Posts tagged homotopy-theory

### Every principal bundle is flat, in the infinity world

22/02/05

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!

### Left adjoints, lenses, and localisation

21/11/26

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.

### JMRA Video

20/09/06

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.

### Hochschild and cyclic homology (briefly)

20/07/15

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!

### Thesis online

20/07/03

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.

### Graded homotopy structures

19/07/29

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.

### Twisting cochains and twisted complexes

19/07/26

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.