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In this paper, we give a detailed introduction to the theory of (curved) L ∞ -algebras and L ∞ -morphisms, avoiding the concept of operads and providing explicit formulas. In particular, we recall the notion of (curved) Maurer–Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of L ∞ -algebras and L ∞ -modules. In particular, one can interpret L ∞ -morphisms and morphisms of L ∞ -modules as Maurer–Cartan elements in certain L ∞ -algebras, and we show that twisting the morphisms with equivalent Maurer–Cartan elements yields homotopic morphisms. We hope that these notes provide an accessible entry point to the theory of L ∞ -algebras. [ABSTRACT FROM AUTHOR]

The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative. We prove that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the statement by proving a more general homotopy equivalence between L∞-morphisms that are twisted with gauge equivalent Maurer-Cartan elements. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we propose a reduction scheme for multivector fields phrased in terms of L∞-morphisms. Using well-known geometric properties of the reduced manifolds, we perform a Taylor expansion of multivector fields, which allows us to build up a suitable deformation retract of differential graded Lie algebras (DGLAs). We first obtained an explicit formula for the L∞- projection and -inclusion of generic DGLA retracts.We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction L∞-morphism. Finally, we perform a comparison with other reduction procedures. [ABSTRACT FROM AUTHOR]