Modular operads, iterated distributive laws and a nerve theorem for circuit algebras

TL;DR

This paper develops a graphical calculus and nerve theorem for circuit algebras, extending modular operad theory and revealing deep structural insights, with implications for quantum topology and algebraic frameworks.

Contribution

It introduces a monad and nerve theorem for circuit algebras, generalizing modular operad results and connecting them to wheeled props and quantum topology.

Findings

Established a graphical calculus for circuit algebras

Proved an abstract nerve theorem for these algebras

Showed equivalence of oriented circuit algebras and wheeled props

Abstract

Circuit algebras are a symmetric version of Jones's planar algebras. They originated in quantum topology as a framework for encoding virtual crossings. This paper extends existing results for modular operads to construct a graphical calculus and monad for general circuit algebras and prove an abstract nerve theorem. The proof relies on a subtle interplay between distributive laws and abstract nerve theory, and provides extra insights into the underlying structures. Oriented circuit algebras are equivalent to wheeled props and specialisations of the results to wheeled props follow as straightforward corollaries.