Briefly: two very short translations concerning geometry, and the final two seminars of FGA (thus finishing a complete first draft of the entire FGA seminar series).

After a pretty long time (over two and half years or so), Mahmoud Zeinalian and I have finished our paper “Simplicial presheaves of Green complexes and twisting cochains” (arXiv:2308.09627). In this blog post I want to give a brief overview of one of the main technical tools that we use, which we call Čech totalisation. The full story involves model categories and homotopy limits and all this sort of machinery, but the main part of this post will try to keep this to a minimum, and just talk about a surprisingly useful roundabout way of describing principal bundles. There might be a second part to this, but for those interested or wanting for more details I recommend just delving into the paper — I spent a lot of time trying to make it as readable as possible!

Hopefully one of my co-authors and I will be uploading a long-awaited preprint on dg-categories, twisting cochains, and homotopy limits to the arXiv “soon”. Until then, here are the small handful of translations that I’ve finished in the year since I last wrote

I now realise two things: firstly, I haven’t posted anything in really quite a while; secondly, I should have written this specific blog post a week ago. Over the past five days, Bryce Clarke and I have been running the Virtual Double Categories Workshop (a “fun” pun, since the workshop was entirely online, and there is a notion of a double category called “virtual”). The speakers were incredible, both in terms of the talks they gave, in their attitude and enthusiasm for the workshop, and in the variety of their interests. You can find abstracts, slides, and recordings of the talks on the workshop webpage, but if you want to hear a bit more about my personal interest in double categories and how this workshop came about, then read on.

For the first time, I have released into the wild a preprint of which I am the sole author, and had no real supervision. This is a scary moment indeed — how do I know that I haven’t written complete made-up nonsense? It’s true that I talked with a couple of close colleagues about the results, and they nodded in vague agreement, but the responsibility of checking the actual formal details is all on me. Even worse, I wanted to include some results about something that I don’t really have any formal experience with. Anyway, I hope the resulting paper is at least mildly “good” (whatever that might mean). It’s called “Various notions of (co)simplicial (pre)sheaves”, and is now on the arXiv: 2205.15185.

One thing that the past few years have taught me is that I am not good at doing maths all by myself. In fact, I would go as far as to say I am completely useless and unmotivated. I do much better when I have co-authors to give me deadlines and friends to talk to, but, for obvious reasons, the past two years have not been good for this. Not really the ideal time for first postdocs, but alas, that’s life.

I recently found an old notebook with some vague questions and research ideas in it, and then realised that I have had no motivation to work on any of these alone, so why not put them out there for other people to see?

Just a very short post (you don’t even need to click “Continue reading” if you’re looking at this on my blog archive) — I finally managed to do some more maths (but only because I had some very hard working and very good coauthors), and I’ve written about it on the Topos blog as a two-part series: Part 1 and Part 2.

I would love to write more, but after having written these I’m afraid I’m all blogged out for the minute, so ciao for now!

One of the many reasons that teaching is fun is because you get to look back at things that you haven’t seen in a while and try to understand them in light of what you’ve learnt in the meantime. This means that you sometimes have the unexpected joy of having to teach something that always used to confuse you, but that now seems so much more straightforward! I experienced this last year when teaching an algebraic topology course: I remember being super lost when it came to the graded ring structure of cohomology and getting very annoyed at Hatcher’s book; now I look back and realise that it’s really neat! This post has a slightly different intended audience than normal: I’m just gonna assume that you know a bit about rings in the first half; the second half is aimed for somebody who’s a reasonable way through a first course on algebraic topology (e.g. knows what the cup product in cohomology is).

Earlier today, Mahmoud Zeinalian explained something to me that Dennis Sullivan once explained to him, and it’s been sitting in my brain ever since then. In an attempt to empty out my thoughts, and also preserve what little understanding I currently believe to have of the story, I thought I’d write a little blog post about it. It’s going to move quite quickly, because I don’t want to spend time developing the prerequisites — the main purpose is for this to jog my brain two weeks down the line when I forget all the details!

I haven’t blogged about it in a while, but I’ve been working on just making my translations a bit better, both in terms of content and accessibility. Let’s have a look at what I’ve done, shall we?

This blog now has support for comments! I haven’t had a chance to properly test things yet, and there are still some kinks left to iron out, but please do use this as an excuse to browse back through old posts and say nice things.

We’ll end this series (for now) by talking about two things that we really should talk about: Lie algebra homology, and relative Lie algebra (co)homology. (We will work over a field k, but really we are thinking only of the case k=\mathbb{C}).

This is really just a cross-post announcement: I wrote a post over on the Topos blog, but it’s something I’ve been thinking about a lot, so I wanted to share a link to it here as well. It’s basically the result of me, knowing a bit about derived categories and model categories, trying to digest this lovely bit of Australian category theory by Bryce Clarke concerning internal lenses. In fact, it’s really a fermented and distilled version of an old blog post from here, namely Cauchy completion and profunctors.

Continuing on from last time, let’s now take a look at some actual computational methods for Lie algebra cohomology, as well as some applications and important results. We’ll study the cohomology of semisimple Lie algebras, finite dimensional nilpotent Lie algebras, and then take a little detour to talk about the Borel–Weil–Bott theorem.

As part of a reading group I’m in on Kac–Moody groups (something I know nothing about), I had to give a talk introducing the basics of Lie algebra cohomology (something I know very little about), along with some sort of motivation and intuition, and some worked examples. Since I’ve written these notes anyway, I figured I might as well put them up on this blog, and then, when I inevitably forget all I ever once knew, I can refer back to here.

Over the past couple of months, between moving countries, teaching, and a bunch of administrative faff, I’ve managed to write two short preprints with David Spivak. I blogged about them over on the Topos blog, but I’ll just write a little crossover post here with some links (so that people don’t think that this blog is entirely dead).

Despite being an analytic/algebraic geometer by name (and title, and qualification, and academic upbringing, and …), there are so many gaps in my knowledge, even when it comes to simple foundational things. One thing which I have always tried to do during my academic “career”, however, is to be the person who asks the first stupid question, so that others can feel less nervous about asking their (certainly less stupid) questions. Thus: this blog post.

I am going to explain what I do know, talk about what I don’t, and then ask some semi-concrete questions that I’m hoping people will be able to help me out with!

Hello. Yet again I haven’t been blogging in quite a long while, which is, yet again, due to an unexpectedly hectic couple of months. Let’s have a look at what I’ve been doing, shall we?

I haven’t posted anything here in well over a couple of months, so I thought it’d be a nice idea to just say “hello, I am still here, and here’s why I’ve been too tired to post anything”, whence this post.

I’ve had a bit of free time during various quarantines and lockdowns as of late, so I’ve been able to do some more translations. Here’s a quick summary of what I’ve uploaded to thosgood.com/translations recently, as well as what I’m now working on.

I recently submitted a video to the Junior Mathematician Research Archive that gives a brief overview of the work in my PhD thesis. You can watch the video here but, be warned, it’s really probably not the most coherent (ha ha) narrative.

For the sake of it, or in case you’re the sort of person who prefers to read instead of watch, I’ve included the transcript of the whole video below.

I’ve been having quite a bit of fun translating some maths papers into English recently. Of course, I’m still chugging away at the EGA translation project, having just finished another section of EGA II, leaving “only” about 60 pages left before that chapter will be done as well! However, I’ve started putting some of my other translations all together, and they can now be found at https://thosgood.com/translations. There are a lot still in progress, but here’s a summary of what I’ve done so far.

Once again I feel the urge to type something nice, but have nothing new of my own to share. I did recently find, however, some notes I once wrote after hearing the phrase “Hochschild homology” for what felt like the hundredth time, so I thought I’d share them here. They’re not particularly enlightening, and I can’t claim to add any insight, but I often use my own blog as a reference for definitions that I once knew but later forgot, so this will at least serve that purpose!

Having finished my defence (entirely virtually, and hence, of course, plagued with many technical difficulties), I have now been able to share my thesis with the world. Most of the technical content can already be found in my two preprints (part I and part II), but I think the thesis is a much more self-contained and leisurely read, with a lot more examples and (hopefully) helpful appendices. It also has geese! You can find a copy of it on here on TEL.

Edit. I’m in the process of moving over from YouTube to Vimeo (no ads, less tracking, and no Google), so you can find my videos there now instead. In particular, the connections and curvature videos can be found here.

This is a very short post just to say that I’ve started uploading some maths-related videos to my YouTube page. At the moment there’s a talk I recently gave at a graduate seminar about connections and curvature from the point of view of somebody trying to avoid differential geometry, and a series I’ve been working on called Nice Analytic Sheaves For All which aims to discuss motivations for coherence conditions of complex-analytic sheaves from various points of view. Hopefully there will be more updates to the latter soon!

I am a big fan of Stephen Bruce Sontz’s book Principal bundles: The Classical Case, and cannot recommend it enough, as somebody who usually finds differential geometry (a) dull, and (b) incomprehensible. Anyway, there’s a lovely quote from this book about how confusing the terminology surrounding connections can be, so let’s try to clear some of that up today.

This week just gone I uploaded two preprints to the arXiv:

1. Simplicial Chern-Weil theory for coherent analytic sheaves, part I;
2. Simplicial Chern-Weil theory for coherent analytic sheaves, part II.

Both have been extracted from my PhD thesis (which I’ve just sent off to the referees) and contain about 90% of the main mathematical content of my thesis, but with about 90% fewer inane footnotes and digressions. There are also a few appendices in my thesis which explain the background of some of the subjects in a bit more detail, which I think are quite nice, but I’ll probably turn the good ones into blog posts at some point in the coming months.

So what are these two papers about? And why is it split into two parts?

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 affair.

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!

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.

On a particularly bad day, when I’d gone into the maths department on a public holiday by accident¹, I saw nobody around. Coming down the staircase to leave, however, I saw a blue tit trapped inside, loitering by the photo of Grothendieck.

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).

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.

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.

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.

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.

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 here.

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.

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]).

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.

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!

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.

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 therein¹ 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.

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.

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.

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.

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?

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.

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.

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 subobject¹ 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.

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.

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.

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.