  It is believed that most (perhaps all) gapped phases of matter can be
described at long distances by Topological Quantum Field Theory (TQFT). On the
other hand, it has been rigorously established that in 1+1d ground states of
gapped Hamiltonians can be approximated by Matrix Product States (MPS). We show
that the state-sum construction of 2d TQFT naturally leads to MPS in their
standard form. In the case of systems with a global symmetry $G$, this leads to
a classification of gapped phases in 1+1d in terms of Morita-equivalence
classes of $G$-equivariant algebras. Non-uniqueness of the MPS representation
is traced to the freedom of choosing an algebra in a particular Morita class.
In the case of Short-Range Entangled phases, we recover the group cohomology
classification of SPT phases.
