  Quantization needs evaluation of all of states of a quantized object rather
than its stationary states with respect to its energy. In this paper, we have
investigated moduli $\CMeP$ of a quantized elastica, a quantized loop with an
energy functional associated with the Schwarz derivative, on a Riemann sphere
$\PP$. Then it is proved that its moduli space is decomposed to a set of
equivalent classes determined by flows obeying the Korteweg-de Vries (KdV)
hierarchy which conserve the energy. Since the flow obeying the KdV hierarchy
has a natural topology, it induces topology in the moduli space $\CMeP$. Using
the topology, $\CMeP$ is classified. Studies on a loop space in the category of
topological spaces $\Top$ are well-established and its cohomological properties
are well-known. As the moduli space of a quantized elastica can be regarded as
a loop space in the category of differential geometry $\DGeom$, we also proved
an existence of a functor between a triangle category related to a loop space
in {\bf Top} and that in $\DGeom$ using the induced topology. As Euler
investigated the elliptic integrals and its moduli by observing a shape of
classical elastica on $\CC$, this paper devotes relations between hyperelliptic
curves and a quantized elastica on $\PP$ as an extension of Euler's perspective
of elastica.
