The fundamental group of a (universal) branched covering in topos theory
Marta Bunge – 10 October 2001
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 [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. [BD] M.Bunge and E.Dubuc, Constructive Theory of Galois Toposes, http://www.math.mcgill.ca/~bunge/CTGT.dvi (.ps) [BF] M.Bunge and J.Funk, Spreads and the Symmetric Topos II, J. Pure Appl. Alg.130 (1998) 49-84. [BF2] M.Bunge and J.Funk, On a bicomma object condition for KZ-doctrines, J. Pure Appl.Alg.143(1999) 69-105. [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. [BN] M.Bunge and S.B.Niefield, Exponentiability and single universes, J. Pure Appl. Algebra 148-3 (2000) 217-250. [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. [Fu] J.Funk, On branched covers in topos theory, Theory and Applications of Categories 7-1 (2000) 1-22. [Ma] W.S.Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991. [M] I.Moerdijk, Continuous fibrations and inverse limits of toposes, Compositio Math. 58 (1986) 45-72. [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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