Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T11:53:08.202Z Has data issue: false hasContentIssue false
  • English
  • Français

Principal bundles as Frobenius adjunctions with application to geometric morphisms

Published online by Cambridge University Press:  12 August 2015

CHRISTOPHER TOWNSEND
CHRISTOPHER TOWNSEND*
Affiliation:
8 Gordon Villas, Aylesbury Road, Tring, Hertfordshire, HP23 4DJ. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Using a suitable notion of principal G-bundle, defined relative to an arbitrary cartesian category, it is shown that principal bundles can be characterised as adjunctions that stably satisfy Frobenius reciprocity. The result extends from internal groups to internal groupoids. Since geometric morphisms can be described as certain adjunctions that are stably Frobenius, as an application it is proved that all geometric morphisms, from a localic topos to a bounded topos, can be characterised as principal bundles.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 433 - 444
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[B90] Bunge, M. An application of descent to a classification theorem for toposes. Math. Proc. Camb. Phil. Soc. 107 (1990), pp 59–79.Google Scholar
[BLV11] Bruguieres, A., Lack, S. and Virelizier, A. Hopf monads on monoidal categories. Adv. Math. 227 Issue 2, 1 (2011), pp. 745–800.Google Scholar
[J02] Johnstone, P. T. Sketches of an elephant: a topos theory compendium. Vols 1, 2, Oxford Logic Guides 43, 44 (Oxford Science Publications, 2002).Google Scholar
[JT84] Joyal, A. and Tierney, M. An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc. 309 (1984).Google Scholar
[K89] Kock, A. Fibre bundles in general categories. J. Pure Appl. Algebra. 56, 3 (1989), 233–245.Google Scholar
[I90] Moerdijk, I. The classifying topos of a continuous groupoid. II. Cah. Topol. Géom. Differ. Catég. 31, No. 2 (1990), 137–168.Google Scholar
[I91] Moerdijk, I. Classifying toposes and foliations. Ann. Inst. fourier (Grenoble). 41, no. 1 (1991), p. 189–209.Google Scholar
[I96] Moerdijk, I. Classifying Spaces and Classifying Topoi. 1616 (Springer Verlag, 1995).Google Scholar
[P97] Plewe, T. Localic triquotient maps are effective descent maps. Math. Proc. Camb. Phil. Soc. 122, No. 01 (Cambridge University Press, 1997), 17–43.Google Scholar
[T06] Townsend, C. F. On the parallel between the suplattice and preframe approaches to locale theory. Ann. Pure Appl. Logic. 137, 1–3 (2006), 391–412 Google Scholar
[T10a] Townsend, C. F. An axiomatic account of weak triquotient assignments in locale theory. J. Pure Appl. Algebra. 6, 214 (2010), 729–739.Google Scholar
[T10b] Townsend, C. F. A representation theorem for geometric morphisms. Appl. Categ. Structures. 18 (2010), 573–583.Google Scholar