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November 3, 2017

Magnitude Homology is Hochschild Homology

Posted by Mike Shulman

Magnitude homology, like magnitude, was born on this blog. Now there is a paper about it on the arXiv:

  • Tom Leinster and Mike Shulman, Magnitude homology of enriched categories and metric spaces, arXiv:1711.00802

I’m also giving a talk about magnitude homology this Saturday at the AMS sectional meeting at UC Riverside (this is the same meeting where John is running a session about applied category theory, but my talk will be in the Homotopy Theory session, 3 pm on Saturday afternoon). Here are my slides.

This paper contains basically everything that’s been said about magnitude homology so far on the blog (somewhat cleaned up), plus several new things. Below the fold I’ll briefly summarize what’s new, for the benefit of a (hypothetical?) reader who remembers all the previous posts. But if you don’t remember the old posts at all, then I suggest just starting directly with the preprint (or the slides for my talk).

I also have a request for help with terminology at the end.

  1. We’ve reverted to regarding the length scale \ell in magnitude homology of a metric space as a grading rather than as the coefficients. Thus, magnitude homology of a metric space is a single homology theory that takes values in [0,][0,\infty]-graded groups.

  2. The reason for that reversion is that it makes the abelianization/coefficients functor strong monoidal, so that the construction of the “magnitude nerve” can be decomposed into (1) applying a strong monoidal functor to get a category enriched over [0,][0,\infty]-graded chain complexes, and (2) constructing a kind of nerve of the result. It then turns out that (2) is something already known: a two-sided bar construction, whose homology is the Hochschild homology with coefficients constant at the unit object. In general, if VV is semicartesian and Σ:VW\Sigma:V\to W is strong monoidal, we can define the magnitude homology of a VV-category by applying Σ\Sigma and then taking Hochschild homology with constant coefficients — semicartesianness of VV is what makes “constant coefficients” make sense.

  3. What makes this special kind of Hochschild homology deserve the new name “magnitude homology” is that it categorifies magnitude, and we now have a rigorous proof of that (not just a formal calculation). The first ingredient is the idea that quotienting out the degeneracies in forming the normalized chain complex of a simplicial abelian group is a categorification of the “inclusion-exclusion” formula for nondegenerate simplices that arises from taking powers of a matrix (ZId)(Z-Id). This applies in the generality of a semicartesian VV, a strong monoidal functor Σ:VW\Sigma:V\to W, and a “formal Euler characteristic” on WW, but requires a strong finiteness restriction on the VV-category in question (for an ordinary category, having finitely many nondegenerate simplices in its nerve). This finiteness is never satisfied by a metric space, so the second ingredient is the use of formal Hahn series in that case to deal with the resulting convergence problems.

  4. On the blog, Tom showed that H 1(X)=0H_1(X)=0 if and only if XX is Menger convex, which in particular includes closed convex subsets of Euclidean space, and that in the latter case also H 2(X)=0H_2(X)=0. The paper includes a sufficient condition for H 2(X)=0H_2(X)=0 that’s a little more general, isolating different ways in which H 2(X)H_2(X) can be nonzero: intuitively, if XX is not convex or if it is convex but some points are connected by more than one distinct geodesic.

To state the last point, we needed some nonce definitions, for which we had to invent some names. But some of these seem like they might be, or be related to, known properties of metric spaces; has anyone seen them before? If not, what would you call them?

  • A point yy of a metric space is said to be between two other points xx and zz if d(x,y)+d(y,z)=d(x,z)d(x,y)+d(y,z)=d(x,z), i.e. the triangle inequality is an equality. If additionally xyzx\neq y\neq z then it is strictly between. These are reasonably solid definitions, though I’m not sure whether they appear in the published literature (do they?). More questionable is what to call a pair of points x,zx,z such that there is, or is not, a point strictly between them. The current preprint follows Tom’s suggestion to call xx and zz adjacent if there is no such point, since in a graph with the shortest-path metric this is precisely the adjacency relation of the graph; but maybe there is a better name.

  • Another standard definition, rephrased in this language: a metric space is Menger convex if there do not exist any pair of distinct adjacent points. Thus, a Menger convex space is “as far from being a graph as possible”.

  • What do you call a metric space in which d(x,y 1)+d(y 1,y 2)=d(x,y 2)d(x,y_1)+d(y_1,y_2)=d(x,y_2) and d(y 1,y 2)+d(y 2,z)=d(y 1,z)d(y_1,y_2)+d(y_2,z)=d(y_1,z) together imply d(x,z)=d(x,y 1)+d(y 1,y 2)+d(y 2,z)d(x,z) = d(x,y_1)+d(y_1,y_2)+d(y_2,z)? The current preprint calls this being treelike, since (unless there is a mistake) a graph has this property if and only if it is a tree. (Edit: there is an error; see comments below.) But it also applies to familiar spaces like any subspace of n\mathbb{R}^n, which it’s not clear are “treelike” in any intuitive sense, so maybe there is a better word.

  • What do you call a metric space such that whenever y 1y_1 and y 2y_2 are both between xx and zz, then either y 1y_1 is between xx and y 2y_2 while y 2y_2 is between y 1y_1 and zz, or else y 2y_2 is between xx and y 1y_1 while y 1y_1 is between y 2y_2 and zz? The current preprint calls this being geodetic, since (again, unless there is a mistake) a graph has this property if and only if any two points are connected by a unique shortest path, a property which apparently goes by the name “geodetic”. But again, it also applies to familiar spaces like subspaces of n\mathbb{R}^n, so maybe there is a better word.

The sufficient vanishing condition for magnitude H 2(X)H_2(X) is then that XX is Menger convex, treelike, and geodetic. This is essentially an abstraction of Tom’s proof for convex subsets of n\mathbb{R}^n.

Ideally, I would like to be able to fit conditions of this sort into some hierarchy of definitions that starts with some kind of convexity and generalizes to an nn-dimensional condition that’s relevant to magnitude H nH_n. For a while I was proposing “XX is Menger 2-convex” instead of “XX is geodetic”, since that condition seems somehow like a “Menger version” of the uniqueness of geodesics. But I’ve backed off from that for now, since I’m not confident that that is the right condition to fit into an nn-level hierarchy. Does any such hierarchy of “higher convexity” already exist?

Posted at November 3, 2017 6:24 AM UTC

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Re: Magnitude homology is Hochschild homology

One standard category of study in Metric Geometry are “Length Spaces”, in which the Distance Function is what you’d want to call the Shortest Path Length — in which points at finite distance are connected by Some Geodesic.

It would seem that the Irrationals Among the Usual Cantor Fractal are a Menger-Convex space? It is also totally-diconnected, making it far from being a Length Space. Hm.

In any metric space you can ask for inf yd(x,y)+d(y,z)inf_y d(x,y)+d(y,z) and whether it is realized for some y\{x,z}y\in\!\!\!\!\backslash\{x,z\}; and whether including more interpolants admits a bounded limiting inf; and whether the interpolants can be made arbitrarily close. So, some metric spaces can be given improved Mengery-metrics and some among those can be improved even to Length Metrics; This doesn’t work, e.g., for the Koch Snowflake with the induced Euclidean Metric, though it does have a fairly-natural Haar-like metric. (which should, I think, be bi-lipschitz to the Euclidean Metric)

Posted by: Jesse C. McKeown on November 3, 2017 6:45 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

Is a “length space” the same as what we call a geodesic metric space, i.e. a metric space such that for any x,yx,y there is an isometry γ:[0,a]X\gamma : [0,a] \to X with γ(0)=x\gamma(0)=x and γ(a)=y\gamma(a) =y (hence necessarily a=d(x,y)a=d(x,y))? Menger convexity is sort of a “discrete” or “incomplete” version of this property: a “proper” metric space (one in which closed bounded subsets are compact) is geodesic if and only if it is Menger convex. But for non-proper metric spaces Menger convexity is much weaker than geodesy, e.g. any open subset of n\mathbb{R}^n is Menger convex.

Posted by: Mike Shulman on November 3, 2017 7:02 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

When I was introduced to “Length Spaces”, it wasn’t clear that they needed to be Complete; the Euclidean metric on n\mathbb{Q}^n then seemed to be an Example; but it seems that WikiPedia does indeed require this sort of Completeness. It may have become an over-loaded term. )-:

I think the Irrationals in the Cantor Set is more dramatic than Any Open Set… but I see what you mean. (In fact, any monotone function on a Dense Linear Order will do, and these can have large gaps in them…)

Posted by: Jesse C. McKeown on November 3, 2017 8:25 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

To me, a length space is simply one where for any x, y, the distance d(x,y) is the infimum of the lengths of the paths joining x and y. For instance, \R^2 \setminus {0} with metric induced from Euclidean \R^2 is a length space but is not geodesic in your sense. Another example is the metric space consisting of \N copies of [0, 1] where one identifies all the 0’s and all the 1’s, with metric so that the length of the n^th segment [0, 1] is 1+1/n. This is a complete length space which is not geodesic in your sense. The Hopf–Rinow theorem generalizes to prove that a complete locally compact length space is geodesic in your sense (see wikipedia).

Posted by: Benoit on November 3, 2017 8:40 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

Yes, I believe that’s the standard definition. It’s certainly the one given in the book of Papadopoulos that we cite.

Posted by: Tom Leinster on November 3, 2017 10:04 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

The definition of “geodesic metric space” that we’re using is also from Papadopoulos’s book, so it’s not “our sense”; sorry if that wasn’t clear.

Posted by: Mike Shulman on November 3, 2017 10:29 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

Don’t you need the condition y 1y 2y_1 \neq y_2 in your definition of treelike ? (The current definition makes “treelike” pretty restrictive, it seems.) But then, for graphs, the condition is “no cycle of length at least four” ? (I refer to the definition in the blog post, I haven’t read the article yet.)

Posted by: Benoit on November 3, 2017 9:29 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

Yes, you’re right, thanks! We need to assume y 1y 2y_1\neq y_2 in that definition, and for graphs it only excludes cycles of length >3\gt 3. So “treelike” is not actually a very good name. Have you got any suggestions?

Posted by: Mike Shulman on November 3, 2017 9:55 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

Actually, does it even exclude all cycles of length >3\gt 3? It seems to me now that any complete graph has this property.

Posted by: Mike Shulman on November 3, 2017 10:01 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

I think it excludes minimal cycles of length at least four. I don’t know if “minimal” is standard terminology, but I mean “cycle with no proper subcycle”.

Posted by: Benoit on November 3, 2017 11:27 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

Yes, I agree with that. I think that’s what the argument in Example 7.16 of the paper actually proves. Although it’s not clear to me right now whether the converse holds: does a graph with no minimal cycles of length >3\gt 3 have this property?

Posted by: Mike Shulman on November 3, 2017 11:46 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

I think so. A sketch of proof of the contrapositive: let x, y, z, t satisfy the hypotheses but not the conclusion of the “treelike” property, wlog all four distinct. Let a, b, c be shortest paths connecting them (a connects x to y, etc.). By minimality, there is no repeated vertex. Let d be a shortest path connecting x to t. If d has a vertex in common with a, say x’, then by minimality it can be chosen to coincide with a up to x’. Similarly if d has a vertex in common with b, by the first hypothesis of “treelike” and minimality, it can be chosen to coincide with a(concat)b up to that vertex, and by the second hypothesis, with b(concat)c from that vertex. Similarly for c. But we know that (for lengths) d < a + b + c. Therefore (…) we can find a prime cycle of length at least four.

As for the name, maybe “afocal”? At first glance, this property looks related to conjugate and focal points in Riemannian geometry.

Posted by: Benoit on November 4, 2017 12:28 AM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

What do you mean in this proof whenever you say “by minimality”? For instance, you say “By minimality, there is no repeated vertex” — do you mean by the assumption that a,b,ca,b,c are shortest paths? I don’t think that implies that their concatenation has no repeated vertices even under the “treelike” hypotheses, e.g. aa could share a vertex with cc.

Posted by: Mike Shulman on November 5, 2017 7:05 AM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

aa cannot share a vertex with cc : if for instance uu were a common vertex, then the length of aa from uu to yy should be both strictly larger and strictly smaller than then length of cc from zz to uu, by the two “betweenness” conditions. But I agree the whole thing does not constitute a proof and this is very sloppy… Yes, by “by minimality”, I meant: by the assumptions that these are shortest paths and by the betweenness conditions.

Posted by: Benoit on November 5, 2017 10:58 AM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

It’ll take me a while to fill in the details of that argument. Can you explain why it seems related to conjugate/focal points? Offhand I would have expected those to be more related to the property that I called “geodetic” above, since it is roughly about the nonexistence of multiple geodesics connecting the same two points.

Posted by: Mike Shulman on November 4, 2017 5:54 AM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

You’re right. Actually, it looks like for Riemannian manifolds, “treelike” is equivalent to “empty cut locus”, equivalently “all geodesics are minimizing”. Indeed, let x, y, z, t be distinct and satisfy the hypotheses of “treelike”. By these hypotheses, the four points lie on a geodesic which is minimizing on [x, z] and on [y, t] (take minimizing geodesics from x to y, y to z, and z to t; by minimality (i.e., hypotheses of “treelike”), their concatenation is a geodesic from x to t). The conclusion of “treelike” says that the geodesic is minimizing on [x, t].

Note that a Riemannian manifold with empty cut locus is diffeomorphic to \R^n by the exponential map at some point. It also implies that minimizing geodesics connecting two given points are unique.

Posted by: Benoit on November 4, 2017 12:41 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

How do you know that if yy is between xx and zz then the concatenation of a minimizing geodesic from xx to yy and a minimizing geodesic from yy to zz is again a geodesic? Is it some kind of variational argument, e.g. if it weren’t a geodesic then we could deform it to be shorter? Sorry, my Riemannian geometry is a little rusty.

It sounds to me like “empty cut locus” is a Riemannian geometry version of both “treelike” and “geodetic”; both of them are “discrete” ways of saying something about the uniqueness of connecting geodesics, which in the smooth Riemannian case end up coinciding. The notion of focal/conjugate point seems to be more local/infinitesimal, which has even less of an analogue in a general metric space. Or am I misunderstanding?

Posted by: Mike Shulman on November 5, 2017 6:06 AM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

For the first paragraph: we take minimizing geodesics a from x to y and b from y to z. Their concatenation has length l(a) + l(b) = d(x, y) + d(y, z) = d(x, z) by the first hypothesis of “treelike”, so the concatenation a@b minimizes the distance, so it is a (smooth) geodesic. That’s what I meant clumsily above by “by minimality”. Yes, it’s by variational principles (variation of energy/length), and intuitively, if the concatenation weren’t smooth, it could be shortened close to the angular point.

Note: I supposed you can take minimizing geodesics, which is true for instance in complete Riemannian manifolds by Hopf–Rinow. In the general case, I guess you can take a minimizing sequence, but I’ll have to look more carefully.

Yes, I think you’re right, and conjugate/focal is not relevant here. For Riemannian manifolds (at least for complete ones, by the previous note), it looks like geodetic and treelike are equivalent (in the last sentence of my previous comment, “minimizing” should be removed).

Posted by: Benoit on November 5, 2017 10:14 AM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

Okay, makes sense. So maybe it would be reasonable to use a term derived from “cut locus” for “treelike”, since we have “geodetic” from graph theory for the other notion (assuming the argument there is correct). Is there a more concise way to say “empty cut locus”?

Posted by: Mike Shulman on November 6, 2017 12:16 AM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

By induction, the property “treelike” implies that for any (x i) 0in(x_i)_{0 \leq i \leq n} with x ix i+1x_i \neq x_{i+1} for 1in21 \leq i \leq n-2 satisfying the obvious n1n-1 triangular equalities, one has d(x 0,x n)= i=1 n...d(x_0, x_n) = \sum_{i=1}^n .... (This was used above, with the fact that geodesics are locally minimizing, to prove the equivalence with “empty cut locus” for complete Riemannian manifolds). So “treelike” is equivalent to “no shortcut” (or more precisely, “no non-obvious shortcut”). As for “geodetic”, the blog post says “existence of a unique”, but it is rather “at most one” (or maybe I missed that you assume your graphs connected)? As we saw above, both properties are equivalent for complete Riemannian manifolds, and are equivalent to “empty cut locus” (for which I do not know a more concise phrase), so I think none of these two properties should be named after the Riemannian case, after all…

About the non-complete case: I haven’t thought about it, but studying the cut locus in the non-complete case seems peculiar, and its definition should be adapted (and it also looks like magnitude homology is better behaved for complete metric spaces). For instance, R 2{0}\R^2 \setminus \{0\} with standard metric is “treelike” and “geodetic”, but what is the cut locus of (1,0)(-1,0)? One is tempted to say it is R Unknown character0×{0}\R_{> 0} \times \{0\}, hence non-empty.

Posted by: Benoit on November 6, 2017 2:37 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

In fact, I originally called this property “no shortcut”, before I thought there was some connection with trees. But now that name doesn’t seem right either, since it doesn’t exclude 3-cycles, which certainly have “shortcuts” in an intuitive sense. Right now I’m leaning towards a name like “no 4-cuts”, which is kind of ugly, but I don’t mind because I really regard this as a very important property in its own right.

Posted by: Mike Shulman on November 6, 2017 5:08 PM | Permalink | Reply to this

Re: Magnitude homology is Hochschild homology

Yes, the “no non-obvious shortcut” above was to deal with the 3-cycles. I think you can call the treelike property “no shortcut”, and just say once that it excludes the obvious/trivial shortcuts that the 3-cycles are.

Posted by: Benoit on November 6, 2017 10:45 PM | Permalink | Reply to this

Re: Magnitude Homology is Hochschild Homology

The talk seemed to be well-received; I didn’t quite get through all my slides but I got to use some of them to answer questions at the end. I’ve updated the post with a link to the slides.

Posted by: Mike Shulman on November 5, 2017 5:00 AM | Permalink | Reply to this

Re: Magnitude Homology is Hochschild Homology

Trivial typo: Corollary 7.10 should have H 1 ΣH_1^\Sigma instead of H 0 ΣH_0^\Sigma.

Posted by: Mark Meckes on November 9, 2017 6:55 PM | Permalink | Reply to this

Re: Magnitude Homology is Hochschild Homology

Here’s a possibly very basic question — it should be just a matter of understanding the definitions properly, but I don’t yet. Suppose I want to think of the magnitude of a finite metric space XX in the old, simple-minded way as a number that implicitly builds in a fixed choice of scale, as opposed to as a function in (q )\mathbb{Q}(q^{\mathbb{R}}). Do Corollary 6.28 and Theorem 6.29 imply that n=0 t nsum[(Z XId) n], \sum_{n=0}^\infty t^n sum[(Z_X - Id)^n], converges in a neighborhood of 00 to a meromorphic function of tt, whose value at t=1t = -1 is the magnitude of XX?

Posted by: Mark Meckes on November 9, 2017 7:12 PM | Permalink | Reply to this

Re: Magnitude Homology is Hochschild Homology

I believe that is a true statement, but I don’t believe it follows from the statements of Corollary 6.28 and Theorem 6.29. The latter are formulated assuming that #:V𝕜\# : \mathbf{V} \to \mathbb{k} is induced as a composite χΣ\chi \circ \Sigma, but when 𝕜=\mathbb{k}=\mathbb{R} I don’t know any way to express #=e \# \ell = e^{-\ell} in that way; indeed that was the whole reason for introducing ((q ))\mathbb{Q}((q^{\mathbb{R}})) in the first place.

However, I think your statement (at least, if you add the assumption that XX has a magnitude) follows from the same proof as Theorem 6.29: just leave out the part that connects Z XZ_X to the ww’s (the latter being only defined in the context of χΣ\chi\circ\Sigma). This is also basically the same as the theory in Tom’s paper with Clemens Berger (BL08 in the bibliography), only for V=[0,]\mathbf{V}=[0,\infty] instead of FinSetFinSet.

Posted by: Mike Shulman on November 9, 2017 7:57 PM | Permalink | Reply to this

Re: Magnitude Homology is Hochschild Homology

You’re right. In fact when working over \mathbb{R} (or \mathbb{C}), the matrix-valued series n=0 t n(ZId) n \sum_{n=0}^\infty t^n (Z-Id)^n has radius of convergence equal to the reciprocal of the spectral radius of ZIdZ-Id. The fact that it has a unique analytic continuation which gives Z 1Z^{-1} (assuming it exists) at t=1t=-1 follows from the second displayed equation in the proof of Theorem 6.29.

Posted by: Mark Meckes on November 10, 2017 2:12 PM | Permalink | Reply to this

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