category theory | Chris Aldrich

📖 Read pages i-8 of Category Theory for the Sciences by David I. Spivak

📖 Read pages i-8, front matter and Chapter 1: Introduction of Category Theory for the Sciences by David I. Spivak.

Highlights, Quotes, Annotations, & Marginalia

But it is easy to think we are in agreement, when we really are not. Modeling our thoughts on heuristics and graphics may be convenient for quick travel down the road, but we are liable to miss our turnoff at the first mile. The danger is in mistaking convenient conceptualizations for what is actually there.

A functor is like a conductor of mathematical truth.

The answer is that when we formalize our ideas, our understanding is clarified.

Creativity demands clarity of thinking, and to think clearly about a subject requires an organized understanding of how its pieces fit together. Organization and clarity also lead to better communication with others. Academics often say they are paid to think and understand, but that is not the whole truth. They are paid to think, understand, and communicate their findings.

👓 nPOV | nLab

Read POV (ncatlab.org)

Wikipedia enforces its entries to adopt an NPOV – a neutral point of view . This is appropriate for an encyclopedia.

However, the nLab is not Wikipedia, nor is it an encyclopedia, although it does aspire to provide a useful reference in many areas (among its other purposes). In particular, the $n$ Lab has a particular point of view, which we may call the $n$ POV or the n- categorical point of view .

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🔖 Introduction to Category Theory | UCLA Continuing Education

Bookmarked Introduction to Category Theory (UCLA Continuing Education)

This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra.

Category theory, since its development in the 1940s, has assumed an increasingly center-stage role in formalizing mathematics and providing tools to diverse scientific disciplines, most notably computer science. A category is fundamentally a family of mathematical obejcts (e.g., numbers, vector spaces, groups, topological spaces) along with “mappings” (so-called morphisms) between these objects that, in some defined sense, preserve structure. Taking it one step further, one can consider morphisms (so-called functors) between categories. This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra. Topics to be discussed include: isomorphism; products and coproducts; dual categories; covariant, contravariant, and adjoint functors; abelian and additive categories; and the Yoneda Lemma. The course should appeal to devotees of mathematical reasoning, computer scientists, and those wishing to gain basic insights into a hot area of mathematics.

January 8, 2019 - March 19, 2019
Tuesday 7:00PM - 10:00PM
Location: UCLA
Instructor: Michael Miller
Fee: $453.00

The new catalog is out today and Mike Miller’s Winter class in Category Theory has been officially announced.

Oddly, it wasn’t listed in the published physical catalog, but it’s available online. I hope that those interested in mathematics will register as well as those who are interested in computer science.

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Gems And Astonishments of Mathematics: Past and Present—Lecture One

Last night was the first lecture of Dr. Miller’s Gems And Astonishments of Mathematics: Past and Present class at UCLA Extension. There are a good 15 or so people in the class, so there’s still room (and time) to register if you’re interested. While Dr. Miller typically lectures on one broad topic for a quarter (or sometimes two) in which the treatment continually builds heavy complexity over time, this class will cover 1-2 much smaller particular mathematical problems each week. Thus week 11 won’t rely on knowing all the material from the prior weeks, which may make things easier for some who are overly busy. If you have the time on Tuesday nights and are interested in math or love solving problems, this is an excellent class to consider. If you’re unsure, stop by one of the first lectures on Tuesday nights from 7-10 to check them out before registering.

Lecture notes

For those who may have missed last night’s first lecture, I’m linking to a Livescribe PDF document which includes the written notes as well as the accompanying audio from the lecture. If you view it in Acrobat Reader version X (or higher), you should be able to access the audio portion of the lecture and experience it in real time almost as if you had been present in person. (Instructions for using Livescribe PDF documents.)

We’ve covered the following topics:

Class Introduction
Erdős Discrepancy Problem
- n-cubes
- Hilbert’s Cube Lemma (1892)
- Schur (1916)
- Van der Waerden (1927)
Sylvester’s Line Problem (partial coverage to be finished in the next lecture)
- Ramsey Theory
- Erdős (1943)
- Gallai (1944)
- Steinberg’s alternate (1944)
- DeBruijn and Erdős (1948)
- Motzkin (1951)
- Dirac (1951)
- Kelly & Moser (1958)
- Tao-Green Proof
Homework 1 (homeworks are generally not graded)

Over the coming days and months, I’ll likely bookmark some related papers and research on these and other topics in the class using the class identifier MATHX451.44 as a tag in addition to topic specific tags.

Course Description

Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.

Suggested Prerequisites

Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory. Most in the class are taking the course for “fun” and the enjoyment of learning, so there is a huge breadth of mathematical abilities represented–don’t not take the course because you feel you’ll get lost.

I’ve written some general thoughts, hints, and tips on these courses in the past.

Renovated Classrooms

I’d complained to the UCLA administration before about how dirty the windows were in the Math Sciences Building, but they went even further than I expected in fixing the problem. Not only did they clean the windows they put in new flooring, brand new modern chairs, wood paneling on the walls, new projection, and new white boards! I particularly love the new swivel chairs, and it’s nice to have such a lovely new environment in which to study math.

Category Theory for Winter 2019

As I mentioned the other day, Dr. Miller has also announced (and reiterated last night) that he’ll be teaching a course on the topic of Category Theory for the Winter quarter coming up. Thus if you’re interested in abstract mathematics or areas of computer programming that use it, start getting ready!

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As I get amped up for the start of Mike Miller’s Fall math class Gems and Astonishments of Mathematics, which is still open for registration, I’m even more excited that he’s emailed me to say that he’ll be teaching Category Theory for the Winter Quarter in 2019!!

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Following Ilyas Khan

Followed Ilyas Khan (LinkedIn)

Co-Founder and CEO at Cambridge Quantum Computing

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.

Reply to A (very) gentle comment on Algebraic Geometry for the faint-hearted | Ilyas Khan

Replied to A (very) gentle comment on Algebraic Geometry for the faint-hearted by

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 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/

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🔖 [1803.05316] Seven Sketches in Compositionality: An Invitation to Applied Category Theory

Bookmarked Seven Sketches in Compositionality: An Invitation to Applied Category Theory by Brendan Fong, David I. Spivak (arxiv.org)

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]

This is the textbook that John Carlos Baez is going to use for his online course in Applied Category Theory.

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