:

“What is math? How exactly does it work? And what do three siblings trying to share a cake have to do with it? In How to Bake Pi, math professor Eugenia Cheng provides an accessible introduction to the logic and beauty of mathematics, powered, unexpectedly, by insights from the kitchen: we learn, for example, how the béchamel in a lasagna can be a lot like the number 5, and why making a good custard proves that math is easy but life is hard. Of course, it’s not all cooking; we’ll also run the New York and Chicago marathons, pay visits to Cinderella and Lewis Carroll, and even get to the bottom of a tomato’s identity as a vegetable. This is not the math of our high school classes: mathematics, Cheng shows us, is less about numbers and formulas and more about how we know, believe, and understand anything, including whether our brother took too much cake.

At the heart of How to Bake Pi is Cheng’s work on category theory—a cutting-edge “mathematics of mathematics.” Cheng combines her theory work with her enthusiasm for cooking both to shed new light on the fundamentals of mathematics and to give readers a tour of a vast territory no popular book on math has explored before. Lively, funny, and clear, How to Bake Pi will dazzle the initiated while amusing and enlightening even the most hardened math-phobe.”

Dr. Cheng recently appeared on NPR’s Science Friday with Ira Flatow to discuss her book.  You can listen to the interview below. Most of the interview is about her new book. Specific discussion of category theory begins about 14 minutes into the conversation.

Dr. Eugenia Cheng can be followed on Twitter @DrEugeniaCheng. References to her new book as well as some of her syllabi and writings on category theory have been added to our resources pages for download/reading.

Eugenia Cheng, mathematician
Eugenia Cheng, mathematician

Lurie and Categorifying the Fourier Transform | Peter Woit (Not Even Wrong)

Lurie and Categorifying the Fourier Transform | Peter Woit (Not Even Wrong)

Commutative Diagrams in LaTeX


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.

The Category Theory Site Is Now Live

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.

[Editor’s Note: This post was originally posted at]