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

23/09/08Earlier 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 →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 →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 →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 →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.