  Motivated by recent developments of $\infty$-categorical theories related to
differential graded (dg for short) Lie algebras, we develop a general framework
for locally finite $\infty$-$\mathfrak{g}$-modules over a dg Lie algebra
$\mathfrak{g}$. We show that the category of such locally finite
$\infty$-$\mathfrak{g}$-modules is almost a model category in the sense of
Vallette. As a homotopy theoretical generalization of Loday and Pirashvili's
Lie algebra objects in the tensor category of linear maps, we further study
weak Loday-Pirashvili modules consisting of $\infty$-morphisms from locally
finite $\infty$-$\mathfrak{g}$-modules to the adjoint module $\mathfrak{g}$.
From the category of such weak Loday-Pirashvili modules over $\mathfrak{g}$, we
find a functor that maps to the category of Leibniz$_\infty$ algebras enriched
over the Chevalley-Eilenberg dg algebra of $\mathfrak{g}$. This functor can be
regarded as the homotopy lifting of Loday and Pirashvili's original method to
realize Leibniz algebras from Lie algebra objects in the category of linear
maps.
