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

2024/12/28 by Sophie Raynor, Raynor, Sophie
Computer Science · Mathematics · #18M85 (Primary) 18C15 #57K16 (Secondary) #Advanced Algebra and Logic #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Logic, programming, and type systems #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2412.20262

openalex publication_date 2024/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

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.

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