The n-Category Café

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 $[0,\infty]$-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 $[0,\infty]$-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 $V$ is semicartesian and $\Sigma:V\to W$ is strong monoidal, we can define the magnitude homology of a $V$-category by applying $\Sigma$ and then taking Hochschild homology with constant coefficients — semicartesianness of $V$ 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 $(Z-Id)$. This applies in the generality of a semicartesian $V$, a strong monoidal functor $\Sigma:V\to W$, and a “formal Euler characteristic” on $W$, but requires a strong finiteness restriction on the $V$-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 $H_1(X)=0$ if and only if $X$ is Menger convex, which in particular includes closed convex subsets of Euclidean space, and that in the latter case also $H_2(X)=0$. The paper includes a sufficient condition for $H_2(X)=0$ that’s a little more general, isolating different ways in which $H_2(X)$ can be nonzero: intuitively, if $X$ 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 $y$ of a metric space is said to be between two other points $x$ and $z$ if $d(x,y)+d(y,z)=d(x,z)$, i.e. the triangle inequality is an equality. If additionally $x\neq y\neq z$ 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 $x,z$ such that there is, or is not, a point strictly between them. The current preprint follows Tom’s suggestion to call $x$ and $z$ 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 $d(x,y_1)+d(y_1,y_2)=d(x,y_2)$ and $d(y_1,y_2)+d(y_2,z)=d(y_1,z)$ together imply $d(x,z) = d(x,y_1)+d(y_1,y_2)+d(y_2,z)$? 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 $\mathbb{R}^n$, 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 $y_1$ and $y_2$ are both between $x$ and $z$, then either $y_1$ is between $x$ and $y_2$ while $y_2$ is between $y_1$ and $z$, or else $y_2$ is between $x$ and $y_1$ while $y_1$ is between $y_2$ and $z$? 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 $\mathbb{R}^n$, so maybe there is a better word.

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

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 $n$-dimensional condition that’s relevant to magnitude $H_n$. For a while I was proposing “$X$ is Menger 2-convex” instead of “$X$ 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 $n$-level hierarchy. Does any such hierarchy of “higher convexity” already exist?