Maths & Stats / Seminars / AusCat Seminar / 011010bunge

The fundamental group of a (universal) branched covering in topos theory

Marta Bunge (10/10/01)

  • Background Branched coverings in topos theory [BN] [Fu] may be described (in several ways with varying degrees of abstraction) in terms of the more general notion of a complete spread [BF]. The goal of this talk is to establish a theorem relating the fundamental group of a (universal) branched covering of a topos with that of the (universal) covering of its unbranched part.

  • Motivation. The main motivation comes from theorem of R.H. Fox [Fox] (Section 7) for locally finite complexes, where, however, the fundamental group of a branched covering is directly given in terms of paths rather than by exploiting the Galois theory that is inherent in the given (branched and unbranched) coverings. I set out to prove one key ingredient of this theorem in the general context of an admissible KZ-doctrine in the sense of [BF2], satisfying an additional axiom of "interior" (or "density").

  • Definitions. Context: an admissible KZ-doctrine M satisfying an axiom of interior on a 2-category K.

    (a) A "branched covering" over an admissible object E relative to a fully faithful (and final) discrete opfibration f: X --> E is a discrete fibration h: D --> E whose interior g = d(h): Y --> E factors (necessarily uniquely) through f: X --> E, by means of an "unbranched covering", meaning a 1-cell k:Y--->X which is at the same time a discrete fibration and a discrete opfibration.

    (b) A branched covering h: D -->E relative to f:X-->E is said to be a "universal branched covering" if its associated unbranched covering k:Y -->X has the property that every endomorphism of k is an automorphism. Note: It follows that any endomorphism of h is an automorphism.

    Theorem. Let M be an admissible KZ-doctrine in a 2-category K satisfying an axiom of interior. Let E be an admissible object of K and let h:D-->E be a universal branched covering of E relative to a fully faithful discrete opfibration f: X-->E. Let k:Y-->X be the associated universal (unbranched) covering. Denote by Aut(h) and Aut(k) the corresponding (discrete) groups of automorphisms. Then there is given a canonical group isomorphism Aut(h)-->> Aut(k).

    Proof. The proof of this theorem involves, besides some properties of the admissible KZ-doctrine M with an axiom of interior, just the comprehensive factorization in K relative to M and the given definitions.

  • Examples. The motivating example for the above definitions and theorem is that of the 2-category K of Grothendieck toposes with M the "symmetric KZ-doctrine", where the axiom of interior is satisfied on account of the existing notion of density of a distribution [BF}. In this context the notions and theorem stated above can be shown to have familiar interpretations. Other examples from [BF2] are in principle amenable to a similar analysis.

  • Acknowledgements. I gratefully acknowledge the interest and helpful remarks and questions from the members of the Australian Category Seminar, in particular from Mark Weber, Ross Street and Steve Lack (temporal order).

    References

    1. [B] M.Bunge, Classifying toposes and fundamental localic groupoids, in R.A.G.Seely, ed., Categroy Theory '91, CMS Conference proceedings 13 (1992) 73-96.
    2. [BD] M.Bunge and E.Dubuc, Constructive Theory of Galois Toposes, http://www.math.mcgill.ca/~bunge/CTGT.dvi (.ps)
    3. [BF] M.Bunge and J.Funk, Spreads and the Symmetric Topos II, J. Pure Appl. Alg.130 (1998) 49-84.
    4. [BF2] M.Bunge and J.Funk, On a bicomma object condition for KZ-doctrines, J. Pure Appl.Alg.143(1999) 69-105.
    5. [BM] M.Bunge and I.Moerdijk, On the construction of the Grothendieck fundamental group of a topos by paths, J. Pure Appl. Alg.116 (1997) 99-113.
    6. [BN] M.Bunge and S.B.Niefield, Exponentiability and single universes, J. Pure Appl. Algebra 148-3 (2000) 217-250.
    7. [Fox] R.H.Fox, Covering spaces with singularities, in R.H.Fox et al. (Eds.), Algebraic Geometry and Topology: A Symposium in Honor of S.Lefschetz, Princeton University Press (1957) 243--257.
    8. [Fu] J.Funk, On branched covers in topos theory, Theory and Applications of Categories 7-1 (2000) 1-22.
    9. [Ma] W.S.Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991.
    10. [M] I.Moerdijk, Continuous fibrations and inverse limits of toposes, Compositio Math. 58 (1986) 45-72.
    11. [PS] V.V.Prasolov and A.B.Sossinsky, Knots, Links, Braids and 3-Manifolds, An introduction to the new invariants in low-dimensional topology, Translations of Mathematical Monographs 154, Amer. Math. Soc. 1991.

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    Last modified: Mon Nov 12 08:44:35 EST 2001