Last summer I did an eight week mathematics research project, as mentioned in my previous post. This was my first real taste of mathematical research, and so I tried to write up some of my thoughts on lessons learned while they were still fresh in my mind. I just recently got around to editing this into a publishable form.

I’m sure very little of this is new to those of you who have experience in research – but I find it hard to remember what was difficult once I’m very used to something, so I hope there’s value in me writing this.

Read more » ]]>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 mathematics students who don’t know any category theory, so rather than taking a category-heavy point of view, I instead 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.

Read more » ]]>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.

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.

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.

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.

Read more » ]]>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.

Read more » ]]>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.

The paper is available from John Baez’s website.

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.

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.

Read more » ]]>For some reason in the Cambridge Part II course we only prove this for curves — I think the general case is much more enlightening. This proof is based on Hulek’s Elementary Algebraic Geometry (Theorem 3.14), although somewhat expanded.

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