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I’ve started cleaning up and adding stuff at fibration in a 2-category, but it’s bedtime now, so I’ll finish it tomorrow.
Okay, ’I’ll finish it tomorrow’ was a bit ambitious, but I’m now finished with fibration in a 2-category for the moment.
There’s a bit of ambiguity at fibration in a 2-category – are we talking about strict Grothendieck fibrations or weak Street fibrations? I believe all the same facts are true in either case, as long as we make the correct choices about what to interpret strictly or weakly in each case. Since “2-category” means “weak 2-category” by default on the nLab, in which case the Street version is the only sensible one, it might make sense to focus on that version – although we would then have to clarify that fibrations in the 2-category Cat are not actually Grothendieck fibrations but Street ones.
I tried to clarify the difference a bit in the “Idea” and “Definition” sections, but I still need to look at the “Details” section.
That’s a good point, which I didn’t think about when I added the stuff from Fibrations in bicategories. It’s definitely worth clarifying. I like what you’ve added so far.
I modified the first lemma on the page to work with Street fibrations and non-strict slice 2-categories, but now I have to go. However, I’m not actually sure where that lemma should go – it’s not really about fibrations in a 2-category, but is just an alternate characterization of ordinary fibrations in Cat. So maybe it should go at Grothendieck fibration and/or Street fibration (in the two versions) – or maybe at cleavage? (By the way, I edited cleavage some too, and in particular I added the version for Street fibrations.)
I put the strict version of the lemma at Grothendieck fibration and the non-strict version at Street fibration.
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