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 -bundles.
How can be an identity functor when that is not contravariant? Maybe each is a covariant functor but the mapping is a contravariant functor? But then it's automatic that (and furthermore that ). —Toby
David Roberts: whoops! I didn’t pick that up. I think you are partly right: it should be some sort of contravariant assignment , 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 of a space and a system of objects (morphisms) over each such that and agree over , then there is a unique common extension of the over
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 is a morphism in such that each restriction for a numerable open cover of is a homotopy equivalence, then is a homotopy equivalence. If , then the restrictions are homotopy equivalent (for objects) or homotopic (for morphisms)
(Mapping cylinder axiom) If is a homotopy equivalence, then there is an object which serves as a mapping cylinder for . That is, restricts to at and to at with a characterising homotopy equivalence which restricts to , respectively .
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 is a category of fibrant objects or a Quillen model category or -category or a category with an interval objects or something. One could even ask for a subcategory of which is closed under some conditions. In that instance, something needs to be said about the compatibility of homotopies etc with the functors .
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!