  The notion of Poisson dialgebras was introduced by Loday. In this article, we
propose a new definition with some modifications that is supported by several
canonical examples coming from Poisson algebra modules, averaging operators on
Poisson algebras, and differential Poisson algebras. We show that a Poisson
object in the category of linear maps has an associated Poisson dialgebra
structure. Conversely, starting from a Poisson dialgebra we describe a Poisson
object in the category of linear maps. These constructions yield a pair of
adjoint functors between the category of Poisson objects in the category of
linear maps and the category of Poisson dialgebras. There is a Lie $2$-algebra
associated with any Leibniz algebra. Here, we first obtain an associative
$2$-algebra starting from a dialgebra. Then, for a Poisson dialgebra, we
construct a graded space that inherits both a Lie $2$-algebra and an
associative $2$-algebra structure. In a particular case of Poisson dialgebras,
which we call `reduced Poisson dialgebra', we obtain an associated $2$-term
homotopy Poisson algebra (of degree $0$).
