I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose:

- objects are groupoids;
- 1-morphisms are (left-principal?) bibundles;
- 2-morphisms are bibundle homomorphisms.

On the right, I have the 2-category ALG, whose:

- objects are algebras (over $\mathbb C$, say);
- 1-morphisms are (adjectives?) bimodules;
- 2-morphisms are bimodule homomorphisms.

And probably I should go through and add "in TOP" to every word on the left and "C-star" to every word on the right.

(I have the impression that ALG is two-equivalent to another category, which I will put on the far right, whose:

- objects are cocomplete VECT-enriched categories (with extra adjectives?);
- 1-morphisms are cocontinuous VECT-enriched functors;
- 2-morphisms are VECT-enriched natural transformations.

In one direction, the functor takes an algebra to its full category of modules. In the other direction, there might be extra adjectives needed, and I mean to appeal to the Mitchell embedding theorem; but I'm pretty sure an equivalence exists if I insist that every object on the far right comes with a cocontinuous faithful VECT-enriched functor to VECT, and my idea is that Mitchell says that every category admits such a functor.

So anyway, the point is that either category on the right or far right is a sort of "algebraic" category, as opposed to the more "geometric" category on the left.)

Then I've been told on numerous occasions that there is a close relationship between GPOID and ALG. See, for example, the discussion at Geometric interpretation of group rings? — in fact, it's reasonable to think of the present question as a follow-up on that one.

The relationship is something like the following. To each (locally compact Hausdorff) topological, say, groupoid, we can associate a C-star, say, algebra — the construction restricts in various special cases to: a (locally compact Hausdorff) topological space $X$ going to its algebra of continuous vanishing-at-infinity functions $C_0(X)$; a finite group $G$ going to its group ring $\mathbb C G$; etc. The construction extends to 1- and 2-morphisms to build a (contra)functor. At least if I get all the adjectives right, the functor should be a two-equivalence.

Question:What's the precisification of what I have said above? What exactly is the two-functor from groupoids to algebras, and which adjectives make it into an equivalence of two-categories? Groupoids have natural "disjoint union" and "product" constructions; these presumably correspond to Cartesian product and tensor product (?!? that's not the coproduct in the category of algebras, but maybe in this two-category it is?) on the algebraic side?

Let me end with an example to illustrate my confusion, which I brought up in Op. cit.. Let $G$ be a finite abelian group; then it has a Pontryagin dual $\hat G$. Now, there is a canonical way to think of $G$ as a groupoid: it is the groupoid $\{\text{pt}\}//G$ with only one object and with $G$ many morphisms. If I'm not mistaken, the corresponding algebra should be the group algebra $\mathbb C G$. But $\mathbb C G$ is also the algebra $C_0[\hat G]$ of functions on the *space* $\hat G$. And if there is one thing I am certain of, it is that the underlying space of $\hat G$ (a groupoid with no non-identity morphisms) and the one-object groupoid $\{\text{pt}\}//G$ are not equivalent as groupoids. And yet their "function algebras" are the same. So clearly I am confused.