Tbilisi Mathematical Journal

Hilsum–Skandalis maps as Frobenius adjunctions with application to geometric morphisms

Christopher Townsend

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Abstract

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.

Article information

Source
Tbilisi Math. J., Volume 10, Issue 3 (2017), 83-120.

Dates
Received: 15 August 2017
Revised: 13 October 2017
First available in Project Euclid: 20 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1524232076

Digital Object Identifier
doi:10.1515/tmj-2017-0104

Mathematical Reviews number (MathSciNet)
MR3724481

Zentralblatt MATH identifier
1376.18003

Citation

Townsend, Christopher. Hilsum–Skandalis maps as Frobenius adjunctions with application to geometric morphisms. Tbilisi Math. J. 10 (2017), no. 3, 83--120. doi:10.1515/tmj-2017-0104. https://projecteuclid.org/euclid.tbilisi/1524232076


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