• # Slides on algebraic theories

I spent a couple months this summer doing a very enjoyable research project supervised by Martin Hyland, looking at several topics to do with algebraic theories. It was a great opportunity to get some idea of what research can be like, and to learn from someone with much more experience.

I recently gave a short talk about the project, and I’ve uploaded the slides. The talk was aimed at people who don’t know any category theory, so I spent the first half explaining equational theories in some detail, while in the second half I tried to give a more intuitive description of the problems that I was actually working on.

• # What is a quasi-category?

A category describes a collection of objects with composable morphisms between them. In a 2-category you also have 2-morphisms between morphisms with matching domain and codomain. For example there is a 2-category $$\text{Cat}$$ with objects given by (small) categories, 1-morphisms given by functors between categories, and 2-morphisms given by natural transformations between functors. Continuing this pattern gives the idea of an $$n$$-category, where you have objects, 1-morphisms, 2-morphisms, and so on up to $$n$$-morphisms. The ultimate such structure is an $$\infty$$-category, with a never ending sequence of morphisms between morphisms between … between morphisms.

In a weak $$n$$- or $$\infty$$-category, associativity and identity laws only hold up to (coherent) isomorphism. In fact we don’t even need to require composition of morphisms to be uniquely defined! We only ask that all possible composites are uniquely isomorphic (where uniquely is interpreted in an appropriate higher categorical sense).

Weak higher categories are generally hard to understand, or even define correctly, and are currently an area of active research. However an especially well-behaved special case is that of a weak $$(\infty, 1)$$-category, where all $$k$$-morphisms for $$k > 1$$ are equivalences. Quasi-categories are one of the earliest and most important ways to formalise the idea of a weak $$(\infty, 1)$$-category. The definition of a quasi-category turns out to be less horrendously complicated than you might expect! The key is to understand simplicial sets, which are a kind of combinatorial approach to shape. Then a quasi-category is a simplicial set satisfying a condition that asks for certain “compositions of morphisms” to exist. We’ll start with a quick overview of simplicial sets, emphasising geometric intuition, before explaining the precise definition of a quasi-category.

## Simplicial sets

The idea of a simplicial set is to describe a shape by building it up out of simplices – points, lines, triangles, and so on. Moreover each simplex should come with an ordering on its vertices, so for example edges have a direction and triangles have an orientation.

• # Creative perspective

Tags: Art, Perspective

I’ve just started reading Ernest W. Watson’s Creative Perspective for Artists and Illustrators. I found that it cleared up several confusions I had about how perspective works, why correct perspective can seem distorted, and when and in what ways you might want to bend the rules.

I’ll follow Watson’s convention and refer to scientific perspective when I want to emphasise the idea of sticking precisely to the standard rules.

## What is perspective?

Stand in front of a window, and (keeping your head still and closing one eye) use a marker to trace what you see on the window. Then the resulting 2-dimensional picture will be in scientific perspective. The resulting image will match the image captured by an idealised camera.

• # Three proofs that limits commute with limits

A very useful fact in category theory is that limits commute with limits (and dually colimits commute with colimits). That is, given a functor $$F : I \times J \to C$$ we have $\lim_i \lim_j F(i, j) \cong \lim_j \lim_i F(i, j)$ under appropriate conditions.

In this post I will explain the precise statement of this theorem, and describe three proofs. The first proof directly uses universal properties. The second is a standard argument using the Yoneda embedding to reduce to the case of $$\text{Set}$$, which we will see is really the same proof in disguise. The third proof is an elegant argument using uniqueness of adjoints.

• # Getting started with category theory

So you want to learn category theory - where should you start? I’ve spent the past six months trying to get a background in the essentials, and so it’s a good time to write up a reading list to point others along the same path. This is more of a record of my journey than an exhaustive guide, although I’ll try to mention all the popular resources I know of.

## Introductory resources

### Physics, Topology, Logic and Computation: A Rosetta Stone — John Baez and Michael Stay

If you’re like me, you’ll want to get excited about the subject before wading through a couple hundred pages of textbook. This paper is perfect for that — it describes how category theory draws beautiful analogies between different areas of physics, maths and computer science, all assuming no prerequisites.

### Category Theory for Programmers — Bartosz Milewski

The first part of this is one of the best gentle introductions to category theory that I know of. He tries to avoid too much mathematical notation and technical detail, and instead gives intuition with cute drawings and keeps things grounded with snippets of Haskell code. If you have a formal training in mathematics you might prefer the next item on the list; on the other hand if you want to learn category theory in order to understand Haskell then this might be the only thing you need to read.

The posts are online here — that page also links to a pdf version, a hardcover book and a series of YouTube videos.

### Category Theory in Context — Emily Riehl

This was my choice of introductory category theory textbook. It’s accessible to a later year maths undergrad without needing to know an excessive amount of abstract algebra or other prerequisites, and the writing is clear and of a consistently high quality. The content is evidently very well thought through: every time I skipped something I would find that it became important later, and need to go back and learn it properly! It’s a fairly dense book but not needlessly so — all the key intuition and examples are there.

You can read the book for free online, and the paperback version is inexpensive.