I have often encountered definitions of the kind "stacks are equivalence classes of groupoids under Morita equivalence" in topological or differentiable context, with the notion of Morita equivalence appropriate for the context. 


I have two questions:


1) can one define Deligne-Mumford or Artin stacks this way, as Morita equivalence classes of algebraic groupoids?


2) if yes, what is the connection between the corresponding notion of Morita equivalence and the one for the rings (i.e. rings are called Morita equivalent if they have equivalent categories of modules over them)?



stacks as Morita equivalence classes
Regarding your question relating Morita equivalence as defined for internal groupoids and as defined for rings:


Given an internal groupoid $G$ (say, Lie, topological or algebraic), it defines a presheaf of groupoids on the ambient category ($Diff$, $Top$ or $Sch$). The stackification of this presheaf is the category of (right, say) principal $G$-bundles. Thus two groupoids define equivalent stacks (isomorphic in the correct 2-categorical sense) whenever their categories of principal bundles are equivalent. This is the groupoid version of 'Morita equivalence' a la rings. But when these two groupoids present equivalent stacks, they are 'Morita equivalent' in the sense there is a span between them of 'essential equivalences' - fully faithful essentially surjective (-appropriately interpreted) functors. So the two notions coincide, at least when the ambient category has enough quotients of the right sort of reflexive coequalisers (as the examples I list do). In the more general case with no quotients, as I mention in my comment to the original question, I'm not sure what happens.




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EDIT: It's been a while since I answered this, but now I know what happens in the case I mention in the final sentence above the line. One shouldn't take the category of principal bundles as being the stackification of the internal groupoid $G$, rather take the category of [internal anafunctors](http://arxiv.org/abs/1101.2363) with codomain $G$ and with domain a groupoid with no non-identity arrows. The arrows are a little tricky to describe, but one gets a stack over the base category with fibre over $X$ the hom-category $Hom(X,G)$ in the bicategory of internal groupoids, anafunctors and transformations. This will be covered in a forthcoming paper of mine. 


