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Motivated by recent developments of $\infty$-categorical theories related todifferential graded (dg for short) Lie algebras, we develop a general frameworkfor 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 ofVallette. As a homotopy theoretical generalization of Loday and Pirashvili'sLie algebra objects in the tensor category of linear maps, we further studyweak Loday-Pirashvili modules consisting of $\infty$-morphisms from locallyfinite $\infty$-$\mathfrak{g}$-modules to the adjoint module $\mathfrak{g}$.From the category of such weak Loday-Pirashvili modules over $\mathfrak{g}$, wefind a functor that maps to the category of Leibniz$_\infty$ algebras enrichedover the Chevalley-Eilenberg dg algebra of $\mathfrak{g}$. This functor can beregarded as the homotopy lifting of Loday and Pirashvili's original method torealize Leibniz algebras from Lie algebra objects in the category of linearmaps.

Locally finite infinity-modules and weak Loday-Pirashvili modules over differential graded Lie algebras