This talk deals with families of gapped Hamiltonians that depend continuously on some external parameters and examines their ability to be represented by matrix product states (MPS). Recently, a notion of higher Chern number applicable to many-body systems has been defined in order to determine whether families of such systems can be deformed to a constant family of product states. Here, we show that, for one-dimensional systems, a non-zero higher Chern number can also be understood as an obstruction to representing a continuous family of ground states as MPS whose tensors are continuous in the external parameters. This result parallels the usual Chern number for zero-dimensional systems, which represents an obstruction to continuously choosing the phase of the ground state wavefunction. The obstruction for MPS comes from the phase ambiguity that exists in gauge transformations between two tensors representing the same state. This leads to the mathematical structure of a gerbe, which necessarily goes beyond the familiar principle-bundle structure of MPS, and the higher Chern number precisely classifies the global structure of this gerbe. Thus, our results give new physical intuition to the higher Chern number, and also demonstrate new structures in the geometry of states representable by MPS.