Higher dimensional algebras via colored PROPs

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.

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