The n-Category Café

I’ll explain twice: a short version, then a long version. After that, there’s a section going into some of the details that I wanted to keep tucked out of the way. Choose the level of detail you want!

So that I don’t have to keep saying it, almost everything here that’s new is due to Mike Shulman, who put these ideas together on the other thread. Some aspects were present in work that Aaron Greenspan did during his master’s year with me (2014–15); you can read his MSc thesis here. But Aaron and I didn’t get very far, and it was Mike who made the decisive contributions and to whom this theory should be attributed.

The short version

I won’t actually give the definition here — I’ll just sketch its shape.

Let $V$ be a semicartesian monoidal category. Semicartesian means that the unit object of $V$ is terminal. This isn’t as unnatural a condition as it might seem!

Let $X$ be a small $V$-category ($=$ category enriched in $V$). Small means that the collection of objects of $X$ is small (a set).

Let $A\colon V \to Ab$ be a small functor. In this context, small means that $A$ is the left Kan extension of its restriction to some small full subcategory of $V$. This condition holds automatically if the category $V$ is small, as it often will be for us.

From this data, we define a sequence $\bigl( H_n(X; A) \bigr)_{n \geq 0}$ of abelian groups, called the (magnitude) homology of $X$ with coefficients in $A$. Dually, given instead a contravariant functor $A \colon V^{op} \to Ab$, there is a sequence $\bigl( H^n(X; A) \bigr)_{n \geq 0}$ of cohomology groups. But we’ll concentrate on homology.

As for any notion of homology, we can attempt to form the Euler characteristic

$$\chi(X; A) = \sum_{n \geq 0} (-1)^n rank(H_n(X; A)).$$

Depending on $X$ and $A$, it may or may not be possible to make sense of this infinite sum.

Examples:

• When $V = Set$ and $A$ is chosen suitably, we recover the notion of homology and Euler characteristic of an ordinary category. What do “homology” and “Euler characteristic” mean for an ordinary category? There are several equivalent answers; one is that they’re just the homology and Euler characteristic of the topological space associated to the category, called its geometric realization or classifying space. The Euler characteristic of a category is also called its magnitude.

• When $V$ is the poset $(\mathbb{N} \cup \{\infty\}, \geq)$, made monoidal by taking $\otimes$ to be addition, graphs can be understood as special $V$-categories. By choosing suitable values of $A$, we obtain Hepworth and Willerton’s magnitude homology of a graph. Its Euler characteristic is the magnitude of a graph.

• When $V$ is the poset $([0, \infty], \geq)$, made monoidal by taking $\otimes$ to be addition, metric spaces can be understood as special $V$-categories. By choosing suitable values of $A$, we obtain a new notion of the magnitude homology of a metric space. Subject to convergence issues that haven’t been fully worked out yet, its Euler characteristic is the magnitude of a metric space.

The long version

Again, let’s start by fixing a semicartesian monoidal category $V$. I’ll use the letter $\ell$ for a typical object of $V$, because an important motivating case is where $V = [0, \infty]$, and in that case the objects of $V$ are thought of as lengths.

Aside   Actually, you can be a bit more general and work with an arbitrary monoidal category $V$ equipped with an augmentation, as described here, or you can do something more general still. But I’ll stick with the simpler hypothesis of semicartesianness.

Step 1   Let $X$ be a small $V$-category. We define a kind of nerve $N(X)$. The nerve of an ordinary category is a single simplicial set, but for us $N(X)$ will be a functor $V^{op} \to sSet$ into the category $sSet$ of simplicial sets. For $\ell \in V$, the simplicial set $N(X)(\ell)$ is defined by

$$N(X)(\ell)_n = \coprod_{x_0, \ldots, x_n \in X} V\bigl(\ell, X(x_0, x_1) \otimes \cdots \otimes X(x_{n - 1}, x_n)\bigr)$$

($n \geq 0$). The degeneracy maps are given by inserting identities. The inner face maps are given by composition. The outer face maps are defined using the unique maps from the first factor $X(x_0, x_1)$ and the last factor $X(x_{n - 1}, x_n)$ to the unit object of $V$. (There are unique such maps because $V$ is semicartesian.)

Mike wrote $MS$ instead of $N$. I guess he intended the M to stand for magnitude and the S to stand for simplicial. I’m using $N$ because I want to emphasize that it’s a kind of nerve. Still, half of me regrets removing the notation MS from a construction described by Mike Shulman.

Steps 2 and 3   Let $C(X)$ be the composite functor

$$V^{op} \stackrel{N(X)}{\longrightarrow} sSet \stackrel{\mathbb{Z} \cdot -}{\longrightarrow} sAb \longrightarrow Ch.$$

Here $sAb$ is the category of simplicial abelian groups, $Ch$ is the category of chain complexes of abelian groups, and the functor $\mathbb{Z} \cdot - \colon sSet \to sAb$ is induced by the free abelian group functor $\mathbb{Z} \cdot - \colon Set \to Ab$. The unlabelled functor $sAb \to Ch$ sends a simplicial abelian group to either its unnormalized chain complex or its normalized chain complex. It won’t matter which we use, for reasons I’ll explain in the details section below.

Notice that $C(X)$ isn’t a single chain complex; it’s a functor into the category of chain complexes. There’s one chain complex $C(X)(\ell)$ for each object $\ell$ of $V$.

Step 4   Now we bring in the other piece of data: a small functor $A \colon V \to Ab$, which I’ll call the functor of coefficients. Actually, everything that follows makes sense in the more general context of a functor $A \colon V \to Ch$, where $Ab$ is thought of as a subcategory of $Ch$ by viewing an abelian group as a chain complex concentrated in degree zero. But we don’t seem to have found a purpose for that extra generality, so I’ll stick with $Ab$.

We form the tensor product of $C(X)\colon V^{op} \to Ch$ with $A \colon V \to Ab$. By definition, this is the chain complex defined by the coend formula

$$C(X) \otimes_V A = \int^{\ell \in V} C(X)(\ell) \otimes A(\ell).$$

The tensor product on the right-hand side is the tensor product of chain complexes. Under our assumption that $A(\ell)$ is concentrated in degree zero, its $n$th component is simply $C(X)(\ell)_n \otimes A(\ell)$.

Explicitly, this coend is the coproduct over all $\ell \in V$ of the chain complexes $C(X)(\ell) \otimes A(\ell)$, quotiented out by one relation for each map $\ell \to \ell'$ in $V$. Which relation? Well, given such a map, you can write down two maps from $C(X)(\ell') \otimes A(\ell)$ to the coproduct I just mentioned, and the relation states that they’re equal.

This coend exists because of the smallness assumption on $A$. Indeed, by definition of small functor, there exists some small full subcategory $W$ of $V$ such that $A$ is the left Kan extension of $A|_W$ along the inclusion $W \hookrightarrow V$. Then $C(X)|_W \otimes_W A|_W$ exists because $Ch$ has small colimits, and you can show that it has the defining universal property of the coend above. So $C(X) \otimes_V A$ exists and is equal to $C(X)|_W \otimes_W A|_W$.

We have now constructed from $X$ and $A$ a single chain complex $C(X) \otimes_V A$.

If you choose to use unnormalized chains, you can unwind the coend formula to get a simple explicit formula for $C(X) \otimes_V A$:

$$(C(X) \otimes_V A)_n = \coprod_{x_0, \ldots, x_n \in X} A\bigl(X(x_0, x_1) \otimes \cdots \otimes X(x_{n - 1}, x_n)\bigr)$$

with the differential that you’d guess. (This formula does assume that $A$ is a functor from $A$ into $Ab$ rather than $Ch$. For $Ch$-valued $A$, the formula becomes slightly more complicated.) I don’t think there’s such a simple formula for normalized chains, at least for general $V$.

Step 5   The (magnitude) homology of $X$ with coefficients in $A$, written as $H_\ast(X; A)$, is the homology of the chain complex $C(X) \otimes_V A$. In other words, $H_n(X; A)$ is the $n$th homology group of $C(X) \otimes_V A$, for $n \geq 0$.

For the definition of cohomology, let $A$ instead be a small contravariant functor $V^{op} \to Ab$. Then we can form the chain complex

$$Hom(C(X), A)_V = \int_{\ell \in V} Hom(C(X)(\ell), A(\ell)).$$

The $Hom$ on the right-hand side denotes the closed structure on the monoidal category of chain complexes. And $H^\ast(X; A)$, the cohomology of $X$ with coefficients in $A$, is defined as the homology of the chain complex $Hom(C(X), A)_V$.

Everything is functorial in the way it should be: homology $H_\ast(X; A)$ is covariant in $X$, cohomology $H^\ast(X; A)$ is contravariant in $X$, and both are covariant in the functor $A$ of coefficients.

Example: ordinary categories

When $V = Set$, a small $V$-category is just a small category $X$.

The functor $N(X) \colon Set^{op} \to sSet$ sends $\ell \in Set$ to the $\ell$th power of the ordinary nerve. So, we might suggestively write $N(X)(\ell)$ as $N(X)^\ell$ instead.

Now let’s think about the functor of coefficients, which is some small functor $A \colon Set \to Ab$. For $A$ to be small means exactly that there is some small full subcategory $W$ of $Set$ such that $A$ is the left Kan extension of $A|_W$ along the inclusion $W \hookrightarrow Set$. For instance, choose an abelian group $B$ and define $A(\ell)$ to be the coproduct $\ell \cdot B$ of $\ell$ copies of $B$. Then $A$ is small, since if we take $W \subset Set$ to be the full subcategory consisting of just the one-element set then $A$ is the left Kan extension of its restriction to $W$. Let’s write $A$ as $- \cdot B \colon Set \to Ab$.

The general definition gives us homology groups $H_\ast(X; -\cdot B)$ for every small category $X$ and abelian group $B$. These homology groups, more normally written as $H_\ast(X; B)$, are actually something familiar. In simplicial terms, they’re simply the homology of the ordinary nerve of $X$ (with coefficients in $B$). In terms of topological spaces, they’re just the homology of the geometric realization (classifying space) of $X$.

Example: graphs

Let $V = (\mathbb{N} \cup \{\infty\}, \geq)$, a poset seen as a category. The objects of $V$ are the natural numbers together with $\infty$, there’s exactly one map $\ell \to m$ when $\ell \geq m$, and there are no maps $\ell \to m$ when $\ell \lt m$. It’s a monoidal category under addition. Any graph $X$ can be seen as a $V$-category: the objects are the vertices, and $X(x, y) \in V$ is the number of edges in a shortest path from $x$ to $y$ (understood to be $\infty$ if there is no such path at all).

So, we’re going to get a homology theory of graphs.

What about the coefficients? Well, the first point is that we don’t have to worry about the smallness condition. The category $V$ is small, so it’s automatic that any functor on $V$ is small too.

The second, important, point is that every object $\ell$ of $V$ gives rise to a functor $\delta_\ell \colon V \to Ab$, defined by

$$\delta_\ell(m) = \begin{cases} \mathbb{Z} &\text{if }\,\, m = \ell,\\ 0 &\text{if }\,\, m \neq \ell. \end{cases}$$

($m \in V$). We’re going to use $\delta_\ell$ as our functor of coefficients.

So, for any graph $X$ and natural number $\ell$, we get homology groups $H_\ast(X; \delta_\ell)$. It turns out that $H_n(X; \delta_\ell)$ is exactly what Richard Hepworth and Simon Willerton called the magnitude homology group $MH_{n, \ell}(X)$.

I’ll repeat Richard and Simon’s definition here, so that you can see concretely what Mike’s general theory actually produces in a specific situation. Let $X$ be a graph. For integers $n, \ell \geq 0$, let $MC_{n, \ell}(X)$ be the free abelian group on the set

$$\{ (x_0, \ldots, x_n) \in X^{n + 1} \,\,:\,\, x_0 \neq x_1 \neq \cdots \neq x_n, \,\, d(x_0, x_1) + \cdots + d(x_{n - 1}, x_n) = \ell \}.$$

For $1 \leq i \leq n - 1$, define $\partial_i \colon MC_{n, \ell}(X) \to MC_{n - 1, \ell}(X)$ by

$$\partial_i(x_0, \ldots, x_n) = \begin{cases} (x_0, \ldots, x_{i - 1}, x_{i + 1}, \ldots, x_n) & \text{if }\,\, d(x_{i - 1}, x_{i + 1}) = d(x_{i - 1}, x_i) + d(x_i, x_{i + 1}), \\ 0 & \text{otherwise}. \end{cases}$$

Then define $\partial \colon MC_{n, \ell}(X) \to MC_{n - 1, \ell}(X)$ by $\partial = \sum_{i = 1}^{n - 1} (-1)^i \partial_i$. This gives a chain complex $MC_{\ast, \ell}(X)$ for each natural number $\ell$. The Hepworth–Willerton magnitude homology group $MH_{n, \ell}(X)$ is defined to be its $n$th homology.

So, this two-case formula for the differential, involving the triangle inequality, somehow comes out of Mike’s general definition. I’ll explain how in the details section below.

Incidentally, Richard and Simon proved a Künneth theorem, an excision theorem and a Mayer–Vietoris theorems for their magnitude homology of graphs. Can these be generalized magnitude homology of arbitrary enriched categories?

Example: metric spaces

Let $V$ be the poset $([0, \infty], \geq)$, made into a monoidal category in the same way that $\mathbb{N} \cup \{\infty\}$ was. As Lawvere pointed out long ago, any metric space can be seen as a $V$-category.

So, we get a homology theory of metric spaces. More exactly, we have a graded abelian group $H_\ast(X; A)$ for each metric space $X$ and functor $A \colon [0, \infty] \to Ab$. Exactly as for graphs, every element $\ell \in [0, \infty]$ gives rise to a functor $\delta_\ell \colon [0, \infty] \to \Ab$, taking value $\mathbb{Z}$ at $\ell$ and $0$ elsewhere. So we get a group $H_n(X; \delta_\ell)$ for each $n \in \mathbb{N}$ and $\ell \in [0, \infty]$.

Explicitly, this group $H_n(X, \delta_\ell)$ turns out to be the same as the group $MH_{n, \ell}(X)$ that you get from Hepworth and Willerton’s definition above by simply crossing out the word “graph” and replacing it by “metric space”, and letting $\ell$ range over $[0, \infty]$ rather than $\mathbb{N} \cup \{\infty\}$.

But here’s the thing. There are some metric spaces, including most finite ones, where the triangle inequality is never an equality (except in the obvious trivial situations). For such spaces, the Hepworth–Willerton differential $\partial$ is always $0$. Hence the homology groups are the same as the chain groups, which tend to be rather large. For instance, that’s almost always the case when $X$ is a random finite collection of points in Euclidean space. So homology fails to do its usual job of summarizing useful information about the space.

In that situation, we might prefer to use different coefficients. So, let’s think again about the construction of the functor $\delta_\ell \colon V \to Ab$ from the object $\ell \in V$. This construction makes sense for any partially ordered set $V$, and it also makes sense not only for single elements (objects) of $V$, but arbitrary intervals in $V$.

What I mean is the following. An interval $J$ in a poset $V$ is a subset with the property that if $\ell_1 \leq \ell_2 \leq \ell_3$ in $V$ with $\ell_1, \ell_3 \in J$ then $\ell_2 \in J$. For any interval $J \subseteq V$, there’s a functor $\delta_J \colon V \to Ab$ defined on objects by

$$\delta_J(\ell) = \begin{cases} \mathbb{Z} &\text{if }\,\, \ell \in J, \\ 0 &\text{otherwise}. \end{cases}$$

It’s defined on maps by sending everything to either a zero map or the identity on $\mathbb{Z}$. For instance, if $J$ is a trivial interval $\{\ell\}$ then $\delta_J$ is the functor $\delta_\ell$ that we met before.

I observed a few paragraphs back that when $X$ is a finite metric space, $H_\ast(X; \delta_\ell)$ typically isn’t very interesting. However, it seems likely that $H_\ast(X; \delta_J)$ is more interesting for nontrivial intervals $J \subseteq [0, \infty]$. The idea is that it introduces some blurring, to compensate for the fact that the triangle inequality is never exactly an equality. And here we get into territory that seems close to that of persistent homology… but this connection still needs to be explored!

Decategorification: from homology to magnitude

For any homology theory of any kind of object $X$, we can attempt to define the Euler characteristic of $X$ as the alternating sum of the ranks of the homology groups. We immediately have to ask whether that sum makes sense.

It may be that only finitely many of the homology groups are nontrivial, in which case there’s no problem. Or it may be that infinitely many of the groups are nontrivial, but the Euler characteristic can be made sense of using one or other technique for summing divergent series. Or, it may be that the sum is beyond salvation. Typically, if you want the Euler characteristic to make sense — or even just in order for the ranks to be finite — you’ll need to impose some sort of finiteness condition on the object that you’re taking the homology of.

The idea — perhaps the entire point of magnitude homology — is that its Euler characteristic should be equal to magnitude. For some enriching categories $V$, we have a theorem saying exactly that. For others, we don’t… but we do have some formal calculations suggesting that there’s a theorem waiting to be found. We haven’t got to the bottom of this yet.

I’ll say something about the general situation, then I’ll explain the state of the art in the three examples above.

In general, for a semicartesian monoidal category $V$, a small $V$-category $X$, and a small functor $A \colon V \to Ab$, we want to define the Euler characteristic of $X$ with coefficients in $A$ as

$$\chi(X; A) = \sum_{n \geq 0} (-1)^n rank(H_n(X; A)).$$

Here’s how it looks in our three running examples: categories, graphs and metric spaces.

• In the case $V = Set$, we’re talking about the Euler characteristic of a category $X$. Take $A = -\cdot \mathbb{Z}$, as defined above. Then the homology group $H_n(X; A)$ is equal to $H_n(X; \mathbb{Z})$, the $n$th homology of the category $X$ with coefficients in $\mathbb{Z}$. That’s the same as the $n$th homology of the nerve (or its geometric realization).

  To make sense of $\chi(X; \mathbb{Z})$, we impose a finiteness condition. Assume that the category $X$ is finite, skeletal, and contains no nontrivial endomorphisms. Then the nerve of $X$ has only finitely many nondegenerate simplices, from which it follows that only finitely many of the homology groups are nontrivial. So, the sum is finite and $\chi(X; \mathbb{Z})$ makes sense.

  Under these finiteness hypotheses, what actually is $\chi(X; \mathbb{Z})$? Since $H_n(X; \mathbb{Z})$ is the $n$th homology of the nerve of $X$ with integer coefficients, $\chi(X; \mathbb{Z})$ is the ordinary (simplicial/topological) Euler characteristic of the nerve of $X$. And it’s a theorem that this is equal to the Euler characteristic of the category $X$, defined combinatorially and also called the “magnitude” of $X$.

  So for a small category $X$, the Euler characteristic of the magnitude homology $H_\ast(X; -\cdot\mathbb{Z})$ is indeed the magnitude of $X$. In other words: magnitude homology categorifies magnitude.

• Take a graph $X$, seen as a category enriched in $V = (\mathbb{N} \cup \{\infty\}, \geq)$. For each natural number $\ell$, we can try to define the Euler characteristic

  $$\chi(X; \delta_\ell) = \sum_{n \geq 0} (-1)^n rank(H_n(X; \delta_\ell)).$$

  I said earlier that these homology groups are the same as Hepworth and Willerton’s homology groups $MH_{n, \ell}(X)$, and I described them explicitly.

  To make sure that the ranks are all finite, let’s assume that the graph $X$ is finite. That alone is enough to guarantee that the sum defining $\chi(X; \delta_\ell)$ is finite. Why? Well, from the definition of the chain groups $MC_{n, \ell}(X)$, it’s clear that $MC_{n, \ell}(X)$ is trivial when $n \gt \ell$. Hence the same is true of $MH_{n, \ell}(X)$, which means that the sum defining $\chi(X; \delta_\ell)$ might as well run only from $n = 0$ to $n = \ell$.

  At the moment, our graph has not one Euler characteristic but an infinite sequence of them:

  $$\chi(X; \delta_0), \,\, \chi(X; \delta_1), \,\, \chi(X; \delta_2), \,\, \cdots$$

  Let’s assemble them into a single formal power series over $\mathbb{Z}$:

  $$\chi(X) := \sum_{\ell \in \mathbb{N}} \chi(X; \delta_\ell) q^\ell = \sum_{n \geq 0} (-1)^n \sum_{\ell \in \mathbb{N}} rank(H_n(X; \delta_\ell)) q^\ell,$$

  where $q$ is a formal variable. (You might wonder what’s happened to $\ell = \infty$. In principle, it should be present in the sum. However, if we adopt the convention that $q^\infty = 0$ then it might as well not be. It will become clear when we look at metric spaces that this is the right convention to adopt.)

  On the other hand, viewing graphs as enriched categories leads to the notion of the magnitude of a graph. The magnitude of a finite graph $X$ is a formal expression in a variable $q$, and can be understood either as a rational function in $q$ or as a power series in $q$. Hepworth and Willerton showed that the power series $\chi(X)$ above is precisely the magnitude of $X$, seen as a power series.

  So in the case of graphs too, magnitude homology categorifies magnitude.

• Finally, consider a metric space $X$, viewed as a category enriched in $V = ([0, \infty], \geq)$. For each $\ell \in V$, we want to define

  $$\chi(X; \delta_\ell) = \sum_{n \geq 0} (-1)^n rank(H_n(X; \delta_\ell)).$$

  I have no idea what these homology groups look like when $X$ is a familiar geometric object such as a disk or line, so I don’t know how often these ranks are finite. But they’re certainly finite if $X$ has only finitely many points, so let’s assume that.

  The sum on the right-hand side is, then, automatically finite. To see this, the argument is almost the same as for graphs. For graphs, we used the fact that the distance between two distinct vertices is always at least $1$, from which it followed that the homology groups $H_n(X; \delta_\ell)$ can only be nonzero when $n \leq \ell$. Now in a finite metric space, distances can of course be less than $1$, but finiteness implies that there’s a minimal nonzero distance: $\eta$, say. Then $H_n(X; \delta_\ell)$ can only be nonzero when $n \leq \ell/\eta$. That’s why the sum is finite.

  We’ve now assigned to our metric space not one Euler characteristic but a one-parameter family of them. That is, we’ve got an integer $\chi(X, \delta_\ell)$ for each $\ell \in [0, \infty]$. Actually, all but countably many of these integers are zero. Better still, for each real $m$ there are only finitely many $\ell \leq m$ such that $\chi(X, \delta_\ell) \neq 0$. (I’ll explain why in the details section.) So, it’s not too crazy to write down the formal expression

  $$\chi(X) = \sum_{\ell \in [0, \infty)} \chi(X; \delta_\ell) q^\ell.$$

  There are a couple of ways to think about the expression on the right-hand side. You can treat $q$ as a formal variable and the expression as a Hahn series (like a power series, but with non-integer real powers allowed). Or you can (attempt to) evaluate at a particular value of $q$ in $\mathbb{R}$ or $\mathbb{C}$ or some other setting where the sum makes analytic sense.

  So far no one knows how exactly we should proceed from here, but it looks as if the story goes something like this.

  Remember, we’re trying to show that magnitude homology categorifies magnitude, which in this instance means that $\chi(X)$ should be equal to the magnitude of a metric space $X$. That’s a real number, and it’s defined in terms of negative exponentials $e^{-d}$ of distances $d$, so let’s put $q = e^{-1}$. (This explains why we can ignore $\ell = \infty$, since then $q^\ell = e^{-\infty} = 0$.) I’m not claiming that anything converges! You can treat $e^{-1}$ as a formal variable for the time being, although at some stage we’ll want to interpret it as an actual real number.

  It’s a useful little lemma that when you have a bounded chain complex $C$, the alternating sum of the ranks of the groups $C_n$ is equal to the alternating sum of the homology groups $H_n(C)$. So,

  $$\chi(X; \delta_\ell) = \sum_{n \geq 0} (-1)^n rank(MC_{n, \ell}(X))$$

  where $MC$ denotes the Hepworth–Willerton chain groups that I defined earlier. Substituting this into the definition of $\chi(X)$ gives

  $$\chi(X) = \sum_{n \geq 0} (-1)^n \sum_{\ell \in [0, \infty)} rank(MC_{n, \ell}(X)) e^{-\ell}.$$

  That’s potentially a doubly infinite sum. But we can do some formal calculations leading to the conclusion that $\chi(X)$ is indeed equal to the magnitude of the metric space $X$ (that is, the sum of all the entries of the inverse of the matrix $(e^{-d(x, y)})_{x, y \in X}$). Again, that’s deferred to the details section below. It’s not clear how to make rigorous sense of it, but I’m confident that it can somehow be done.

So, magnitude homology categorifies magnitude in all three of our examples… well, definitely in the first two cases, and tentatively in the third. Of course, we’d like to make a general statement to the effect that homology categorifies magnitude over an arbitrary base category $V$. The metric space case illustrates some of the difficulties that we might expect to encounter in making a general statement.

Details and proofs

The rest of this post mostly consists of supporting details that we figured out in the other thread. I’ve mostly only bothered to include the points that weren’t immediately obvious to us (or me, at least).

If you’ve read this far, bravo! You can think of what follows as an appendix.

From simplicial abelian groups to chain complexes

The relationship between simplicial abelian groups and chain complexes is a classical part of homological algebra, but there’s at least one aspect of it that some of us in the old thread hadn’t previously appreciated.

First, the definitions. Let $G$ be a simplicial abelian group. The unnormalized chain complex $C(G)$ is defined by $C_n(G) = G_n$, the differentials being the alternating sums of the face maps. The degenerate elements of $C_n(G)$ generate a subgroup $D_n(G)$, which assemble to give a subcomplex of $C(G)$. The normalized chain complex is $C(G)/D(G)$.

Now here are two facts. First, there’s an isomorphism of chain complexes $C(G) \cong D(G) \oplus \frac{C(G)}{D(G)}$, natural in $G$. Second, the projection and inclusion maps between $C(G)$ and $C(G)/D(G)$ are mutually inverse up to a chain homotopy that is natural in $G$ (in the obvious sense). That naturality will be crucial for us. We therefore say that $C(G)$ and $C(G)/D(G)$ are naturally chain homotopy equivalent.

The functoriality of the nerve construction

Given a monoidal category $V$ and a small $V$-category $X$, we defined a functor

$$N(X): V^{op} \to sSet$$

by

$$N(X)(\ell)_n = \coprod_{x_0, \ldots, x_n \in X} V \bigl(\ell, X(x_0, x_1) \otimes \cdots \otimes X(x_{n - 1}, x_n)\bigr).$$

Obviously $N$ is functorial in $X$: any $V$-functor $F \colon X \to Y$ induces a map of simplicial sets $N(F)_\ell \colon N(X)(\ell) \to N(Y)(\ell)$ for each $\ell \in V$, and this map is natural in $\ell$.

Less obvious is that $N$ is functorial in the following 2-dimensional sense. Take $V$-functors

$$F, G \colon X \rightrightarrows Y$$

and a $V$-natural transformation $\alpha \colon F \to G$. The claim is that for each $\ell \in V$, there’s an induced simplicial homotopy $\alpha_\ell$ from $N(F)_\ell$ to $N(G)_\ell$. Moreover, $\alpha_\ell$ is natural in $\ell$.

How does this work? I’m pretty much a klutz with things simplicial, so let me explain it in a concrete way and refer to this comment of Mike’s for a more abstract perspective.

Fix $\ell$. We have our two maps

$$N(F)_\ell, N(G)_\ell \colon N(X)(\ell) \rightrightarrows N(Y)_\ell.$$

By definition, a simplicial homotopy from $N(F)_\ell$ to $N(G)_\ell$ is a map $h \colon N(X)(\ell) \times \Delta^1 \to N(Y)(\ell)$ of simplicial sets that satisfies the appropriate boundary conditions. Here $\Delta^1$ means the representable simplicial set $\Delta(-, [1])$. There are two maps from the terminal simplicial set $1 = \Delta^0$ to $\Delta^1$, corresponding to the two maps $[0] \rightrightarrows [1]$ in $\Delta$. The “boundary conditions” are that the two composites in the diagram

$$N(X)(\ell) \cong N(X)(\ell) \times 1 \rightrightarrows N(X)(\ell) \times \Delta^1 \stackrel{h}{\longrightarrow} N(Y)(\ell)$$

are equal to $N(F)_\ell$ and $N(G)_\ell$.

Concretely, a simplicial homotopy $h$ from $N(F)_\ell$ to $N(G)_\ell$ consists of a map of sets

$$h_\phi \colon N(X)(\ell)_n \to N(Y)(\ell)_n$$

for each map $\phi \colon [n] \to [1] = \{0, 1\}$ in $\Delta$. When $\phi$ is the map with constant value $0$,   $h_\phi$ is required to be equal to $N(F)_{\ell, n}$, and when $h$ has constant value $1$,   $h_\phi$ is required to be equal to $N(G)_{\ell, n}$. The maps $h_\phi$ also have to satisfy some other equations which I don’t need to mention.

There are $n + 2$ maps $[n] \to [1]$ in $\Delta$, so what this means really explicitly is that a simplicial homotopy from $N(F)$ to $N(G)$ consists of an ordered list of $n + 2$ functions $N(X)(\ell)_n \to N(Y)(\ell)_n$ for each $n \geq 0$. The first has to be $N(F)_{\ell, n}$, the last has to be $N(G)_{\ell, n}$, and the whole lot have to hang together in some reasonable way. So roughly speaking, a simplicial homotopy is a kind of discrete path between two simplicial maps, as you’d probably expect.

We’re supposed to be building a simplicial homotopy from $N(F)$ to $N(G)$ out of a $V$-natural transformation $\alpha \colon F \to G$. So, let’s recall what a $V$-natural transformation actually is. More or less by definition, $\alpha$ consists of a map

$$\alpha_{x, x'} \colon X(x, x') \to Y(F(x), G(x'))$$

in $V$ for each $x, x' \in X$ (subject to some axioms). For instance, when $V = Set$, this map sends $f \in X(x, x')$ to the diagonal of the naturality square for $f$.

Now let $n \geq 0$. For any objects $x_0, \ldots, x_n$ of $X$, we can build from $F$, $G$ and $\alpha$ a sequence of $n + 2$ maps in $V$, which for ease of typesetting I’ll show for $n = 3$ (and you’ll guess the general pattern):

$$\begin{aligned} C(x_0, x_1) \otimes C(x_1, x_2) \otimes C(x_2, x_3) & \,\,\longrightarrow\,\, D(F x_0, F x_1) \otimes D(F x_1, F x_2) \otimes D(F x_2, F x_3), \\ C(x_0, x_1) \otimes C(x_1, x_2) \otimes C(x_2, x_3) & \,\,\longrightarrow\,\, D(F x_0, F x_1) \otimes D(F x_1, F x_2) \otimes D(F x_2, G x_3), \\ C(x_0, x_1) \otimes C(x_1, x_2) \otimes C(x_2, x_3) & \,\,\longrightarrow\,\, D(F x_0, F x_1) \otimes D(F x_1, G x_2) \otimes D(G x_2, G x_3), \\ C(x_0, x_1) \otimes C(x_1, x_2) \otimes C(x_2, x_3) & \,\,\longrightarrow\,\, D(F x_0, G x_1) \otimes D(G x_1, G x_2) \otimes D(G x_2, G x_3), \\ C(x_0, x_1) \otimes C(x_1, x_2) \otimes C(x_2, x_3) & \,\,\longrightarrow\,\, D(G x_0, G x_1) \otimes D(G x_1, G x_2) \otimes D(G x_2, G x_3). \end{aligned}$$

These $n + 2$ maps in $V$ induce, in the obvious way, $n + 2$ maps of sets

$$\coprod_{x_0, \ldots, x_n \in X} V(\ell, C(x_0, x_1) \otimes \cdots \otimes C(x_{n-1}, x_n)) \,\,\longrightarrow\,\, \coprod_{y_0, \ldots, y_n \in Y} V(\ell, D(y_0, y_1) \otimes \cdots \otimes D(y_{n-1}, y_n))$$

for each $\ell \in V$. The domain and codomain here are just $N(X)(\ell)_n$ and $N(Y)(\ell)_n$: so we have $n + 2$ maps $N(X)(\ell)_n \to N(Y)(\ell)_n$. The first of these maps is $N(F)_{\ell, n}$ and the last is $N(G)_{\ell, n}$. Some checking reveals that these maps, taken over all $n$, do indeed determine a simplicial homotopy from $N(F)_\ell$ to $N(G)_\ell$. Moreover, everything is obviously natural in $\ell$. So that’s our natural simplicial homotopy!

Functoriality of the tensor product

Let $A \colon V \to Ch$ be a small functor. For any functor $B \colon V^{op} \to Ch$, we can form the tensor product

$$B \otimes_V A = \int^{\ell \in V} B(\ell) \otimes A(\ell),$$

which is a chain complex. Obviously this determines a functor

$$- \otimes_V A \colon [V^{op}, Ch] \to Ch.$$

A little less obviously, $- \otimes_V A$ transforms any natural chain homotopy into a chain homotopy.

In other words, take functors $P, Q \colon V^{op} \rightrightarrows Ch$ and natural transformations $\kappa, \lambda \colon P \rightrightarrows Q$. (So, $\kappa$ and $\lambda$ consist of chain maps $\kappa_\ell, \lambda_\ell \colon P(\ell) \rightrightarrows Q(\ell)$ for each $\ell \in V$, natural in $\ell$.) Suppose we also have a chain homotopy $h_\ell \colon \kappa_\ell \to \lambda_\ell$ for each $\ell$, and that $h_\ell$ is natural in $\ell$. The claim is that there’s an induced chain homotopy $h \otimes_V A$ between the chain maps

$$\kappa \otimes_V A, \, \lambda \otimes_V A \colon P \otimes_V A \rightrightarrows Q \otimes_V A.$$

To show this, the key point is that a chain homotopy between the chain maps $\kappa_\ell, \lambda_\ell \colon P(\ell) \to Q(\ell)$ can be understood as a chain map $P(\ell) \otimes \mathbf{I} \to Q(\ell)$ satisfying appropriate boundary conditions. Here $\mathbf{I}$ (for “interval”) is the chain complex

$$\cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{id}{\to} \mathbb{Z} \to 0 \to 0 \to \cdots$$

with the two copies of $\mathbb{Z}$ in degrees $0$ and $1$. Once you adopt this viewpoint, it’s straightforward to prove the claim, using only the associativity of $\otimes$ and the fact that $\otimes$ distributes over colimits.

An important consequence is that if two functors $B, B' \colon V^{op} \rightrightarrows Ch$ are naturally chain homotopy equivalent, then the complexes $B \otimes_V A$ and $B' \otimes_V A$ are chain homotopy equivalent.

It doesn’t matter whether you normalize your chains

Let $X$ be a small $V$-category. The functor $C(X) \colon V^{op} \to Ch$ was defined by first building from $X$ a certain functor $\mathbb{Z}\cdot N(X) \colon V^{op} \to sAb$, then turning simplicial sets into chain complexes. I (or rather Mike) said that it doesn’t matter whether you do that last step with unnormalized or normalized chains. Why not?

Earlier in this “details” section, I recalled the fact that the two chain complexes coming from a simplicial abelian group $G$ are not only chain homotopy equivalent, but chain homotopy equivalent in a way that’s natural in $G$. We can apply this fact to the simplicial abelian group $\mathbb{Z}\cdot N(X)(\ell)$, for each $\ell \in V$. It implies that the two chain complexes coming from $\mathbb{Z}\cdot N(X)(\ell)$ are chain homotopy equivalent naturally in $\ell$. Or, said another way, the two versions of $C(X) \colon V^{op} \to Ch$ that you get by choosing the “unnormalized” or “normalized” option are naturally chain homotopy equivalent.

But we just saw that when two functors $B, B' \colon V^{op} \rightrightarrows Ch$ are naturally chain homotopy equivalent, their tensor products with $A$ are chain homotopy equivalent. So, the two versions of $C(X)$ have the same tensor product with $A$, up to chain homotopy equivalence. In other words, the chain homotopy equivalence class of $C(X) \otimes_V A$ is unaffected by which version of $C(X)$ you choose to use. The homology $H_\ast(C; A)$ of that chain complex is, therefore, also unaffected by this choice.

Invariance of magnitude homology under equivalence of categories

It’s a fact that the magnitude of an enriched category is invariant not only under equivalence, but even under the existence of an adjunction (at least, if both magnitudes are well-defined). Something similar is true for magnitude homology, as follows.

Let $F, G \colon X \rightrightarrows Y$ be $V$-functors between small $V$-categories. We’ll show that if there exists a $V$-natural transformation from $F$ to $G$ then the maps

$$H_\ast(X; A) \rightrightarrows H_\ast(Y; A)$$

induced by $F$ and $G$ are equal (for any coefficients $A$). It will follow that whenever you have $V$-categories that are equivalent, or even just connected by an adjunction, their homologies are isomorphic. (Even “adjunction” can be weakened further, but I’ll leave that as an exercise.)

The proof is mostly a matter of assembling previous observations. Take a $V$-natural transformation $\alpha \colon F \to G \colon X \to Y$. We have functors

$$N(X), N(Y) \colon V^{op} \rightrightarrows sSet,$$

natural transformations

$$N(F), N(G) \colon N(X) \rightrightarrows N(Y),$$

and (as we saw previously) a natural simplicial homotopy from $N(F) \to N(G)$ induced by $\alpha$. When we pass from simplicial sets to chain complexes, this natural simplicial homotopy turns into a natural chain homotopy (Lemma 8.3.13 of Weibel’s book). So, the natural transformations $C(F)$ and $C(G)$ between the functors

$$C(X), C(Y) \colon V^{op} \rightrightarrows Ch$$

are naturally chain homotopic. It follows from another of the previous observations that the chain maps

$$C(F) \otimes_V A, \,\,\, C(G) \otimes_V A\,\, \colon \,\, C(X) \otimes_V A \rightrightarrows C(Y) \otimes_V A$$

are chain homotopic. Hence they induce the same map $H_\ast(X; A) \to H_\ast(Y; A)$ on homology, as claimed.

Homology of graphs and of metric spaces

Earlier, I claimed that Mike’s general theory of homology of enriched categories reproduces Richard Hepworth and Simon Willerton’s theory of magnitude homology of graphs, by choosing the coefficients suitably. It’s trivial to extend Richard and Simon’s theory from graphs to metric spaces, as I did earlier; and I claimed that this too is captured by the general theory.

I’ll prove this now in the case of metric spaces. It will then be completely clear how it works for graphs.

Let $X$ be a metric space, seen as a category enriched in $V = [0, \infty]$. Let $\ell \geq 0$ be a real number, and recall the functor $\delta_\ell \colon V \to Ab$ from earlier. The aim is to show that the groups $H_n(X, \delta_\ell)$ and $MH_{n, \ell}(X)$ are isomorphic, where the latter is defined à la Hepworth–Willerton.

The nerve functor $N(X) \colon V^{op} \to sSet$ is given by

$$N(X)(\ell')_n = \{ (x_0, \ldots, x_n) : d(x_0, x_1) + \cdots + d(x_{n-1}, x_n) \leq \ell' \}.$$

The unnormalized chain group $C(X)(\ell')_n$ is simply the free abelian group on this set, but in order to make the connection with Richard and Simon’s definition, we’re going to use the normalized version of $C(X)$. It’s not too hard to see that the normalized $C(X)(\ell')_n$ is the free abelian group on the set

$$\{ (x_0, \ldots, x_n) : \, x_0 \neq x_1 \neq \cdots \neq x_n, \,\, d(x_0, x_1) + \cdots + d(x_{n-1}, x_n) \leq \ell' \}$$

with differentials $\partial = \sum_{i = 0}^n (-1)^i \partial_i$, where

$$\partial_i(x_0, \ldots, x_n) = \begin{cases} (x_0, \ldots, x_{i - 1}, x_{i + 1}, \ldots, x_n) & \text{if } \,\, i = 0 \,\,\text{ or }\,\, i = k \,\,\text{or}\,\, x_{i - 1} \neq x_{i + 1}, \\ 0 & \text{otherwise }. \end{cases}$$

Now we have to compute $C(X) \otimes_V \delta_\ell$. I know two ways to do this. You can use the definition of coend directly, as Mike does here. Alternatively, note that for any functor of coefficients $A \colon V \to Ab$,

$$\begin{aligned} (C(X) \otimes_V A)_n & = \int^{\ell' \in V} (C(X)(\ell') \otimes A(\ell'))_n \\ & = \int^{\ell' \in V} C(X)(\ell')_n \otimes A(\ell') \\ & = \int^{\ell' \in V} \coprod_{x_0 \neq \cdots \neq x_n} \mathbb{Z} \cdot V(\ell', d(x_0, x_1) + \cdots + d(x_{n-1}, x_n)) \otimes A(\ell') \\ & = \coprod_{x_0 \neq \cdots \neq x_n} \int^{\ell' \in V} \mathbb{Z} \cdot V(\ell', d(x_0, x_1) + \cdots + d(x_{n-1}, x_n)) \otimes A(\ell') \\ & = \coprod_{x_0 \neq \cdots \neq x_n} A(d(x_0, x_1) + \cdots + d(x_{n-1}, x_n)), \end{aligned}$$

where the last step is by the density formula. We’re interested in the case $A = \delta_\ell$, and then the expression $A(\cdots)$ in the last line is either $\mathbb{Z}$ if the distances sum to $\ell$, or $0$ if not. So

$$(C(X) \otimes_V \delta_\ell)_n = \{ (x_0, \ldots, x_n) : \, x_0 \neq x_1 \neq \cdots \neq x_n, \,\, d(x_0, x_1) + \cdots + d(x_{n-1}, x_n) = \ell \}.$$

That’s exactly Richard and Simon’s chain group $MC_{n, \ell}(X)$. With a little more thought, you can see that the differentials agree too. Thus, the chain complexes $C(X) \otimes_V \delta_\ell$ and $MC_{\ast, \ell}(X)$ are isomorphic. It follows that their homologies are isomorphic, as claimed.

Decategorification for metric spaces

The final stretch of this marathon post is devoted to finite metric spaces — specifically, how the magnitude of a finite metric space can be obtained as the Euler characteristic of its magnitude homology. Here’s where there are some gaps.

Let $X$ be a finite metric space. For each $\ell \in V$, we have the Euler characteristic

$$\chi(X; \delta_\ell) = \sum_{n \geq 0} (-1)^n rank(H_n(X; \delta_\ell)).$$

The ranks here are finite because the sets $N(X)(\ell)_n$ are manifestly finite. We saw earlier that the sum itself is finite, but let me repeat the argument slightly more carefully. First, these homology groups are the same as the Hepworth–Willerton homology groups. Second, the Hepworth–Willerton chain groups $MC_{n, \ell}(X)$ are trivial when $n \gt \ell/\eta$, where $\eta$ is the minimum nonzero distance occurring in $X$. So, the same is true of the homology groups $MH_{n, \ell}(X) = H_n(X; \delta_\ell)$.

Let $\mathbb{L}_X \subseteq [0, \infty]$ be the set of (extended) real numbers occurring as finite sums $d(x_0, x_1) + \cdots + d(x_{n - 1}, x_n)$ of distances in $X$. Although this set is usually infinite, it’s always countable. Better still, $\mathbb{L}_X \cap [0, L]$ is finite for all real $L \geq 0$. It’s easy to prove this, again using the fact that there’s a minimum nonzero distance.

For a number $\ell$ that’s not in $\mathbb{L}_X$, the Hepworth–Willerton chain groups $MC_{\ast, \ell}(X)$ are trivial, so the homology groups $MH_{\ast, \ell}(X) = H_\ast(X; \delta_\ell)$ are trivial too. Hence $\chi(X; \delta_\ell) = 0$. Or in other words: $\chi(X; \delta_\ell)$ only stands a chance of being nonzero if $\ell$ belongs to the countable set $\mathbb{L}_X$.

So, in the definition

$$\chi(X) = \sum_{\ell \in [0, \infty]} \chi(X; \delta_\ell) e^{-\ell},$$

that scary-looking sum over all $\ell \in [0, \infty]$ might as well only be over the relatively tame range $\ell \in \mathbb{L}_X$.

Now let’s do a formal calculation. Back in the main part of the post (just before the start of this “details” section), I observed that

$$\chi(X) = \sum_{n \geq 0} (-1)^n \sum_{\ell \in [0, \infty)} rank(MC_{n, \ell}(X)) e^{-\ell}.$$

Now $MC_{n, \ell}(X)$ is the free abelian group on the set

$$\{ (x_0, \ldots, x_n) : \, x_0 \neq x_1 \neq \cdots \neq x_n, \,\, d(x_0, x_1) + \cdots + d(x_{n-1}, x_n) = \ell \},$$

so $rank(MC_{n, \ell}(X))$ is the cardinality of this set. Hence, working formally,

$$\sum_{\ell \in [0, \infty)} rank(MC_{n, \ell}(X)) e^{-\ell} = \sum_{x_0 \neq \cdots \neq x_n} e^{-d(x_0, x_1)} e^{-d(x_1, x_2)} \cdots e^{-d(x_{n-1}, x_n)}.$$

Let $Z_X$ be the square matrix with rows and columns indexed by the points of $X$, and entries $Z_X(x, y) = e^{-d(x, y)}$. Write $I$ for the $X \times X$ identity matrix, and write $sum(M)$ for the sum of all the entries of a matrix $M$. Then

$$\sum_{x_0 \neq \cdots \neq x_n} e^{-d(x_0, x_1)} \cdots e^{-d(x_{n-1}, x_n)} = sum((Z_X - I)^n).$$

So our earlier formula

$$\chi(X) = \sum_{n \geq 0} (-1)^n \sum_{\ell \in [0, \infty)} rank(MC_{n, \ell}(X)) e^{-\ell}$$

now gives

$$\chi(X) = \sum_{n \geq 0} (-1)^n sum(Z_X - I)^n = sum \biggl( \sum_{n \geq 0} (I - Z_X)^n \biggr).$$

Again formally speaking, the part inside the brackets is a geometric series whose sum is $Z_X^{-1}$. So, the conclusion is that

$$\chi(X) = sum(Z_X^{-1}).$$

The right-hand side is by definition the magnitude of the metric space $X$ (at least, assuming that $Z_X$ is invertible).

So, using non-rigorous formal methods, we’ve achieved our goal. That is, we’ve shown that the magnitude of a finite metric space is the Euler characteristic of its magnitude homology.

We know how to make some of this rigorous. The basic idea is that to sum a possibly-divergent series $\sum_{n \geq 0} (-1)^n a_n$, we “vary the value of $-1$” by replacing it with a formal variable $t$. Thus, we define the formal power series $f(t) = \sum_{n \geq 0} a_n t^n$, hope that $f$ is formally equal to a rational function, hope that the rational function $f$ doesn’t have a pole at $-1$, and if not, interpret $\sum_{n \geq 0} (-1)^n a_n$ as $f(-1)$.

That’s a time-honoured technique for summing divergent series. To apply it in this situation, here’s a little theorem about matrices that essentially appears in a paper by Clemens Berger and me:

Theorem Let $Z$ be a square matrix of real numbers. Then:

• The formal power series $f(t) = \sum_{n \geq 0} sum((Z - I)^n) \cdot t^n$ is rational.

• If $Z$ is invertible, the value of the rational function $f$ at $-1$ is (defined and) equal to $sum(Z^{-1})$.

This result provides a respectable way to interpret the last part of the unrigorous argument presented above — the bit about the geometric series. But the earlier parts remain to be made rigorous.