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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]