Co-Founder and CEO at Cambridge Quantum Computing
Ilyas Khan (LinkedIn)This short article is the result of various conversations over the course of the past year or so that arose on the back of two articles/blog pieces that I have previously written about Category Theory (here and here). One of my objectives with such articles, whether they be on aspects of quantum computing or about aspects of maths, is to try and de-mystify as much of the associated jargon as possible, and bring some of the stunning beauty and wonder of the subject to as wide an audience as possible. Whilst it is clearly not possible to become an expert overnight, and it is certainly not my objective to try and provide more than an introduction (hopefully stimulating further research and study), I remain convinced that with a little effort, non-specialists and even self confessed math-phobes can grasp some of the core concepts. In the case of my articles on Category Theory, I felt that even if I could generate one small gasp of excited comprehension where there was previously only confusion, then the articles were worth writing.
I’ve enjoyed your prior articles on category theory which have spurred me to delve deeper into the area. For others who are interested, I thought I’d also mention that physicist and information theorist John Carlos Baez at UCR has recently started an applied category theory online course which I suspect is a bit more accessible than most of the higher graduate level texts and courses currently out. For more details, I’d suggest starting here: https://johncarlosbaez.wordpress.com/2018/03/26/seven-sketches-in-compositionality/
This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration. The tour takes place over seven sketches, each pairing an evocative application, such as databases, electric circuits, or dynamical systems, with the exploration of a categorical structure, such as adjoint functors, enriched categories, or toposes. No prior knowledge of category theory is assumed. [.pdf]
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/
Twitter: @math3ma
Instagram: @math3ma
YouTube series: PBS Infinite Series
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.
Lecture one of six in an introductory set of lectures on category theory.
Take Away from the lecture: Morphisms are just as important as the objects that they morph. Many different types of mathematical constructions are best described using morphisms instead of elements. (This isn’t how things are typically taught however.)
Would have been nice to have some more discussion of the required background for those new to the broader concept. There were a tremendous number of examples from many areas of higher math that many viewers wouldn’t have previously had. I think it’s important for them to know that if they don’t understand a particular example, they can move on without much loss as long as they can attempt to apply the ideas to an area of math they are familiar with. Having at least a background in linear algebra and/or group theory are a reasonable start here.
In some of the intro examples it would have been nice to have seen at least one more fully fleshed out to better demonstrate the point before heading on to the multiple others which encourage the viewer to prove some of the others on their own.
Thanks for these Steven, I hope you keep making more! There’s such a dearth of good advanced math lectures on the web, I hope these encourage others to make some of their own as well.
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.
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.
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.
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 […]
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
Yellow–general highlights and highlights which don’t fit under another category below
Orange–Vocabulary word; interesting and/or rare word
Green–Reference to read
Blue–Interesting Quote
Gray–Typography Problem
Red–Example to work through
Jeremy Kun (Twitter)Sub things and glue things and red things and blue things! https://twitter.com/math3ma/status/948290058930581505
This one has a little car. Say, what a lot of categories there are!
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.
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.
https://www.instagram.com/p/Bdd8vmfBQvG/
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.
h/t John Carlos Baez
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.
h/t to John Carlos Baez for the notice:
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:
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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.
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.
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.
Based on a cursory look of his website(s), I’m going to have to start following more of this work.
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 starsWhile 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…
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.
