Which are the morphisms for sets with semimetrics?

Question

A semimetric is due to Wikipedia defined as a metric but without the triangle inequality. Intuitively it seems possible to define some kind of continuity for functions between such sets $f:A\to B$, a "small change" in input will result in only a "small change" of output: $$\forall\epsilon>\!0\,\exists\delta>0: d(a,a')<\delta\implies d(f(a),f(a'))<\epsilon$$ but generally the triangle inequality is needed to show that the composition of two continuous functions is continuous.

So which are the morphisms in the category of sets with semimetrics?

Answer 1

A modulus of continuity is a function $[0,\infty]\xrightarrow{\omega}[0,\infty]$ so that

1. $r_1\leq r_2$ implies $\omega(r_1)\leq\omega(r_2)$.
2. $\omega(0)=0$ and $\omega$ is continuous at $0$.

Fix an order-preserving function $[0,\infty]\times[0,\infty]\xrightarrow{\oplus}[0,\infty]$. A $\oplus$-metric on a space $A$ is a function $A\times A\xrightarrow{d_A}[0,\infty]$ so that

1. $d(a,a)=0$
2. $d(a,c)\leq d(a,b)\oplus d(a,c)$

Notice that the case where $[0,\infty]\times[0,\infty]\xrightarrow{\oplus}[0,\infty]$ is identically $\infty$ is exactly the case of a "premetric space''.

If $A$ and $B$ are $\oplus$-metric spaces, then a function $A\xrightarrow{f}B$ has modulus of continuity $[0,\infty]\xrightarrow{\omega_a}[0,\infty]$ at $a\in A$ if

• $d_B(f(a),f(a')\leq\omega_a(d_A(a,a'))$

A morphism between such $\oplus$-metric spaces is then a function $A\xrightarrow{f}B$ equipped with a modulus of continuity $[0,\infty]\xrightarrow{\omega_{f,a}}[0,\infty]$ at every point $a\in A$.

Note that with this definition

1. uniformly continuous functions (which you defined) are morphisms that have the same modulus of continuity at every point.
2. non-expansive maps (i.e. $d_B(f(a),f(a'))<d_A(a,a)$) are morphisms with modulus of continuity the identity function.
3. continuous maps are morphisms that have forgotten what their moduli of continuity are.

As far as I know, category theorists have studied extensively the category of metric spaces and non-expansive maps between them because there $0$ is a two-sided identity for $\oplus=+$, and metric spaces end up being enriched categories with non-expansive functions corresponding to enriched functors.