The notion of fibration theory was created by James Wirth in his PhD thesis in 1965. It allows classification of what would now be called $\infty$-bundles.
A fibration theory $E$ is an assignment of a category $E(B)$ to each topological space $B$ and a contravariant functor $f^*:E(C) \to E(B)$ to each continuous map $f:B \to C$ such that $id^*$ is the identity functor. $E$ is required to satisfy the following
How can $id^*$ be an identity functor when that is not contravariant? Maybe each $f^*$ is a covariant functor but the mapping $f \mapsto f^*$ is a contravariant functor? But then it's automatic that $id^* = id$ (and furthermore that $(f ; g)^* = f^* \circ g^*$). —Toby
David Roberts: whoops! I didn’t pick that up. I think you are partly right: it should be some sort of contravariant assignment $f\mapsto f^*$, but maybe not functorial (since I believe that category should be replaced as I said below). The protoypical example, AFAICS, is assigning the category of locally homotopy trival fibrations over the given space. It is not spelled out in detail in the paper.
For a numerable open cover $U = \coprod U_i$ of a space $B$ and a system of objects (morphisms) $E_i$ over each $U_i$ such that $E_i$ and $E_j$ agree over $U_i\cap U_j$, then there is a unique common extension of the $E_i$ over $B$
What makes an open cover ‘numerable’? —Toby
David Roberts: A cover is numerable if it admits a subordinate partition of unity. Numerable open covers form a site. The axiom is there to link locally homotopically trivial fibrations and Dold fibrations (see theorem 2.3 in Wirth-Stasheff, due to Dold.)
Also, the uniqueness should at least be demoted to unique-up-to-isomorphism.
If $\phi$ is a morphism in $E(B)$ such that each restriction $\phi|_{U_i}$ for a numerable open cover $U$ of $B$ is a homotopy equivalence, then $\phi$ is a homotopy equivalence. If $H\in E(I\times B)$, then the restrictions $H|_{\{t\}\times B}$ are homotopy equivalent (for objects) or homotopic (for morphisms)
(Mapping cylinder axiom) If $\phi:F \to F' \in E(B)$ is a homotopy equivalence, then there is an object $M(\phi)\in E(I\times B)$ which serves as a mapping cylinder for $\phi$. That is, $M(\phi)$ restricts to $F$ at $t=0$ and to $F'$ at $t=1$ with a characterising homotopy equivalence $\psi_M:M(\phi) \to I\times F'$ which restricts to $\{0\}\times\phi$, respectively $\{1\}\times id$.
David Roberts: The axioms are just copied from Wirth–Stasheff JHRS 1(1) 2006, p 273. They need to be clarified a little, as the notion of homotopy and homotopic are undefined. We could ask that $E(B)$ is a category of fibrant objects or a Quillen model category or $(\infty,1)$-category or a category with an interval objects or something. One could even ask for a subcategory of $Top$ which is closed under some conditions. In that instance, something needs to be said about the compatibility of homotopies etc with the functors $f^*$.
Toby: I know that you're just copying things, so maybe you don't know the answers to my questions, but so far I don't even understand the parts that I should be able to understand!