Tbilisi Mathematical Journal

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

Christopher Townsend

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Export citation


  • Hilsum, M. and Skandalis, G. Morphismes K-orientes d'espace de feuilles et functorialite en theorie de Kasparov. Ann. Scient. Ec. Norm. Sup. 20 (1987) 325-390.
  • Johnstone, P.T. Sketches of an elephant: A topos theory compendium. Vols 1, 2, Oxford Logic Guides 43, 44, Oxford Science Publications, 2002.
  • Joyal, A. and Tierney, M. An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society 309, 1984.
  • Lawvere, F. W. Equality in Hyperdoctrines and comprehension schema as an adjoint functor in Applications of Categorical Algebra, Proc. of Symposia in Pure Math., AMS, vol. XVII (1970) 1-14.
  • Moerdijk, I. The classifying topos of a continuous groupoid. I, Transactions of the American Mathematical Society, Volume 310, Number 2, (1988) 629-668.
  • Moerdijk, I. Morita equivalence for continuous groups, Math. Proc. Camb. Phil. Soc. 103 (1988) 97-115.
  • Moerdijk, I., Mrcun, J. Lie groupoids, sheaves and cohomology. Part Three of `Poisson Geometry, Deformation Quantisation and Group Representations' London Math. Soc. Lecture Note Ser. 323, Cambridge University Press, Cambridge, (2005) 145-272.
  • Mrcun, J. Stability and invariants of Hilsum–Skandalis maps, Ph.D. thesis, Utrecht University, (1996).
  • Townsend, C.F. A representation theorem for geometric morphisms. Applied Categorical Structures. 18 (2010) 573-583.
  • Townsend, C.F. Aspects of slice stability in Locale Theory Georgian Mathematical Journal. Vol. 19, Issue 2, (2012) 317-374.
  • Townsend, C.F. A short proof of the localic groupoid representation of Grothendieck toposes. Proceedings of the American Mathematical Society 142.3 (2014) 859-866.
  • Townsend, C.F. Principal bundles as Frobenius adjunctions with application to geometric morphisms. Math. Proc. Camb. Phi. 159 (2015), 433-444.
  • Townsend, C.F. Stability of Properties of Locales Under Groups. Appl Categor Struct Vol. 25 Issue 3 (2017), 363-380.