This blog is a collection of reading notes of things that I try to learn. Often they are nothing more than summaries of various pages on the nLab, but collated, (over)simplified, and reordered to be more easily understood by, e.g., me. There will almost certainly be mistakes and bad viewpoints, so caveat lector.

• ## More-than-one-but-less-than-three-categories

What with all the wild applications of, and progress in, the theory of $\infty$-categories, I had really neglected studying any kind of lower higher-category theory. But, as in many other ways, CT2019 opened my eyes somewhat, and now I’m trying to catch up on the theory of 2-categories, which have some really beautiful structure and examples.

• ## Cauchy completion and profunctors

An idea that came up in a few talks at CT2019 was that of ‘spans whose left leg is a left adjoint’. I managed (luckily) to get a chance to ask Mike Shulman a few questions about this, as well as post in #math.CT:matrix.org. What follows are some things that I learnt (mostly from [BD86]).

• ## CT2019

I have just come back from CT2019 in Edinburgh, and it was a fantastic week. There were a bunch of really interesting talks, and I had a chance to meet some lovely people. I also got to tell people about #math.CT:matrix.org, and so hopefully that will start to pick up in the not-too-distant future.

• ## Skomer island

I haven’t posted anything in a while, and rather than trying to write about maths, I wanted to just share some lovely photos of Skomer island (which I recently visited). I am even less knowledgeable about birds than I am about maths, but I do love them, and this was the first time in my life that I’d actually seen a puffin in the flesh!

• ## Twisting cochains and arbitrary dg-categories

Having recently been thinking about twisting cochains (a major part of my thesis) a bit more, I think I better understand one reason why they are very useful (and why they were first introduced by Bondal and Kapranov), and that’s in ‘fixing’ a small quirk about dg-categories that I didn’t quite understand way back when I wrote this post about derived, dg-, and $A_\infty$-categories and their role in ‘homotopy things’.

This isn’t a long post and could probably instead be a tweet but then this blog would be a veritable ghost town.

I haven't posted anything in forever, and one of my supervisor's strong pedagogical beliefs is that 'affine vector spaces should be understood as $G$-torsors, where $G$ is the underlying vector space acting via translation', which makes a nice short topic of discussion, whence this post.