Moduli space of fibrations in the category of simplicial presheaves

TL;DR

This paper characterizes the moduli space of fibrations within the category of simplicial presheaves, extending prior results and providing new classifications for M-bundles and their relation to generalized cohomology theories.

Contribution

It generalizes existing results on moduli spaces to simplicial presheaves and introduces a classification of M-bundles via classifying spaces of submonoids.

Findings

Describes the moduli space of extensions in simplicial presheaves.

Classifies M-bundles over a fixed space using classifying spaces of submonoids.

Provides a categorical model for classifying spaces BG and EG for simplicial monoid groups.

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.