# Summary

The base-$\beta$ logarithm gives an isomorphism of topological spaces $$ \log_\beta\colon\mathbb{R}_{\geq0}\xrightarrow{\cong}[-\infty,\infty). $$ This continuous map preserves some algebraic structure, e.g. sending multiplication to addition. As is well-known in tropical geometry, it also approximately sends the binary operation addition to the binary operation $\max$, $$\log_\beta(a+b)\approx\max\left(\log_\beta(a),\log_\beta(b)\right),$$ and this approximation gets better as the base $\beta\in(1,\infty)$ increases.

**Question:** Is there a category-theoretic way to tell this story?

# Background

A commutative *rig* ("ring without negatives") is a tuple $(S, 0, +, 1, *)$, where $(S,0,+)$ and $(S,1,*)$ are both commutative monoids, and where multiplication distributes over addition, $a*(b+c)=(a*b)+(a*c)$. Both
$$
\big(\mathbb{R}_{\geq0},0,+,1,*\big)\quad\text{and}\quad\big([-\infty,\infty), -\infty, \max, 0, +\big)
$$
are rigs; in fact, they are topological rigs. That is, each of the sets $\mathbb{R}_{\geq0}$ and $[-\infty,\infty)$ has a well-known topology (e.g. in $[-\infty,\infty)$, the open sets containing $-\infty$ are generated under union by those of the form $[-\infty,x)$ for $x\in \mathbb{R}$), and in each case the two binary operations, namely $+,*$ and $\max,+$, are continuous with respect to that topology.

As mentioned above, the $\log_\beta$ function is an isomorphism of topological spaces, but it is not a homomorphism of rigs. It preserves almost all of the structure ($0\mapsto-\infty, 1\mapsto 0, *\mapsto +$), the one exception being that the addition operation is not preserved.

However, we do have simple bounds on the difference between $\log_\beta(a+b)$ and $\max(\log_\beta a,\log_\beta b)$ for all $a,b\in\mathbb{R}_{\geq0}$. One checks easily (using the two exhaustive cases $a\leq b$ and $b\leq a$) that $$ 0\leq \log_\beta(a+b)-\max\big(\log_\beta(a),~\log_\beta(b)\big)\leq\log_\beta 2. $$ Thus as $\beta$ gets bigger—one might think of $\beta$ physically as "coldness", so "as the system gets colder"—the approximation gets better: $\log_\beta$ becomes closer and closer to an isomorphism of topological rigs.

# Motivation

Tropical geometry is widely studied field, and I imagine that nothing in this section is particularly novel in that context. But let me just say what motivates me.

Having an almost-isomorphism (asking precisely what that should mean is the goal of this post) between these two topological rigs is interesting because, for one thing, $\max,+$ matrix multiplication is much easier for humans and faster to compute than $+,*$ matrix multiplication. Moreover, lots of ideas port over.

For example, one can discuss what I might call "log-stochastic matrices" where the max of each column is 0. These form a category that is similar in certain ways to the usual category **Stoch** of finite sets and stochastic maps, but of course having key differences. Working a bit in this category, I could imagine it being relevant in behavioral economics, where there may be multiple "best choices" in a given situation (multiple 0's in a given column), and calculations need to be easy (again max, + being easier than +,*).

# Categorical issues

My goal is to be able to say, category-theoretically, both that $\log_\beta$ is an approximate rig map for each $\beta\in(1,\infty)$, and also that these approximations get better as $\beta$ increases. To tell this story categorically, one needs to define what "better approximation" means.

One attempt could be to give something like a distance between $\log(a+b)$ and $\max(\log a, \log b)$, and for this we would need something like a metric. The usual category of metric spaces has short maps (distance non-decreasing functions) as its morphisms. This doesn't work too well for us, because $(\mathbb{R},0,+,1,*)$ is not an internal rig in that category; in particular, the multiplication operation $*\colon\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq0}\to\mathbb{R}_{\geq 0}$ is not a short map. In fact, it is not even Lipschitz. But once one moves to the category of topological spaces or sets—where our two structures do become internal rigs—how will one measure the distances between their distance addition operations?

The goal is not to necessarily stick with the viewpoint of the previous paragraph, but to come up with a categorical viewpoint in which the successive approximations story is most at home.