  We prove that the homotopy theory of parametrized spaces embeds fully and
faithfully in the homotopy theory of simplicial presheaves, and that its
essential image consists of the locally homotopically constant objects. This
gives a homotopy-theoretic version of the classical identification of covering
spaces with locally constant sheaves. We also prove a new version of the
classical result that spaces parametrized over X are equivalent to spaces with
an action of the loop space of X. This gives a homotopy-theoretic version of
the correspondence between covering spaces over X and sets with an action of
the fundamental group of X. We then use these two equivalences to study base
change functors for parametrized spaces.
