Some time ago I wrote notes constructing in a purely 2-categorical language a bicategory of anafunctors, starting from a 2-category $K$ equipped with a notion of cover, which in the original setting that Makkai studied, reduces to a trivial isofibration of categories. I always wanted to do something more with the construction, and I still do, but I thought it worthwhile to get the notes into a shape suitable for public consumption (at one point I had changed notational convention, and I found this week that the transition was half-way through a diagram!). So here they are:
Abstract: These notes give an elementary and formal 2-categorical construction of the bicategory of anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful.
As always, comments welcome.
Added 21 August: In my paper Internal Categories, Anafunctors and Localisation (ICAL), published in TAC in 2012, I mentioned a ‘sequel’, which I then called Strict 2-sites, J-spans and localisations, in obvious parallel with ICAL. The notes above are essentially that sequel, but I have more ideas I want to add to the above before it is is finalised and I submit it for publication.