I’ve found something interesting:
Having a permutation group (in a set of permutation groups, such as the set of all (small) Euclidean spaces), we apparently can construct a space-in-general (as spaces are defined in this work):
Let $latex \pi$ be a set of permutation groups $\latex G$ (on set $latex M_G$). For spaces such as Euclidean, $M_G$ is the set of subsets of the space.
It induces ordered semigroup action with elements (spaces) $latex \mu$ determined by $latex G$ such that
$latex \langle\mu\rangle S=\{\rsupfun{f}S \mid f\in G,S\in\subsets M_G\}$, where arguments~$latex S$ are ordered by set-inclusion
and spaces~$latex \mu$ are ordered by
$latex \mu_0\leq\mu_1\Leftrightarrow\forall G\in\pi, S\in M_G:\langle\mu_0\rangle S\leq\langle\mu_1\rangle S. \]
(TODO: Prove it is an OSA.)
(For Euclidean spaces that is the set of all moves of sets of sets.)
So, we can go from groups to spaces-in-general. Can we go in the reverse direction, making an isomorphism between groups and spaces-in-general?