The n-Category Café

Hi Jamie,

I don’t really believe that anything in mathematics is pathological. There’s just stuff we understand and stuff we don’t.

It can seem surprising at first that one particular scale can play a special role for a metric space. But it’s actually quite common. We’ve been talking about persistent homology a bit recently, so here’s an example of that flavour.

Take the metric space $X$ consisting of ten points spaced equally around a circle of radius $r$ in $\mathbb{R}^2$. Around each point, draw the disk of radius $\varepsilon$, where $\varepsilon$ is small (at least to begin with). Consider what happens to the homotopy type of the union of the disks as $\varepsilon$ grows.

At first, the union of the ten disks is disjoint, so it has the homotopy type of ten points. But for a certain value $\varepsilon_0$ of $\varepsilon$, each disk meets its neighbour and their union suddenly acquires the homotopy type of a circle. Then later, when $\varepsilon$ reaches $r$, the disks all merge into one, and thereafter the homotopy type is that of a point.

So associated with our original finite metric space are two special scales, $\varepsilon_0$ and $r$, where the two “catastrophes” (changes of homotopy type) occur.

I can’t give any such appealing interpretation of the special role of the scale $\log\sqrt{2}$ for this particular space. In fact, metric spaces corresponding to graphs are about as ungeometric as it’s possible to be. If I’m ever wondering whether a true statement about subsets of $\mathbb{R}^n$ is also true for general metric spaces, the first place I’ll look for counterexamples is in the class of graphs. From that point of view, maybe it’s not surprising that there’s no easy geometric explanation here.

What do these singularities ‘mean’?

I don’t know, and I’d really like to. For instance, the graph

has magnitude function

$$\frac{5 + 5q - 4q^2}{(1 + q)(1 + 2q)}$$

where $q = e^{-t}$. (For graphs, it’s useful to make that substitution so that the magnitude is always a rational function.) It therefore has singularities at $q = -1$ and $q = -1/2$. How should we interpret those numbers? I spent a while thinking about this general question, and I simply don’t know.

Is the magnitude still defined for these metric spaces by some other method?

Their magnitude as a real number probably isn’t defined, but if you’ll allow the magnitude to live elsewhere then you’re probably OK.

Let $A$ be a finite metric space (which for convenience I’ll assume satisfies the traditional axiom of symmetry). Note that $[0, \infty)$ is a rig in the obvious way, so we can talk about commutative algebras over it. Let $R_A$ be the commutative $[0, \infty)$-algebra generated by one generator $w_a$ for each $a \in A$, subject to the equation

$$\sum_b e^{-d(a, b)} w_b = 1$$

for each $a \in A$. Define $[A]$, the “formal magnitude” of $A$, to be the element $\sum_a w_a$ of $R_A$.

If there exists a $[0, \infty)$-algebra homomorphism $\phi \colon R_A \to \mathbb{R}$ then $\phi([A]) \in \mathbb{R}$ is independent of the choice of $\phi$. (That’s true for any $[0, \infty)$-algebra, not just $\mathbb{R}$.) Moreover, in that case, $\phi([A])$ is the ordinary magnitude $|A|$ of $A$. So the question shifts from “Is the magnitude defined?” to “In what algebras is the magnitude defined?”

Are these particular numbers like $\log \sqrt{2}$ actually determined in an invariant way, or are they just an artifact of the particular choice of the base $e$ for the exponentials appearing in the matrix $Z_A$? I mean, of course $e$ is a somewhat canonical choice of a base for exponentials, but it’s still a choice, and in some contexts (e.g. the homological one) it seems more natural to take the base to be a formal variable.

Here’s a slightly different take on Tom’s response to Jamie’s question. The definition of magnitude as stated above involves an arbitrary choice of scale. We can avoid this unpleasant arbitrariness in at least three ways, which are subtly different perspectives on the same thing:

1. Instead of looking just at $A$, look at the whole family $\{t A : t \in (0, \infty)\}$ of rescalings of $A$, and then take the magnitudes of those. This is how Tom defined the magnitude function above, and the last part of Jamie’s question seems to be coming at the issue from this perspective: what’s going on with this one particular rescaling of $K_{3,2}$?

2. Use all possible scales to define magnitude, so that the magnitude of $A$ is a (partially-defined) function instead of a real number. Forget the definitions of magnitude and magnitude function above, and suppose we instead start by defining a family of matrices $Z_{A, t}(a,b) = e^{-t d(a,b)}$ for $t \in (0,\infty)$. Now define the magnitude function of $A$ by $$M_A(t) = \sum_{a,b \in A} Z_{A, t}^{-1}(a, b)$$ for all $t \in (a,b)$ for which $Z_{A,t}$ is invertible. Notice the difference? Of course $M_A(t) = | t A |$, so this gives the same thing as before, but in this approach, I never mentioned rescaling the space $A$ itself.

3. Replace the base of the exponent with a formal variable, as Mike suggests. As Tom alludes to, whether perspective 2 or 3 is more useful seems to depend on context. For graphs, 3 is nice because then magnitude is a rational function. For infinite spaces, 2 is nice because we can use analytic tools, and it’s not clear whether the magnitude function is at all nice algebraically.

From either of the latter two perspectives, there’s no reason even to ask “What’s so pathological about $(\log \sqrt{2}) K_{3,2}$?” On the other hand, Jamie’s first question:

What do these singularities ‘mean’?

becomes even more interesting. The fact that the magnitude function of $K_{3,2}$ has a singularity at $\log \sqrt{2}$ depends on the fact that we say the edges of $K_{3,2}$ have length 1; change that basic length and the location of the singularity changes. But the facts that there is a singularity in the magnitude function, and that there is only one in $(0,\infty)$ (but three if you let the base be a formal variable, or look at the meromorphic continuation), are intrinsic facts about $K_{3,2}$, and about any rescaling of it. In the geometry of metric spaces, people are usually interested in properties which are invariant (or at least behave predictably) under rescaling, and these singularities give us a lot of such properties. (Going farther, from the formal-variable/rational-function perspective, we could start looking at degrees of poles, residues, etc.)

So what does it “mean” that the magnitude function of $K_{3,2}$ has a singularity in $(0,\infty)$? What does it “mean” that, if you broaden your perspective sufficiently, it has three singularities? What does it “mean” that those singularities are they types that they are?

I don’t know answers to all those questions. One thing we know is that if the magnitude function of a finite metric space $A$ has a singularity in $(0,\infty)$, then $A$ is not of “negative type”, which implies in particular that $A$ cannot be isometrically embedded in a Euclidean space, or in $L^1$ (a stronger statement, since $L^2$ embeds isometrically in $L^1$). These facts can be proved without using magnitude, but they’re enough to convince me that Jamie’s question is worth thinking about.

What stands in the way of a (co)homology of enriched categories?

Is Thomason cohomology of categories state-of-the-art on (co)homology of ordinary categories? (An impressively old average year of reference in bibliography!)

There’s a notion of internal nerve.

There’s a geometric nerve of a bicategory.

We also have nerve and realization in some kind of enriched setting.

Would some adequate kind of ‘enriched nerve’ be enough?

What stands in the way of a (co)homology of enriched categories?

Simply that we don’t know how to define it. Or I don’t, anyway.

Of course, if we had a good notion of the nerve of an enriched category, and whatever kind of thing that nerve was, it was something we could take the (co)homology of, then we’d be home. But I don’t know of such a construction, and I’m not immediately seeing one in the links you give.

Maybe that’s just my ignorance speaking. If someone else can see a way through, I’d love to know what it is and what (co)homology theory it gives for metric spaces.

I suppose one difficulty is that to cover metric spaces, we need our theory of enriched (co)homology to work for the enriching category $V = [0, \infty]$. Although this is symmetric monoidal closed, and complete and cocomplete, in other ways it’s quite unlike the category of sets, which is everyone’s starting point.

Here’s one possible generalization, based on this comment and the ensuing discussion. Let $(V,\le,\otimes,1)$ be a semicartesian monoidal poset, such as $([0,\infty],\ge,+,0)$ or $(\mathbb{N},\ge,+,0)$ or $(\mathbf{2}=\{\bot\le \top\},\wedge,\top)$. We intend to write down something like a homology theory for finite $V$-enriched categories, which in these examples are (related to) metric spaces, graphs, and ordinary posets.

Let $C$ be a finite $V$-category, and let $MS(C)$ be the codiscrete simplicial set on the objects of $C$, so that $MS_k(C) = C_0^{k+1}$. This simplicial set is filtered by $V$: given any $\ell \in V$, define a subset $F_\ell MS_k(C)$ to consist of those $k$-simplices $(x_0,\dots,x_k)$ such that there is a map $\ell \le C(x_{k-1},x_k) \otimes \cdots \otimes C(x_0,x_1)$ in $V$. The identities $1\le C(x,x)$ in $C$ imply that degeneracies preserve the filtration, the composition $C(y,z) \otimes C(x,y) \le C(x,z)$ implies that inner faces preserve the filtration, and semicartesianness $C(x,y) \le 1$ implies that the outer faces do too.

Now if $\ell\le m$ in $V$, then $F_m MS(C) \subseteq F_{\ell} MS(C)$. Thus we can consider the relative homology of this pair of spaces, which can be computed concretely by generating chain complexes of free abelian groups from both of them, taking the quotient chain complex, and then its homology. This gives us a homology theory graded by pairs $\ell\le m$ (in addition to the usual homological grading $k$). We should probably also include the homologies of the individual spaces $F_\ell MS(C)$ as part of the theory.

In the case of $(\mathbb{N},\ge,+,0)$, I think the homology groups $H_{k,\ell\le\ell+1}(C)$ are the Hepworth-Willerton homology groups. (There is something to say about normalization, but I think that was mostly answered in the previous discussion.) And in the case of $\mathbf{2}$, the space $F_\top MS(C)$ is just the nerve of $C$, so the groups $H_{k,\top} (C)$ are the ordinary poset homology of $C$. (Since $F_\bot MS(C) = MS(C)$ is contractible, in this case the other homology groups $H_{k,\bot\le\top}(C)$ carry no additional information.)

Does this give anything interesting for $([0,\infty],\ge,+,0)$? Does it have anything to do with the magnitude?

One could also, of course, imagine various generalizations of this. If $V$ is a (semicartesian monoidal) category rather than a mere poset, it would be natural to define $F_\ell MS_k(C)$ to consist of $k$-simplices $(x_0,\dots,x_k)$ together with a morphism $\ell \to C(x_{k-1},x_k) \otimes \cdots \otimes C(x_0,x_1)$. Then it’s no longer a subset of $MS_k(C)$, but that’s not a problem; the associativity and identity axioms for the $V$-category $C$ should ensure that the simplicial identities still hold. The resulting homology theory will be graded by morphisms in $V$ (and also objects of $V$), and when $V=Set$ it should recover the usual homology of a category. I’m not sure whether we can do without semicartesianness, though.

Quick question: how much of the theory of magnitude of graphs (including Hepworth-Willerton homology) depends on having a graph, and how much of it depends only on having a metric space whose distances are all integers?

Here’s another approach to categorifying magnitude, which seems to more or less fail. But I had fun thinking about it.

Tom didn’t say much about it in this post, but the magnitude of a metric space is a special case of the magnitude of an enriched category. The latter is defined relative to any monoidal category $V$ equipped with a monoid homomorphism $|\cdot |$ from $\pi_0(V)$, its monoid of isomorphism classes of objects, to the multiplicative monoid of some rig $k$. The magnitude of a $V$-category is then (when defined) an element of $k$: we apply $|\cdot |$ to the hom-objects to get a $k$-valued matrix, then add up the entries of any weighting or coweighting if it has both (or the elements of its inverse, if it has one).

Of course there is a universal choice of $k$, namely the “monoid rig” (the inverse-free version of a “group ring”) of $\pi_0(V)$, whose elements are $\mathbb{N}$-linear combinations of elements of $\pi_0(V)$. (For instance, if $V = (\mathbb{N},\ge)$ for “metric spaces with integer distances” including graphs, then this monoid rig is the rig of polynomials in one variable with natural number coefficients.) Note that if we define magnitude using this universal choice of $k$, then it will be a finite $\mathbb{N}$-linear combination of isomorphism classes of objects of $V$, or equivalently a finitely-supported function from $\pi_0(V)$ to $\mathbb{N}$. Thus, the magnitude is naturally already “graded by objects of $V$”, so it makes sense that a categorification of it would be too.

Now, if we want magnitude to be the “Euler characteristic” of a homology theory, we ought to be over a ring rather than a rig, since Euler characteristic involves $(-1)^k$. Thus, let’s consider instead the “monoid ring” $\mathbb{Z}[\pi_0(V)]$, which for simplicity we may write $\mathbb{Z}[V]$.

If we also want to generalize some of Tom’s arguments about graph magnitude, we need an analogue of “power series”. For this purpose, suppose that $V$ contains no nontrivial invertible objects; that is, if $X\otimes Y \cong I$, then $X\cong I$ and $Y\cong I$, where $I$ is the unit object. Then the non-unit isomorphism classes generate a proper ideal $\mathcal{I}$ in $\mathbb{Z}[V]$. Let $k$ be the completion of $\mathbb{Z}[V]$ at that ideal. Informally, in $k$ we are allowed to formally add up infinite $\mathbb{Z}$-linear combinations of objects of $V$ as long as each of those objects is decomposed as a nontrivial (i.e. not involving the unit object) $n$-ary tensor product in such a way that each $n$ appears only finitely many times.

Now we can define magnitude of $V$-categories relative to $k$. It seems that we should then be able to reproduce the proof of Proposition 3.9 from here, leading to a similar formula for the magnitude of a $V$-category in terms of tuples of objects with adjacent ones distinct for which the tensor product of their hom-objects belongs to a particular isomorphism class. One could then try to mimic the Hepworth-Willerton construction on such tuples, and get a “categorification” of magnitude for cardinality reasons as Tom mentioned.

However, this doesn’t help for metric spaces, because the completion can actually destroys a lot of information. In particular, if a single object of $V$ can be decomposed as a nontrivial $n$-ary tensor product for all $n$, then it maps to $0$ in $k$. (Formally this is because it lies in $\bigcap_n \mathcal{I}^n$; informally this property would mean we could add up infinitely many copies of it, leading to an Eilenberg swindle.) This is unfortunate because $x = \frac{x}{n} + \cdots + \frac{x}{n}$ in $[0,\infty]$ for any $x$, so that when $V=[0,\infty]$ we have $k=\mathbb{Z}$ with all nonzero $x\in[0,\infty]$ mapping to $0\in\mathbb{Z}$. So it seems that this approach isn’t going to tell us anything about the magnitude of metric spaces.

Thanks for writing that out, Tom! I had something similar going around in my head, but couldn’t see it as clearly as that.

I think that for $\ell\in\mathbb{L}_A$, your $c_{k,\ell}$ is the rank of my chain group $MS_{k,\ell\ge \ell-\varepsilon}$, where $\varepsilon$ is small enough that $(\ell-\varepsilon,\ell] \cap \mathbb{L}_A = \{\ell\}$. Thus $\sum_{k\ge 0} (-1)^k c_{k,\ell}$ is the Euler characteristic of the chain complex $MS_{\ell\ge \ell-\varepsilon}$, and hence also of the homology $MH_{\ast,\ell\ge\ell-\varepsilon}$. So insofar as your formal calculation is meaningful, it should mean that the magnitude of a finite metric space is indeed an Euler characteristic of my proposed homology theory.

The other thought I had is that maybe we could make this formal calculation meaningful by approximating a finite metric space by ones we understand better. For instance, it seems to me that at least in the “generic” case when the triangle inequality is never an equality (i.e. there are no “collinear triples”), we should be able to deform a finite metric space slightly to make its distances all integer multiples of some fixed small positive real number, and that the formal-power-series calculation that works for graphs ought also to work for metric spaces with that property. Does that seem plausible?

Thus $\sum_{k \geq 0} (-1)^k c_{k, \ell}$ is the Euler characteristic of […] the homology $MH_{*, \ell \geq \ell - \varepsilon}$.

Yes, absolutely. Reflecting on your original comment, I had in my head the rather melodramatic notation $\lim_{\varepsilon \to 0+} MH_{\ast, \ell \geq \ell - \varepsilon}$. Perhaps it wouldn’t be too misleading to write $MH_{\ast, \ell-}$, much as people sometimes write $f(x-)$ to mean $\lim_{\varepsilon \to 0+} f(x - \varepsilon)$.

(I can’t help mentioning a possible connection with something else, though maybe it’s a red herring. In a post about the Euler calculus — Integrating against the Euler characteristic — I mentioned the formula $\int f \,d\chi = \sum_{x \in \mathbb{R}} (f(x) - f(x-))$ for the integral against the Euler characteristic of a function $f$ on $\mathbb{R}$. I think Euler characteristic is playing a different role there, and I can’t see any definite relevance of this formula, but neither can I resist mentioning it.)

Regarding your last paragraph, I see what you mean. The hypothesis that all distances are integer multiples of some fixed real was something Aaron and I occasionally adopted. Incidentally, one crucial thing we didn’t do, which you did, was think about relative homology.

One potential problem, depending on what one is going to do with this integer multiples idea, is that we’re short on stability theorems for magnitude. For instance, we don’t know whether the magnitude of finite subsets of $\mathbb{R}^n$ is continuous with respect to the Hausdorff metric. It’s fairly clear that it’s continuous if you keep the number of points the same, but it’s not so clear when the number of points changes.

I think that for $\ell \in \mathbb{L}_A$, your $c_{k, \ell}$ is the rank of my chain group $MS_{k, \ell \geq \ell - \varepsilon}$

I initially nodded along to this, because I’d been through some similar thought process, but now I’m confused.

We have a simplicial set $MS$, and a filtration $(F_\ell MS)_{\ell \geq 0}$ on it. I’d be happy to write $MS_{k, \ell}$ for $F_\ell MS_k$, and in fact I’ve been doing exactly that in private. We can form the free abelian group on it, $\mathbb{Z} MS_{k, \ell}$, and that gives a chain complex $\mathbb{Z} MS_{\ast, \ell}$ for each $\ell$. It has a subcomplex $\mathbb{Z} MS_{\ast, \ell - \varepsilon}$, and we can form the cokernel of the inclusion:

$$0 \to \mathbb{Z} MS_{\ast, \ell - \varepsilon} \to \mathbb{Z} MS_{\ast, \ell} \to C_{\ast, \ell} \to 0.$$

Is it $C_{\ast, \ell}$ you meant by $MS_{k, \ell \geq \ell - \varepsilon}$?

If so, I’m not sure its rank is $c_{k, \ell}$, because of the question of whether our tuples $(x_0, \ldots, x_k)$ have to satisfy $x_0 \neq x_1 \neq \cdots \neq x_k$.

One potential problem, depending on what one is going to do with this integer multiples idea, is that we’re short on stability theorems for magnitude. For instance, we don’t know whether the magnitude of finite subsets of $\mathbb{R}^n$ is continuous with respect to the Hausdorff metric. It’s fairly clear that it’s continuous if you keep the number of points the same, but it’s not so clear when the number of points changes.

What I had in mind would be keeping the number of points the same. If all the triangle inequalities are strict, then there is a minimum nonzero value of $d(x,y) + d(y,z) - d(x,z)$, say $\eta$. So if we modify all distances by less than $\eta/3$, the triangle inequalities will still all hold. Now for any $\varepsilon \lt \eta/3$, let $d_\varepsilon(x,y)$ be the integer multiple of $\varepsilon$ closest to $d(x,y)$.

I feel like it should be possible to deal with triangle equalities as well, since in that case we essentially have to simultaneously approximate a finite number of (potentially) irrational numbers by rational ones, but I’m not sure exactly how to write it out.

Here’s something that may be useful in seeking to interpret our possibly-divergent infinite sums.

Write $sum(M)$ for the sum of all the entries of a matrix $M$. I believe the following is true.

Theorem   Let $Z$ be an invertible matrix over a field (or maybe any commutative ring). Then:

1. The power series $f_Z(t) = \sum_{k = 0}^\infty sum((Z - I)^k) t^k$ is a rational function.

2. $f_Z$ does not have a pole at $-1$.

3. $f_Z(-1) = |Z|$.

In other words, we sum the divergent series $\sum_k (-1)^k sum((Z - I)^k)$ by changing $-1$ to a formal variable $t$, noting that the formal power series is formally equal to a rational function, and evaluating that function at $-1$.

Part 1 doesn’t need the invertibility hypothesis, and it’s definitely true — or it had better be, because it’s Lemma 2.1 of The Euler characteristic of a category as the sum of a divergent series.

I think parts 2 and 3 are also true. It’s basically Theorem 3.2 of that paper, but I’d need to go back through it in detail in order to be sure.

(Somewhat technical footnote follows.)

Actually, Theorem 3.2 suggests something a bit more general. If you ask whether the matrix $Z$ of an (enriched) category is invertible, the answer isn’t invariant under equivalence. In fact, if a category isn’t skeletal then its matrix has two identical rows and is therefore not invertible. So the result above looks a bit uncategorical.

Let’s say that a square matrix is essentially invertible if it can be obtained from an invertible matrix by repeatedly duplicating rows and columns, but always duplicating the same rows as columns. For instance, we might start with an invertible $6 \times 6$ matrix, insert a new $2\frac{1}{2}$th row identical to the existing 4th row, and insert a new $2\frac{1}{2}$th column identical to the existing 4th column. The resulting $7 \times 7$ matrix would be essentially invertible.

The point is that if a category is equivalent to one whose matrix is invertible, then its own matrix is essentially invertible. And I think the “Theorem” stated above is true for essentially invertible matrices, not just invertible ones. That would be a more categorically relevant result.

You probably won’t be surprised to hear that I think our attention should focus on $S_{k,\le\ell}$; everything else can be understood in relation to it. Note that it is actually a simplicial set, not just a semisimplicial one: if we don’t demand nondegeneracy, then we can duplicate points to give degeneracies.

On the other hand, I don’t actually believe that any of the other three are even semisimplicial sets until you either add a basepoint or take free abelian groups. Definition 2 of Hepworth-Willerton uses “0” in the second case, which doesn’t exist until you do one of those things. They don’t say explicitly what happens if $x_0 = x_2$ and your face map is omitting $x_1$, but since in that case the distance would no longer be $\ell$ it’s also covered by the second case.

So actually,all we have to start with is one family of (unpointed) simplicial sets $S_{\ast,\le\ell}$, which gives rise by free abelian groups to a simplicial abelian group $\mathbb{Z} S_{\ast,\le\ell}$. Now there are several ways to make a chain complex out of a simplicial abelian group, which are summarized here. One is the “alternating sum of face maps” (which only sees the underlying semisimplicial structure), which gives your $C_{\ast,\le\ell}$. Another is the quotient of this by all the degenerate simplicies, which is probably what you have in mind for $C_{\ast,\le\ell}^{nd}$.