  Sponges were recently proposed as a generalization of lattices, focussing on
joins/meets of sets, while letting go of associativity/transitivity. In this
work we provide tools for characterizing and constructing sponges on metric
spaces and groups. These are then used in a characterization of epigraph
sponges: a new class of sponges on Hilbert spaces whose sets of left/right
bounds are formed by the epigraph of a rotationally symmetric function. We also
show that the so-called hyperbolic sponge generalizes to more than two
dimensions.
