I add one more example: for any sort of algebraic/topological/differentiable stack one cares to use, they are representable by internal groupoids and maps between such stacks are given by anafunctors, and they are nice to work with geometrically. And although I haven't looked into it, I'm sure anafunctors work in the enriched context as well. Best, David On 23 January 2011 08:31, David Roberts <droberts@maths.adelaide.edu.au> wrote:

Hi Jean,

I reply only to your point 3, which I reproduce:

...

3- You also say:

"An anafunctor is really a simple thing: a morphism in the bicategory of fractions obtained from Cat by inverting the functors which are fully faithful and essentially surjective".

Woaoo, you call this a simple thing! Ordinary categories of fractions are very complicated, unless you have a calculus of right (or left) fractions. Is there, precisely defined, and without neglecting the coherence of canonical isomorphisms, such a "calculus" defined. Does it apply to the "simple thing" of anafunctors.

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