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

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.

💡 Research Summary

This paper develops a comprehensive categorical framework for circuit algebras, which are symmetric analogues of Jones’s planar algebras and were originally introduced to encode virtual crossings in quantum topology. The author builds on earlier work on modular operads to construct a graphical calculus, a monad, and an abstract nerve theorem for circuit algebras, and shows how oriented circuit algebras are equivalent to wheeled props.

The core technical contribution is the construction of three interacting monads on the category of graphs GS:

* L governs the monoidal (tensor) product of graphs,
* D adds unit (loop) operations, and
* T encodes graph substitution (the basic composition of wiring diagrams).

Using Beck’s theory of distributive laws, the paper first exhibits a distributive law λ₁ : D ∘ T ⇒ T ∘ D, which yields the composite monad DT (the modular operad monad of