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!

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!

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!

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.

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.

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.

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.