  Given an infinity-category C, one can naturally construct an
infinity-category Fam(C) of families of objects in C indexed by
infinity-groupoids. An ordinary categorical version of this construction was
used by Borceux and Janelidze in the study of generalized covering maps in
categorical Galois theory. In this paper, we develop the homotopy theory of
such "parametrized families" as generalization of the classical homotopy theory
of spaces. In particular, we study homotopy-theoretical constructions that
arise from the fundamental infinity-groupoids of families in an
infinity-category. In the same spirit, we show that Fam(C) admits a
Grothendieck topology which generalizes the canonical/epimorphism topology on
the infinity-topos of infinity-groupoids in the sense of Carchedi.
