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Strict pretopologies on 2-categories and anafunctorshttps://nforum.ncatlab.org/discussion/2784/2011-06-03T02:42:31+00:002012-10-04T05:09:12+00:00DavidRobertshttps://nforum.ncatlab.org/account/42/
Hi all. One thing I am thinking about is a 'synthetic' approach to anafunctors. Makkai, in his paper, takes a very elementary approach, using elements at will. In my paper (and in Toby's too), taking ...
Hi all. One thing I am thinking about is a 'synthetic' approach to anafunctors. Makkai, in his paper, takes a very elementary approach, using elements at will. In my paper (and in Toby's too), taking the internal road, I can't do that so blithely, but it essentially treats the theory of anafunctors as a special sort of sketch (not too dissimilar perhaps to FOLDS), but not from that point of view. However, I don't supply all proofs, neither does Toby and for that matter neither does Makkai (a fair bit of 'reader can supply the details'). I think the easiest way to see what is actually going on is to think of Cat(S) as a 2-category K (with cotensors with 2 = {0 --> 1}) equipped with a special class of 1-arrows viz a strict singleton subcanonical Grothendieck pretopology J (essentially just a pretopology on the underlying 1-category). This needs to satisfy an additional condition (**) which is 2-categorical: given fg = h where h,f in J, then g has an anafunctor pseudo-section (this condition is satisfied in a very natural way in K=Cat(S), and I would like to see if it follows from existing assumptions in general). Then one can think about spans where the left leg is a cover as 1-arrows in a bicategory K_ana, and the usual thing for 2-arrows (take the pullback of covers etc). There should be a canonical inclusion K --> K_ana, and we could even talk about the [[proarrow equipment]] that arises.

This is a different approach - more restrictive, I believe - than Mike's work at [[michaelshulman:exact completion of a 2-category]], where he talks about anafunctors in a 2-category (weak, by default, for him). There he talks about anafunctors in the 2-category of 2-congruences in a 2-site, and thinking of them more along the lines as in Cat(S). But I'm very interested in the relation between the two, especially if one could be derived from the other.

One spin-off of this is that I would like to provide another model for the localisation of a 2-category. Here J needs to be weakly cofinal in the class W one wants to invert. One point of my anafunctors paper was to show that the localisation of a 2-category of internal categories had a better model that the default one constructed by Pronk, and this theorem should go through, namely K[W^-1] ~ K_ana. Note that this is (2,2)-category localisation, not (2,1)-category localisation. (As an aside, the approach to localisation via bibundles, which is even simpler to describe, wouldn't work here because that assumes one is in a (2,1)-category.)

The one point which is a bit restrictive is that one needs covers to be an [[ff morphism]] in order to define the bicategory K_ana of anafunctors in K. (This reminds me somewhat of talking about S-local maps in a model theoretic setup, at least when the pretopology J is morally like a cover by open balls or affine schemes. But I haven't thought about this too much yet.)

One direction this may go is if the whole game can be phrased in a suitably 2-categorical way, then perhaps similar techniques could be used to talk about localisation of higher categories (say simplicial categories), at least in special cases. For example, defining weak maps between strict higher categories or something. This is complete speculation, and not a short-term goal by any means.