Simplicial principal bundles in parametrized spaces

Abstract

In this paper we study the classifying theory of principal bundles in the parametrized setting, motivated by recent interest in higher gauge theory. Using simplicial techniques, we construct a product-preserving classifying space functor for groups in the category of spaces over a fixed space B. Additionally, we prove that the fiberwise geometric realization functor sends a large class of simplicial parametrized principal bundles to ordinary parametrized principal bundles. As an application we show that the fiberwise geometric realization of the universal simplicial principal bundle for a simplicial group G in the category of spaces over B gives rise to a parametrized principal bundle with structure group |G|.

v2 Corrected errors in discussion of latching objects; re-organized sections 4,5 & 6; 30 pages; uses xy-pic. v3 fixes some technical LaTeX issues preventing arrows from rendering. v4 Final version, to appear in New York J. Math

References in corpus (7)

• Higher Topos Theory
• Higher gauge theory I: 2-Bundles
• Homotopy Theory of Orbispaces
• On the construction of functorial factorizations for model categories
• The inner automorphism 3-group of a strict 2-group
• Classifying theory for simplicial parametrized groups
• The universal simplicial bundle is a simplicial group

Cited by in corpus (6)

• Principal infinity-bundles - Presentations
• Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation
• The character map in (twisted differential) non-abelian cohomology
• Classifying theory for simplicial parametrized groups
• The universal simplicial bundle is a simplicial group
• Homotopical Algebra in Categories with Enough Projectives