The n-Category Café

Indeed they are, and as the corresponding nLab page shows, this is a topic well-treated by categorists (though sadly the link to the relevant discussion on the categories mailing list appears to be dead). Harrison on page 23 of his book attributes this construction to Steve Schanuel, who taught it to Ross Street, from whom I learned it. (I include this last link because it is missing from that nLab page.)

Arthan 2004, The Eudoxus Real Numbers, says that John Harrison learned of Schanuel’s construction and used a variant in his 1996 thesis. The backmatter to his thesis book is online and I see no cites to Schanuel or Ross.

Maybe Harrison is in some sense claiming primacy because his thesis variant doesn’t have the completeness problems of Ross’s 1985 presentation which weren’t fixed until the early 2000s.

I wonder whether there’s also a higher level of abstraction at which to study this.

We know how to apply the “approximate endomorphism” construction to the monoid $\mathbb{N}$ and the group $\mathbb{Z}$. Also, for an arbitrary metric space $X$, one can take the monoid of equivalence classes of approximate endomorphisms of $X$, using the usual notion of boundedness. I guess the same construction works for general bornological sets, i.e. sets equipped with a bornology.

Is there a unified theory that includes all these constructions? Maybe it would involve bornologies compatible with an algebraic structure. Has this been explored? Are we in the territory of coarse geometry here?

All right, I think the following works.

Take a finitely generated abelian group $A$. If we choose a finite set of generators then we obtain a metric on $A$, the distance between two points being the length of the shortest path between them on the Cayley graph. That is, if $a - b$ can be expressed as $\lambda_1 x_1 + \cdots + \lambda_n x_n$, where the $\lambda_i$ are integers and the $x_i$ are generators, then $d(a, b)$ is at most $\sum |\lambda_i|$ — “at most” because $d(a, b)$ is the infimum over all such ways of expressing $a - b$ as a combination of generators.

This gives a notion of what it means for a subset of $A$ to be bounded. Although the metric itself depends on the choice of generators, I think the notion of boundedness does not. It’s intrinsic to the group $A$.

We can now say what it means for a map of sets $f: A \to A$ to be an approximate endomorphism ($\{f(a + b) - f(a) - f(b): a, b \in A\}$ is bounded) and for two approximate homomorphisms $f, g: A \to A$ to be equivalent ($\{f(a) - g(a): a \in A\}$ is bounded). And I believe the equivalence classes of approximate endomorphisms of $A$ form a ring, under pointwise addition and composition.

When $A = \mathbb{Z}$, this construction should give $\mathbb{R}$. When $A = \mathbb{Z}^n$, I guess we get the ring $End(\mathbb{R}^n)$ of linear endomorphisms of $\mathbb{R}^n$, or equivalently, the ring of $n \times n$ real matrices.

Of course, there aren’t many finitely generated abelian groups! Every finitely generated abelian group can be expressed as $\mathbb{Z}^n \oplus B$ for some natural number $n$ and finite abelian group $B$. My guess is that applying our construction to $\mathbb{Z}^n \oplus B$ just gives $End(\mathbb{R}^n)$ again.

In a sense, we haven’t got much new relative to our starting point of obtaining $\mathbb{R}$ from $\mathbb{Z}$. So, it would be nice to extend the construction above from finitely generated abelian groups to some more general context: perhaps finitely generated commutative monoids, or perhaps finitely generated nonabelian groups.

I haven’t tried either of these. The latter option seems to fit very much into the circle of ideas involving coarse geometry, the Gromov–Hausdorff metric, word metrics, and Gromov’s results on growth of groups. I wonder if it’s already been done.

Very nice reflections! Some quick remarks/thoughts:

a) If we just ignore the fact that one cannot really subtract in $\mathbb{N}^{+}$ (e.g. assume that we have a monomorphism into some group), one certainly recovers the construction Emily described. I have not thoroughly worked through the construction for $\mathbb{Z}$, but I’d assume it works there too.

b) I think your guess about $\mathbb{Z}^{n} \oplus B$ must be correct, as the (Cayley) metric space corresponding to a finite group is coarsely equivalent to a point.

c) A first thing to check in attempting to generalise to monoids (e.g. commutative) would be whether it is still true that the coarse geometry is independent of the choice of generators. Of course, even if that is not true, one could always just try to work with a chosen set of generators and the corresponding metric (and notion of boundedness).

d) It would be really nice to be able to obtain the $p$-adic numbers in this way. One could try to start with $\mathbb{Z}_{(p)}$, the localisation of $\mathbb{Z}$ at $p\mathbb{Z}$, but it is not finitely generated. Is finite generation really necessary, though? One has to choose a set of generators, but let us say that we do so, say $\left\{ \frac{1}{q} \right\}$ such that $q$ is prime and not equal to $p$, or $q=1$. Take the corresponding metric and notion of boundedness. Do we recover the $p$-adic numbers? I haven’t tried to work it out, but intuitively it doesn’t seem impossible. If not, what do we get?

Just to throw out a few more thoughts (sorry, this got me interested!), it seems to me that any example where almost-endomorphisms are not all that obviously related to Cauchy sequences would likely be very instructive. If one is to have any hope of relating the rings which one obtains to completeness, I suspect one will have to replace the role of Cauchy sequences in defining completeness by something else, possibly the notion of a net.

In particular, an easier example than the $p$-adic numbers might be to consider any non-finitely generated abelian sub-group of $\mathbb{R}$, for example $\mathbb{Q}$. Take the generating set $\left\{ \frac{1}{m} \right\}_{m \in \mathbb{Z}}$. Can one somehow relate the notion of boundedness coming from this, and the resulting ring, to $\mathbb{R}$? More specifically, what is a good way to cook up a real number from an almost-endomorphism in this setting? Does the boundedness condition allow one to form the limit in $\mathbb{R}$ of some kind of net, that limit being our desired real number?

And should this work in one way or another, one could of course ask whether can prove by an abstraction of the construction that the resulting ring is complete. Alternatively, one could try to look for examples where one obtains something which is not complete, but this may be a bit self-defeating, as, if there is to be some kind of general theory at all, it would not be surprising if one needs some kind of conditions on one’s starting abelian group/monoid/etc, e.g. to ensure that one’s almost-endomorphisms give rise to nets, or something else which one can take a limit of.