In my paper *ICAL*, I dealt extensively with internal categories and groupoids, and how to localise 2-categories of these at the fully faithful essentially ‘surjective’ functors (‘surjective’ in the sense there is some well-behaved class of regular epimorphisms playing the rôle of surjective maps). Even for 1-categories of ordinary small groupoids or categories one can consider this as a variant on the canonical model structure on groupoids or on categories, and formally invert the fully faithful, essentially surjective functors, which aren’t invertible in the 1-category sense. Actually it really only uses the category of fibrant objects structure, and *ICAL* (and more in generality, these notes) can be seen as a way to make this work in a way that doesn’t use the full category of fibrant objects structure, but the 2-category structure instead. There is another way to think about this process for the case of small groupoids (and more generally from some *cofibration* category), which I learned from a paper of Land, Nikolaus and Szumiło, *Localization of Cofibration Categories and Groupoid C*-algebras* (publisher, arXiv) (cofibrations of groupoids are functors that are injective on objects). I want to outline how one can repeat the 2-categorical, extra weak story for internal groupoids (at least; I haven’t thought about the case of internal categories) using this type of approach. In some respects, I can get more general results, in that I don’t require that I’m localising a small cofibration category with good cylinders, but also kinda less general in the sense that the universal property I get is only for bicategories, not more general -categories (presented by something Quillen equivalent to a combinatorial model category). Below the fold I set up some basic assumptions, and then outline the constructions that will go into the hypotheses of the construction.

First up, recall the definition of a superextensive site , whose underlying category is lextensive. Let me denote by the corresponding coverage with singleton covering families given by the coproducts of -covering families. I will also assume that -covers are stable under coproducts (which I will write as sums), that coequalisers of equivalence relations with the induced a -cover exist, such that the quotient is a -cover, and such that restricted 2-out-of-3 holds: if and are -covers, then is a -cover (or equivalently, a map is a -cover if -locally on the domain it is a -cover).

I will denote by the 2-category of groupoids internal to such that their source and target maps are -covers (this is the setup considered by Meyer and Zhu, for instance), functors such that and is the coproduct inclusion, and arbitrary natural transformations. Such functors will be called *complemented cofibrations*. This seems like a massive restriction on what is possible, but in the end, one can recover many other functors, if not quite all, depending on the properties of the site in question; for a pretopos with the canonical topology, for instance, one can recover all functors. Lastly, recall that a functor is fully faithful if , and essentially -surjective if is a -cover. Now define a *trivial complemented cofibration* to be a functor in that is fully faithful and essentially -surjective. The class of such will be denoted .

**Proposition:** The class of maps is a co-pretopology and consists of fully faithful and co-fully faithful morphisms in the 2-category (co-ff in the sense that they are ff in the opposite 2-category).

The only difficult part is to construct the pushout of a trivial complemented cofibration along an arbitrary complemented cofibration in such a way that it gives a groupoid whose source and target maps are -covers. The hypotheses on are chosen so this is possible, and essentially one builds the objects of the pushout groupoid as an ordinary pushout of coproduct inclusions in , which works fine, and then builds the arrows of the pushout groupoid as a coproduct of components, some of which are quotients of well-behaved equivalence relations. The functor is again a trivial complemented cofibration, as required. Trivial complemented cofibrations are closed under composition and contain the isomorphisms, and so one has a co-pretopology. As the functors are themselves fully faithful, they are fully faithful morphisms, and they are co-fully faithful morphisms since coproduct inclusions are regular monomorphisms.

**Corollary:** The 2-category admits a localisation by a bicategory of fractions at , which can be constructed using the opposite of the bicategory of formal anafunctors in the opposite 2-category .

The proof of the corollary is the formal opposite of Theorem 3.19 in *The elementary construction of formal anafunctors.* The cospans that are the 1-arrows in the localisation are essentially collages of certain representable internal profunctors, but it is nice to know the construction can be done in a fair amount of generality, and that the construction follows from the formal 2-categorical result that is Theorem 3.19.

Now we want to see how the 2-category internal groupoids, (almost) all functors and natural transformations fits into this picture. In the Land–Nikolaus–Szumiło paper, they get the result that the full category of small groupoids maps into the localisation of the category with only cofibrations as maps. Here we can get close, but the site structure is important, as it dictates what sorts of functors ‘work’. Firstly, recall that for a groupoid and a map (in a suitably complete ambient category) there is a groupoid with object of objects and equipped with a fully faithful functor , whose object component is . We can factorise (functorially) *any* functor (internal to a lextensive category) between groupoids and as , where is the identity on and on . Now is a complemented cofibration. Notice also that since splits , there is an induced splitting , which is fully faithful (by construction) and which is also *nearly* essentially -surjective. If it is indeed so, then we have constructed, for a given , a formal anafunctor (or rather, the cospan version) from to . In fact that is a little bit of a lie, in that we also need to have suitable source and target maps. Thankfully, one condition will solve both problems, namely that we ask that is a -cover, or more generally that it is an element of a -covering family such that is a covering family. This might seem like an odd condition, but there are many pretopologies where this is immediate: jointly surjective families of open embeddings, local isomorphisms, submersions, open maps,… In the case that the ambient category is a pretopos and the pretopology is the canonical one (i.e. the singleton pretopology of pullback-stable regular epimorpisms), then *all* morphisms satisfy this technical condition, so it is no restriction at all, and for , we recover a variant on the Land–Nikolaus–Szumiło result.

At this point, we still don’t have a 2-functor from the 2-category of internal groupoids (as defined here), with nearly general functors as above as 1-arrows, to the localisation, since we haven’t specified what to do with the natural transformations (let alone showing functoriality). Given a natural transformation , and cospans and corresponding to and respectively, then the pushout of can be constructed as . Here I am using to denote as an object over via the (object component of the) functor , and similarly for . The 2-arrow in the localised bicategory is then given by the unique lift of through the fully faithful functor , through which and factor. The uniqueness of this lifting ensures that this assignment is functorial for composition of 2-arrows. Now at least we have a functor on hom-groupoids; it still remains to show that it preserves the composition functors, up to natural isomorphism. And, indeed, composition of 1-arrows is not strictly preserved, but I will address that in another, soon-to-appear post.

I hope this example shows the power of the formal construction of anafunctors, in that the hypothesis on the pretopology on the 2-category are very nearly self-dual, the only part where it is not is that one can swap stability under strict pullbacks and stability under strict pushouts, and then instead of a localisation by spans, one has a localisation by cospans. The results are then not quite the same, in that when using spans to localise, one can consider *all* functors, but when using cospans, if the ambient category has few assumptions on it then there is a bit of a restriction as to what functors can ultimately be used.

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