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August 5, 2017

The Rise and Spread of Algebraic Topology

Posted by John Baez

People have been using algebraic topology in data analysis these days, so we’re starting to see conferences like this:

I’m giving the first talk at this one. I’ve done a lot of work on applied category theory, but only a bit on on applied algebraic topology. It was tempting to smuggle in some categories, operads and props under the guise of algebraic topology. But I decided it would be more useful, as a kind of prelude to the conference, to say a bit about the overall history of algebraic topology, and its inner logic: how it was inevitably driven to categories, and then 2-categories, and then \infty-categories.

This may be the least ‘applied’ of all the talks at this conference, but I’m hoping it will at least trigger some interesting thoughts. We don’t want the ‘applied’ folks to forget the grand view that algebraic topology has to offer!

Here are my talk slides:

Abstract. As algebraic topology becomes more important in applied mathematics it is worth looking back to see how this subject has changed our outlook on mathematics in general. When Noether moved from working with Betti numbers to homology groups, she forced a new outlook on topological invariants: namely, they are often functors, with two invariants counting as ‘the same’ if they are naturally isomorphic. To formalize this it was necessary to invent categories, and to formalize the analogy between natural isomorphisms between functors and homotopies between maps it was necessary to invent 2-categories. These are just the first steps in the ‘homotopification’ of mathematics, a trend in which algebra more and more comes to resemble topology, and ultimately abstract ‘spaces’ (for example, homotopy types) are considered as fundamental as sets. It is natural to wonder whether topological data analysis is a step in the spread of these ideas into applied mathematics, and how the importance of ‘robustness’ in applications will influence algebraic topology.

I thank Mike Shulman with some help on model categories and quasicategories. Any mistakes are, of course, my own fault.

Posted at August 5, 2017 8:21 AM UTC

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Re: The Rise and Spread of Algebraic Topology

I’ll be there too, and I’m very much looking forward to the conference, to my first ever visit to Japan, and to seeing old friends. I’ll be speaking on magnitude homology.

Magnitude itself is first of all an invariant of finite metric spaces (though it can be extended to most compact spaces). Topological data analysis is especially interested in finite metric spaces, which often arise as data sets. So, that’s how magnitude connects with applied topology. More specifically, magnitude seems to give us good information about the “effective number of points” and “effective dimension” of a finite space at different scales.

Magnitude homology is a homology theory of enriched categories, but it can be applied, in particular, to finite metric spaces. Another homology theory of finite metric spaces is persistent homology — which is a major theme in applied algebraic topology. So, it’s tempting to compare the two. No one has yet done a proper comparison as far as I know, but I hope my talk will tempt someone to do one.

Posted by: Tom Leinster on August 5, 2017 11:06 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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.

Posted by: John Baez on August 6, 2017 10:55 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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

Posted by: David Corfield on August 6, 2017 12:57 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

I won’t be there, but I was at a nice applied topology in Bedlewo, Poland in June. I spoke about instantaneous dimension (or the dimension profile) of finite metric spaces, which was my way of relating magnitude to applied topology. I should put a post up about it!

I found that “the ‘applied’ folks” were often ‘pure’ folks who were interested in applying things! I thought you, John, would be the first to disparage the dichotomy of ‘pure’ versus ‘applied’. :-)

Posted by: Simon Willerton on August 5, 2017 12:44 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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 WW 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 (,1)(\infty,1)-topos.

Posted by: John Baez on August 6, 2017 7:21 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

The last slide says “If all the maps F_ε,δ: F_ε→F_δ are inclusions of Kan complexes such a functor is also cofibrant.”

This is false, as one can clearly see from the necessary and sufficient criterion for cofibrancy in the projective model structure, see https://mathoverflow.net/questions/97690/necessary-conditions-for-cofibrancy-in-global-projective-model-structure-on-simp/127187#127187.

An explicit counterexample is provided by the constant functor that sends any x∈(0,∞) to a single point, Δ^0. This functor is not cofibrant because it is not a coproduct of retracts of corepresentable functors: the coproduct family must have cardinality 1, and the corepresentable functor of some a∈(0,∞) sends x≥a to Δ^0 and x

Posted by: Dmitri Pavlov on August 5, 2017 4:40 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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,)(0,\infty) by [0,)[0,\infty) with its usual order? I think then the functor F:[0,)KanF : [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 opsSetF : C^{op} \to sSet and a simplicial sphere Δ nF(c)\partial\Delta^n \to F(c) at some object cCc \in C, we attach a filler Δ nF(c)\Delta^n \to F(c) and all of its images under the action of morphisms in CC 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,)KanF : (0,\infty) \to Kan as a ‘time-dependent Kan complex’. I’m imposing the condition that each Kan complex F(t)F(t) is included in F(t)F(t') whenever ttt \le t'. But you’re saying that for FF to be cofibrant we also need each new cell in this time-dependent Kan complex to come into existence at a specific time t(0,)t \in (0,\infty). This can’t happen if a cell is there for all times t(0,)t \in (0,\infty), simply because (0,)(0,\infty) doesn’t have a minimal element.

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

Posted by: John Baez on August 6, 2017 4:14 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Sorry, my message was actually finished,
but it was forcibly truncated
by the forum software:
as you can see, it stopped right at x<a,
and it appears that the < sign was interpreted
as an HTML tag or a hyperlink.

Yes, requiring that every simplex is created
at a particular time is precisely the cofibrancy condition
as described by Dugger.

This cofibrancy issue also arose in my work with Owen Gwilliam
on filtered objects in stable ∞-categories
(https://arxiv.org/abs/1602.01515),
which is how I noticed the problem in the first place.

Posted by: Dmitri Pavlov on August 6, 2017 11:44 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Great, thanks for the confirmation!

So, if we define the category [0,][0,\infty] to be the poset [0,][0,\infty] with its usual ordering and then define the Rips complex of a metric space XX to be the functor

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

such that F(t)F(t) consists of ordered (n+1)(n+1)-tuples of points in XX such that the distance between any two points is t\le t, we see that F(t)F(t) is a Kan complex for each t[0,]t \in [0,\infty], F(s)F(s) is a sub-complex of F(t)F(t) when sts \le t, and each simplex in F()F(\infty) first appears at some particular t[0,]t \in [0,\infty]. So, it’s cofibrant in the global projective model structure on SSet [0,]SSet^{[0,\infty]}.

It’s nice to include \infty in the poset [0,][0,\infty], but it’s necessary to include 00, since the 00-simplices all show up at t=0t = 0. It’s also necessary to use \le in the condition on distances when defining F(t)F(t), and not just because a carelessly typed < sign is interpreted as an HTML command here on the nn-Café: we need it to make sure each simplex shows up at a particular time, not just all times after a particular time.

Fans of Lawvere metric spaces will be happy to note this all works for those too, if we decree that (x 0,,x n)(x_0, \dots, x_n) is an nn-simplex in F(t)F(t) iff d(x i,x j)td(x_i, x_j) \le t for all iji \le j. But in this case, including \infty in the poset [0,][0,\infty] is not just nice—it’s necessary.

(By the way, one way to type a less than sign here without getting into trouble is to use the HTML code

&lt;

This avoids the problem you ran into. Similarly with ‘greater than’. I’m sorry it’s so hard!)

Posted by: John Baez on August 6, 2017 12:38 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Alas, the above comment is wrong: the simplicial set F(t)F(t) is not a Kan complex, because not all 2-dimensional horns have fillers. Oddly, horns of all every other dimension have fillers.

So, the Rips complex is not a fibrant object in the projective global model structure on SSet [0,]SSet^{[0,\infty]}.

I’ve fixed my talk to reflect this.

Has anyone seen something that’s like a Kan complex except the 2-dimensional horns may lack fillers?

Posted by: John Baez on August 7, 2017 12:57 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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

How important is the infinitely divisible nature of [0,][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 Gpd [0,]\infty Gpd^{[0,\infty]}. But it does work (or more precisely, it’s closer to working) if we replace [0,][0,\infty] by some reverse-well-founded sub-poset such as {1nn}\{\frac{1}{n} \mid n\in\mathbb{N}\}.

Posted by: Mike Shulman on August 6, 2017 11:43 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

I misled myself: you gave me the facts and I didn’t quite use them right.

How important is the infinitely divisible nature of [0,][0,\infty]?

I don’t think it’s important at all. For the stuff I’m doing, I believe we could replace this poset of ‘distances’ by any poset PP with a bottom element, and define a Rips complex for any set XX equipped with a function d:X×XPd : X \times X \to P such that d(x,x)d(x,x) is the bottom element, straightforwardly mimicking the definition I gave for P=[0,]P = [0,\infty], and then this Rips complex would be a fibrant and cofibrant object in SSet PSSet^P with its projective global model structure.

Posted by: John Baez on August 6, 2017 12:54 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Sorry, the Rips complex is not fibrant, nor is this generalization. I’ve fixed my talk to reflect this.

But maybe what you meant was: do people working on the Rips complex really care about the infinitely divisible nature of [0,][0,\infty]? I think the answer is largely “no”. For computational work, which is big in this field, it’s fine to use an ordered set like this:

{0,ϵ,2ϵ,,Nϵ}\{0, \epsilon, 2 \epsilon, \dots , N \epsilon\}

since we typically measure distances only up to some precision. So, abstractly, we’re using a finite ordinal as our poset.

Posted by: John Baez on August 7, 2017 12:57 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

> 600 page letter to Quillen.

The letter is no more than 15 pages, and AG writes the rest of PS not to Quillen. But perhaps this is a bit too finicky?

PS a torus is a K(Z^2,2).

Posted by: David Roberts on August 5, 2017 5:35 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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.

Posted by: John Baez on August 6, 2017 3:47 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

I agree with David, and do not think it finickity. There is a definite end to his letter (on page 3’ he clearly signs off `very cordially yours’; page 3’ is the back of the third sheet so is page 6).

I also agree with John that it reads like a letter, but that does not mean that it was `to someone’. It is more like a diary and diary entries can be like a letter from the writer to the diary. As Ronnie Brown details on his website, the copies were sent to him and a smallish set of others. I do not know the full list and it was not constant. There were a large number of letters written at about the same time and of which there is mention in the 600 pages. These include ones to Ronnie and me but there were others to including, I think, André Joyal and Hans Baues. There is a lot of correspondence to and fro. Earlier he had outlined the theory in letters to Larry Breen.

Posted by: Tim Porter on August 6, 2017 7:22 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

He definitely seemed to be talking to someone in particular when he was discussing the recipe for kimchi near the very end of Pursuing Stacks. Now I want to look at it again, but I don’t have my copy at hand.

Posted by: John Baez on August 6, 2017 7:42 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

See section 131 here.

Posted by: David Roberts on August 6, 2017 11:44 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Recall that there’s a TeX’ed version, which is searchable. The kimchi recipe is referenced in the beginning of section 131, but not actually included in the typescript. (Apparently it is 10 pages!)

Posted by: Ulrik Buchholtz on August 6, 2017 12:01 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Ah, yes, I realised you meant the physical object, not the surface. Though I do drink coffee out of (a colleague’s) one of these coffee cups, so perhaps you’ll forgive the confusion :-P

Posted by: David Roberts on August 6, 2017 11:51 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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 (,1)(\infty,1)-categories, and how model categories were for many years considerably more popular than any more explicit approach to (,1)(\infty,1)-categories (e.g. quasicategories or simplicially enriched categories).

Posted by: John Baez on August 6, 2017 8:02 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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)”.

Posted by: David Roberts on August 6, 2017 11:48 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

I’m fairly sure that I had `corrected’ John before, but there are so many things to remember. Maybe it was over breakfast in Barcelona, so before John had got into high speed mode for the day!

What is sure, as David says, is that the letter itself is just five and a half pages long. There is a scanned version on the web I think, so checking on recipes is possible (although tedious as you cannot search).

Posted by: Tim Porter on August 7, 2017 9:56 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Ulrik Buchholtz pointed out the relevant passage in a comment above; it’s the start of Section 131 on page 498 here. It’s not actually a recipe for kimchi; it’s a remark about a recipe. Grothendieck writes:

Yesterday again, I didn’t do any mathematics — instead, I have been writing a ten pages typed report on the preparation and use of kimchi, the traditional Korean basic food of fermented vegetables, which I have been practicing now for over six years. Very often friends ask me for instructions for preparing kimchi, and a few times already I promised to put it down in writing, which is done now. Besides this, I wrote to Larry Breen to tell him a few words about my present ponderings as he is the one person I would think of for whom my rambling reflection on schematization and on schematic homotopy types may make sense.

This remark tended to make me think that the whole manuscript was a letter to a friend, but now I see that Grothendieck was just sort of muttering to himself, as in a kind of diary.

Posted by: John Baez on August 7, 2017 10:26 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

… a Lab-diary but not as well structured as the n-Lab. ;-)

Posted by: Tim Porter on August 7, 2017 10:53 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

You wrote on the slides

Unfortunately homotopy type theory does not solve all our problems. Homotopy coherence is not ‘built in’.

It’s true that homotopy coherence is not completely built in to HoTT. But sometimes it is, and we continue to find new ways to build it in. So I think this flat negative statement is a little misleading.

Posted by: Mike Shulman on August 6, 2017 11:55 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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’.

Posted by: John Baez on August 6, 2017 12:57 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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.

Posted by: John Baez on August 6, 2017 1:18 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Thanks John, nice overview. NB. Slides 18 and 19 appear to be duplicates. I’ve recently started working with neuroscientists to see what algebraic topology might do for their data - there are already some applications and I see a few related people and papers at the conference.

Posted by: Matt Earnshaw on August 6, 2017 10:40 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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.

Posted by: John Baez on August 7, 2017 10:32 AM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

Thanks for your interest. As yet, it’s very early days.

Posted by: Matt Earnshaw on August 8, 2017 2:48 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

We can also notice that “persistence modules” (the primary algebraic objects of study in TDA) are functors, [0,]Vect[0,\infty] \to \text{Vect}. I don’t know whether these can also be seen to live in an (,1)(\infty,1)-category but presumably we could get a kind of “parameterized homology functor” from the category John constructs … also note that we can generalize by replacing [0,][0, \infty] with various other categories such as n\mathbb{R}^n (“multidimensional persistence”) or zigzag diagrams, etc. (cf. for example arXiv:1312.3829v3) … hard to resist the suggestion of passing then to slices of Cat\text{Cat} over SSet\text{SSet} and Vect\text{Vect} (?) but that may be rather useless.

Posted by: Matt Earnshaw on August 8, 2017 2:24 PM | Permalink | Reply to this

Re: The Rise and Spread of Algebraic Topology

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

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

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

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

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

F:[0,]SSetF : [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 (,1)(\infty,1)-category, while if we just used vector spaces we wouldn’t have those (or more precisely, they’d be trivial).

Posted by: John Baez on August 9, 2017 8:07 AM | Permalink | Reply to this

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