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We generalize the notion of semi-universality in the classical deformationproblems to the context of derived deformation theories. A criterion for aformal moduli problem to be semi-prorepresentable is produced. This can be seenas an analogue of Schlessinger's conditions for a functor of Artinian rings tohave a semi-universal element. We also give a sufficient condition for asemi-prorepresentable formal moduli problem to admit a $G$-equivariantstructure in a sense specified below, where $G$ is a linearly reductive group.Finally, by making use of these criteria, we derive many classical resultsincluding the existence of ($G$-equivariant) formal semi-universal deformationsof algebraic schemes and that of complex compact manifolds.

Semi-prorepresentability of formal moduli problems and equivariant structures