  We define an exactly solvable model for 2+1D topological phases of matter on
a triangulated surface derived from a crossed module of semisimple
finite-dimensional Hopf algebras, the `Hopf-algebraic higher Kitaev model'.
This model generalizes both the Kitaev quantum double model for a semisimple
Hopf algebra and the full higher Kitaev model derived from a 2-group, and can
hence be interpreted as a Hopf-algebraic discrete higher gauge theory.
  We construct a family of crossed modules of semisimple Hopf algebras,
$(\mathscr{F}_{\mathbb{C}}(X) \otimes \mathbb{C}E \xrightarrow{\partial}
\mathscr{F}_{\mathbb{C}}(Y) \rtimes \mathbb{C} G, \triangleright)$, that
depends on four finite groups, $E,G,X$ and $Y$. We calculate the ground-state
spaces of the resulting model on a triangulated surface when $G=E=\{1\}$ and
when $Y=\{1\}$, prove that those ground-state spaces are canonically
independent of the triangulations, and so depend only on the underlying
surface; and moreover we find a 2+1D TQFT whose state spaces on surfaces give
the ground-state spaces. These TQFTs are particular cases of Quinn's finite
total homotopy TQFT and hence the state spaces assigned to surfaces are free
vector spaces on sets of homotopy classes of maps from a surface to homotopy
finite spaces, in this case obtained as classifying spaces of finite groupoids
and finite crossed modules of groupoids.
  We leave it as an open problem whether the ground-state space of the
Hopf-algebraic higher Kitaev model on a triangulated surface is independent of
the triangulation for general crossed modules of semisimple Hopf algebras,
whether a TQFT always exists whose state space on a surface gives the
ground-state space of the model, and whether the ground-state space of the
model obtained from $E,G,X,Y$ can always be given a homotopical explanation.
