  Sheaves are objects of a local nature: a global section is determined by how
it looks locally. Hence, a sheaf cannot describe mathematical structures which
contain global or nonlocal geometric information. To fill this gap, we
introduce the theory of "gleaves", which are presheaves equipped with an
additional "gluing operation" of compatible pairs of local sections. This
generalizes the conditional product structures of Dawid and Studen\'y, which
correspond to gleaves on distributive lattices. Our examples include the gleaf
of metric spaces and the gleaf of joint probability distributions. A result of
Johnstone shows that a category of gleaves can have a subobject classifier
despite not being cartesian closed.
  Gleaves over the simplex category $\Delta$, which we call compositories, can
be interpreted as a new kind of higher category in which the composition of an
$m$-morphism and an $n$-morphism along a common $k$-morphism face results in an
$(m+n-k)$-morphism. The distinctive feature of this composition operation is
that the original morphisms can be recovered from the composite morphism as
initial and final faces. Examples of compositories include nerves of categories
and compositories of higher spans.
