Moduli space of fibrations in the category of simplicial presheaves

Abstract: We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of extensions treat the equivariant and the nonequivariant case in the same setting. As a new result, we describe the moduli space of M-bundles over a fixed space X, when M is a simplicial monoid. Moreover, the moduli space of M-bundles is classified by the classifying space of the simplicial submonoid generated by homotopy invertible elements of M. We give a general interpretation of generalized cohomology theories (connective) in terms of classification of principle bundles. We also construct categorical model for the classifying space BG and EG when G is a simplicial (topological) monoid group like.