Higher dimensional algebras via colored PROPs

TL;DR

This paper introduces a new framework for higher dimensional algebras using colored PROPs, defining categories of shapes called P-propertopes and studying their associated propertopic sets, which generalize various higher categorical structures.

Contribution

It develops a novel approach to higher algebra by constructing P-propertopes and propertopic sets from unital colored PROPs, enabling the study of higher categorical structures.

Findings

Defined P-propertopes and P-propertopic sets from unital colored PROPs

Established a framework for n-time categorified P-algebras

Unified various higher categorical structures like polycategories and TQFTs

Abstract

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.