  We study celestial chiral algebras appearing in celestial holography, using
the light-cone gauge formulation of self-dual Yang-Mills theory and self-dual
gravity, and explore also a deformation of the latter. The recently discussed
$w_{1+\infty}$ algebra in self-dual gravity arises from the soft expansion of
an area-preserving diffeomorphism algebra, which plays the role of the
kinematic algebra in the colour-kinematics duality and the double copy relation
between the self-dual theories. The $W_{1+\infty}$ deformation of
$w_{1+\infty}$ arises from a Moyal deformation of self-dual gravity. This
theory is interpreted as a constrained chiral higher-spin gravity, where the
field is a tower of higher-spin components fully constrained by the graviton
component. In all these theories, the chiral structure of the operator-product
expansion exhibits the colour-kinematics duality: the implicit `left algebra'
is the self-dual kinematic algebra, while the `right algebra' provides the
structure constants of the operator-product expansion, ensuring its
associativity at tree level. In a scattering amplitudes version of the Ward
conjecture, the left algebra ensures the classical integrability of this type
of theories. In particular, it enforces the vanishing of the tree-level
amplitudes via the double copy.
