it's a beautiful paper (or at least section 1 is, that's the main part i've spent any time with), you should check it out if you haven't
roughly, a distributor is an inclusion $X \subset Y$ of $\infty$-categories which is a generalization of an $\infty$-topos
it allows for a robust notion of "an $\infty$-category enriched in $Y$, whose "object of objects" is an object of $X$"
as i said, i'd like to verify that $\mathrm{Spaces} \subset s\mathrm{Space}$ is a distributor (Def 1.2.1), presumably via the characterization of Cor 1.2.5. but i just can't quite wrap my head around it.
(however, this only allows for enriching with respect to the cartesian symmetric monoidal structure on $Y$)
(which should be believable, since it's all a generalization of the ordinary theory of complete segal spaces)
