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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