On the one hand, as mentioned here, basically "everything" in algebraic geometry could be seen in the context of "moduli problems" - on the other hand, Grothendieck's few remarks on a possible "tame topology" tell that he wondered about some general principle behind the stratifications of known moduli "spaces".
This makes me wonder:
When did re-interpretations as moduli problems turn out to be helpfull?
How is "tame topology" used then?
... and how develops tame topology, e.g. Grothendieck mentions in his "Esquisse" something on "tubular neighbourhoods of subtopoi" - what's that?
Which role play moduli problems in derived or else generalised geometry?
