Hi Theo. This is something I have been working on lately. In Makkai's original paper on anafunctors he defines a condition on an anafunctor which makes it saturated. His motivations are logical, in that he wants the 'image' of a point in the domain category (pullback and pushdown along the span) to be closed under isomorphism. This is satisfied for his examples of (co)limit anafunctors, or an anafunctor arising from a universal property. In the case that he talks about saturated anafunctors between groupoids, this is the same as a right principal bibundle (in Set). When we replace groupoids by categories, then he defines a saturated anafunctor to be an anafunctor such that the underlying span between the cores is saturated. Thus the 'cover' (in Makkai's case, a surjection of sets) is a right principal bibundle for the underlying groupoids.

This is done so that the canonical 2-functor $Cat \to Cat_{sat.ana}$, where we do not assume choice, sends fully faithful, essentially surjective functors to equivalences in the bicategory $Cat_{ana}$. Actually I'm fudging here, because Makkai defines $Cat_{sat.ana}$ as a an anabicategory - a category weakly enriched over $Cat_{ana}$.

Note also that Street at one point defines in his Oberwolfach descent notes a definition of a 'torsor for a category' which is just the same as Makkai's definition. He correctly notes that there are non-invertible maps between such 'torsors', unlike the group(oid) case.

So to cut a long story short, it is possible to define a saturated anafunctor for internal groupoids and hence categories. (email me if you would like some notes on this)

But! This is not the same as a torsor for an internal category, as one could extract or otherwise from various places, e.g. Moerdijk's or Johnstone. See this answer. The two definitions are aiming at very different things. For example, a saturated anafunctor with codomain a topological monoid (as a one-object category) is trivial, but a torsor for the same is not. In general internal saturated anafunctors are about inverting weak equivalences between internal categories (fully faithful and essentially 'surjective': i.e. some map admitting local sections) but internal torsors are about characterising maps between topoi and classifying topoi and stuff (you can tell I know less about the latter).

Hi Theo. This is something I have been working on lately. In Makkai's original paper on anafunctors he defines a condition on an anafunctor which makes it saturated. His motivations are logical, in that he wants the 'image' of a point in the domain category (pullback and pushdown along the span) to be closed under isomorphism. This is satisfied for his examples of (co)limit anafunctors, or an anafunctor arising from a universal property. In the case that he talks about saturated anafunctors between groupoids, this is the same as a right principal bibundle (in Set). When we replace groupoids by categories, then he defines a saturated anafunctor to be an anafunctor such that the underlying span between the cores is saturated. Thus the 'cover' (in Makkai's case, a surjection of sets) is a right principal bibundle for the underlying groupoids.

This is done so that the canonical 2-functor $Cat \to Cat_{sat.ana}$, where we do not assume choice, sends fully faithful, essentially surjective functors to equivalences in the bicategory $Cat_{ana}$. Actually I'm fudging here, because Makkai defines $Cat_{sat.ana}$ as a an anabicategory - a category weakly enriched over $Cat_{ana}$.

Note also that Street at one point defines in his Oberwolfach descent notes a definition of a 'torsor for a category' which is just the same as Makkai's definition. He correctly notes that there are non-invertible maps between such 'torsors', unlike the group(oid) case.

So to cut a long story short, it is possible to define a saturated anafunctor for internal groupoids and hence categories. (email me if you would like some notes on this)

But! This is not the same as a torsor for an internal category, as one could extract or otherwise from various places, e.g. Moerdijk's or Johnstone. See this answer. The two definitions are aiming at very different things. For example, a saturated anafunctor with codomain a topological monoid (as a one-object category) is trivial, but a torsor for the same is not. In general internal saturated anafunctors are about inverting weak equivalences between internal categories (fully faithful and essentially 'surjective': i.e. some map admitting local sections) but internal torsors are about characterising maps between topoi and classifying topoi and stuff (you can tell I know less about the latter).