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We describe the moduli space of extensions in the model category ofsimplicial presheaves. This article can be seen as a generalization ofBlomgren-Chacholski results in the case of simplicial sets. Our description ofthe moduli space of extensions treat the equivariant and the nonequivariantcase in the same setting. As a new result, we describe the moduli space ofM-bundles over a fixed space X, when M is a simplicial monoid. Moreover, themoduli space of M-bundles is classified by the classifying space of thesimplicial submonoid generated by homotopy invertible elements of M. We give ageneral interpretation of generalized cohomology theories (connective) in termsof classification of principle bundles. We also construct categorical model forthe classifying space BG and EG when G is a simplicial (topological) monoidgroup like.

Moduli space of fibrations in the category of simplicial presheaves