In my previous post I described a construction whereby a 2-category of groupoids internal to a lextensive category whose 1-arrows are functors with object component a coproduct inclusion can be localised at the fully faithful essentially -surjective functors. The 1-arrows of are called *complemented cofibrations*, and the 1-arrows of the localisation were then certain cospans of such functors. This was rather formal and relied on these notes of mine. However, for the case that the ambient category is that of sets, Land, Nikolaus and Szumiło showed that this localisation is equivalent to localising the category of all small groupoids at the fully faithful essentially surjective functors (although they worked in the setting of -categories, the result is still just a -category). I started sketching how one should construct a 2-functor from a 2-category of internal groupoids with more general functors to the localisation by cospans. (The functors need to satisfy a condition related to the pretopology one is using on the lextensive category, which in the case of a pretopos with the canonical topology turns out to be always true, but there are interesting cases where it is a real restriction.) That said, one can use Pronk’s comparison theorem to show the stronger result that , which I will sketch below the fold.

The condition that one must impose on the functors in is that the object components need to be ‘partial -covers’, in the sense that they are morphisms in that are an element of a -covering family , such that if is any -covering family, then so is . For any pretopology whose covering families are defined as ‘jointly surjective collection of X-maps’, the X-maps will be partial -covers. With this condition, plus the standing assumptions on the pretopology from last time, if one has an anafunctor (a 1-arrow in ), then it is possible to replace it by an isomorphic anafunctor where is a right principal -bundle and is a left -space over . This is what Makkai refers to as a *saturated* anafunctor, and the object with its two actions is a right principal bibundle, another approach to localising the 2-category of groupoids. There is a small subtlety here in that one needs to verify that is a groupoid in the 2-category we are working with (recall that the source and target maps are always assumed to be in ), and the object component of is a partial -cover, but this works fine, given the assumptions. Then one can form the weak pushout of the span to get a cospan of complemented cofibrations, where the condition on the object components now means that is a groupoid in our 2-category. In the case that the anafunctor we started with was an ordinary functor, this gives the construction I have at the end of my last post.

Now we need to see how to apply Pronk’s comparison theorem, which is the following

**Theorem (Pronk):** Let be a localisation of the bicategory by (right) fractions and let be a 2-functor sending 1-arrows in to equivalences. Then if

- is essentially surjective;
- Every 1-arrow in is isomorphic to a factorisation for 1-arrows in with ;
- is locally fully faithful;

then is also a localisation of at , so then .

We are actually going to apply the *opposite* of this theorem with the composite , since we have used a cospan construction (hence *left* fractions) to get . This means that condition 2. of the theorem is instead

- Every 1-arrow in is isomorphic to a factorisation for 1-arrows in with ;

Now we can use the construction above of a cospan from an anafunctor to factorise the latter as isomorphic to the composite of anafunctors and , where the latter anafunctor is a quasi-inverse to the complemented cofibration . This proves (the opposite of) 2. above. Condition 1. follows since all the 2-functors here are the identity map on objects, and condition 3. follows since is a full sub-2-category of (i.e. an isomorphic on hom-groupoids) and is locally fully faithful. Thus

**Theorem: **There is a 2-functor which is a localisation of the source at the class of fully faithful, essentially -surjective functors.

One such 2-functor would be the composite of the functor I incompletely defined in the previous post, with the localisation functor for . Note also that here I’m not assuming is a small category, so that could be a large 2-category. If a suitable weak smallness condition holds for (that is, a version of WISC), then the localisation is still locally essentially small.

One reason for forming the localisation this way is that every complemented cofibration can be factorised as where the right functor is the inclusion of a disjoint subgroupoid, and the left functor has the identity map as its object component. Such identity-on-objects functors give rise to another kind of generalised map of groupoids, called an *actor*, due to Buneci and Stachura in the case of topological groupoids, and more generally to Meyer and Zhu for internal groupoids as I’ve been working with here (and not to be confused with a morphism also called an actor, defined by Pradines). This means one can also factor a more general functor , with a partial -cover as object component, as a cospan , each leg of which factors as above into two special functors as above. The right leg in the copan is fully faithful and essentially -surjective, and the identity-on-objects functor in the factorisation of *that* is the identity functor. In the setting of topological groupoids, actors have good functoriality properties when forming the associated groupoid -algebras, hence the interest here.

## 4 thoughts on “Localisation by cospans redux”