What is the correct notion of Morita Equivalence between topological groupoids Hello,
Morita Equivalences occur in various categories, such as rings, operator algebras, homotopical categories, groupoids, etc. I'd like to know: What is the correct and precise definition of Morita Equivalence and what is the fundamental concept behind it. Here I want to restrict to (possibly étale, i.e. r-discrete) topological groupoids. From the literature, I have collected the following concepts (roughly stated, probably incorrect):
1) Two (topological) groupoids are Morita equiv. iff the associated stacks are isomorphic.
2) Two groupoids are Morita equiv. iff their representation categories (representation of G = groupoid morphism into the frame groupoid associated to a vector bundle over the space of objects of G) are equivalent.
3) Referring to David Roberts: Internal Categories, anafunctors and localisations http://arxiv.org/abs/1101.2363. There is a notion of canonical weak equivalence in the category of groupoids internal to TOP after choosing a certain Grothendieck pretopology. The literature somehow suggests étale surjections (étale=local homeos) as singleton covers in the topology, but I don't know the conceptual reason behind this. Anyway, those weak equivalences satisfy a sort of calculus of right fractions and the localisation can be described as a category of anafunctors. Following http://ncatlab.org/nlab/show/anafunctor, anafunctor isos are spans $G\leftarrow A\rightarrow H$ where both arrows are étale surjections on objects and fully faithful in the internal sense. This gives one possible precise definition of Morita equivalence.
4) In the context of homotopical category theory, it is a span $G\leftarrow A\rightarrow H$ such that both arrows are acyclic fibrations. But what are the weak equivalences and the fibrations? As weak equivalences we can choose the same as in 3 above and as fibrations the étale surjections on objects, then the notion of Morita equiv. would be the same in both cases. The question is, what homotopical structure do those weak equivalences and fibrations satisfy? It is not a category of fibrant objects since not every top. groupoid is fibrant in this sense. Anyway if some sort of factorization lemma is true in this category (like the one in a category of fibrant objects), this would explain the following statement which I have found several times in the literature (between the lines): Every Morita equivalence (span of acyclic fibrations) is equivalent to a single weak equivalence.
So here are a bunch of questions: What is the correct unifying concept of Morita equiv.? What is the explicit definition of Morita equiv. in the case of groupoids internal to TOP? Is it the one in 3,4? Why étale surjections? Is the last statement in 4 correct? What axioms does this homotopical structure satisfy?
 A: I'll attempt to answer some of your questions. First off, 1) and 3) are equivalent. This is because the bicategory of fractions in 3) is the Morita bicategory of topological groupoids, which is equivalent to  the bicategory of topological stacks. What maps are you inverting in 3)? Well, if you have an internal functor $F:G \to H$ of topological groupoids, you basically want to know when is this functor morally an equivalence, but I can be more precise. If you have a functor of categories, it is an equivalence if and only if it is essentially surjective and full and faithful. Note that this uses the axiom of choices, namely, that every epimorphism splits.
An internal functor $F:G \to H$ is a Morita equivalence if 
1) it is essentially surjective in the following sense:
The canonical map $$t \circ pr_1:H_1 \times_{H_0}  G_0 \to H_0$$
which sends a pair $(h,x),$ such that $h$ is an arrow with source $F(x),$ to the target of $h,$ is an surjective local homoemorphism 'etale surjection)
2) $G_1$ is homeomorphic to the pullback $(G_0 \times G_0) \times_{H_0 \times H_0} H_1,$ which is literally a diagramatic way of saying full and faithful.
You asked in 1), why 'etale surjection? Because this is what makes the map, when viewed as a map in in the topos $Sh(Top)$ an epimorphism, since the  Grothendieck topology on topological spaces can be generated by surjective local homeomorphisms. If you want to use another Grothendieck topology (e.g. the compacty generated one, as I do in one of my papers), you must adjust accordingly.
Anyway, $F$ satisfies 1) and 2) if and only if the induced map between the associated topological stacks is an equivalence. Note though, that surjective local homeomorphisms don't always split (if they did, then every Morita equivalence would have an inverse internal functor). Hence, we have to use spans to represent morphisms, where one leg "ought to be" invertible.
Finally, you say that $G$ and $H$ are Morita equivalent if there is a diagram $G \leftarrow K \to H$ of Morita equivalences. This is the standard definition of Morita equivalent. Since the bicategory of fractions with respect to Morita equivalences is equivalent to topological stacks, one can also say that $G$ and $H$ are Morita equivalent of the have equivalent topological stacks.
Now, I'll respond to some of the comments:
@Zhen: If $G$ and $H$ are 'etale (or more generally 'etale complete) then they have equivalent classifying topoi if and only if they have equivalent stacks. In fact, there is an equivalence of bicategories between 'etale topological stacks, and the topoi which are classifying topoi of 'etale topological groupoids ('etendue). For more general topological groupoids however, there is information lost when passing to their classifying topoi.
@Ben: I believe this is related to Zhen's question. The definition in the way you stated it, usually appears in topos literature, and is related to the fact that open surjections of topoi are of effective descent. This is not a good concept when the groupoids in question are not etale. To deal with torsors one really wants to have some version of local sections.
A: David C answered the question pretty thoroughly, but let me add a few more details. You mention that you want to restrict to etale groupoids. This is an important distinction. Without some sort of condition on the source and target maps you do not have the connection to topological stacks. And given this condition, it is important that this is compatible with the pretopology $J$ that you are using, in the sense that given a topological groupoid $X$ of the kind you are interested in (etale, or Hurewicz, or proper, or...) and a $J$-cover $U\to X_0$, the induced groupoid with objects $U$ and arrows $U^2\times_{X_0^2} X_1$ needs to also be of the kind you are interested in. For conditions like properness (i.e. $(s,t)$ is a proper map) this is automatic. Otherwise there is a subtle interplay between the sort of maps the source and target are, and the pretopology.
In terms of 4), given a finitely complete site with a subcanonical singleton pretopology, the category of groupoids in it where the source and target maps are covers, forms a category of fibrant objects. The fibrations are the functors $f\colon E\to B$ for which the canonical map
$$
E_1 \to E_0 \times_{f,B_0,s} B_1
$$
is a cover. Then every object is fibrant, because this reduces to the defining condition on the groupoids you are considering. In the case you mention, you are dealing with $Top$ with the pretopology of etale surjections. (This is another reason to restrict to a full sub-2-category of the 2-category of all topological groupoids). Acyclic fibrations are then fully faithful functors where the object component is a cover, and this recovers your characterisation of Morita equivalences in terms of the category of fibrant objects structure.
[Aside:
This generalises a result by Everaert, Kieboom and van der Linden, where they show that given certain conditions on the ambient category, one gets a Quillen model structure on the category of internal groupoids (this extends earlier work by Joyal and Tierney, I believe). Isomorphism in the homotopy category is the same as being equivalent in the weak 2-category of anafunctors. That anafunctors work in the absence of all finite limits is one motivation for me to consider them.
]
Notice you can weaken the condition that the map $t\circ pr_1$ is an etale surjection in David C's definition and still get the same result. It can be a map from any subcanonical singleton pretopology in which open covers are cofinal. Essentially you need to have local sections of $t\circ pr_1$, and you pick the right sort of maps for the application at hand.
I should mention there is a newer version of my paper available from my nlab page, which I hope will soon be finished being rewritten and then updated on the arXiv.
