TITLE: What is the correct notion of Morita Equivalence between topological groupoids
QUESTION [11 upvotes]: 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?

REPLY [2 votes]: 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.