Patrick Stevens

Another short post to highlight the existence of an article about the Monadicity Theorems, in which I prove one direction of both the Crude and Precise versions.

Just a post to draw attention to my new article about the General Adjoint Functor Theorem.

In the Part III Topics in Set Theory course, we have used forcing to show the consistency of the Continuum Hypothesis, and we are about to show the consistency of its negation.

As an exercise in understanding the definitions involved, I find the Eilenberg-Moore category of a certain functor.

Now that we’ve had the definition of an exponential, we move on to the Heyting algebra, pages 129 through 131 of Awodey.

These are located on pages 114 through 118 of Awodey.

Now we come to Chapter 6 of Awodey, on exponentials, pages 119 through 128.

It has been proposed to me that if one is to play the National Lottery, one should be sure to select one’s own numbers instead of allowing the machine to select them for you.

We’ve seen how limits are formed, and that they exist iff products and equalisers do.

We have just had the definition of a pullback; now in Awodey pages 95 through 100 we’ll see some more about them, and after that we’ll get the more general unifying idea of the limit in pages 100 through 105.

I’m going to skip pages 85 through 88 of Awodey for the moment, because time is starting to get short and I want to make sure I’m doing stuff which is relevant to the Part III course on category theory.

I go into this chapter hoping that it will be on things I already know about group theory.

Exercise 1 is easy: at the end of Chapter 2 the corresponding products statement was proved, and the obvious dual statement turns out to be this one.

This is pages 62 through 71 of Awodey, on equalisers and coequalisers.

I don’t have strong preconceptions about this chapter.

This is on pages 48 through 52 of Awodey, covering the hom-sets section and the exercises at the end of Chapter 2.

This is on pages 38 through 48 of Awodey.

This is pages 33 to 38 of Awodey.

This post is on pages 29 through 33 of Awodey.

Here I explain proof by contradiction so that anyone who has ever done a sudoku and seen algebra may understand it.

Here, I will be going through the Free Categories and Foundations sections of Awodey’s Category Theory - pages 18 through 25.

Here, I will be going through the Isomorphisms and Constructions sections of Awodey’s Category Theory - pages 12 through 17.

This post will cover the initial “examples” section of Category Theory.

In the summer of 2015, I worked through Awodey’s Category Theory, and I produced a large collection of posts as I tried to understand its contents.

The next few posts will be following me on my journey through the book Category Theory, by Steve Awodey.

In which I am a wizard.

I’m clearing out my computer, and found a file which may as well be here.

I recently saw a problem from an Indian maths olympiad: There is a square arrangement made out of n elements on each side (n^2 elements total).

I’ve just come back from seeing Interstellar, a film of peril and physics.

In which I provide my favourite carols and my favourite renditions of them.

*Wherein I detail the most beautiful proof of a theorem I’ve ever seen, in a bite-size form suitable for an Anki deck.

A very brief post about the solution to a problem I came across in Python.

One day, a group of investors came to Bezos in the Temple and begged of him, “You are known throughout the land for your wisdom.

I have a limited form of perfect (absolute) pitch, which I am sometimes asked about.

A couple of weeks ago, someone opined to me that there was a type of person who was just able to sit down and play at the piano, without sheet music.

In my activities on The Student Room, a student forum, someone (let’s call em Entity, because I like that word) recently asked me about the following question.

Recently, a friend re-introduced me to the joys of the nonogram (variously known as “hanjie” or “griddler”).

I have seen many glowing reviews of Soylent, and many vitriolic naturalistic arguments against it.

This comes up quite frequently, but I’ve been stuck for an easy memory-friendly way to do this.

This is part of what has become a series on discovering some fairly basic mathematical results, and/or discovering their proofs.

Prof Körner told us during the IB Metric and Topological Spaces course that the real meat of the course (indeed, its hardest theorem) was “a metric space is sequentially compact iff it is compact”.

This is to detail a much easier proof (at least, I find it so) of Cayley-Hamilton than the ones which appear on the Wikipedia page.

As part of the recent series on how I approach maths problems, I give another one here (question 14 on the Maths Tripos IB 2007 paper 4).

This post is to be a list of conformal mappings, so that I can get better at answering questions like “Find a conformal mapping from <this domain> to <this domain>”.

I’m running through my Analysis proofs, trying to work out which ones are genuinely hard and which follow straightforwardly from my general knowledge base.

A little while ago I set myself the exercise of stating and proving the Contraction Mapping Theorem.

It has been commented to me that it’s quite hard to find out (on the Internet) what different games involve.

I feel that I can write a sonnet well.

The aim of this post is twofold: to find out whether a certain mental habit of mine is common, and to draw parallels between that habit and the writing of essays.

Recently, I was browsing the Wolfram Community forum, and I came across the following question: What are the symbols @, #, / in Mathematica?