Monoidal topology and coarse spaces

Question

Is there a description of (quasi-)coarse spaces that is analogous to the description of (quasi-)uniform spaces as lax algebras?

Answer 1

Apologies if this won't be an answer, but I can't put much more headspace in this question other than brainstorm in public. Let's see if we can reverse engineer the definition!

Coarse structures in a nutshell:

Fix a set $X$; a coarse structure on $X$ collection of relations $\cal E$ such that

• the diagonal $\Delta$ is in $\cal E$;
• $\cal E$ is closed under composition of relation;
• $\cal E$ is downward closed;
• $\cal E$ is closed under taking inverse relation and union of relations.

Monoidal topology in a nutshell:

Fix a monad $T$ on $Set$, a quantale $\cal V$ and define, for each lax lifting of $T$ to ${\cal V}\text{-}Rel$ (the usual category of relations, just "V-valued"), called $\hat T$, a bicategory $(\hat T,V)\text{-}Rel$ where objects are sets, and a 1-cell $X\to Y$ is a function $TX\times Y \to \cal V$.

For suitable choices of T and V, a reflexive and transitive $(\hat T,V)$-relation is one of the spaces you want to talk about.

What could T and V be, in order to describe coarse spaces?

We can try taking as $\cal V$ just {0,1}; then, $\cal E$ is a subset of $Rel(X,X)$, i.e. a map $E:Rel(X,X)\to 2$.

I would now try to see whether this simple request

$E$ is a homomorphism of quantales $Rel(X,X)^{op}\to \{0\le 1\}$

recovers the properties above. First of all, monotonicity is condition 3, and since composition of relation is the quantale operation on $Rel(X,X)$, $E(id_X)=1$ and $E(R\circ S)=E(R)\land E(S)$ is condition 1+2.

I think where things start to diverge a little bit is condition 4, since $E(R^\ast)$ now must not be the dual of $E(R)$ in $\{0\le 1\}$. Also, closure under union is stated in terms of finite unions, and requiring that $E$ is a map of quantales would instead ask preservation of arbitrary large suprema.

At least this is a very concise packaging of almost all the properties!

If I were you, I would scan the literature on allegories (but probably you know it better than me already!) looking for some inspiration. I feel this discussion went a bit astray from monoidal topology, meaning that I can't see a $(T,V)$-algebra around here...