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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 overX $X$we associate a gerbe, which generalizes the line bundle of ground states in zero-dimensional families ( in few-body quantum mechanics). The nontriviality of the gerbe is measured by a class inH^3(X, \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 onX $X$. We illustrate our construction with two examples of nontrivial parametrized systems overX=S^3 $X=S^3$andX = \R P^2 × 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 overX = S^4 $X=S^4$.