  We apply an algebraic double copy construction of gravity from gauge theory
to three-dimensional (3D) Chern-Simons theory. The kinematic algebra ${\cal K}$
is the 3D de Rham complex of forms equipped, for a choice of metric, with a
graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on
polyvector fields. The double copied gravity is defined on a subspace of ${\cal
K}\otimes \bar{\cal K}$ and yields a topological double field theory for a
generalized metric perturbation and two 2-forms. This local and gauge invariant
theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality.
Upon fixing a gauge this reduces to the double copy of Chern-Simons theory
previously proposed by Ben-Shahar and Johansson. Furthermore, using complex
coordinates in $\mathbb{C}^3$ this theory is related to six-dimensional (6D)
Kodaira-Spencer gravity in that truncating the two 2-forms and one equation
yields the Kodaira-Spencer equations on a 3D real slice of $\mathbb{C}^3$. The
full 6D Kodaira-Spencer theory can instead be obtained as a consistent
truncation of a chiral double copy.
