Koszul Operads Governing Props and Wheeled Props

TL;DR

This paper develops a new algebraic framework for props and wheeled props using Koszul operads, extending operad theory with groupoid colourings, and explores their homotopy and formality properties.

Contribution

It introduces groupoid coloured operads governing props and wheeled props, proves their Koszulity, and applies homotopy transfer to study their higher structures and formality.

Findings

Props and wheeled props are governed by Koszul operads.

Homotopy (wheeled) props are characterized via the Koszul machine.

Massey products reveal the formality and higher homotopies of these structures.

Abstract

In this paper, we construct groupoid coloured operads governing props and wheeled props, and show they are Koszul. This is accomplished by new biased definitions for (wheeled) props, and an extension of the theory of Groebner bases for operads to apply to groupoid coloured operads. Using the Koszul machine, we define homotopy (wheeled) props, and show they are not formed by polytope based models. Finally, using homotopy transfer theory, we construct Massey products for (wheeled) props, show these products characterise the formality of these structures, and re-obtain a theorem of Mac Lane on the existence of higher homotopies of (co)commutative Hopf algebras.