  Matrix Product Operators (MPOs) are tensor networks representing operators
acting on 1D systems. They model a wide variety of situations, including
communication channels with memory effects, quantum cellular automata, mixed
states in 1D quantum systems, or holographic boundary models associated to 2D
quantum systems. A scenario where MPOs have proven particularly useful is to
represent algebras of non-trivial symmetries. Concretely, the boundary of both
symmetry protected and topologically ordered phases in 2D quantum systems
exhibit symmetries in the form of MPOs.
  In this paper, we develop a theory of MPOs as representations of algebraic
structures. We establish a dictionary between algebra and MPO properties which
allows to transfer results between both setups, covering the cases of
pre-bialgebras, weak bialgebras, and weak Hopf algebras. We define the notion
of pulling-through algebras, which abstracts the minimal requirements needed to
define topologically ordered 2D tensor networks from MPO algebras. We show, as
one of our main results, that any semisimple pivotal weak Hopf algebra is a
pulling-trough algebra. We demonstrate the power of this framework by showing
that they can be used to construct Kitaev's quantum double models for Hopf
algebras solely from an MPO representation of the Hopf algebra, in the exact
same way as MPO symmetries obtained from fusion categories can be used to
construct Levin-Wen string-net models, and to explain all their topological
features; it thus allows to describe both Kitaev and string-net models on the
same formal footing.
