  We present a sheaf-theoretic construction of shape space -- the space of all
shapes. We do this by describing a homotopy sheaf on the poset category of
constructible sets, where each set is mapped to its Persistent Homology
Transform (PHT). Recent results that build on fundamental work of Schapira have
shown that this transform is injective, thus making the PHT a good summary
object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of
different shapes together to build up the PHT of a larger shape. In the case
where our shape is a polyhedron we prove a generalized nerve lemma for the PHT.
Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we
show that we can reliably approximate the PHT of a manifold by a polyhedron up
to arbitrary precision.
