Some categorical remarks on coarse subspaces of coarse spaces

In this paper, we provide a categorical framework for understanding coarse subspaces of coarse spaces.

First, we introduce the notion of a controlled total relation between coarse spaces and show that the category whose morphisms are closeness classes of controlled total relations is isomorphic to the conventional category of coarse spaces defined using closeness classes of controlled maps.

Next, we show that the assignment associating to each coarse space the finite-join partially ordered set of its coarse subspaces is functorial, and prove that this partially ordered set is naturally isomorphic to the poset of subobjects in the category of coarse spaces.

Furthermore, we formulate asymptotic disjointness between coarse subspaces and show that mono-morphisms preserve this relation.

These results provide a categorical interpretation of the framework of coarse subspaces introduced by Leitner--Vigolo [Lecture Notes in Math.~(2023)] and characterize coarse subspaces as objects intrinsic to the category of coarse spaces.

They also provide a foundation for a coarse-geometric interpretation of the properness criterion established by Kobayashi [Math.~Ann.~(1989); J.~Lie Theory (1996)] and Benoist [Ann.~of Math.~(1996)] (cf.~Nagaya--Ogawa--Okuda [Proc.~Japan Acad.~Ser.~A (2025)]).