Is there a categorical definition of submetry?

(Updated to include effective epimorphism.)

This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use category theory.

Consider the category CpltMet in which the objects are complete metric spaces and morphisms are 1-Lipschitz maps; the maps $f:X\to Y$ such that $d_Y(f(a),f(b))\le d_X(a,b)$ for all $a,b\in X$. Note that in this category monomorphisms are injective 1-Lipschitz maps, and epimorphisms are 1-Lipschitz maps with dense range, but not necessarily surjective.

An isometric embedding is a map $f:X\to Y$ such that $d_Y(f(a),f(b)) = d_X(a,b)$ for all $a,b\in X$. I can describe such maps in the arrow-speak as follows. $f:X\to Y$ is an isometric embedding iff the following holds: whenever $f$ factors through an epimorphism $g:X\to Z$ (meaning $f=h\circ g$ for some $h:Z\to Y$), $g$ is an isomorphism. (Proof is given in Note 1.) If there is a better categorical description of isometric embeddings, I'd like to see it.

A submetry is a map $f:X\to Y$ such that for every $a\in X$ and every $r\ge 0$ we have $f(B_X(a,r))=B_Y(f(a),r)$ where $B$ denotes a closed ball. (See Note 2 about the definition). To appreciate this definition, consider the following.

• isometric embeddings are characterized by the condition $f^{-1}(B_Y(f(a),r))=B_X(a,r)$, mirroring the definition of submetry.
• among 1-Lipschitz maps, submetries are characterized by the 2-point lifting property:
  for every $y_0,y_1\in Y$ and every $x_0\in f^{-1}(y_0)$ there exists $x_1\in f^{-1}(y_1)$ such that $d_X(x_0,x_1)=d_Y(y_0,y_1)$.
• for linear operators between Banach spaces, the adjoint of an isometric embedding is a submetry (the proof is an exercise with Hahn-Banach).

My question is: can the submetries be defined categorically, preferably in a way that makes them a dual class to isometric embeddings?

The problem is that reversing the arrows in the above definition of an isometric embedding gives a wider class of maps than submetries. Indeed, the reversed definition is: $f:Y\to X$ does not factor through any monomorphism $g:Z\to X$ unless $g$ is an isomorphism. But this holds, for example, for the function $f:\mathbb R\to \mathbb R$ defined by $$ (*)\qquad \qquad f(x)=\begin{cases} x+1,\quad &x\le -1 \\ 0,\quad &|x|\le 1 \\ x-1,\quad &x\ge 1 \end{cases}$$ which is not a submetry in the standard metric of the real line. (See Note 3 for the proof.) Maybe I'm reversing arrows in a "wrong" description of isometric embeddings.

Note 1. If $f$ preserves distances, then so does $g$; having dense range, $g$ must be onto because $X$ is complete; hence, $g$ is an isomorphism. Conversely, if $f$ decreases distances somewhere, let $Z$ be the same set as $X$ with the metric $(d_X(a,b)+d_Y(f(a),f(b)))/2$. The identity map $g:X\to Z$ is an epimorphism, $f$ factors through it, but $g$ is not an isomorphism.

Note 2. I am following the original definition of submetry given by Berestovskii ("Submetries of space-forms of nonnegative curvature", 1987). If one uses open balls instead of closed, the class is enlarged to weak submetries. In Riemannian Geometry by Petersen the term submetry is used for more general maps, which I would call weak local submetries.

Note 3. Proof: Suppose $f=g\circ h$ where $g: Z\to \mathbb R$ is a monomorphism. Then $h$ maps $[-1,1]$ into a single point $z\in Z$. When $a\le -1$ and $b\ge 1$, the triangle inequality yields $d_Z(h(a),h(b))\le |a-b|-2=|f(a)-f(b)|$. Hence, $g$ must be an isomorphism.

Note 4. Following the immersion:submersion terminology of differential geometry, I'd like to call an isometric embedding an immetry, but I'm not sure that the neologism would catch on.

Following the suggestion by @t.b., I considered the concept of an effective epimorphism. Unfortunately, the undesirable map defined by (*) appears to be effective. Indeed, let $R=\{(x,y)\in\mathbb R^2: f(x)=f(y)\}$. The orthogonal projections $\pi_x,\pi_y : R\to \mathbb R$ are 1-Lipschitz and satisfy $f\circ \pi_x=f\circ \pi_y$ by construction. As far as I can tell, $\pi_x$ and $\pi_y$ qualify as a kernel pair for which $f$ is a coequalizer.

A common problem, whenever one tries to express analysis concepts in terms of category theory, is that analysis is done with inequalities rather than equalities. So, in general, ordinary category theory is not very well-suited to talk about analysis. On the other hand, a slight generalization of categories, namely order-enriched categories (or even quantale-enriched) gives a much better language to talk about analysis. It can be thought of as a "category theory with inequalities". However, the translation often requires some work.

Now to the question. In the context of enriched category theory, a metric space can be seen as a particular enriched category (see for example here). Complete metric spaces can be described categorically in quite an elegant way (here), and the correspondent enriched notion of functor gives exactly 1-Lipschitz maps. (In enriched category theory, distances are usually not required to be symmetric or finite, but of course one can make that requirement, and basically all the results restrict to ordinary metric spaces.)

Treating then your category as a category of enriched categories, submetries are precisely the proper maps in the sense of Definition 3.1.1 of this book:

• Gavin J. Seal, Dirk Hofmann and Walter Tholen, Monoidal Topology, Cambridge University Press, 2014.

(Not to be confused with the different, but related, notion of proper map in topology.)

I won't copy the definition since it requires quite some buildup - but let's see what it means for our case. I paraphrase from the book above, Chapter V, Example 3.13(3):

For metric spaces, a 1-Lipschitz map $f:X\to Y$ is proper if and only if $$ d_Y(f(x),y) \;=\; \inf \big\{d_X(x,x')\;|\;x'\in f^{-1}(y)\big\} . $$ for all $x\in X$ and $y\in Y$.

The two notions agree for $f:X\to Y$ 1-Lipschitz, let's show this explicitly.

• Suppose that $f$ is a submetry. Let $r>0$, $x\in X$ and $y\in Y$. If $d_Y(f(x),y)\le r$, then $$ y \;\in\; B_Y(f(x),r) \;=\; f(B_X(x,r)) , $$ therefore for all $x'$ with $f(x')=y$, we have $d_X(x,x')\le r$. In other words, $$ \inf \big\{d_X(x,x')\;|\;x'\in f^{-1}(y)\big\} \;\le\; r. $$ This is true for all $r$, so $$ \inf \big\{d_X(x,x')\;|\;x'\in f^{-1}(y)\big\} \;\le\; d_Y(f(x),y). $$ The reverse inequality holds by the fact that $f$ is 1-Lipschitz. Therefore $f$ is proper in the sense above.

• Conversely, suppose that $f$ is proper in the sense given above. Then for $x\in X$ and $r>0$, $$ f(B_X(x,r)) \;\subseteq B(f(x),r). $$ To prove equality suppose that $y\in B(f(x),r)$, i.e. that $d_Y(f(x),y)\le r$. Then $$ \inf \big\{d_X(x,x')\;|\;x'\in f^{-1}(y)\big\} \;\le\; r, $$ so for all $x'$ such that $f(x')=y$, $d_X(x,x')\le r$. So $y\in f(B_X(x,r))$. This means that $f$ is a submetry.

(You can also see what happens if we drop the 1-Lipschitz requirement.)

By the way, we've noticed this equivalence of notions during our work (arXiv:1808.09898, see Appendix A therein), which indeed is about a categorical study of some structures in analysis and functional analysis (such as the stochastic order of probability measures and ordered Banach spaces).

(Thanks to Tobias Fritz, Slava Matveev and Walter Tholen for the pointers.)