  Circuit algebras, introduced by Bar-Natan and the first author, are a
generalization of Jones's planar algebras, in which one drops the planarity
condition on "connection diagrams". They provide a useful language for the
study of virtual and welded tangles in low-dimensional topology. In this note,
we present the circuit algebra analogue of the well-known classification of
planar algebras as pivotal categories with a self-dual generator. Our main
theorem is that there is an equivalence of categories between circuit algebras
and the category of linear wheeled props - a type of strict symmetric tensor
category with duals that arises in homotopy theory, deformation theory and the
Batalin-Vilkovisky quantization formalism.
