Even more ancient: Parallel transport in fibre spaces," Bol. Soc. Mat. Mexicana (1968), 68-86. Unfortunately that's a hard paper to get a hold of somewhat related is Associated fibre spaces," Michigan Math. Journal 15 (1968), 457-470. and at the survey level H-spaces and classifying spaces, I-IV", AMS Proc. Symp. Pure Math. 22 (1971), 247-272. Of course, as you might expect, I describe things in terms of A_\infty-morphisms from the space of loops into Aut(F) of homotopy equivalences of $F$ Now that some of us are comfortable with A_\infty-cats, categborification should proceed perhaps with some technical details. jim Peter May wrote:
In his posting today, John Baez advertised the slogan:
FIBRATIONS OVER THE BASE SPACE B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F.
He hedged it with a ``dose of vagueness'', but in fact I proved a completely precise and general version of exactly this result in ``Classifying spaces and fibrations'', Memoirs AMS 155, Jan. 1975. Using Moore loops on B, LB, one has a topological monoid, and one also has the topological monoid Aut(F) of homotopy equivalences of $F$. A ``transport'' is a homomorphism of topological monoids from LB to Aut(F). Allowing F to vary by a homotopy equivalence, one can define an equivalence relation on transports such that the equivalence classes are in natural bijective correspondence with the equivalence classes of `fibrations over the base space B with fiber F'. One can generalize the context by allowing fibers in some nice category and prove the same result. See opus cit, Theorem 14.2, page 83. That was over 30 years ago, so naturally I wasn't thinking about categorification, but I would imagine that the methods categorify.