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!

### Čech totalisation

23/09/08One 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).

Continue reading →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!

Continue reading →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!

Continue reading →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.

Continue reading →### Torsors and principal bundles

18/10/31In 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.

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 →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 →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.