Brauer diagrams, modular operads, and a graphical nerve theorem for circuit algebras

TL;DR

This work develops a complete categorical framework for circuit algebras by describing them as algebras over a composite monad ${\mathbb L}{\mathbb D}{\mathbb T}$ on graphical species, built via iterated distributive laws. It introduces Brauer-diagram categories (monochrome and coloured) as a conduit between circuit algebras and modular operads, and proves a Weber-style nerve theorem: circuit algebras embed fully faithfully into presheaves on a graph-category $\Xi^\times$ satisfying a Segal condition. Central to the method are the monads ${\mathbb L}$ (free external product), ${\mathbb D}$ (pointing), and ${\mathbb T}$ (modular/circuit-ops), together with liftings that yield a coherent nerve theory, extending the modular-operad nerve framework to circuit-operads. The results unify representations of classical groups via Brauer categories with operadic/wiring-diagram formalisms, providing a graphical calculus and a robust algebraic bridge to modular operads, wheeled PROPs, and representation theory. The nerve construction identifies circuit algebras with Segal presheaves on a graphical category, offering a pathway toward $(\infty,1)$-circuit algebras and future model-category developments. Empirically, this renders circuit algebras amenable to operadic and representation-theoretic techniques, enabling transfer of modular-operad machinery to the study of finite-type knot invariants and related invariants.

Abstract

Circuit algebras, used in the study of finite-type knot invariants, are a symmetric analogue of Jones's planar algebras. They are very closely related to circuit operads, which are a variation of modular operads admitting an extra monoidal product. This paper gives a description of circuit algebras in terms categories of Brauer diagrams. An abstract nerve theorem for circuit operads -- and hence circuit algebras -- is proved using an iterated distributive law, and an existing nerve theorem for modular operads.

Figures (22)

• Figure 1: The triangle identities.
• Figure 2: (a) ${\lfloor{f}\rfloor}\colon y^* \otimes x \to I$; (b)${\lceil{f}\rceil}\colon I \to y \otimes x^*$; (c) $f^*\colon y^*\to x^*$ .
• Figure 3: (a) Composition of pairings on $X \amalg Y$ and $Y \amalg Z$; (c) the resulting pairing on $X \amalg Z$, together with the single closed component formed in the composition.
• Figure 4: The triangle identities in $\mathsf{BD}$ with $n = 3$.
• Figure 5: Composing coloured pairings.

Theorems & Definitions (150)

• Theorem 1.1
• Theorem 1.2: \ref{['iterated law']}
• Theorem 1.3: \ref{['nerve theorem']}
• Theorem 1.4: \ref{['prop. iterated law']} & \ref{['iterated law']}
• Definition 2.1
• Example 2.2
• Example 2.3
• Definition 2.6
• Example 2.7
• Example 2.8