I’m (slowly) working on writing an introduction for chapter 1 of my thesis so it can go on the arXiv as a standalone paper, then for submission to TAC. (The working title is ’Internal anafunctors I’, but if this seems rather drab or a turn-off, I’ll work through some more options - I’m open to suggestions). I know why *I* care about internal anafunctors, but I know there are other points of view, and I want to at least touch on them to catch peoples’ interest. Here are some examples, sometimes only talking about special cases.

anafunctors correspond to:

- maps between stacks on a large site, themselves represented by internal groupoids in the site
- maps between presheaf categories (I think - this one I need help on)
- localisation of the 2-category of internal categories at weak equivalences
- presentation of maps in the homotopy category of internal categories
- something like internal profunctors (saturated internal anafunctors $C\to D$ are internal discrete opfibrations on $Core(C\times D)$)
- bimodules/bibundles of internal groupoids
- (hence) Hilsum-Skandalis morphisms between Lie groupoids
- good models for maps between orbifolds
- a good replacement for a functor in categories without choice (especially with reference to the internal logic of a catgory)

Is this a decent enough list to catch the eye of the casual category theorist? Can anyone think of any more?

There is more material on internal anafunctors floating around my notes and in my head, at least for an IA2, and I hope to eventually get it out in the public eye, but this needs doing “now” so it can be reference by someone’s upcoming paper on butterflies.

I think I’d also like (and this is directed at Cafe hosts) to talk about anafunctors in a post on the cafe, perhaps covering some of the above examples - if I can garner a few more examples/applications that way I’d be happy.

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