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

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.