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?

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