Cofibrations of internal groupoids and localisations by cospans

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 \infty-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 (S,J), whose underlying category S is lextensive. Let me denote by \amalg J the corresponding coverage with singleton covering families given by the coproducts of J-covering families. I will also assume that \amalg J-covers are stable under coproducts (which I will write as sums), that coequalisers of equivalence relations R \subset X\times X with the induced R \to X a \amalg J-cover exist, such that the quotient X \to X/R is a \amalg J-cover, and such that restricted 2-out-of-3 holds: if f and g\circ f are \amalg J-covers, then g is a \amalg J-cover (or equivalently, a map is a J-cover if J-locally on the domain it is a J-cover).

I will denote by Gpd^{ccof}_J(S) the 2-category of groupoids internal to S such that their source and target maps are \amalg J-covers (this is the setup considered by Meyer and Zhu, for instance), functors f\colon X\to Y such that Y_0 \simeq X_0 + Y'_0 and f_0 = in_L 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 f\colon X\to Y is fully faithful if X_1 \simeq X_0^2 \times_{Y_0^2} Y_1, and essentially \amalg J-surjective if X_0\times_{Y_0,s}Y_1 \to Y_1 \stackrel{t}{\to} Y_0 is a \amalg J-cover. Now define a trivial complemented cofibration to be a functor in Gpd^{ccof}_J(S) that is fully faithful and essentially \amalg J-surjective. The class of such will be denoted ccof_{triv}.

Proposition: The class of maps ccof_{triv} is a co-pretopology and consists of fully faithful and co-fully faithful morphisms in the 2-category Gpd^{ccof}_J(S) (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 j\colon X\to Y along an arbitrary complemented cofibration f\colon X\to Z in such a way that it gives a groupoid Y+_X Z whose source and target maps are \amalg J-covers. The hypotheses on J are chosen so this is possible, and essentially one builds the objects of the pushout groupoid as an ordinary pushout of coproduct inclusions in S, 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 Z \to Y +_X Z 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 Gpd^{ccof}_J(S) admits a localisation by a bicategory of fractions at ccof_{triv}, which can be constructed using the opposite of the bicategory of formal anafunctors in the opposite 2-category Gpd^{ccof}_J(S)^{op}.

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 LandNikolausSzumiło 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 Y and a map p\colon A \to Y_0 (in a suitably complete ambient category) there is a groupoid Y[A] with object of objects A and equipped with a fully faithful functor Y[A] \to Y, whose object component is p. We can factorise (functorially) any functor (internal to a lextensive category) f\colon X\to Y between groupoids X and Y as X \to Y[X_0 + Y_0] \to Y, where X_0 + Y_0 \to Y_0 is the identity on Y_0 and f_0 on X_0. Now X\to Y[X_0 + Y_0] is a complemented cofibration. Notice also that since Y_0 \to X_0 + Y_0 splits X_0 + Y_0 \to Y_0, there is an induced splitting Y = Y[X_0 + Y_0][Y_0] \to Y[X_0 + Y_0], which is fully faithful (by construction) and which is also nearly essentially \amalg J-surjective. If it is indeed so, then we have constructed, for a given f, a formal anafunctor (or rather, the cospan version) X \to Y[X_0+Y_0] \leftarrow Y from X to Y. In fact that is a little bit of a lie, in that we also need Y[X_0+Y_0] to have suitable source and target maps. Thankfully, one condition will solve both problems, namely that we ask that f_0 is a \amalg J-cover, or more generally that it is an element of a J-covering family such that \{f_0, id_{Y_0}\} 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 S=Set, we recover a variant on the LandNikolausSzumiło 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 a\colon f\Rightarrow g \colon X\to Y, and cospans X \to Y[X_0^f+Y_0] \leftarrow Y and X \to Y[X_0^g+Y_0] \leftarrow Y corresponding to f and g respectively, then the pushout of Y[X_0^f+Y_0] \leftarrow Y \to Y[X_0^g+Y_0] can be constructed as Y[X_0^f + Y_0 + X_0^g]. Here I am using X_0^f to denote X_0 as an object over Y_0 via the (object component of the) functor f, and similarly for g. The 2-arrow in the localised bicategory is then given by the unique lift of a through the fully faithful functor Y[X_0^f + Y_0 + X_0^g] \to Y, through which f and g 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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