nLab fibration theory

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.

The axioms

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.

1. 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.

2. 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)

3. (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!