For any dg algebra A we construct a closed model category structure on dg A-modules such that the corresponding homotopy category is compactly generated by dg A-modules that are finitely generated and free over A (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a so-called extended bar construction of A. This generalises and complements certain aspects of dg Koszul duality for associative algebras.

We introduce a notion of left homotopy for Maurer-Cartan elements in L∞algebras and A∞-algebras, and show that it corresponds to gauge equivalence in the differential graded case. From this we deduce a short formula for gauge equivalence, and provide an entirely homotopical proof to Schlessinger-Stasheff's theorem. As an application, we answer a question of T. Voronov, proving a non-abelian Poincaré lemma for differential forms taking values in an L∞-algebra.

In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements. The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential. Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the resulting differential graded Lie or associative algebras. We also recall the relation between the cohomologies and the differential graded Lie and associative algebras that control the deformations.

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We introduce a notion of left homotopy for Maurer-Cartan elements in L∞-algebras and A∞-algebras, and show that it corresponds to gauge equivalence in the differential graded case. From this we deduce a short formula for gauge equivalence, and provide an entirely homotopical proof to Schlessinger-Stasheff's theorem. As an application, we answer a question of T. Voronov, proving a non-abelian Poincaré lemma for differential forms taking values in an L∞-algebra.