  There is a well-established homotopy theory of simplicial objects in a
Grothendieck topos, and folklore says that the weak equivalences are
axiomatisable in the geometric fragment of $L_{\omega_1, \omega}$. We show that
it is in fact a theory of presheaf type, i.e. classified by a presheaf topos.
As a corollary, we obtain a new proof of the fact that the local Kan fibrations
of simplicial presheaves that are local weak homotopy equivalences are
precisely the morphisms with the expected local lifting property.
