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We study several homotopical and geometric properties of Maurer–Cartan spaces for \(L_{\infty }\)-algebras which are not nilpotent, but only filtered in a suitable way. Such algebras play a key role especially in the deformation theory of algebraic structures. In particular, we prove that the Maurer–Cartan simplicial set preserves fibrations and quasi-isomorphisms. Then we present an algebraic geometry viewpoint on Maurer–Cartan moduli sets, and we compute the tangent complex of the associated algebraic stack.

Maurer–Cartan spaces of filtered $$L_{\infty }$$ L ∞ -algebras