Edit: Having now completed the full draft of this post, it turns out that I’m actually just going to talk about sheaves and holomorphic functions for a really long time. Stay tuned for the next part in this series, where I actually will get back to talking about cohomology!

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!

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!

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.

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!

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?

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.

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.

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.

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.

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?

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.