  Field theories with kinematic Lie algebras, such as field theories featuring
colour-kinematics duality, possess an underlying algebraic structure known as
BV${}^{\color{gray} \blacksquare}$-algebra. If, additionally, matter fields are
present, this structure is supplemented by a module for the BV${}^{\color{gray}
\blacksquare}$-algebra. We explain this perspective, expanding on our previous
work and providing many additional mathematical details. We also show how the
tensor product of two metric BV${}^{\color{gray} \blacksquare}$-algebras yields
the action of a new syngamy field theory, a construction which comprises the
familiar double copy construction. As examples, we discuss various scalar field
theories, Chern-Simons theory, self-dual Yang-Mills theory, and the pure spinor
formulations of both M2-brane models and supersymmetric Yang-Mills theory. The
latter leads to a new cubic pure spinor action for ten-dimensional
supergravity. We also give a homotopy-algebraic perspective on
colour-flavour-stripping, obtain a new restricted tensor product over a wide
class of bialgebras, and we show that any field theory (even one without
colour-kinematics duality) comes with a kinematic $L_\infty$-algebra.
