Varieties of Coarse Spaces

A class $\mathfrak{M}$ $$ of coarse spaces is called a variety if $\mathfrak{M}$ $$ is closed under formation of subspaces, coarse images and products. We classify the varieties of coarse spaces and, in particular, show that if a variety $\mathfrak{M}$ $$ contains an unbounded metric space then $\mathfrak{M}$ is the variety of all coarse spaces.