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Hilsum-Skandalis maps as Frobenius adjunctions with application to geometric morphisms. (English) Zbl 1376.18003
Summary: Hilsum-Skandalis maps, from differential geometry, are studied in the context of a cartesian category. It is shown that Hilsum-Skandalis maps can be represented as stably Frobenius adjunctions. This leads to a new and more general proof that Hilsum-Skandalis maps represent a universal way of inverting essential equivalences between internal groupoids.
To prove the representation theorem, a new characterisation of the connected components adjunction of any internal groupoid is given. The characterisation is that the adjunction is covered by a stable Frobenius adjunction that is a slice and whose right adjoint is monadic.
Geometric morphisms can be represented as stably Frobenius adjunctions. As applications of the study we show how it is easy to recover properties of geometric morphisms, seeing them as aspects of properties of stably Frobenius adjunctions.
MSC:
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
58H05 Pseudogroups and differentiable groupoids
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