After my previous post, I got the chance to *(a)* spend a bit of time thinking about things, and *(b)* talk to Jade Master on Twitter. Rather than going back to edit my original post, I decided to turn this into more of a series. Here are some new thoughts (but not really any *visible* progress) on the whole kit and caboodle.^{1}

This blog is mainly a collection of notes of things that I'm trying to understand. The posts are often nothing more than summaries of various textbooks, papers, and nLab pages, but collated, (over)simplified, and reordered to be more easily understood by e.g. me.

There will almost certainly be mistakes and bad points of view, so caveat lector.

Jan 20

#### Categorical translation for mathematical language (Part 2)

Dec 19

#### The determinant bundle

As always, when I haven’t written anything on here in a while, I’m going to write about something entirely irrelevant to anything I’ve written about before (instead of continuing any of the many series of posts I’ve started or planned to start). Today we’re going to look at a ‘fun’ little calculation, which might be the most algebraic-geometry-like calculation I’ve done in a long time: we’re going to compute the determinant of the cotangent sheaf of the complex projective sphere. Note that even the statement of this problem makes me uneasy: I am nowhere near as comfortable with classical algebraic geometry as I should be!

Continue reading →Nov 19

#### Categorical translation for mathematical language (Part 1)

Being unable to ever properly finish any project that I start, but loving starting new projects, has made getting around to typing up this blog post quite an effort. Not only that, but it’s also unsatisfying to me how much I’ve failed to understand the categorical framework behind my translation project, so it’s mildly intimidating (to say the least) to present this stuff to the whole internet (although, in actuality, it’s really just to the one (mabye two) reader(s) of this blog), but I’m doing so in the hopes that somebody who actually knows about this sort of applied category theory can help me get somewhat closer to a solid understanding.

Continue reading →Nov 19

#### Bird rescue

On a particularly bad day, when I’d gone into the maths department on a public holiday by accident^{1}, I saw nobody around. Coming down the staircase to leave, however, I saw a blue tit trapped inside, loitering by the photo of Grothendieck.

Oct 19

#### Matrix: open, secure, and decentralised

Anybody who follows me on Twitter will have heard me go on about Matrix, and Riot, and maybe a whole host of other such things. I understand that most people probably won’t really care much about it all, but I thought that I’d try to explain a bit why I’m quite so passionate about this stuff, and why I’m bothering to try to work on ‘LaTeX’ support (with scare quotes for good reason, as explained later).

Coincidentally, I took a sleeping pill just before writing this. The pill was to make me sleep, but deciding to write this just after was more of an experiment into brain activity and cohesiveness and also just seemed like a bit of a funny idea to me.

Continue reading →Sep 19

#### Weighted limits, ends, and Day convolution (Part 3)

Finally, I find myself with enough motivation to start writing the last part to this series. It’s been a while, but hopefully nobody has actually been waiting… This is where we will finally see some of the exciting applications of (co)ends, including tensor products, geometric realisation, and Day convolution. One reason I’ve got around to writing this post is because coends (or, really, cowedges) appeared to me recently in a tweet about Stokes’ theorem, which I found pretty neat indeed — more details can be found in this post.

Continue reading →Sep 19

#### Everything is Stokes'; everything is coend

For category theorists, the idea that “everything is a Kan extension” is a familiar one, as is the slightly more abstract version “everything is a (co)end”. For differential geometers, the idea that “everything is Stokes’ theorem” is sort of the equivalent adage. In an entirely typical turn of events, it seems that these two seemingly unrelated aphorisms can be linked together, as I found out today on Twitter.

Continue reading →Aug 19

#### Mathematical motivation and meagre contributions

I find myself in a mathematical rut more often than I would like. It is very easy, especially as a PhD student, I think, to become disillusioned with maths, the work that one can do, and academia as a whole. I realised only the other day, after talking to my family, some of the things that can contribute towards this. Hopefully this post can serve as a reminder to myself that I am human being who is trying to be a mathematician, and not the other way around.

Continue reading →Jul 19

#### Graded homotopy structures

As I mentioned in a previous post, I recently saw a talk by Rachel Hardeman on the A-homotopy theory of graphs, and it really intrigued me. In particular, it seemed to me that there was some nice structure that could be abstractified: that of a “graded homotopy structure”, as I’ve been calling it in my head. Rather than trying to type out everything in `#math.CT:matrix.org`

, I’ve decided to post it here, in the hope that I might be able to get some answers.

Jul 19

#### Twisting cochains and twisted complexes

This week was the Young Topologists Meeting at EPFL in Lausanne, and had courses by Julie Bergner (on 1- and 2-Segal spaces) and Vidit Nanda (on the topology of data), as well as a bunch of great talks by various postdocs and PhD students. I was lucky enough to get the chance to talk about twisting cochains and twisted complexes for half an hour, and you can find the slides on my GitHub here.

Continue reading →Jul 19

#### 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.

Continue reading →Jul 19

#### 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]).

Jul 19

#### 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.

Apr 19

#### 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!

Continue reading →Dec 18

#### 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.

Continue reading →Oct 18

#### Torsors and principal bundles

In my thesis, switching between vector bundles and principal \mathrm{GL}_r-bundles has often made certain problems easier (or harder) to understand. Due to my innate fear of all things differentially geometric, I often prefer working with principal bundles, and since reading Stephen Sontz’s (absolutely fantastic) book Principal Bundles — The Classical Case, I’ve really grown quite fond of bundles, especially when you start talking about all the lovely \mathbb{B}G and \mathbb{E}G things therein^{1} Point is, 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’*,^{2} which makes a nice short topic of discussion, whence this post.

Aug 18

#### Localisation and model categories (Part 1)

After some exceptionally enlightening discussions with Eduard Balzin recently, I’ve made some notes on the links between model categories, homotopy categories, and localisation, and how they all tie in together. There’s nothing particularly riveting or original here, but hopefully these notes can help somebody else who was lost in this mire of ideas.

Continue reading →Aug 18

#### Categorication of the Dold-Kan correspondence

So I’m currently at the Max Planck Institute for Mathematics in Bonn, Germany, for a conference on ‘Higher algebra and mathematical physics’. Lots of the talks have gone entirely above my head (reminding me how far behind my physics education has fallen), but have still been very interesting.

Continue reading →Jul 18

#### Nothing really that new

Just a small post to point out that I’ve uploaded some new notes, including some I took at the Derived Algebraic Geometry in Toulouse (DAGIT) conference last year. I’ve been hard at work on thesis things, so haven’t been able to write up all the blog stuff that I’ve wanted to, but hopefully will get a chance sometime in the near future.

Continue reading →Apr 18

#### Derived, DG, triangulated, and infinity-categories

This post assumes that you have seen the construction of derived categories and maybe the definitions of dg- and A_\infty-categories, and wondered how they all linked together. In particular, as an undergraduate I was always confused as to what the difference was between the two steps of constructing the derived category of chain complexes was: taking equivalence classes of chain homotopic complexes; and then formally inverting all quasi-isomorphisms. Both of them seemed to be some sort of quotienting/equivalence-class-like action, so why not do them at the same time? What different roles were played by each step?

Continue reading →Apr 18

#### Triangulations of products of triangulations

At a conference this week, I ended up having a conversation with Nicolas Vichery and Eduard Balzin about why simplices are the prevalent choice of geometric shape for higher structure, as opposed to e.g. cubes or globes.

Continue reading →Mar 18

#### Quantum circuits (Part 1)

I am not at all a physicist, and my knowledge of quantum physics in particular comes solely from undergraduate courses that I followed years ago, and any reading I can get done when feeling mathematical but not inclined to work on my thesis. However, after scanning through some papers by Bartlett, Baez, Lauda, and Lurie, my interest in quantum physics, and quantum computing especially, has come back with a vengeance.

Continue reading →Mar 18

#### Loop spaces, spectra, and operads (Part 3)

This post is a weird one: it’s not really aimed at any one audience, but is more of a dump of a bunch of information that I’m trying to process.

Continue reading →Feb 18

#### Weighted limits, ends, and Day convolution (Part 2)

Using the idea of weighted limits, defined in the last post, we can now talk about **ends**. The idea of an end is that, given some functor F\colon \mathcal{C}^\mathrm{op}\times\mathcal{C}\to\mathcal{D}, which we can think of as defining both a left and a right action on \prod_{c\in\mathcal{C}}F(c,c), we wish to construct some sort of universal subobject^{1} where the two actions coincide. Dually, a motivation behind the **coend** is in asking for some universal quotient of \coprod_{c\in\mathcal{C}}F(c,c) that *forces* the two actions to agree.

Dec 17

#### Loop spaces, spectra, and operads (Part 2)

In the previous post of this series, I talked a bit about basic loop space stuff and how this gave birth to the idea of ‘homotopically-associative algebras’. I’m going to detour slightly from what I was going to delve into next and speak about **delooping** for a bit first. Then I’ll introduce **spectra** as sort of a generalisation of infinite deloopings. I’ll probably leave the stuff about E_\infty-algebras for another post, but will definitely at least mention about how it ties in to all this stuff.

Dec 17

#### Loop spaces, spectra, and operads (Part 1)

I have been reading recently about spectra and their use in defining cohomology theories. Something that came up quite a lot was the idea of **E_\infty-algebras**, which I knew roughly corresponded to some commutative version of A_\infty-algebras, but beyond that I knew nothing. After some enlightening discussions with one of my supervisors, I feel like I’m starting to see how the ideas of spectra, E_\infty-algebras, and operads all fit together. In an attempt to solidify this understanding and pinpoint any difficulties, I’m going to try to write up what I ‘understand’ so far.

Dec 17

#### Weighted limits, ends, and Day convolution (Part 1)

A motto of category theory is that ‘Kan extensions are everywhere’. As a simplification of this, ‘(co)limits are in a lot of places’. By rephrasing the definition of a limit we end up with something that looks invitingly generalisable. This is how we can stumble across the idea of a **weighted limit**. In this post I’m going to assume that you are already convinced of the usefulness and omnipresence of limits and not talk too much (if at all) about why they are interesting in their own right.