  One of the central notions to emerge from the study of persistent homology is
that of interleaving distance. It has found recent applications in symplectic
and contact geometry, sheaf theory, computational geometry, and phylogenetics.
Here we present a general study of this topic. We define interleaving of
functors with common codomain as solutions to an extension problem. In order to
define interleaving distance in this setting we are led to categorical
generalizations of Hausdorff distance, Gromov-Hausdorff distance, and the space
of metric spaces. We obtain comparisons with previous notions of interleaving
via the study of future equivalences. As an application we recover a definition
of shift equivalences of discrete dynamical systems.
