### Internal Categories, Anafunctors and Localisations

#### Posted by Mike Shulman

*guest post by David Roberts*

This post is about my forthcoming paper, extracted from chapter 1 of my thesis:

Internal categories, anafunctors and localisations

and is also a bit of a call for examples from $n$-category cafe visitors (see also these MO questions). I am most familiar with topological and Lie groupoids, but many interesting examples come from the world of schemes and algebraic stacks. I would like to know about these, but first I need to explain what I’m looking for. I would also appreciate to have any typos or inaccuracies pointed out. Please note that I haven’t written a final abstract yet, so what is there is just a placeholder!

Recall the notion of internal category. This comes with a natural definition of internal functor, and internal transformation, leading to a 2-category $Cat(S)$ of categories internal to $S$. However there are a number of settings where there are not enough 1-arrows between a pair of objects. One of these is when $S = Grp$, the category of groups. Then categories internal to $Grp$ are algebraic representations of pointed connected homotopy 2-types, but the natural hom-groupoid in $Cat(Grp)$ (yes, it’s a groupoid, as internal categories=internal groupoids here) does not represent the homotopy type of the mapping space.

Another example, which is rather generic, is when you want to represent smooth stacks of groupoids (or topological stacks, or algebraic stacks) by Lie groupoids (or topological, etc. groupoids). Then the hom-groupoid between stacks is not the same as the hom-groupoid between the Lie groupoids (etc.).

A third example is given by orbifolds. These are well-known, by now, to be the same as proper etale Lie groupoids. There is a very well-argued review paper, *Orbifolds as stacks?*, that talks about the ‘correct’ 2-category of orbifolds as being a localisation of the 2-category of proper etale Lie groupoids at the class of ‘weak equivalences’.

What is a weak equivalence? I hear you ask, as there are many types of weak equivalence. here we call an internal functor $f:X \to Y$ an $E$-equivalence if this diagram is a pullback $\begin{array}{ccc} X_1 & \to & Y_1 \\ \downarrow && \downarrow\\ X_0^2 & \rightarrow & Y_0^2 \end{array}$ and the composite map $X_0 \times_{Y_0} Y_1^{iso} \to Y_1^{iso} \to Y_0$ is in $E$, a specified class of maps satisfying some properties. In practice $E$ will usually be the class of maps that admit local sections for a given Grothendieck pretopology $J$. These were introduced by Bunge and Par'e for $E$=regular epimorphisms and $S$ a finitely complete regular category. We denote the class of $E$-equivalences by $W_E$.

Now I don’t need to assume so much on $S$, only binary products and the existence of $E$. I rather move my assumptions to the 2-category $Cat(S)$, or rather a full sub-2-category $Cat'(S)$ (which could be $Gpd(S)$). I assume that the objects of $Cat'(S)$, which are internal categories, are such that pullbacks of the source and target maps exist. This is a familiar assumption from the theory of Lie groupoids, where source and target maps are assumed to be submersions. Let $E$ be a class of maps in $S$ which satisfies the following conditions:

- $E$ is a Grothendieck pretopology
- $E$ contains the split epimorphisms
- ‘Condition (S)’: If $A\to B$ is a split epimorphism and the composite $A \to B \to C$ is in $E$, then $B\to C$ is in $E$

A class of maps which satisfies these conditions is called *admissible*. An example is the class of maps in a finitely complete category admitting local sections where ‘local’ means for a given pretopology $J$.

The first main result of the paper is this:

**Theorem 1:** If $Cat'(S)$ admits weak pullbacks and admits base change along arrows in $E$, then $Cat'(S)$ admits a calculus of fractions for $W_E$, the class of $E$-equivalences.

The calculus of fractions here is a bicategorical localisation, as covered in Pronk’s article. ‘Base change along arrows in $E$’ may sound mysterious, but it is a simple construction. Consider an internal category $X_1 \rightrightarrows X_0$ and a map $M \to X_0$. There is (when the displayed pullback below exists) another category denoted $X[M]$ with objects $M$ and arrows the pullback $\begin{array}{ccc} M^2\times_{X_0^2}X_1 & \to & X_1 \\ \downarrow && \downarrow\\ M^2 & \rightarrow & X_0^2 \end{array}$ Often we know that this pullback exists, but we don’t know a priori that $X[M]$ is an object of $Cat'(S)$. Example for when it does is when $Cat'(S)$ is the 2-category of proper, etale Lie groupoids. Properness is a condition on $(s,t)$, but etale-ness is a condition on the source and target maps, $s,t$. When this category $X[M]$ is an object of $Cat'(S)$ for all maps in $E$, we say $Cat'(S)$ admits base change along arrows in $E$.

My first query is this: what classes of maps $P$ (for your favourite category) can we take the source and target maps of $X$ to belong to so that the source and target of $X[M]$ also belong to $P$? I’m sure there are some examples arising in the theory of algebraic stacks, but I don’t know enough algebraic geometry to figure them out.

Now once we know this localisation exists, how do we calculate it? Pronk gives a construction, but it is, to me, quite complicated. This is where anafunctors come in. (Anafunctors have had a long history at the n-Cafe!) There are all sorts of fancy foundational motivations for anafunctors, but we are only interested in the version for internal categories (although the formalism is exactly the same, the applications are different). I like to think of an anafunctor $X ⇸ Y$, which is a span from $X$ to $Y$, as a map from a resolution of $X$ to $Y$. More accurately, what we need on our ambient category $S$ is a subcanonical singleton Grothendieck pretopology (‘singleton’ means that covering families are just single maps) $J$. An anafunctor $X ⇸ Y$ is then a span $X \leftarrow X[U] \to Y$, where $U \to X_0$ is a $J$-cover.

There is a bicategory $Cat_{ana}(S)$ of internal categories, anafunctors and transformations of anafunctors, the construction of which was detailed by Toby Bartels in his thesis, generalises very slightly to give us a sub-bicategory $Cat'_{ana}(S)$ corresponding to $Cat'(S)$ and a canonical strict 2-functor $Cat'(S) \to Cat'_{ana}(S)$. The second result of the paper is this:

**Theorem 2:** Let $Cat'(S)$ and $E$ be as in theorem 1, let $J$ be a subcanonical singleton pretopology on $S$ which is cofinal in $E$ and let $Cat'(S)$ admit base change along arrows in $J$. Then there is an equivalence of bicategories
$Cat'_{ana}(S) \simeq Cat'(S)[W_E^{-1}]$
which, up to equivalence, the identity on $Cat'(S)$.

This allows us to focus on anafunctors if we want to invert weak equivalences.

As far as I know, theorem 1 covers pretty much all examples of localising a 2-category of internal categories or groupoids in the literature. Some of these use Hilsum-Skandalis maps/right principal bibundles instead of Pronk’s construction or anafunctors, but all these give equivalent bicategories in the end.

The paper ends with a string of comparison theorems between localisations arising from different pretopologies, and a section on size considerations. The main point of the latter is to say that if the site $(S,J)$ satisfies (an internal version of) the axiom WISC, then all these localisations are essentially locally small bicategories.

## Re: Internal categories, anafunctors and localisations

Nice! I have one question, about something that’s always confused me. People often formulate this stuff in terms of a Grothendieck

pretopology, as you did. But then they start talking about the maps that “admit local sections” relative to this pretopology, as you also did. Aren’t those just the maps which are coverings in the Grothendiecktopologygenerated by the pretopology?