  We prove that every Q-factorial complete toric variety is a finite quotient
of a poly weighted space (PWS), as defined in our previous work
arXiv:1501.05244. This generalizes the Batyrev-Cox and Conrads description of a
Q-factorial complete toric variety of Picard number 1, as a finite quotient of
a weighted projective space (WPS) \cite[Lemma~2.11]{BC} and
\cite[Prop.~4.7]{Conrads}, to every possible Picard number, by replacing the
covering WPS with a PWS. As a consequence we describe the bases of the subgroup
of Cartier divisors inside the free group of Weil divisors and the bases of the
Picard subgroup inside the class group, respectively, generalizing to every
Q-factorial complete toric variety the description given in arXiv:1501.05244,
Thm. 2.9, for a PWS.
