This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known temporal logics---such as Linear and Metric Temporal Logic (LTL and MTL)---embed within the logic of temporal type theory.
The types in this theory represent "behavior types". The language is rich enough to allow one to define arbitrary hybrid dynamical systems, which are mixtures of continuous dynamics---e.g. as described by a differential equation---and discrete jumps. In particular, the derivative of a continuous real-valued function is internally defined.
We construct a semantics for the temporal type theory in the topos of sheaves on a translation-invariant quotient of the standard interval domain. In fact, domain theory plays a recurring role in both the semantics and the type theory.
Dear god, I wish Ilyas had a traditional blog with a true feed, but I’m willing to put up with the inconvenience of manually looking him up from time to time to see what he’s writing about quantum mechanics, quantum computing, category theory, and other areas of math.
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 just finished a course on Algebraic Geometry through UCLA Extension, which was geared toward non-traditional math students and professionals, and wish I had known about Smith’s textbook when I’d started. I did spend some time with Cox, Little, and O’Shea’s Ideals, Varieties, and Algorithms which is a pretty good introduction to the area, but written a bit more for computer scientists and engineers in mind rather than the pure mathematician, which might recommend it more toward your audience here as well. It’s certainly more accessible than Hartshorne for the faint-of-heart.
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]
I’ve started an informal online course based on this book on the Azimuth Forum. I’m getting pretty sick of the superficial quality of my interactions on social media. This could be a way to do something more interesting.
The idea is that you can read chapters of this book, discuss them, try the exercises in the book, ask and answer questions, and maybe team up to create software that implements some of the ideas. I’ll try to keep things moving forward. For example, I’ll explain some stuff and try to help answer questions that people are stuck on. I may also give some talks or run discussions on Google Hangouts or similar software—but only when I have time: I’m more of a text-based guy. I may get really busy some times, and leave the rest of you alone for a while. But I like writing about math for at least 15 minutes a day, and more when I have time. Furthermore, I’m obsessed with applied category theory and plan to stay that way for at least a few more years.
If this sounds interesting, let me know here—and please visit the Azimuth Forum and register!
Some awesome news just as I’ve wrapped up a class on Algebraic Geometry and was actively looking to delve into some category theory over the summer. John Carlos Baez announced that he’s going to offer an online course in applied category theory. He’s also already posted some videos and details!
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!
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.
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 Green–Reference to read Blue–Interesting Quote Gray–Typography Problem Red–Example to work through
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.
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.
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.
My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory! It’s free:
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:
My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:
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.
T. Leinster, Basic Category Theory, 1st ed. Cambridge University Press, 2014.
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.
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.
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.
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.