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It appears to me that the two definitions of discrete double fibration are equivalent: Lambert studies a notion of fibration equivalent to lax functors into the double category of sets and spans, whereas Fröhlich and Moser study a notion of fibration equivalent to normal lax functors into the double category of categories and distributors. However, it follows from Proposition 5.14 of CS10 that these are equivalent.
Yes, I think that’s right. But I thought the two definitions of non-discrete double fibration were not equivalent?
Are there multiple definitions of non-discrete double fibrations? I thought the only place a definition was proposed was in Double Fibrations.
Perhaps not. Maybe I misunderstood the comments in the Fröhlich-Moser paper about their approach being different.
My understanding of that remark is that the authors do believe their approach to be fundamentally different from that of Lambert (because their fibrations are valued in categories rather than sets) and of Cruttwell–Lambert–Pronk–Szyld (because they describe discrete fibrations rather than arbitrary fibrations). However, I think the universal property of Mod provides the connection they were missing to tie things together. (I sent Fröhlich and Moser an email to clarify, though have not yet received a response.)
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