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created 2-site with the material from Mike’s web (as he suggested). Added pointers to original articles by Ross Street.
…and split it off into a separate 2-sheaf, similarly
There were few errors in the original Street’s paper, there should be note on them. I do not remember where they were corrected exactly.
the above was triggered by me hyperlinking Igor’s upcoming talk.
If anyone feels like expanding the 2-topos part of the nLab, this is now a good occasion…
Of course, Igor knows about the error remarks, but I forgot what he told me.
The main error that I know about is remarked on after the proof of this theorem.
It would be awesome if we could use the nLab to merge the directions that all of the 2-topos people have been thinking. If Igor is interested in contributing that would great. Unfortunately my time is somewhat limited this summer, but I’ll do what I can.
Oh, and thanks for starting to move stuff from my web! A lot of what’s there, like regular 2-category (michaelshulman), could be on the main lab; I put it on my web when I was less sure of it, and less confident about putting newish stuff on the main nLab, than I am now. I’ll start doing some more, when I get a chance, but feel free to continue. I would ask that you not move any of the stuff about logic or fibration classifiers, I think – that’s in a much more fragmentary state at the moment. But all the stuff about regular/coherent 2-categories, factorization systems, and Grothendieck 2-topoi probably deserves to be on the main nLab.
One thing that could use discussing is the terminology for the morphisms occurring. I had been using “ff” and “eso” since that’s what Street uses, but in discussion later with David Rydh I was convinced that it would be better to call these “1-monic” and “strong 1-epic” respectively, with “faithful” and “eso+full” becoming “2-monic” and “strong 2-epic.” I hope this is in line with our numbering schemes at subcategory and elsewhere. Thoughts?
Thoughts?
Assuming that it is in line with the numbering schemes at subcategory (which is a technical question: is a $1$-monic morphism in a $2$-category that happens to be a $1$-category precisely the same thing as a monic morhpism in a category?, etc), then I agree. The numbered terms are much more suggestive.
Oh, and thanks for starting to move stuff from my web!
The pleasure is all mine.
feel free to continue.
I would enjoy to, but I don’t see myself finding the time, soon. I am insanely busy with something else. Or should be. Every now and then I need a distraction of course. What I did here was to supply decent hyperlinking to our little workshop’s little website and it seemed inappropriate to point to a personal web for topics that are under general consideration, if you see what I mean.
But Igor should really get involved here. Maybe I can arrange that he has to work off his travel refunds by $n$Lab labor ;-)
The numbering of these things is like whack-a-mole; it seems impossible to get everything to come out ideally. With the above proposal, a 1-monic morphism in a 2-category that happens to be a 1-category is the same as a monic morphism in a category, and a strong 1-epic morphism in such a 2-category is the same as a strong epic morphism in a category, but a strong 1-epic in C is not necessarily “1-epic” in the sense of being 1-monic in $C^{op}$; all you can say is that it’s 2-monic (a weaker notion) in $C^{op}$. David Rydh and I felt that it was more important for the numbers in “1-monic” and “strong 1-epic” to match up, since those are the two most important ones in 2-topos theory and form a factorization system, and similarly we also have a (2-monic, strong 2-epic) factorization system. Then you can make it true that “strong k-epics are k-epic” if you define a k-epic in an n-category as an (n-k)-monic in the opposite n-category, which I think isn’t so bad, except that it’s not stable under regarding n-categories as (n+1)-categories. However, we’re familiar with the idea that “plain epics” aren’t really well-behaved, and the good and more important notion of “strong k-epic” is stable under that operation.
By the way, I should say that David Rydh and I have been sort of planning to write a “background info paper” on 2-exactness properties and 2-toposes, in particular fixing the error(s?) in and expanding on Street’s paper. Nothing has happened on it for a long time, for the usual reasons. But it seems like something of the sort is really needed, and having good nLab pages on this stuff would also go a long way.
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