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

TL;DR

This work develops a complete categorical framework for circuit algebras by factorising their defining monad into three components via iterated distributive laws, yielding the composite monad $\mathbb{L}\mathbb{D}\mathbb{T}$ whose algebras are circuit algebras. It then establishes a graphical calculus and a Weber-style nerve theorem, proving that circuit algebras are precisely presheaf-nerve objects on a graphical category ${\Xi^{\times}}$, with a Segal condition characterising their essence. The results generalise existing modular-operad/prop theories to a broad, enriched setting and unearth precise connections to wheeled props, Brauer diagrams, and wiring diagrams, while also outlining how these ideas extend to homotopical contexts. The framework provides a unified, operad-like perspective on circuit algebras and their variants, with potential applications in representation theory and quantum topology.

Abstract

Circuit algebras are a symmetric version of Jones's planar algebras. They originated in quantum topology as a framework for encoding virtual crossings. Oriented circuit algebras are equivalent to wheeled props. 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. Specialisations of these results to wheeled props follow as straightforward corollaries. The machinery used to prove these results relies on a subtle interplay between distributive laws and abstract nerve theory, and provides extra insights into the underlying structures.

Figures (13)

• Figure 1: Modular operad axioms (M1)-(M4).
• Figure 2: Circuit algebra axioms (C1) and (C3). For axiom (C2), see \ref{['fig. MO axioms']}(M2).
• Figure 3: Realisations of connected graphs with one edge orbit: (a) the 1-corolla $\mathcal{C}_\mathbf{1}$; (b) the stick graph $(\shortmid)$; (c) the wheel graph $\mathcal{W}$ consists of a single inner edge orbit with one end vertex; (d) a single inner edge orbit with distinct end vertices.
• Figure 4: Realisations of $\mathcal{D}_{X,Y}^{x\ddagger y}$ and $\mathcal{C}_{X}^{x\ddagger y}$ for $X \cong \mathbf{4}, \ Y \cong \mathbf{3}$.
• Figure 5: Line and wheel graphs.

Theorems & Definitions (141)

• Theorem 1.1: \ref{['thm. iterated law']}
• Theorem 1.2: \ref{['nerve theorem']}
• Theorem 1.3: \ref{['prop. iterated law']} & \ref{['thm. iterated law']}
• Definition 2.1
• Example 2.3
• Example 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7
• Remark 2.9