We introduce a category composed of all quantizations of all Poisson
algebras. By the category, we can treat in a unified way the various
quantizations for all Poisson algebras and develop a new classical limit
formulation. This formulation proposes a new method for the inverse problem,
that is, the problem of finding the classical limit from a quantized space.
Equivalence of quantizations is defined by using this category, and the
conditions under which the two quantizations are equivalent are investigated.
Two types of classical limits are defined as the limits in the context of
category theory, and they are determined by giving a sequence of objects. Using
these classical limits, we discuss the inverse problem of determining the
classical limit from some noncommutative Lie algebra. From a Lie algebra, we
construct a sequence of quantized spaces, from which we determine a Poisson
algebra. We also present a method to obtain this sequence of quantizations from
the principle of least action by using matrix regularization. Apart from the
above category of quantizations of all Poisson algebras, a category of
quantizations of a fixed single Poisson algebra is also introduced. In this
category, the other classical limit is defined, and it is automatically
determined for the category.
