# Functors Between Metric-Enriched Categories: Is This A Stupid Idea?

Let Met denote the category whose objects consist of metric spaces (for convenience, we will allow metrics to take the value of infinity) and whose morphisms are (weakly) contractive maps (aka short maps, nonexpansive maps, 1-Lipschitz maps), or more precisely functions satisfying $d(f(x),f(y)) \leq d(x,y)$Met is a monoidal category where $X\otimes Y$ is the product space with metric given by $d((x_1,y_1),(x_2,y_2) = d_X(x_1, x_2) + d(y_1, y_2)$ (the identity object is the singleton).

It is my (perhaps mistaken) opinion that a lot of functional analysis can be done using categories enriched in Met. (Actually, I think that for more generality, one ought to replace metric spaces with semi-metric spaces (aka quasi-metric spaces) where distinct elements can have zero distance. It may also be possible that Lawvere metric spaces are the appropriate choice here, but I’m not yet convinced of this.) Of course, one “trivial” yet important example is using the discrete metric. So for any two objects $A, B$ in a locally small category, we can define a metric on $\operatorname{Hom}(A,B)$ by $d(f,g) = 1$ if $f\neq g$ and $d(f,g) = 0$ if $f = g$.

A category enriched in Met enables one to talk about approximately commuting diagrams. This has been explored in approximate Fraïssé limits, though I don’t know enough logic to understand it.

Given two categories $\mathcal{C}, \mathcal{D}$ enriched in Met, we can define a type of continuity (except that word’s already taken) where a functor $F: \mathcal{C} \to \mathcal{D}$ is “continuous” if for any $\epsilon > 0$, there exists $\delta > 0$ such that for any objects $A,B\in \mathcal{C}$ and any morphisms $f,g\in \operatorname{Hom}(A,B)$, if $d(f,g) < \delta$, then $d(F(f),F(g)) < \epsilon$.

Apologies for the underdeveloped ideas, I have been thinking about this for quite a while (throughout my time as a graduate student) and I have some trouble formulating what I want to say. Talking to a few people about this did not generate a terrible amount of interest, but I was curious if people had any insight as to:

1. Has there been any work done along these lines?
2. Do you think this might be a potentially interesting idea?

My category theory is limited, and what I’ve learned does not seem to have this type of idea in mind, but I’d love to hear some thoughts!