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2 Filteredness And The Point Of Every Galois Topos 2008

For Printing. Recent Issues. The Journal. For this reason,we work with the larger category of locally discrete locales, a category that is also suitable Received by the editors and, in revised form, Transmitted by Ross Street. Published on Key words and phrases: topos, fundamental progroupoid, spreads, zero dimensional locales, coveringprojections. Permission to copy for private use granted. The setting in which we work is that of a category V of locally discrete locales, givenaxiomatically to model the category of zero-dimensional locales.

An outline of the paper follows. A category V of locally discrete locales is introduced axiomatically.

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Its main examples are the categories D of discrete, and Z of zero-dimensional locales. Thekey notion in this section is that of a V-localic geometric morphism in TopS. The moti-vating example for it are the spreads [10]. We use results of [21, 15] in order to obtain, fromthe system of toposes GU E indexed by a generating category of covers in E , a limittopos.

In turn, this notion is given alternative equivalentversions, analogue to the result of [11] for locally constant coverings. In connection withthe locally constant coverings, we comment on an alternative construction of the funda-mental progroupoid of a Grothendieck topos given in [14] and inspired by shape theory[17, 18]. We also establishnew closure and stability theorems for the factors of the comprehensive V-factorization.

In particular, and unlike the locallyconnected case [11], there are no good topological invariants. We remark that this paper does not attempt to be self contained, a task which wouldhave been more suitable for a monograph. On the otherhand, all references that are needed for providing the background that is necessary for afull understanding of this paper are given in the references.

In addition, the systematic use of the universal property of fundamental pushoutconstruction renders this work into a purely categorical one. Rather than collecting remarks and questions in a separate section at the end of thepaper, these are scattered throughout the paper. Some of these remarks constitute ideasfor further research, whereas others are just lose ends which might prove interesting tocertain readers. Denote by TopS the 2-category of S -bounded toposes,geometric morphisms over S , and iso 2-cells.

Fox [16] in topology. Spreads are localic geometric morphisms. For a geometric morphismwhose domain is locally connected, the notion of a spread may be stated in terms ofconnected components [7]. We shall work with a category V of locales in the base topos S , modeled on thecategory Z of zero-dimensional locales. Such functors necessarily would have to forget about 2-cells in V. Thissuggests that we only consider such V in which the partial ordering of the morphisms istrivial.

We shall say that a locale Z is locally discrete if for every locale X thepartial ordering in Loc X, Z is discrete. Let L denote the category of locally discrete locales in S. L is closed under limits, which arecreated in Loc, the 2-category of locales in S.

The category D, of discrete locales isincluded in LD. L has the following additional properties: i If Y G Z is a locally discrete map, and Z is locally discrete, then Y is locally discrete. V is closed under open sublocales. Theinterior of a localic geometric morphism always exists. Let E be any topos in TopS.

For example,a D-determined geometric morphism in TopS is precisely a locally connected geometricmorphism.

D is given as follows. Din F. D has a fully faithful inverse image part, or is connected. Consider the commutative triangle 3. We now state a factorization theorem for V-determined geometric morphisms seeRemark 1. Such a factorization is unique up to equivalence. D is furthermore connected and V-determined, and the second factor is V-localic. Any locally connected geometric morphism in TopS factorsuniquely into a connected locally connected geometric morphism followed by a local home-omorphism. The main examples of a category V satisfying the clauses in Assump-tion 1.

Any discrete locale is locally discrete.

Notice that any open inclusion U X in Loc is etale. Any zero-dimensional locale X is locally discrete since O X has a basis consisting of constructibly complemented opens. In this example, 2 holds since any open sublocale of a spread is a spread. However, 2 cannot be strengthened to any etale p : Y G X in Loc E , since there exists a zero-dimensional space X and an etale map p : Y G X of locales, where Y is not zero-dimensional [9]. We give a proof of 3 below in Proposition 1.

We recall the following example due to Peter Johnstone and includedin [9]. Let X be azero-dimensional space that has the property: if for all etale maps Y G X of spaces, withY regular, Y is zero-dimensional, then X is discrete. The argument is by contradiction. Assume that X is zero-dimensional but not discrete. Construct Y G X etale as above, letting p take the role of 0. The space Y is not zero-dimensional, but by etaleness it is locally zero-dimensional.

The proof clearly reducesto showing that Y locally zero-dimensional and regular implies that Y is globally zero-dimensional, thus contradicting the assumption. This is shown to be the case. It is anopen question whether the analogous statement for arbitrary locales is true and if so inwhat form. This factorization is an instance of Theorem 1.

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Let E be a V-determined topos over S. Let E be a V-determined topos, and let U be a cover in E. The localic groupoid GU is locally discrete. Consider now a similar diagram for the morphism pU : ShS E! Bymethods of classifying toposes [24], we extract from it an etale complete groupoid GU inGU :. It also follows from Remark 1. This establishes the second claim. A morphism of K-torsorsin E corresponds to a natural transformation. This shows the third claim. Then there is an. U are in general sur-jective by Lemma 1. The square brackets in both cases indicate that the morphismsin those Hom-categories are to be taken to be iso 2-cells in TopS.