  In this paper, we first construct the controlling algebras of embedding
tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and
an $L_\infty$-algebra respectively. Then we introduce representations and
cohomologies of embedding tensors and Lie-Leibniz triples, and show that there
is a long exact sequence connecting various cohomologies. As applications, we
classify infinitesimal deformations and central extensions using the second
cohomology groups. Finally, we introduce the notion of a homotopy embedding
tensor which will induce a Leibniz$_\infty$-algebra. We realize Kotov and
Strobl's construction of an $L_\infty$-algebra from an embedding tensor, to a
functor from the category of homotopy embedding tensors to that of
Leibniz$_\infty$-algebras, and a functor further to that of
$L_\infty$-algebras.
