  We study and construct maps to toric varieties. In the process, we generalize
torus embeddings to the non-projective case. Moreover, we give an analog of
Cox's construction of toric varieties as GIT quotients of affine spaces for the
non-normal case after T. Kajiwara.
  The main focus of the paper is an application to toric degenerations,
(proper) families whose special fibers are not-necessarily-normal toric
varieties. We give a negative answer to a question of I. Dolgachev and K. Kaveh
as to whether a toric degeneration can be constructed as a degeneration by
projection. In the classical topology over the complex numbers, we recover an
alternative construction of integral systems as was done by Harada--Kaveh in M.
Harada and K. Kaveh. "Integrable systems, toric degenerations and okounkov
bodie" using a deformation retract. In particular, we have an analogue of a
moment map from a variety admitting a toric degeneration to its
Newton--Okounkov polytope.
