Cocycle condition on bundles as some equivalence relation?

Question

The three cocycle conditions on the transition maps of a fiber bundle look alot like the three condition defining equivalence relation - refelxivity, symmetry, and transitivity. I was wondering whether the cocycle condition(s) actually do come from some equivalence relation, and if so, which one?

Answer 1

Too long for a comment, but also not really an answer.

These conditions are more about the question of when partially defined vector bundles can be glued together to form a vector bundle on a larger space, so "compatibility" rather than "equivalence". The moral of these conditions is that if you stay in the intersection of a bunch of opens above which we have defined bundles, and run around on some loop of those bundles by transition functions, when you return home everything is the way you expected it to be (the function you have changed by is $\phi_{ii} = id$). Thus, locally, your glued construction looks the same as one of these partial constructions - which is good, because checking that something is a fiber bundle can be done locally once you have a global construction.

So to summarize - you can think of them like this:

1) What happens if I stay where I am? $(\phi_{ii} = 1)$

2) What happens if I hop over to a neighbor and then come right back? ($\phi_{ij} = \phi{ji}^{-1}$)

3) What happens if I go on a long journey and then come back? ($\phi_{ij} \phi_{jk} = \phi_{ik}$)

The same conditions and moral applies in many situations: gluing manifolds, gluing schemes, gluing sheaves, abstract descent data.