  Anyon condensation forms a mechanism which allows to relate different
topological phases. We study anyon condensation in the framework of Projected
Entangled Pair States (PEPS) where topological order is characterized through
local symmetries of the entanglement. We show that anyon condensation is in
one-to-one correspondence to the behavior of the virtual entanglement state at
the boundary (i.e., the entanglement spectrum) under those symmetries, which
encompasses both symmetry breaking and symmetry protected (SPT) order, and we
use this to characterize all anyon condensations for abelian double models
through the structure of their entanglement spectrum. We illustrate our
findings with the Z4 double model, which can give rise to both Toric Code and
Doubled Semion order through condensation, distinguished by the SPT structure
of their entanglement. Using the ability of our framework to directly measure
order parameters for condensation and deconfinement, we numerically study the
phase diagram of the model, including direct phase transitions between the
Doubled Semion and the Toric Code phase which are not described by anyon
condensation.
