Homotopy Transition Cocycles

James Wirth and Jim Stasheff

For locally homotopy trivial fibrations, one can define transition functions \[ g\dab : U\da\cap U\db \to H = H(F) \] 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 $U\da\cap U\db\cap U\dgam$ 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 \cite{wirth: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'.

Journal of Homotopy and Related Structures, Vol. 1(2006), No. 1, pp. 273-283