## August 11, 2017

### Magnitude Homology in Sapporo

#### Posted by Tom Leinster

John and I are currently enjoying Applied Algebraic Topology 2017 in the city of Sapporo, on the northern Japanese island of Hokkaido.

I spoke about magnitude homology of metric spaces. A central concept in applied topology is persistent homology, which is also a homology theory of metric spaces. But magnitude homology is different.

It was brought into being one year ago on this very blog, principally by Mike Shulman, though Richard Hepworth and Simon Willerton had worked out a special case before. You can read a long post of mine about it from a year ago, which in turn refers back to a very long comments thread of an earlier post.

But for a short account, try my talk slides. They introduce both magnitude itself (including some exciting new developments) and magnitude homology. Both are defined in the wide generality of enriched categories, but I concentrated on the case of metric spaces.

Of course, John’s favourite slide was the one shown.

Posted at August 11, 2017 9:05 PM UTC

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### Re: Magnitude Homology in Sapporo

Nice slides! I’m curious what you said to accompany slides 3 and 4.

It would be nice to write up the work on magnitude homology into something publishable at some point. But it feels like before doing that we ought to plug the hole in the decategorification by finding some sense in which those formal series manipulations make sense. Also it would be nice to have more example applications along the lines of “$H_1$ detects convexity”, and maybe a Kunneth or Mayer-Vietoris theorem.

I don’t really have time to be thinking about this now, but since you posted these slides I might as well ask: have you thought any more about those open problems in the past year?

Posted by: Mike Shulman on August 12, 2017 7:13 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

Nice slides! I’m curious what you said to accompany slides 3 and 4.

Thanks! I just said that although I wasn’t going to assume much category theory, when I did, I’d be proud (like the little boy) rather than being apologetic and blaming John. I also said that category theory isn’t just a language or a way of organizing things, but also a guide as to what to do next.

It would be nice to write up the work on magnitude homology

Definitely! There were lots of requests for a write-up. This community would certainly be interested.

Re the open problems, I haven’t really given them much thought in the last year.

The one exception is the question of what happens if you take coefficients in some other functor $A: [0, \infty] \to Ab$, and my “progress” there has been negative/puzzling. I’m failing to see what interest there is in taking $A$ to be anything other than the “Dirac deltas” called $A_\ell$ in my slides. There’s a structure theorem for functors $[0, \infty] \to Ab$ (or at least $[0, \infty] \to Vect$), which maybe I’ll write about another day, but it puts limits on what such functors there are.

Posted by: Tom Leinster on August 12, 2017 10:47 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

I’d be interested to hear anything you can say about why non-Dirac-delta coefficients seem uninteresting. Is it at all related to my observation last year that for magnitude homology with the obvious interval-parametrized coefficients $A_J$ (with $A_J(\ell)=\mathbb{Z}$ iff $\ell\in J$) the blurring is “non-uniform” in that points near the middle of $J$ are blurred more than the ends?

Posted by: Mike Shulman on August 14, 2017 5:54 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

I’m very impressed that your ‘exciting new developments’ included a link to a paper that appeared on the arXiv the same morning! :-)

Anyway, here’s an observation: magnitude homology has a persistence module as an input, whereas persistent homology has a persistence module as an output. Thoughts?

Posted by: Simon Willerton on August 12, 2017 9:57 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

It’s amazing how many benefits there are to writing one’s slides at the last minute…

In question time, someone raised the possibility mentioned in your second paragraph: homology with coefficients in homology. We had a bit of a laugh about it. But it occurred to me that it might be exactly as wacky and way-out an idea as considering categories enriched in Cat: how crazy would that be!

Posted by: Tom Leinster on August 12, 2017 10:35 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

exactly as wacky and way-out an idea as considering categories enriched in Cat: how crazy would that be!

Or maybe a spectral sequence that converges to the first page of another spectral sequence, or chain complexes with values in (a certain subcategory of) a triangulated category? (-:

Posted by: Mike Shulman on August 12, 2017 10:42 PM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

You seem to have neglected to put in a photo of Richard Hepworth. Let me remedy that.

Posted by: Simon Willerton on August 12, 2017 10:02 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

Very good; thank you!

Decent photos of Mike were hard to find. In the end I used the $64 \times 64$ pixel one from the front page of this blog, so he comes out looking noticeably blurrier than everyone else, which isn’t true in real life.

Posted by: Tom Leinster on August 12, 2017 10:59 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

Decent photos of Mike were hard to find.

Here is a slightly bigger one.

Posted by: Mike Shulman on August 12, 2017 10:37 PM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

It was a great talk! But, for the record, cohomology with coefficients in homology is a staple of sheaf theory…

Posted by: Robert Ghrist on August 13, 2017 12:17 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

cohomology with coefficients in homology is a staple of sheaf theory…

Yeah. And even for someone who doesn’t know about sheaves, there’s the $E_2$ page of the Serre spectral sequence.

Posted by: Mike Shulman on August 13, 2017 5:29 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

It’s odd; when I read (sounds like red) “[co]homology with coefficients in homology” what jumped to mind was that $\vee$ should be an $E_\infty$ operation on spectra, so that “twisted homology theories on $X$” is itself a fairly-decent kind of homology theory on $X$… and wasn’t there a TWF where John rejoiced in someone’s (a student’s?) paper about “the cohomology theory of cohomology theories”?

Posted by: Jesse C. McKeown on August 14, 2017 9:46 PM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

• Charles Rezk, A model for the homotopy theory of homotopy theory.

Abstract. We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more precisely that the category of such models has a well-behaved internal hom-object.

Posted by: John Baez on August 18, 2017 4:16 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

… I don’t think so… but I haven’t had luck with the search-engines, either.

I hadn’t yet heard of TWF in 1998, but of course that doesn’t mean you hadn’t mentioned it later.

Posted by: Jesse C. McKeown on August 18, 2017 3:18 PM | Permalink | Reply to this

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