## 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

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2977

### 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

Yes, that’s pretty much it.

Actually, when I last did the calculations, I thought the blurring was greatest at the bottom end of the interval rather than the middle. But I was doing it in my head in an airport and could easily have made a mistake.

Posted by: Tom Leinster on August 25, 2017 1:13 PM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

Maybe it’s misleading to speak of “more” and “less”: the way it seems to me right now is that points at different places in the interval are just blurred differently. On one side of the interval, there are more cycles and more boundaries, while on the other side of the interval there are fewer cycles and also fewer boundaries.

Posted by: Mike Shulman on August 26, 2017 4:20 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

Here’s one point of view from which the importance of the Dirac-delta coefficients is clearer. Instead of considering each $A_\ell$ in isolation as a functor $[0,\infty] \to Ab$, let’s consider them all together as a functor $[0,\infty] \to Ab^{[0,\infty]_0}$. Here $[0,\infty]_0$ denotes the underlying set without regard to the ordering, so that $Ab^{[0,\infty]_0}$ is just an uncountable product of copies of $Ab$, $\prod_{\ell\in[0,\infty]} Ab$; we could say that its elements are “abelian groups graded by $[0,\infty]$” (which I guess is taking us back a bit to an earlier viewpoint on the status of $\ell$). For each $\ell$, we have a Dirac-delta object (or maybe “Kronecker delta” would be better) $B_\ell \in Ab^{[0,\infty]_0}$, and the functor $A:[0,\infty] \to Ab^{[0,\infty]_0}$ sends each $\ell$ to $B_\ell$ (and each nonidentity map to the zero map, which is the only one that exists between distinct $B_\ell$’s).

Since $Ab^{[0,\infty]_0}$ is an abelian category, we can proceed to define magnitude homology with coefficients in this module $A$, and the result will be computed pointwise, consisting of all the magnitude homology groups $H_n(X;A_\ell)$ assembled together into a graded object of $Ab^{[0,\infty]_0}$. But the point is that the functor $A:[0,\infty] \to Ab^{[0,\infty]_0}$, unlike the individual functors $A_\ell :[0,\infty] \to Ab$, is strong monoidal if we equip $Ab^{[0,\infty]_0}$ with the evident graded tensor product so that $B_{\ell_1} \otimes B_{\ell_2} = B_{\ell_1+\ell_2}$.

In particular, this functor $A$ is a much more direct categorification of the “size” function that we use to define magnitude itself, which is supposed to be a monoid homomorphism from $[0,\infty]_0$ to the multiplicative monoid of some semiring. (The addition on the semiring corresponds to the abelian structure of $Ab^{[0,\infty]_0}$.) If we treat $q$ as a formal variable, then the natural semiring target is the field of Hahn series, with $\ell \mapsto q^\ell$, and such a Hahn series is in turn also the natural target for a “dimension” function on a suitable subcategory of “locally finite” objects of $Ab^{[0,\infty]_0}$. So it makes sense that to categorify magnitude-like features we would be looking only at $A$-coefficients.

With this perspective I feel like I’m on the cusp of understanding how magnitude homology is related to magnitude for arbitrary bases of enrichment, but I’m still a little confused by semicartesianness and divergent series.

Posted by: Mike Shulman on August 26, 2017 10:45 PM | 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

### Re: Magnitude Homology in Sapporo

I just noticed/remembered that back here you listed a bunch of other “dial positions” for enriched category theory for which we could say something about magnitude. We should think about what magnitude homology looks like in these cases!

After a quick calculation I think that in the case of $FinSet^{\mathbb{N}}$ (at least, when applied to free categories) it isn’t going to tell us any more than the magnitude itself. But your formula for the magnitude of the category of projective indecomposables over a Koszul ring with finite dimension and global dimension is clearly crying out to be the Euler characteristic of a homology theory; but I’m not sure how to go about computing its magnitude homology, since $FDVect$ isn’t semicartesian. And I haven’t thought at all about the case of ultrametric spaces.

Posted by: Mike Shulman on August 26, 2017 10:50 PM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

Yes indeed, we should definitely look at magnitude homology for other values of $\mathcal{V}$!

But I thought non-semicartesian $\mathcal{V}$s held no fear for you? In a previous discussion of magnitude homology, you seemed happy to drop the semicartesian hypothesis and replace it by a 2-sided thingy involving a richer system of coefficients. I confess I haven’t got to proper grips with that construction.

Re the result on projective indecomposables, contrary to what we originally thought, the Koszul hypothesis can be dropped. Joe Chuang, Alastair King and I wrote it up as a little paper here. Algebraists refer to the magnitude of this linear category as the “Euler form” of a certain module (called $S$ there).

So, the question is whether the Euler form is the Euler characteristic of some homology theory. Presumably yes! But I don’t know.

Ultrametric spaces: I wrote in the dial post that Mark Meckes’s working-out of “ultramagnitude” was unpublished. It later appeared as section 8 of this paper. But I haven’t thought about the homology.

Posted by: Tom Leinster on August 27, 2017 8:44 PM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

But I thought non-semicartesian $\mathcal{V}$s held no fear for you? In a previous discussion of magnitude homology, you seemed happy to drop the semicartesian hypothesis and replace it by a 2-sided thingy involving a richer system of coefficients.

Yes; what I should have said is that I’m not sure how to go about choosing the coefficients for a magnitude homology of $FDVect$-categories so as to make it decategorify to magnitude. In general the type of those richer coefficients depends on the particular category we are taking the homology of (like how group homology takes coefficients in a $G$-module), so we can’t just define them once and for all; we would at least need a way to construct such a coefficient system for the category of indecomposable projective $A$-modules for any $A$.

Posted by: Mike Shulman on August 28, 2017 7:57 AM | Permalink | Reply to this

### Re: Magnitude Homology in Sapporo

Okay, here is something of an answer to the indecomposable projectives case. In general, the extra coefficients for magnitude homology of a category $X$ consist of a functor $X^{op} \otimes X \to Ab$ (or chain complexes, etc.). Now the category $IP(A)$ of indecomposable projectives in $A$ is Morita equivalent to $A$, so such a functor is essentially determined by an $A$-$A$-bimodule $P$, in which case the magnitude homology should be the Hochschild homology $HH_\ast^K(A;P)$ (where $K$ is the ground field). The appearance of $Ext$ in your formula, however, suggests that we should instead look at magnitude cohomology (which we have noted the existence of, but heretofor otherwise ignored), which in this case should dually give the Hochschild cohomology $HH^\ast_K(A;P)$.

Now, for any two left $A$-modules $M$ and $N$, we have an $A$-$A$-bimodule structure on $Hom_K(M,N)$, and my reference for homological algebra (Weibel) tells me that we have $HH^\ast_K(A;Hom_K(M,N)) = Ext^\ast_{A/K}(M,N)$. And while I don’t see this explicitly in Weibel, I think that because $K$ is a field, the “relative Ext” should coincide with the absolute one $Ext^\ast_{A}(M,N)$. Thus, assuming I didn’t mess something up, the magnitude cohomology of $IP(A)$ with coefficients in $Hom_K(S,S)$ categorifies its magnitude.

This is admittedly a bit unsatisfying; the coefficients $Hom_K(S,S)$ seem chosen somewhat tautologically to make this true. I’d like to understand better what the $IP(A)$-bimodule corresponding to $Hom_K(S,S)$ looks like; maybe it does look like some kind of replacement for a “constant functor at $1$”?

It’s also interesting because the same argument would work with $IP(A)$ replaced by any category Morita equivalent to it, such as the one-object category $A$ itself, or the category of all finitely generated projective modules. Indeed, in general I believe magnitude homology should be Morita invariant, whereas magnitude is not. I suspect that this difference arises from some condition in the presumptive general decategorification theorem that makes it apply to $IP(A)$ but not to $A$, or perhaps a difference in ways of summing a divergent series (in particular, the magnitude of $A$ itself is $1/dim(A)$, which can never arise as a finite alternating sum of dimensions of homology groups).

A different question is whether there are any coefficients for magnitude homology of $IP(A)$ that reproduce the magnitude. Is there any way to express the magnitude in terms of $Tor$ functors (or more generally Hochschild homology) instead of $Ext$?

Posted by: Mike Shulman on August 28, 2017 9:54 AM | Permalink | Reply to this

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