cleavages and choice

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

Dear Jean,

of course, you are right when emphasizing that one need choice for
classes to endow an "anonymous" fibration with a cleavage.
But that applies also to catgeories with say binary products. One
needs choice for classes in order to choose a product cone for every
pair of objects.
In many instances, however, categories come together with a choice of
products and fibrations come together with a choice of a cleavage.

For example Set comes with a choice of a cleavage. Fibrations arising
from internal categories are even split. Many constructions on fibrations
allow one to choose a cleavage given cleavages for the arguments.
Do you know of any construction on fibrations which is not "cleavage
preserving" in this sense?

Of course, one should not require cartesian functors to preserve
cleavages just as one should not require functors to preserve chosen
products.

Thomas


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