## 📺 Proving Brouwer’s Fixed Point Theorem | PBS Infinite Series on YouTube

Proving Brouwer's Fixed Point Theorem by Tai-Danae Bradley from PBS Infinite Series | YouTube
There is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra. Analogous to the relationship between geometry and algebra, there is a mathematical “portal” from a looser version of geometry -- topology -- to a more “sophisticated” version of algebra. This portal can take problems that are very difficult to solve topologically, and recast them in an algebraic light, where the answers may become easier. Written and Hosted by Tai-Danae Bradley; Produced by Rusty Ward; Graphics by Ray Lux; Assistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington; Made by Kornhaber Brown (www.kornhaberbrown.com)

I had originally started following Tai-Danae Bradley on Instagram having found her account via the #math tag. Turns out she’s burning up the world explaining some incredibly deep and complex mathematics in relatively simple terms. If you’re into math and not following her work already, get with the program. She’s awesome!

Personal Website: http://www.math3ma.com/
Instagram: @math3ma

While this particular video leaves out a masters degree’s worth of detail, it does show some incredibly powerful mathematics by analogy. The overall presentation and descriptions are quite solid for leaving out as much as they do. This may be some of the best math-based science communication I’ve seen in quite a while.

I must say that I have to love and laugh at the depth and breadth of the comments on the video too. At best, this particular video, which seems to me to be geared toward high school or early college viewers and math generalists, aims to introduce come general topics and outline an incredibly complex proof in under 9 minutes. People are taking it to task for omitting “too much”! To completely understand and encapsulate the entirety of the topics at hand one would need coursework including a year’s worth of algebra, a year’s worth of topology including some algebraic topology, and a minimum of a few months worth of category theory. Even with all of these, to fill in all the particular details, I could easily see a professor spending an hour at the chalkboard filling in the remainder without any significant handwaving. The beauty of what she’s done is to give a very motivating high level perspective on the topic to get people more interested in these areas and what can be done with them. For the spirit of the piece, one might take her to task a bit for not giving more credit to the role Category Theory is playing in the picture, but then anyone interested is going to spend some time on her blog to fill in a lot of those holes. I’d challenge any of the comments out there to attempt to do what she’s done in under 9 minutes and do it better.

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## 📗 Read pages i-14 of The Theoretical Minimum: What You Need to Know to Start Doing Physics by Leonard Susskind and George Hrabovsky

📖 Read pages i-14 of The Theoretical Minimum: What You Need to Know to Start Doing Physics by Leonard Susskind and George Hrabovsky (Basic Books, , ISBN: 978-0465028115)

I’d read a portion of this in the past, but thought I’d circle back to it when I saw it sitting on the shelf at the library before the holidays. It naturally helps to have had lots of physics in the past, but this has a phenomenally clear and crisp presentation of just the basics in a way that is seldom if ever seen in actual physics textbooks.

### Highlights, Quotes, & Marginalia

Lecture One: The Nature of Classical Physics

There is a very simple rule to tell when a diagram represents a deterministic reversible law. If every state has a single unique arrow leading into it, and a single arrow leading out of it, then it is a legal deterministic reversible law.

Highlight (yellow) – 1. The Nature of Classical Physics > Page 9

There’s naturally a much more sophisticated and subtle mathematical way of saying this. I feel like I’ve been constantly tempted to go back and look at more category theory, and this may be yet another motivator.
Added on Wednesday, January 4, 2018 late evening

The rule that dynamical laws must be deterministic and reversible is so central to classical physics that we sometimes forget to mention it when teaching the subject. […] minus-first law [: …] undoubtedly the most fundamental of all physics laws–the conservation of information. The conservation of information is simply the rule that every state has one arrow in and one arrow out. It ensures that you never lose track of where you started.

Highlight (yellow) – 1. The Nature of Classical Physics > Page 9-10

This is very simply and naturally stated, but holds a lot of complexity. Again I’d like to come back and do some serious formalization of this and reframe it in a category theory frameork.
Added on Wednesday, January 4, 2018 late evening

There is evan a zeroth law […]

Highlight (gray) – 1. The Nature of Classical Physics > Page 9

spelling should be even; I’m also noticing a lot of subtle typesetting issues within the physical production of the book that are driving me a bit crazy. Spaces where they don’t belong or text not having clear margins at the tops/bottoms of pages. I suspect the math and layout of diagrams and boxes in the text caused a lot of problems in their usual production flow.
Added on Wednesday, January 4, 2018 late evening

##### Guide to highlight colors

Yellow–general highlights and highlights which don’t fit under another category below
Orange–Vocabulary word; interesting and/or rare word
Blue–Interesting Quote
Gray–Typography Problem
Red–Example to work through

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## 👓 Limits and Colimits, Part 1 (Introduction) | Math3ma

Limits and Colimits, Part 1 (Introduction) by Tai-Danae Bradley (Math3ma)
I'd like to embark on yet another mini-series here on the blog. The topic this time? Limits and colimits in category theory! But even if you're not familiar with category theory, I do hope you'll keep reading. Today's post is just an informal, non-technical introduction. And regardless of your categorical background, you've certainly come across many examples of limits and colimits, perhaps without knowing it! They appear everywhere - in topology, set theory, group theory, ring theory, linear algebra, differential geometry, number theory, algebraic geometry. The list goes on. But before diving in, I'd like to start off by answering a few basic questions.

A great little introduction to category theory! Can’t wait to see what the future installments bring.

Interestingly I came across this on Instagram. It may be one of the first times I’ve seen math at this level explained in pictorial form via Instagram.

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## 🔖 Adjoint School, ACT 2018 (Applied Category Theory)

Adjoint School, ACT 2018 (Applied Category Theory)
The Workshop on Applied Category Theory 2018 takes place in May 2018. A principal goal of this workshop is to bring early career researchers into the applied category theory community. Towards this goal, we are organising the Adjoint School. The Adjoint School will run from January to April 2018.

There’s still some time left to apply. And if nothing else, this looks like it’s got some interesting resources.

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Basic Category Theory (arxiv.org)
This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.

Tom Leinster has released a digital e-book copy of his textbook Basic Category Theory on arXiv [1]

My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory! It’s free:

https://arxiv.org/abs/1612.09375

It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.

Huh? What’s a ‘universal property’?

In category theory, we try to describe things by saying what they do, not what they’re made of. The reason is that you can often make things out of different ingredients that still do the same thing! And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do.

A universal property amounts to a precise description of what an object does.

Universal properties show up in three closely connected ways in category theory, and Tom’s book explains these in detail:

through representable functors (which are how you actually hand someone a universal property),

through limits (which are ways of building a new object out of a bunch of old ones),

through adjoint functors (which give ways to ‘freely’ build an object in one category starting from an object in another).

If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom’s book! It’s not easy to learn this stuff – but it’s good for your brain. It literally rewires your neurons.

Here’s what he wrote, over on the category theory mailing list:

…………………………………………………………………..

Dear all,

My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:

https://arxiv.org/abs/1612.09375

It’s also freely editable, under a Creative Commons licence. For instance, if you want to teach a class from it but some of the examples aren’t suitable, you can delete them or add your own. Or if you don’t like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros. Just go the arXiv, download, and edit to your heart’s content.

There are lots of good introductions to category theory out there. The particular features of this one are:
• It’s short.
• It doesn’t assume much.
• It sticks to the basics.

### References

[1]
T. Leinster, Basic Category Theory, 1st ed. Cambridge University Press, 2014.
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## Emily Riehl’s new category theory book has some good company

Instagram filter used: Clarendon

Photo taken at: UCLA Bookstore

I just saw Emily Riehl‘s new book Category Theory in Context on the shelves for the first time. It’s a lovely little volume beautifully made and wonderfully typeset. While she does host a free downloadable copy on her website, the book and the typesetting is just so pretty, I don’t know how one wouldn’t purchase the physical version.

I’ll also point out that this is one of the very first in Dover’s new series Aurora: Dover Modern Math Originals. Dover has one of the greatest reprint collections of math texts out there, I wish them the best in publishing new works with the same quality and great prices as they always have! We need more publishers like this.

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## [1609.02422] What can logic contribute to information theory?

[1609.02422] What can logic contribute to information theory? by David Ellerman (128.84.21.199)
Logical probability theory was developed as a quantitative measure based on Boole's logic of subsets. But information theory was developed into a mature theory by Claude Shannon with no such connection to logic. But a recent development in logic changes this situation. In category theory, the notion of a subset is dual to the notion of a quotient set or partition, and recently the logic of partitions has been developed in a parallel relationship to the Boolean logic of subsets (subset logic is usually mis-specified as the special case of propositional logic). What then is the quantitative measure based on partition logic in the same sense that logical probability theory is based on subset logic? It is a measure of information that is named "logical entropy" in view of that logical basis. This paper develops the notion of logical entropy and the basic notions of the resulting logical information theory. Then an extensive comparison is made with the corresponding notions based on Shannon entropy.

Ellerman is visiting at UC Riverside at the moment. Given the information theory and category theory overlap, I’m curious if he’s working with John Carlos Baez, or what Baez is aware of this.

Based on a cursory look of his website(s), I’m going to have to start following more of this work.

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## Eugenia Cheng, author of How to Bake Pi, on Colbert Tonight

The author of one of the best math (and cooking) books of the year is on Stephen Colbert's show tonight.

Earlier this year, I read Eugenia Cheng’s brilliant book How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics. Tonight she’s appearing (along with Daniel Craig apparently) on the The Late Show with Stephen Colbert. I encourage everyone to watch it and read her book when they get the chance.

You can also read more about her appearance from Category Theorist John Carlos Baez here: Cakes, Custard, Categories and Colbert | The n-Category Café

My brief review of her book on GoodReads.com:

How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics by Eugenia Cheng
My rating: 4 of 5 stars

While most of the book is material I’ve known for a long time, it’s very well structured and presented in a clean and clear manner. Though a small portion is about category theory and gives some of the “flavor” of the subject, the majority is about how abstract mathematics works in general.

I’d recommend this to anyone who wants to have a clear picture of what mathematics really is or how it should be properly thought about and practiced (hint: it’s not the pablum you memorized in high school or even in calculus or linear algebra). Many books talk about the beauty of math, while this one actually makes steps towards actually showing the reader how to appreciate that beauty.

Like many popular books about math, this one actually has very little that goes beyond the 5th grade level, but in examples that are very helpfully illuminating given their elementary nature. The extended food metaphors and recipes throughout the book fit in wonderfully with the abstract nature of math – perhaps this is why I love cooking so much myself.

I wish I’d read this book in high school to have a better picture of the forest of mathematics.

More thoughts to come…

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## How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics | Category Theory

Eugenia Cheng's new book How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics helps to introduce the public to category theory.

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 Category Theory resources pages for download/reading.

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## 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).

## Registration

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.

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.

## Textbook

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.

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.

## Overview

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.

### TikZ-CD: Florêncio Neves

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## Category Theory Summer Study Group 2015

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

## Syllabus

Initial details for putting  the group together can be found at http://boffosocko.com/2015/05/21/category-theory-anyone/.

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 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 $\LaTeX$ 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.

### Meetings

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.

## 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)

• 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

• 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

#### Week Thirteen: August 16 (32 pages)

• 7.2 Categories of functors

## 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 Other:

#### Videoconferences

 Platform Can use Can’t use Prefer Not to Use Google Hangouts Skype ooVoo Other

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

Weekdays:

Weekends:

Weekdays:

Weekends:

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:

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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?

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:

### References

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)

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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