  Starting from any unital colored PROP $P$, we define a category $P(P)$ of
shapes called $P$-propertopes. Presheaves on $P(P)$ are called $P$-propertopic
sets. For $0 \leq n \leq \infty$ we define and study $n$-time categorified
$P$-algebras as $P$-propertopic sets with some lifting properties. Taking
appropriate PROPs $P$, we obtain higher categorical versions of polycategories,
2-fold monoidal categories, topological quantum field theories, and so on.
