In derived algebraic geometry (Lurie's thesis) he defines a notion of cohesiveness for functors $F: SCR \to S$, where $SCR$ is the $\infty$-category of simplicial commutative rings, and $S$ is the $\infty$-category of spaces. Such a functor is cohesive if $F(A \times_C B) \to F(A) \times_{F(C)} F(B)$ is a weak equivalence for all $A \to C \leftarrow B$ with both arrows surjective.
As far as I can tell, if we take simplicial sets as a model for the category $S$, then the fiber product $F(A) \times_{F(C)} F(B)$ is really supposed to be a homotopy fiber product, but the fiber product on the left hand side (i.e. $A \times_C B$) is supposed to be the levelwise fiber product in the category of simplicial commutative rings. Or is it also supposed to be a homotopy fiber product, with respect to a model structure on $SCR$?
I realized that if either $A \to C$ or $B \to C$ is (levelwise) surjective, then the map is a fibration, and since every simplicial ring is fibrant, this means that the ordinary fiber product computes the homotopy fiber product. So I guess in this case the distinction doesn't really matter.
But I guess Lurie is kind of sweeping this under the rug, since $(\infty,1)$-pullbacks are supposed to be homotopy pullbacks, and I'm not sure if he uses an explicit presentation of $SCR$ later as a cofibrantly generated model category to actually compute the fact that the pullback and the homotopy pullback are the same. I guess in his language, there's no ordinary pullback around.