  In order to solve two problems in deformation theory, we establish natural
structures of homotopy Lie algebras and of homotopy associative algebras on
tensor products of algebras of different types and on mapping spaces between
coalgebras and algebras. When considering tensor products, such algebraic
structures extend the Lie algebra or associative algebra structures that can be
obtained by means of the Manin products of operads. These new homotopy algebra
structures are proven by to compatible with the concepts of homotopy theory:
$\infty$-morphisms and the Homotopy Transfer Theorem. We give a conceptual
interpretation of their Maurer-Cartan elements. In the end, this allows us to
construct the deformation complex for morphisms of algebras over an operad and
to represent the deformation $\infty$-groupoid for differential graded Lie
algebras.
