Fibred categories as foundations

Fibred categories as foundations

[Colin McLarty]

Many people have said that fibred categories are more elementary than
indexed categories and achieve what indexed categories do but without
using Grothendieck universes.

I understand how they are more elementary, and I quite like Benabou's
JSL article.  But I do not see how fibred categories achieve what is
done with a universe.

To give an example, a universe U lets me talk about the topos of all
U-sheaves on a given U-small topological space T.  That topos is
indexed over U so I can work with U limits and colimits in the topos
-- while applying all the tools of ZFC inside U, since U models ZFC.

Can I get that effect using fibred categories, without also using a universe U?

Or is the claim only meant to say there is a good elementary theory of
fibred categories, which can then be applied in the context of a set
theory with a universe when we want to use it as foundation for the
SGA or the like?

best, Colin


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