  We employ matrix product states (MPS) and tensor networks to study
topological properties of the space of ground states of gapped many-body
systems. We focus on families of states in one spatial dimension, where each
state can be represented as an injective MPS of finite bond dimension. Such
states are short-range entangled ground states of gapped local Hamiltonians. To
such parametrized families over $X$ we associate a gerbe, which generalizes the
line bundle of ground states in zero-dimensional families (\emph{i.e.} in
few-body quantum mechanics). The nontriviality of the gerbe is measured by a
class in $H^3(X, \mathbb{Z})$, which is believed to classify one-dimensional
parametrized systems. We show that when the gerbe is nontrivial, there is an
obstruction to representing the family of ground states with an MPS tensor that
is continuous everywhere on $X$. We illustrate our construction with two
examples of nontrivial parametrized systems over $X=S^3$ and $X = \mathbb{R}
P^2 \times S^1$. Finally, we sketch using tensor network methods how the
construction extends to higher dimensional parametrized systems, with an
example of a two-dimensional parametrized system that gives rise to a
nontrivial 2-gerbe over $X = S^4$.
