Grothendieck's homotopy hypothesis, is, as the $n$lab states:
Theorem:There is an equivalence of $(∞,1)$-categories $(\Pi⊣|−|): \mathbf{Top} \simeq \mathbf{\infty Grpd}$.
What are the applications of this hypothesis? Why is it so fundamental? Can it be "generalized", perhaps by using the following definition of spaces: a space is simply a sheaf of sets on some site $\mathbf{Loc}$ of local models with a Grothendieck topology $τ$ on it?