The n-Category Café

It’ll be great to see you, Tom! I’ll also see Nina Otter, my former masters’ student now at Oxford… and Kathryn Hess, whom I first met a long long time ago at MIT when she was a grad student… and I’ll meet Robert Ghrist for the first time: my student Brendan Fong has been working with him lately.

Have things moved on, Tom, since that flurry of activity last year on magnitude homology?

Hi! I do disparage it. I’m really just expressing my anxiety at giving the first talk at a conference on ‘applied algebraic topology’, a field I don’t feel knowledgeable about. At first I wanted to talk about my work with Nina Otter, on operads and phylogenetic trees, since it springs from a real-world topic and leads to some very nice math connected to the Boardmann–Vogt $W$ construction on operads. But this is about going from applications to nice pure math: we haven’t made the round trip back to applications. Someone will ask “so how does this help people working on phylogenetic trees?” and I’ll say “I don’t know”.

So then I considered discussing my work on networks of various kinds. But while this uses some category theory originally developed for algebraic topology — like PROPs — it doesn’t really use algebraic topology. Not yet, anyway.

So I decided to talk about the aspect of algebraic topology I love most: the way it’s changed our vision of mathematics, replacing sets by ‘spaces’ and wanting everything to be true only ‘up to coherent homotopy’. This seems to fit in well with the general philosophy of topological data analysis. And it involves math that everyone in algebraic topology should learn—but I hope not all of them already have! I feel this stuff is usually explained in such a complicated, overly technical way that only determined people can fight their way through it.

And then I noticed that the Rips complex seems to want to live in an $(\infty,1)$-topos.

Your message ended rather suddenly — was there more? But I think I get your point.

I’d like to get this straightened out before I give my talk. Such are the dangers of including last-minute research in a talk. But it’s too fun to resist!

Would that particular counterexample go away if I replaced the poset $(0,\infty)$ by $[0,\infty)$ with its usual order? I think then the functor $F : [0,\infty) \to Kan$ sending each number to the same Kan complex is corepresentable.

Mike Shulman told me another way of thinking about cofibrant objects in the projective model category on simplicial presheaves in terms of (abstract) ‘cell complexes’. He said these are built up by attaching cells ‘at an object’. That is, given a diagram $F : C^{op} \to sSet$ and a simplicial sphere $\partial\Delta^n \to F(c)$ at some object $c \in C$, we attach a filler $\Delta^n \to F(c)$ and all of its images under the action of morphisms in $C$ freely, and then repeat this process transfinitely often.

I thought the objects I was describing were cell complexes according to this description. But now I see they’re not. I like to imagine $F : (0,\infty) \to Kan$ as a ‘time-dependent Kan complex’. I’m imposing the condition that each Kan complex $F(t)$ is included in $F(t')$ whenever $t \le t'$. But you’re saying that for $F$ to be cofibrant we also need each new cell in this time-dependent Kan complex to come into existence at a specific time $t \in (0,\infty)$. This can’t happen if a cell is there for all times $t \in (0,\infty)$, simply because $(0,\infty)$ doesn’t have a minimal element.

The main point is not whether we use $[0,\infty)$ or $(0,\infty)$, but let me talk about $[0,\infty)$. I think a functor $F : [0,\infty) \to Kan$ will be cofibrant if $F$ applied to each morphism in the poset $[0,\infty)$ is an inclusion of Kan complexes, and for each cell of $F(t)$ there is some minimal $s \in [0,\infty)$ such that this cell is present in $F(s)$.

I’m sorry that I misled you here. I think your new description is right.

How important is the infinitely divisible nature of $[0,\infty]$? The model theory of the univalence axiom is incompletely developed, so we don’t know for sure yet that the usual version of that axiom has a model in $\infty Gpd^{[0,\infty]}$. But it does work (or more precisely, it’s closer to working) if we replace $[0,\infty]$ by some reverse-well-founded sub-poset such as $\{\frac{1}{n} \mid n\in\mathbb{N}\}$.

The whole Pursuing Stacks seems like a letter to someone: near the end it includes a recipe for kimchi, and chats as if talking to someone. I always assumed it was all sent to Quillen; that’s what people told me. I’d like to know the real story.

Maybe in Australia doughnuts and coffee cups are hollow, but not in the US — so here, they are homotopy equivalent to circles.

I’m a bit surprised at this, because I’ve been running around for over a decade telling everyone (at conferences, in papers, and in This Week’s Finds, etc.) that Pursuing Stacks was a letter to Quillen, and nobody ever contradicted me. So now I’m very curious to re-examine my copy. But anyway, I’ll remove this remark. It was mainly meant as a lead-in to a remark about how Quillen’s model categories can be used to describe $(\infty,1)$-categories, and how model categories were for many years considerably more popular than any more explicit approach to $(\infty,1)$-categories (e.g. quasicategories or simplicially enriched categories).

Actually I’ve corrected this misunderstanding online before, probably on MathOverflow. I can’t recall who was in the question/conversation though. I can’t remember who told me about the myth, but it could have been Tim. Otherwise I realised it myself when I read PS (yes, all of it that was legible in the old djvu scan). You can see for yourself at the start of Part II in the scrivener’s version where Grothendieck writes

The following notes are the continuation of the reflection started in my letter to Daniel Quillen written previous week (19.2 – 23.2), which I will cite by (L) (“letter”). I begin with some corrections and comments to this letter.

Part I is titled “The Take-off (a letter to Daniel Quillen)”.

Okay. I wanted people to realize that there’s interesting work left to do. But maybe I’ll have time and room to stick in the word ‘completely’.

Okay, I did it! I also removed the word ‘unfortunately’.

It’s great having all of you folks help me fix my talk while I wait here in the airport. By Tuesday, having got all the bugs out, I’ll be able to focus on making the talk fun.

Thanks! The duplication there is the result of a slight bug in my procedure for converting talk slides into slides suitable for the web. It’s too complicated to explain, given how boring it ultimately is, but I’ll fix it in the next version.

I’d enjoy hearing about your work sometime. By the end of this week I’ll be sick of hearing about applications of algebraic topology, but I’ll recover quickly I’m sure.

Hello again! There’s got to be an $(\infty,1)$-category arising from functors

$$F: [0,\infty] \to Ch(Vect)$$

where $Ch(Vect)$ is the category of chain complexes of vector spaces. The reason is that $Ch(Vect)$ is equivalent to the category of simplicial objects in $Vect$, so functors

$$F: [0,\infty] \to Ch(Vect)$$

can be seen as ‘simplicial presheaves of vector spaces’ just as functors

$$F : [0,\infty] \to SSet$$

like the Rips complex are called ‘simplicial presheaves’.

More generally, taking homology kills off, or hides, the interesting higher categorical structure that’s hiding in a chain complex.

There’s more to say about this, but the crucial point is that working with chain complexes of vector spaces gives us chain homotopies, chain homotopies between chain homotopies, etc., and thus an interesting $(\infty,1)$-category, while if we just used vector spaces we wouldn’t have those (or more precisely, they’d be trivial).