2006/09/07 by James F. Wirth, Wirth, James, Jim Stasheff +1
Mathematics · #55P65 #55P99 #55R05 #55R10 #55R15 #55R35 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.math/0609220
openalex publication_date 2006/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For locally homotopy trivial fibrations, one can define transition functions g\dab : U\da∩ U\db → H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle condition, one obtains only that g\dab g\dbgam is homotopic to g\dagam as a map of U\da∩ U\db∩ U\dgam into H. Moreover on multiple intersections, higher homotopies arise and are relevant to classifying the fibration. The full theory was worked out by the first author in his 1965 Notre Dame thesis \citewirth:diss. Here we present it using language that has been developed in the interim. We also show how this points a direction `on beyond gerbes'.