  We propose a systematic wave function based approach to construct topological
invariants for families of lattice systems that are short-range entangled using
local parameter spaces. This construction is particularly suitable when given a
family of tensor networks that can be viewed as the ground states of $d$
dimensional lattice systems, for which we construct the closed $(d+2)$-form
higher Berry curvature, which is a generalization of the well known 2-form
Berry curvature. Such $(d+2)$-form higher Berry curvature characterizes a flow
of $(d+1)$-form higher Berry curvature in the system. Our construction is
equally suitable for constructing other higher pumps, such as the (higher)
Thouless pump in the presence of a global on-site $U(1)$ symmetry, which
corresponds to a closed $d$-form. The cohomology classes of such higher
differential forms are topological invariants and are expected to be quantized
for short-range entangled states. We illustrate our construction with exactly
solvable lattice models that are in nontrivial higher Berry classes in $d=2$.
