Locally path simply connected toposes and their fundamental groupoids
Marta Bunge – 14 November 2001
Outline The lecture will consist of two parts, as follows. In the first part we begin by recalling the definitions and properties of certain classes of geometric morphisms, such as connected locally connected (c.l.c.) and surjective local homeomorphisms (s. l.h.). This is followed by a review of the main result of [BM], which consists on establishing the equivalence of a natural comparison map from the paths to the coverings fundamental groups of an unpointed topos E bounded over an elementary base topos S ([MW],[M],[B]). The second part is joint work with Steve Lack [BL]. The general theme of [BL] is to establish a Van Kampen theorem for toposes (bounded over a base topos S) in terms of their fundamental groups (coverings and paths) and to apply it to branched coverings and the fundamental group of a knot, motivated by work of R.H.Fox [Fox]. In this lecture I complement results presented by my co-author Steven Lack, in which, first, a 2-categorical version of a theorem of Brown and Janelidze [BJ] is given, with an application to a van Kampen theorem for covering projections of toposes, and second, on the result that the 2-category Top_S of toposes bounded over S is lextensive as a 2-category. . In this lecture, after reviewing the notion of a locally paths simply connected (l.p.s.c.) topos and recalling the results mentioned above, I turn to a proof of the following results. Theorem.(Van Kampen theorem for toposes) Let E, E_0, E_1 and E_2 be c.l.c. and l.p.s.c. toposes, with E_1 and E_2 subtoposes of E having intersection E_0 in E. Assume that the induced geometric morphism p: E_1 + E_2 ---> E is a surjective local homeomorphism. Then, the induced square of (paths) fundamental group toposes (P_1(E_0), P_1(E_1), P_1(E_2) and P_1(E)) is a pushout of toposes. Corollary. In the context of the above theorem, assume furthermore that the inclusions E_1 >---> E and E_2>--->E are local homeomorphism and that E_2 is paths simply connected. Then, in the pushout of fundamental group toposes, the geometric morphism P_1(E_1) ---> P_1(E) is connected locally connected, and the composite P_1(E_0)--->P_1(E_1)--->P(E) factors though a point S--->P(E) via the structural map P_1(E_0)--->S. In order to derive an application of the above to the fundamental group of a knot in this general context, we recall the following definition, given in [BN] and [Fu] but both following [Fox]. The intuition is that E_1 >--->E is the inclusion of the open complement of a knot Z>--->E in E, and that E_2>---->E is an inclusion into E of a paths simply connected subtopos containing Z>--->E, and that E_1 and E_2 together cover E. Recall that by the fundamental group of the knot embedded in say R^3, it is meant (see [R]) the fundamental group of its complement. In an alternative treatment of (topological) knot invariants, Fox [Fox] suggested using the category Br_Z(E) of branched coverings over E with the knot Z as their branching set. In the context of toposes, this has a precise meaning, as follows. Definition Let E be a connected locally connected topos. Let a given local homeomorphism inclusion E/Y >--->E. Denote by Br_Y(E) the full subcategory of Top_S/E determined by those complete spreads over E with locally connected domain, assumed to be the spread completions of a local homeomorphism E/X--->E which factors through the inclusion E/Y>--->E by means of a covering projection E/X--->E/Y. We now give a justification for using branched covers (instead of paths) to calculate (via automorphisms) the fundamental group of a knot. PropositionIn the context of the above definition, there is a canonical equivalence of Br_Y(E) with P_1(E/Y). In topological applications, a base point p is chosen and assumed to belong to E, E_1, E_2 and (therefore) E_0, but this point p is not to lie on the knot (whose complement in E is E_1). When the toposes are pointed, the fundamental groups of these pointed toposes are (discrete localic) groups. In this setting, we can say something quite specific. Corollary With assumptions as in the Corollary of the van Kampen theorem for toposes, and with the additional assumption that the topos E_0 is pointed by means of a geometric morphism p:S--->E_0, let G(E,p), G(E_1, p) and G(E_0, p) denote the (discrete localic) groups classified by the respective fundamental group toposes. Then, the induced continuous group homomorphism h: G(E_1, p) ---> G(E, p) is surjective, and its kernel is precisely the image N of the continuous homomorphism k: G(E_0, p) ---> G(E_1, p). It follows that G(E, p) is the quotient (discrete localic) group G(E_1, p)/N. 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