This is an open question and it's to find out who is interested in this kind of thing, who can benefit from thinking about this. It is very brief but hopefully will only be unclear to people who are not into this anyway. Briefly, we can think of a Morita category (with object C*-algebras and morphisms bimodules) as a "weak" groupoid. It seems to me one can put a topology on the object space by thinking of it as a Banach bundle or continuous field of C*-algebras, which comes with a topological structure. This should satisfy axioms of a Grothendieck site (a topology on a category). So I think one can use this to define a weak topological groupoid. What about a weak differentiable groupoid or Lie groupoid? We can consider comparing this structure (with a certain sheaf over the site of objects) to a spectral triple with the geometrical data in the Dirac operator. I will stop to keep this short. This should lead to something closely related to Quillen's point of view on cyclic cohomology where the spectral triple algebra is also related to a Grothendieck site. The motivation is coming from Crane's "quantum geometries" for quantum gravity and I think there are unexplored conceptual intersections between category theory and non-commutative geometry. Thanks for reading! Sorry if this is not a math overflow type question but there is an indication that we may ask open questions.

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    $\begingroup$ The ideas in your question sound interesting, but I have a hard time understanding what question you are asking. So I will point you to the following "how to ask" advice and hope that a revised version of your question receives more attention: mathoverflow.net/howtoask $\endgroup$ – Manny Reyes Dec 14 '12 at 16:11
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    $\begingroup$ I'm 'interested in this kind of thing', but, i don't get what you mean... what do you mean by "weak" groupoid. what is for you a 'Morita category', the whole 2-category of C^* algebra - bimodules - map between bimodule ? $\endgroup$ – Simon Henry Dec 14 '12 at 17:58
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    $\begingroup$ This is not really an answer to your question, so it is a comment. I see the problem as to how to use the power of strict higher groupoids in the context you mention - see my question mathoverflow.net/questions/86617/… . In homotopy theory, the use of strict multiple groupoids has given rise to new nonabelian calculations of $n$-ad homotopy groups and so of absolute homotopy groups of some complexes, and of k-invariants. One would seem to need strict structures to make explicit calculations, which is surely what you need for physics (??). $\endgroup$ – Ronnie Brown Dec 14 '12 at 21:46
  • $\begingroup$ Thank you for reading my question and taking the time to leave comments :) In the coarse-grain view you have a category where arrows are Morita equivalence bimodules (objects are C*-algebras). Bertozzini et al called this a Morita category if the bimodules are considered up to isomorphism. This is really like a "weak" groupoid i.e. it is a groupoid but with equalities replaced by isomorphisms. Yes this is really a bicategory :) ... For a generalised notion of path or geodesic in a non-com space, one can consider elements of these bimodules, (now we're strict) in a richer context. $\endgroup$ – Rachel Dec 17 '12 at 16:06
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    $\begingroup$ You should realise the difference between a Grothendieck topology and an ordinary topology - they really are nothing alike. If you start out with a continuous field of $C^*$-algebras, then you might just be defining a topological groupoid outright, but starting from an arbitrary category whose objects are $C^*$-algebras, you don't really have anything to grab onto. In any case, a 'Morita category' is really the shadow of some 2-category of stacks or similar, via the correspondence between (certain) locally compact groupoids and (certain) $C^*$-algebras. $\endgroup$ – David Roberts Jan 23 '13 at 23:29

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