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• CommentRowNumber1.
• CommentAuthorGuest
• CommentTimeJan 8th 2010
Comment at codomain fibration about the suggested categorification, Cat^2 --> Cat. I personally don't think we've got to the bottom of what a 2-fibration is, with the possible exception of Igor Bakovic.

David Roberts
• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJan 8th 2010

Replied.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeJan 8th 2010

The answer is correct, my original statement was placed there in haste. When I studied Hermida's paper, however I was having in mind another construction which I will soon explain in nlab. Namely, one can indeed start with ANY 1-fibration above a cartesian category C, and induce an indexed 2-category over Cat(C) in a canonical way, which I discussed 4 years ago with Jibladze and Hermida. However the ad hoc construction gives something what is just half a 2-fibration, and one needs a Grothendieck construction to get the correct answer. The hint how to do the induced fibratuon to start with is an idea in MacLane-Pare paper from 1980s, I think JPAA, there is however an alternative construction which I worked out once and it gives the same result only after the Grothendieck construction. Roughlz speaking, above internal category in C one takes an appropriate version of internal category in indexed category in MacLane Pare's sense and in my recipe one takes an internal category above in usual sense but requires that the structure maps are made out of cartesian arrows. This is rigid and needs a cofibrant replacement, this is why Grothendieck construction for 2-indexed categories is needed. Nut 2 out of 4 Hermida's universality properties hold in my case even before taking cofibrant replacement.