I apologize for the vagueness of the following.
Informally, in the site of commutative rings, one roughly get the notion of a derived stack by swapping out the commmutative rings with its subcategory of simplicial objects. This is part of the story told by Toen and Vezzosi in their many expository accounts on derived geometry.
Now, there appears to be a fairly well-studied story for stacks defined in the site of topological spaces (topological stacks), notably in a series of papers by Noohi. Then it’s natural to try to obtain a notion of a derived topological stack by replacing the source category by simplicial topological spaces. Has this been done by anyone? Is the story there trivial or uninteresting for some reason?
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