Theory and Applications of Categories, Vol. 20, No. 9, 2008, pp. 186–214.

FUNDAMENTAL PUSHOUT TOPOSES

MARTA BUNGE

Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption.

Introduction

The author [2, 5] introduced certain ‘fundamental pushout toposes’ in the construction and study of the prodiscrete fundamental groupoid of a locally connected topos. The main purpose of this paper is to generalize this construction without the local connectedness assumption.

The key to this program is the theory of spreads in topos theory [7], originating in [16, 25], and motivated therein by branched coverings. Our main tools are the factorization theorems [10] and [8] for geometric morphisms whose domains are definable dominances, a notion which states that the constructively complemented subobjects are well-behaved.

In connection with fundamental pushout toposes we also discuss covering projections and branched coverings without local connectedness by analogy with, but also departing from, the locally connected case [7].

The passage from the discrete to the zero-dimensional is not perfect, as the category of zero-dimensional locales does not have good enough closure properties. For this reason, we work with the larger category of locally discrete locales, a category that is also suitable

Received by the editors 2008-02-22 and, in revised form, 2008-05-27. Transmitted by Ross Street. Published on 2008-06-04. 2000 Mathematics Subject Classification: 18B25, 57M12, 18C15, 06E15. Key words and phrases: topos, fundamental progroupoid, spreads, zero dimensional locales, covering

projections.

\begin{matrix}c \\ ◯ \end{matrix} Marta Bunge, 2008. Permission to copy for private use granted.

for discussing localic reflections.

The setting in which we work is that of a category V of (locally discrete) locales, given axiomatically to model the category of zero-dimensional locales. Since also the category of discrete locales is an instance of such a V, we are able to discuss the differences between the discrete and the non-discrete (e.g., zero-dimensional) cases in a unified way.

An outline of the paper follows.

In § 1 we review and update definitions and results from [9] in order to include geometric morphisms. A category V of locally discrete locales is introduced axiomatically. Its main examples are the categories D of discrete, and Z of zero-dimensional locales. The key notion in this section is that of a V-localic geometric morphism in Top_{𝒮} . The motivating example for it are the spreads [10]. In the discrete case we prove special properties of the V-localic reflection of use in connection with the fundamental progroupoid.

In § 2 we generalize definitions and results of [2, 5] to the V-context. For a Vdetermined topos ℰ, and a cover U in ℰ, the localic V-reflection of ℰ /U determines a ‘fundamental V-pushout’ topos 𝒢_{U}(ℰ). We use results of [21, 15] in order to obtain, from the system of toposes 𝒢_{U}(ℰ) indexed by a generating category of covers in ℰ, a limit topos. Our results show that this topos (or its corresponding progroupoid) represents first ordered cohomology with coefficients in discrete groups. In the discrete case, we obtain the familiar result that the progroupoid may be replaced by a prodiscrete localic groupoid, using results from § 1.

In § 3 we characterize the fundamental V-pushout toposes 𝒢_{U}(ℰ) of § 2 in terms of what we call V-covering projections. In turn, this notion is given alternative equivalent versions, analogue to the result of [11] for locally constant coverings. In connection with the locally constant coverings, we comment on an alternative construction of the fundamental progroupoid of a (Grothendieck) topos given in [14] and inspired by shape theory [17, 18].

In § 4 we first review the V-comprehensive factorization [9] of a geometric morphism. with a V-determined domain into a V-initial geometric morphism followed by a V-fibration. The V-fibrations are an abstraction of the complete spreads [8], and are the geometric counterpart of the V-valued Lawvere distributions [23]. We also establish new closure and stability theorems for the factors of the comprehensive V-factorization.

In § 5 we introduce notions of locally trivial V-coverings and branched V-coverings suitably generalizing the corresponding notions in the locally connected case [7]. The key for the passages from one to the other is given by the analogue of a ‘pullback lemma’ from [13]. Beyond that point, we are faced with the two ‘inconvenient truths’ that affect the non-discrete (or non locally connected) case. In particular, and unlike the locally connected case [11], there are no good topological invariants. Also unlike the locally connected case [7], the ‘ideal knot’ need not exist in the general case.

We remark that this paper does not attempt to be self contained, a task which would have been more suitable for a monograph. A reason for this is that the paper involves a great deal of material from two different lines of research in which the author has been involved, to wit, the fundamental groupoid of a locally connected topos, and complete topos spreads in the non-locally connected case, up until now unrelated. On the other hand, all references that are needed for providing the background that is necessary for a full understanding of this paper are given in the references.

A justification for the work done here is not simply that it provides a construction of the fundamental groupoid of a topos in the non-locally connected case, but primarily that this is done in a conceptually simple manner (to replace connected components by clopen subsets), thereby providing a better understanding of the locally connected case itself. In addition, the systematic use of the universal property of fundamental pushout construction renders this work into a purely categorical one.

Rather than collecting remarks and questions in a separate section at the end of the paper, these are scattered throughout the paper. Some of these remarks constitute ideas for further research, whereas others are just lose ends which might prove interesting to certain readers.

1. V-localic reflections

Let 𝒮 be an elementary topos. Denote by Top_{𝒮} the 2-category of 𝒮 -bounded toposes, geometric morphisms over 𝒮, and iso 2-cells.

Our motivation for the notion of a V-localic geometric morphism in Top_{𝒮} is the notion of a spread (with a definable dominance domain) [10, 7], originally due to R. H. Fox [16] in topology.

A topos f : ℱ [figure] 𝒮 is a definable dominance [10] if the class of definable monos in ℱ [1] is well behaved — that is, classifiable and closed under composition. A geometric morphism over a base topos 𝒮 is said to be a spread if it has a definable generating family [7]. Spreads are localic geometric morphisms. The defining locales of spreads are said to be zero-dimensional. It is shown in [10] that any geometric morphism whose domain is a definable dominance has a unique pure surjection, spread factorization.

Every Grothendieck topos is a definable dominance since, over Set, the definable subobjects of an object are its complemented subobjects. Every locally connected topos is a definable dominance by the characterization theorem in [1]. For a geometric morphism whose domain is locally connected, the notion of a spread may be stated in terms of connected components [7].

We shall work with a category V of locales in the base topos 𝒮, modeled on the category Z of zero-dimensional locales. This generality suits the interplay between the discrete and the zero-dimensional, that is, between the locally connected and the ‘general’ cases, since the category D, of discrete locales, will also be regarded as a model of such a V (with special properties).

Denote by Loc the category of locales in ℰ, a partial ordering of its morphisms given by

m ≤ l : W [figure] X

if m^{∗}U ≤ l^{∗}U, for any U ^{∈} 𝒪(X). It is ℰ -indexed, with Σ satisfying the BCC.

We shall introduce certain functors V [figure] ℱ from a category V of locales to a topos ℱ in Top_{𝒮} . Such functors necessarily would have to forget about 2-cells in V. This suggests that we only consider such V in which the partial ordering of the morphisms is trivial. This feature justifies restricting our attention to locally discrete locales.

1.1. Definition. We shall say that a locale Z is locally discrete if for every locale X the partial ordering in Loc(X, Z) is discrete. Likewise, a map p : Z [figure] B is locally discrete if for every q : X [figure] B, Loc/B(q, p) is discrete.

Let L denote the category of locally discrete locales in 𝒮. It is an 𝒮 -indexed category, with Σ satisfying the BCC, and small hom-objects. L is closed under limits, which are created in Loc, the 2-category of locales in 𝒮. The category D, of discrete locales is included in LD.

1.2. Remark. L has the following additional properties:

(i) If Y [figure] Z is a locally discrete map, and Z is locally discrete, then Y is locally discrete.

(ii) If Y is locally discrete, then any locale morphism Y [figure] Z is locally discrete.

(iii) The pullback of a locally discrete map along another locally discrete map is again locally discrete.

(iv) If Z is locally discrete, then any sublocale S [figure] Z is also locally discrete.

1.3. Assumption. In what follows, V denotes a full 𝒮 -indexed subcategory of Loc such that, in addition, it satisfies

1. D ⊆ V ⊆ L.

2. V is closed under open sublocales.

3. Let

[figure: Y; X; p; W; Y; n; W; Z; q; Z; X; m]

be a pullback in Loc in which p is etale. If m ∈ V then n ∈ V.

The interior of a localic geometric morphism Sh_{ℱ} (Y) [figure] ℱ is an object d(Y) of ℱ such that

[figure: ℱ /d(Y); Sh_{ℱ} (Y); ℱ /d(Y); ℱ; Sh_{ℱ} (Y); ℱ]

commutes, and any ℱ /D [figure] Sh_{ℱ} (Y) over ℱ factors uniquely through ℱ /d(Y). The interior of a localic geometric morphism always exists.

For any topos f : ℱ [figure] 𝒮 in Top_{𝒮} , then there is a functor

F^{∗} : Loc [figure] ℱ

such that for X ∈ Loc, F^{∗}X = d(f^{#}X) is the interior of the locale f^{#}X, where the latter is defined by the bipullback

[figure: ℱ; 𝒮; f; Sh_{ℱ} (f^{#}X); ℱ; Sh_{ℱ} (f^{#}X); Sh_{𝒮} (X); Sh_{𝒮} (X); 𝒮]

in Top_{𝒮} . In particular, for any locale X in 𝒮, there is a canonical geometric morphism

ℱ /F^{∗}X [figure] Sh_{𝒮} (X)

over 𝒮, which we refer to as ‘the projection’.

1.4. Definition. A topos f : ℱ [figure] ℰ over 𝒮 is said to be V-determined if there is an 𝒮 -indexed left adjoint F_{!} ⊣ F^{∗} : V [figure] ℱ, such that ‘the BCC for etales holds, in the sense that for any etale map p : Y [figure] X in V, the transpose (below, right) of a pullback square (below, left) is again a pullback.

[figure: E; F^{∗}X; m; P; E; q; P; F^{∗}Y; F^{∗}Y; F^{∗}X; F^{∗}p; F_{!}E; X; m̂; F_{!}P; F_{!}E; F_{!}q; F_{!}P; Y; Y; X; p].

1.5. Remark. The definition of a V-determined topos can be relativized to any geometric morphism. Let ℰ be any topos in Top_{𝒮} . Denote by Loc(ℰ) the category of locales in ℰ and, similarly, define D(ℰ) ≅ ℰ, L(ℰ), and V(ℰ), the latter axiomatically by analogy with Assumption 1.3. Let ψ : ℱ [figure] ℰ be a morphism in Top_{𝒮} . We say that ψ is a V-determined geometric morphism if there is an ℰ -indexed left adjoint Ψ_{!} ⊣ Ψ^{∗} : V(ℰ) [figure] ℱ, such that ‘the BCC for etales in V(ℰ) holds. (For example, a D-determined geometric morphism in Top_{𝒮} is precisely a locally connected geometric morphism.)

Let ψ : ℱ [figure] ℰ be a V-determined geometric morphism in Top_{𝒮} . For any object D of ℱ, let

ρ_{D} : ℱ /D  [figure] Sh_{ℰ} (Ψ_{!}D)  (1)

be the composite of the projection ℱ /Ψ^{∗}(Ψ_{!}D) [figure] Sh_{ℰ} (Ψ_{!}D) with the unit of adjointness Ψ_{!} ⊣ Ψ^{∗}.

1.6. Lemma. The inverse image part of ρ_{D} : ℱ /D [figure] Sh_{ℰ} (Ψ_{!}D) is given as follows. If p : Y [figure] Ψ_{!}D is an etale map in Loc (ℰ), then (ρ_{D})^{∗}(p) is the left vertical leg of the pullback

[figure: D; Ψ^{∗}Ψ_{!}D; η_{D}; (ρ_{D})^{∗}(Y); D; (ρ_{D})^{∗}(Y); Ψ^{∗}Y; Ψ^{∗}Y; Ψ^{∗}Ψ_{!}D; Ψ^{∗}p]  (2)

in ℱ.

1.7. Lemma. Let ψ : ℱ [figure] ℰ be a V-determined geometric morphism in Top_{𝒮} .

1. Then, for every object D of ℱ, ρ_{D} in (1) is a surjection.

2. If V is such that the BCC holds for all etale morphisms Y [figure] X with X ∈ V(ℰ), then ρ_{D} in (1) is connected and V-determined in the sense of Remark 1.5.

Proof.

1. If Y [figure] Ψ_{!}D is etale, then Y is locally in V(ℰ) by Assumption 1.3(2). For each open inclusion p : U [figure] Ψ_{!}D, the BCC for etales in V(ℰ) implies that the transpose Ψ_{!}ρ_{D}^{∗}(U) [figure] U of the top horizontal in the diagram (2) is an isomorphism. This implies that ρ_{D} is a surjection.

2. For an arbitrary etale map Y [figure] Ψ_{!}D, the BCC implies that Ψ_{!}ρ^{∗}(Y) ≅ Y, so that ρ_{D} : ℱ /D [figure] Sh_{ℰ} (Ψ_{!}D) has a fully faithful inverse image part, or is connected. That ρ_{D} is V-determined follows from the same hypothesis, since in that case Ψ_{!} trivially preserves etaleness.

□

Consider a commutative triangle

[figure: ℱ; 𝒮; f; ℱ; ℰ; π; ℰ; 𝒮; e]  (3)

in Top_{𝒮} . If Z is a locale in V, then there is a geometric morphism ℱ /π^{∗}(E^{∗}Z) [figure] Sh_{ℱ} (Z), which factors through ℱ /F^{∗}(Z) by a morphism π^{∗}(E^{∗}Z) [figure] Ψ^{∗}(Z) in ℱ, since Ψ^{∗}(Z) is the interior of Sh_{ℱ} (ψ^{#}Z). Thus, there is a natural transformation

α : (π^{∗} · E^{∗}) [figure] F^{∗} : V  [figure] ℱ  (4)

1.8. Remark. The 2-cell α in (4) is an isomorphism when restricted to discrete locales, or else when π is a local homeomorphism.

1.9. Definition. Consider the commutative triangle (3).

1. We shall say that ρ : ℱ [figure] ℰ is V–initial if the transpose α̂ : E^{∗} [figure] (ρ_{∗} · F^{∗}) : V [figure] ℰ of (4) under ρ^{∗} ⊣ ρ_{∗} is an isomorphism.

2. We shall say that ρ : ℱ [figure] ℰ is V-localic if there is given Z ∈ V(ℰ) where φ : Sh_{ℰ} (Z) [figure] ℰ has a V-determined domain, and an equivalence σ in the commutative triangle:

[figure: ℱ; ℰ; ρ; ℱ; Sh_{ℰ} (Z); σ; Sh_{ℰ} (Z); ℰ; φ]  (5)

1.10. Remark. If f : ℱ [figure] 𝒳 is a geometric morphism between V-determined toposes, then the canonical geometric morphism ρ in (1) is V-initial iff the canonical 2-cell β : (F_{!} · ρ^{∗}) [figure] X_{!} corresponding to α̂ by taking left adjoints, is an isomorphism.

We now state a factorization theorem for V-determined geometric morphisms (see Remark 1.5).

1.11. Theorem. Let ψ : ℱ [figure] ℰ in Top_{𝒮} be a V-determined geometric morphism. For any object D of ℱ, let

ℱ /D [figure: ρ_{D}] Sh_{ℰ} (Ψ_{!}D) [figure: φ_{D}] ℰ

be the canonical factorization.

1. For a general V, the first factor is surjective V-initial over ℰ, and the second factor is V-localic. Such a factorization is unique up to equivalence.

2. If V is closed under all etale p : Y [figure] X with X ∈ V(ℰ), then the V-initial first factor ρ_{D} : ℱ /D [figure] Sh_{ℰ} (Ψ_{!}D) is (furthermore) connected and V-determined, and the second factor is V-localic. Such a factorization is unique up to equivalence.

1.12. Corollary. Any locally connected geometric morphism in Top_{𝒮} factors uniquely into a connected locally connected geometric morphism followed by a local homeomorphism.

1.13. Example. The main examples of a category V satisfying the clauses in Assumption 1.3 are :

1. V = D. Any discrete locale is locally discrete. Notice that any open inclusion U [figure] X in Loc is etale. In this example, (2) may be strengthened to any etale p : Y [figure] X in Loc.

2. V = Z. 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 [figure] X in Loc(ℰ), since there exists a zero-dimensional space X and an etale map p : Y [figure] X of locales, where Y is not zero-dimensional [9].

As for (3), indeed, in Top_{𝒮} , the pullback of a spread with a definable dominance domain along a local homeomorphism is again one such. We give a proof of (3) below in Proposition 1.14.

1.14. Proposition. Consider a bipullback in Top_{𝒮} , in which ξ is a local homeomorphism:

[figure: ℱ; ℰ; ξ; 𝒳; ℱ; φ; 𝒳; 𝒴; π; 𝒴; ℰ; ψ]

If ψ : 𝒴 [figure] ℰ is a spread then φ is spread .

Proof.

In the bipullback, π is a local homeomorphism since ξ is one. Thus, 𝒳 ≅ 𝒴 /B and π is identified via this equivalence with the canonical local homeomorphism Σ_{B} ⊣ B^{∗} ⊣ Π_{B} : 𝒴 /B [figure] 𝒴 . A monomorphism m in 𝒴 /B is 𝒮 -definable iff Σ_{B}(m) is 𝒮 -definable in 𝒴 . This implies that 𝒴 /B is a definable dominance since 𝒴 is one.

Since 𝒳 is a definable dominance, we may factor φ into its pure surjection ρ : 𝒳 [figure] 𝒲 and a spread τ : 𝒲 [figure] ℱ parts: φ ≅ τ · ρ. Then, since 𝒱 is a definable dominance, we may factor the composite ξ · τ into its pure surjection and spread parts. This is represented in the diagram below, where ζ is uniquely defined using orthogonality, since ψ is a spread and η · ρ is the composite of two pure surjections hence a pure surjection.

[figure: ℱ; ℰ; ξ; 𝒳; ℱ; φ; 𝒳; 𝒴; π; 𝒴; ℰ; ψ; 𝒵; 𝒵; ζ; 𝒲; τ; 𝒲; ρ; η]

Since the outer square is a pullback, there is 𝒱 [figure: θ] 𝒳 such that π · θ ≅ ζ · η and φ · θ ≅ τ. The universal property of the pullback implies that θ · ρ ≅ id_{𝒳} . Hence, ρ · θ · ρ ≅ ρ. Therefore, the two geometric morphisms ρ · θ and id_{𝒲} from the spread τ to itself agree when precomposed with the pure surjection ρ. They must therefore agree: ρ · θ ≅ id_{𝒲} . We have shown that ρ is an equivalence, so that φ is a spread.

□

3. V = L. In this case, (2) and (3) may both be strengthened to any p : Y [figure] X ∈ L.

1.15. Example. We recall the following example (due to Peter Johnstone and included in [9]). Let X be the subspace \{0\}∪\{\frac{1}{n} | n ≥ 1\} of ℝ, and let Y = X +X/ ∼, obtained by identifying the two \frac{1}{n}’s, for every n. The topology on Y is T_{1}, but not Hausdorff. The map Y [figure] X identifying the two 0’s is etale, X is 0-dimensional, but Y is not 0-dimensional.

The converse for spaces (due to Gábor Lukács) holds in the following form. Let X be a zero-dimensional space that has the property: if for all etale maps Y [figure] X of spaces, with Y regular, Y is zero-dimensional, then X is discrete. The argument is by contradiction. Assume that X is zero-dimensional but not discrete. Let p ∈ X be an isolated point. Construct Y [figure] X etale as above, letting p take the role of 0. The space Y is not zerodimensional, but (by etaleness) it is locally zero-dimensional. The proof clearly reduces to showing that Y locally zero-dimensional and regular implies that Y is globally zerodimensional, thus contradicting the assumption. This is shown to be the case. It is an open question whether the analogous statement for arbitrary locales is true and if so in what form.

2. The fundamental V-progroupoid

In this section we abstract the fundamental pushouts of [2, 5] in the V-context of § 1.

We recall that for ℰ locally connected in Top_{𝒮} , the terminology ‘fundamental pushout’ was applied to refer to any pushout

[figure: ℰ /e_{!}U; 𝒢_{U}(ℰ); p_{U}; ℰ /U; ℰ /e_{!}U; ρ_{U}; ℰ /U; ℰ; φ_{U}; ℰ; 𝒢_{U}(ℰ); σ_{U}]  (6)

where U [figure] 1 in ℰ, φ_{U} : ℰ /U [figure] ℰ is the canonical local homeomorphism, and ρ_{U} is the first factor in the canonical factorization of the locally connected geometric morphism

ℰ /U [figure: φ_{U}] ℰ [figure: e] 𝒮

into a connected locally connected geometric morphism followed by a local homeomorphism.

This factorization is an instance of Theorem 1.11(2). This suggests that we give a more general notion of ‘fundamental pushout’ in the V-context, by using Theorem 1.11(1).

2.1. Definition. Let ℰ be a V-determined topos over 𝒮. For an epimorphism U [figure] 1 in ℰ, the following pushout in Top_{𝒮}

[figure: Sh_{ℰ} (E_{!}U); 𝒢_{U}(ℰ); p_{U}; ℰ /U; Sh_{ℰ} (E_{!}U); ρ_{U}; ℰ /U; ℰ; φ_{U}; ℰ; 𝒢_{U}(ℰ); σ_{U}]  (7)

where ψ_{U} : ℰ /U [figure] ℰ is the canonical local homeomorphism, and where ρ_{U} was defined in (1) applied to the V-determined topos ℰ /U [figure: φ_{U}] ℰ [figure: e] 𝒮, is said to be a fundamental V-pushout .

2.2. Theorem. Let ℰ be a V-determined topos, and let U be a cover in ℰ. Then,

1. The topos 𝒢_{U}(ℰ) in the fundamental V-pushout is the classifying topos ℬ(𝔾_{U}) of an etale complete groupoid 𝔾_{U}, by an equivalence

𝒢_{U}(ℰ) ≃ ℬ(𝔾_{U})

which identifies p_{U} with the canonical localic point of ℬ(𝔾_{U}).

2. The localic groupoid 𝔾_{U} is locally discrete.

3. The locally discrete groupoid 𝔾_{U} classifies U-split K-torsors in ℰ for discrete groups K.

Proof.

For the morphism φ_{U} : ℰ /U [figure] ℰ in Top_{𝒮} , there is an induced truncated simplicial diagram, namely the kernel of φ_{U}, denoted Ker(φ_{U}):

\xymatrix{ℰ /U ×_{ℰ} ℰ /U ×_{ℰ} ℰ /U \ar[r]^{π_{01}} \ar[r]^{π_{02}} \ar[r]|{π_{12}} & ℰ /U ×_{ℰ} ℰ /U \ar[r]^{π_{0}} \ar[r]|{π_{1}} & ℰ /U \ar[l]_{i}}

where, for example, ℰ /U ×_{ℰ} ℰ /U denotes the bipullback of φ_{U} along itself.

There is a category 𝒟(φ_{U}), an object of which consists of an object y of ℰ /U equipped with an isomorphism η : π_{0}^{∗}(y) ≅ π_{1}^{∗}(y) satisfying the cocycle and normalization conditions. A morphism in 𝒟(φ_{U}) from (y, η) to (z, ζ) is a morphism f : y [figure] z in ℰ /U compatible in the obvious sense with η and ζ.

Let Φ_{U} : 𝒟(φ_{U}) [figure] ℰ /U be the canonical geometric morphism Since φ_{U} is a locally connected surjection, it is of effective descent. In other words, Φ_{U} is an equivalence.

Consider now a similar diagram for the morphism p_{U} : Sh_{𝒮} (E_{!}U) [figure] 𝒢_{U}(ℰ). By methods of classifying toposes [24], we extract from it an etale complete groupoid 𝔾_{U} in 𝒢_{U} :

\xymatrix{G_{2} \ar[r]^{p} \ar[r]^{m} \ar[r]|{q} & G_{1} \ar[r]^{d} \ar[r]|{c} & G_{0} \ar[l]_{i}}

Denote by 𝒟(p_{U})the descent category for Ker(p_{U}) and by P_{U} : 𝒟(p_{U}) [figure] 𝒢_{U}(ℰ) the canonical geometric morphism. We wish to prove any of the following equivalent statements:

(i) ℬ(𝔾_{U}) ≅ 𝒢_{U}(ℰ)

(ii) p_{U} is of effective descent.

(iii) P_{U} : 𝒟(p_{U}) [figure] 𝒢_{U}(ℰ) is an equivalence.

First, notice that ρ_{U} : ℰ /U [figure] Sh_{𝒮} (E_{!}U), alternatively denoted ρ_{0} : ℰ /U [figure] Sh(G_{0}) induces a morphism of truncated simplicial objects in Top_{𝒮} (in simplified notation) depicted together with the pushout diagram.

[figure: Sh(G_{2}); Sh(G_{1}); (ℰ /U)^{3}; Sh(G_{2}); ρ_{2}; (ℰ /U)^{3}; (ℰ /U)^{2}; (ℰ /U)^{2}; Sh(G_{1}); ρ_{1}; Sh(G_{0}); ℰ /U; ℰ /U; Sh(G_{0}); ρ_{0}; 𝒢_{U}(ℰ); p_{U}; ℰ; φ_{U}; ℰ; 𝒢_{U}(ℰ); σ_{U}]

Let λ : Sh(G_{0}) [figure] 𝒦 satisfy the descent conditions for Ker(p_{U}). Then, the composite

ℰ /U [figure: ρ_{0}] Sh(G_{0}) [figure: λ] 𝒦

satisfies the descent conditions for Ker(φ_{U}). (For instance, for y an object of 𝒦 , and η : d_{0}^{∗}λ^{∗}(y) ≅ c^{∗}λ^{∗}(y) in Sh(G_{1}), then ρ_{0}^{∗}η : ρ_{1}^{∗}d_{0}^{∗}λ^{∗}(y) ≅ ρ_{1}^{∗}d_{1}^{∗}λ^{∗}(y) and, in turn, d_{0}^{∗}ρ_{0}^{∗}λ^{∗}(y) ≅ d_{1}^{∗}ρ_{0}^{∗}λ^{∗}(y).) Since φ_{U} is of effective descent, there exists a unique τ : ℰ [figure] 𝒦 , such that the square

[figure: Sh(G_{0}); 𝒦; λ; ℰ /U; Sh(G_{0}); ρ_{U}; ℰ /U; ℰ; φ_{U}; ℰ; 𝒦; τ]  (8)

commutes up to an iso 2-cell.

By the universal property of the pushout (7), there exists a unique κ : 𝒢_{U} [figure] 𝒦 such that κ·p_{U} ≅ λ and κ·σ_{U} ≅ τ. Since both φ_{U} and ρ_{U} are surjections, we have the following

(κσ_{U} ≅ τ) ⇔ (κσ_{U}φ_{U} ≅ τφ_{U}) ⇔ (κp_{U}ρ_{U} ≅ λρ_{U}) ⇔ (κp_{U} ≅ λ)

Thus, we may rephrase the comment above as follows. There exists a unique κ : 𝒢_{U}(ℰ) [figure] 𝒦 such that κ · p_{U} ≅ λ. This shows that p_{U} is of effective descent and it establishes the first claim.

We observe that since Sh_{𝒢_{U}} (G_{0}) ≅ Sh_{𝒮} (E!U) [figure] 𝒮 is localic defined by a (zerodimensional, hence) locally discrete locale E_{!}U, then also p_{U} : Sh(E_{!}U) [figure] 𝒢_{U}(ℰ) is localic defined by a locally discrete locale G_{0} in 𝒢_{U}(ℰ), by Remark 1.2.

It also follows from Remark 1.2 that all objects and morphisms defining 𝔾_{U} are locally discrete, that is, 𝔾_{U} is a locally discrete groupoid in 𝒢_{U}(ℰ). (For instance, in the pullback

[figure: Sh(G_{0}); 𝒢_{U}(ℰ); p_{U}; Sh(G_{1}); Sh(G_{0}); π_{1}; Sh(G_{1}); Sh(G_{0}); π_{0}; Sh(G_{0}); 𝒢_{U}(ℰ); p_{U}]

G_{1} is locally discrete.) This establishes the second claim.

We now argue that 𝔾_{U} (or, equivalently, ℬ(𝔾_{U})) classifies U-split K torsors for discrete groups K. Crucial for this is that D ⊆ V in Assumption 1.3 and the universal property of the V-reflection.

For any cover 𝒰 = ⟨U [figure] 1, I, U [figure: ζ] e^{∗}I⟩ in a topos ℰ [figure: e] 𝒮, the topos 𝒫_{𝒰} (ℰ) in the following pushout is equivalent to the category Split(𝒰 ) [11]:

𝒮 /I  𝒫_{𝒰} (ℰ)_{p_{𝒰}}  [figure] ℰ /U 𝒮 /I^{λ_{𝒰}} [figure] ℰ /U  ℰ^{φ_{U}}  [figure] ℰ 𝒫_{𝒰} (ℰ)^{σ_{𝒰}} [figure] .  (9)

For a group K in 𝒮, we consider the K-torsors in ℰ split by a cover 𝒰 = ⟨U, ζ⟩, where U [figure] 1 and ζ : U [figure] e^{∗}I in ℰ. By Diaconescu’s theorem, a K-torsor in ℰ is equivalently given by a geometric morphism τ : ℰ [figure] ℬ(K). A morphism of K-torsors in ℰ corresponds to a natural transformation. There is a category

Tors(ℰ; K)^{𝒰}

of K-torsors in ℰ split by 𝒰 .

To say that the K-torsor (represented by) τ is 𝒰 -split is to say that there is a factorization

[figure: 𝒫_{𝒰} (ℰ); ℬ(K); λ; ℰ; 𝒫_{𝒰} (ℰ); σ_{𝒰}; ℰ; ℬ(K); τ]

where the ‘point’ p_{𝒰} : 𝒮 [figure] 𝒫_{𝒰} (ℰ) plays no role.

There is induced a double pushout diagram

[figure: Sh(E_{!}U); 𝒢_{U}(ℰ); q_{U}; ℰ /U; Sh(E_{!}U); ρ_{U}; ℰ /U; ℰ; φ_{U}; ℰ; 𝒢_{U}(ℰ); σ_{U}; 𝒮 /I; 𝒫_{𝒰} (ℰ); p_{𝒰}; 𝒮 /I; κ_{𝒰}; 𝒫_{𝒰} (ℰ); χ_{𝒰}; λ_{𝒰}; τ_{𝒰}]  (10)

where κ_{𝒰} is induced the universal property of the V-localic reflection, and χ_{𝒰} by the pushout property.

It follows from the commutative square

[figure: Sh(E_{!}U); ℬ(K); γ; ℰ /U; Sh(E_{!}U); ρ_{U}; ℰ /U; ℰ; φ_{U}; ℰ; ℬ(K); τ]  (11)

where γ = λ · p_{𝒰} · κ_{𝒰} and the pushout property of (7), that there exists a unique δ : 𝒢_{U}(ℰ) [figure] ℬ(K) such that the diagram

[figure: 𝒢_{U}(ℰ); ℬ(K); δ; ℰ; 𝒢_{U}(ℰ); σ_{U}; ℰ; ℬ(K); τ]

commutes up to an iso 2-cell. This shows the third claim. □

2.3. Corollary. For any group K and any cover U [figure] 1 in a V-determined topos ℰ, there is an equivalence

Top_{𝒮} (ℬ(𝔾_{U}), ℬ(K)) ≃ Tors(ℰ; K)^{U}

natural in K, where we write simply U for the canonical cover 𝒰 = ⟨U [figure] 1, η : E_{!}U, U [figure] E^{∗}E_{!}U⟩.

We proceed to define, for Cov(ℰ) a small generating category of covers in ℰ and morphisms α : U [figure] V witnessing (not uniquely) that U ≤ V, a functor

𝒢 : Cov(ℰ) [figure] Top_{𝒮} .

We let 𝒢 (𝒰 ) = 𝒢_{𝒰} (ℰ). Let U ≤ V, where U and V are covers in ℰ, and let α : U [figure] V be a given morphism in Cov(ℰ) witnessing the relation. Then there is an induced geometric morphism φ_{α} : 𝒢_{U}(ℰ) [figure] 𝒢_{V} (ℰ) over 𝒮 which commutes with the canonical localic points p_{U} and p_{V}.

This is clear from the cube below (in simplified notation), using the pushout property (7) of the back face:

[figure: ℰ /V; ℰ; ℰ /U; ℰ; Sh(E_{!}V); 𝒢_{V}; Sh(E_{!}U); 𝒢_{U}]

Recall that the geometric morphisms ρ_{U} : ℰ /U [figure] Sh(E_{!}U) are (in general) surjective by Lemma 1.7. It follows from the pushout that each σ_{U} : ℰ [figure] 𝒢_{U}(ℰ) is also surjective and, in particular, for each α as above, also the geometric morphisms φ_{α} : 𝒢_{U}(ℰ) [figure] 𝒢_{V} (ℰ) are surjective.

Since each induced geometric morphism φ_{α} : 𝒢_{U}(ℰ) [figure] 𝒢_{V} (ℰ) over 𝒮 commutes with the localic points p_{U} and p_{V}, it may therefore be interpreted as an object of the full subcategory

Top_{𝒮} [ℬ(𝔾_{U}), ℬ(𝔾_{V})]_{+}

of Top_{𝒮} [ℬ(𝔾_{U}, ℬ(𝔾_{V})] whose objects are geometric morphisms commuting with the canonical localic points [3]. The square brackets in both cases indicate that the morphisms in those Hom-categories are to be taken to be iso 2-cells in Top_{𝒮} .

As shown in [5] there is a strong equivalence of categories:

Hom(𝔾_{U}, 𝔾_{V}) ≃ Top_{𝒮} [ℬ(𝔾_{U}), ℬ(𝔾_{V})]_{+}.

We derive from it that there are induced groupoid homomorphisms g_{α} : 𝔾_{U} [figure] 𝔾_{V}, for each α. Furthermore, since the geometric morphisms φ_{α} are surjective, the homomorphisms g_{α} are also surjective.

The system consisting of the locally discrete groupoids 𝔾_{U} and the surjective homomorphisms g_{α} is not filtered in general. Nevertheless, the system is a bifiltered 2-category in the sense of [21] or, equivalently, a 2-filtered 2-category in the sense of [15]. As in [15], the bicolimit of the 2-functor exists.

Let 𝔾 = lim{G_{U} | U ∈ Cov(ℰ)}. This is a locally discrete groupoid. There is a canonical geometric morphism

ℬ(lim{G_{U} | U ∈ 𝒰 }) [figure]lim{ℬ(𝔾_{U}) | U ∈ 𝒰 }

Since the transition geometric morphisms φ_{α} : ℬ(𝔾_{U}) [figure] ℬ(𝔾_{V}) for U ≤ V are only surjective, this canonical geometric morphism need not be an equivalence.

In spite of this, the locally discrete progroupoid ℙ = {𝔾_{U} | U ∈ Cov(ℰ)} represents cohomology of ℰ with coefficients in discrete groups. Explicitly, the claim is that for a discrete group K, there is an isomorphism

H^{1}(ℰ; K) ≃ [ℙ, K].

This is shown as in [14].

2.4. Remark. There is a case of special interest, namely when V = D, the category of discrete locales. A D-determined topos ℰ is precisely a locally connected topos. In this case, there are additional properties of the fundamental progroupoid ℙ = ⟨𝔾_{U} | U ∈ Cov(ℰ)⟩ of ℰ, which we proceed to indicate.

1. Firstly, the fundamental progroupoid ℙ may in this case be equivalently replaced by a prodiscrete localic groupoid G. That the localic groupoids G_{U} are discrete follows from the fact that the p_{U} : 𝒮 /e_{!}U [figure] 𝒢_{U}(ℰ) are local homeomorphisms, in turn since the 𝒮 /e_{!}U [figure] 𝒮 are local homeomorphisms. Further, it follows from Lemma 1.7(2) and properties of the pushouts (7) that each σ_{U}, hence also the transition geometric morphisms φ_{α} : ℬ(G_{U}) [figure] ℬ(G_{V}), are connected. This in turn implies that the canonical geometric morphism

ℬ(lim{G_{U} | U ∈ 𝒰 }) [figure]lim{ℬ(G_{U}) | U ∈ 𝒰 }

is an equivalence. In particular, 𝔾 = lim{G_{U} | U ∈ 𝒰 } is a prodiscrete localic groupoid which represents cohomology of ℰ with coefficients in discrete groups.

2. Secondly, in addition to the fundamental theorem of Galois theory, in the form 𝒢 (ℰ) ≅ ℬ(𝒢), there is implicit a Galois theory in the sense of [19]. From Lemma 1.7(2) we deduce that the ρ_{U} : ℰ /U [figure] 𝒮 /e_{!}U are D-determined, that is, locally connected. This fact, applied to the fundamental pushout (7) gives, using a result from [2], that p_{U} : 𝒮 /e_{!}U [figure] 𝒢_{U}(ℰ) (and σ_{U} : ℰ [figure] 𝒢_{U}(ℰ)) is locally connected and a surjection, hence an 𝒮 -essential surjection, therefore with a representable inverse image part. The representor is a locally constant object ⟨A, σ⟩, which is a ‘normal’ object in the sense of [19]. In particular, each 𝒢_{U}(ℰ) ≅ ℬ(G_{U}) is a Galois topos, from which it follows that also the limit lim{ℬ(G_{U}) | U ∈ 𝒰 } is a Galois topos. The reader is referred to [5] for details.

2.5. Remark. The fundamental pushout definition of the coverings fundamental groupoid of a locally connected topos [2] became a useful tool in the comparison theorems between the coverings and the paths versions of the fundamental groupoid [4, 12]. Such a comparison could not be attempted in this context without a prior generalization of the paths fundamental groupoid to the non-locally connected case. The latter might prove to be an interesting project.

3. V-covering projections

In this section we introduce and discuss a notion of V-covering projection and use it to analyze the fundamental V-pushout toposes of Definition 2.1.

3.1. Definition. Let ℰ be a V-determined topos. An object A of ℰ is said to be a V-covering projection object split by a cover 𝒰 = ⟨U [figure] 1, X, η : U [figure] E^{∗}U⟩ in ℰ if A is part of a 3-tuple ⟨A, α, θ⟩ where α is an etale morphism Y [figure] X in Loc, with X ∈ V, and θ is an isomorphism

E^{∗}Y ×_{E^{∗}X} U [figure: θ] A × U

over U. We call the corresponding geometric morphism ℰ /A [figure] ℰ a V-covering projection split by 𝒰 .

A morphism ⟨A, α, θ⟩ [figure] ⟨B, β, κ⟩ between V-covering projection objects split by 𝒰 = ⟨U [figure] 1, X, η : U [figure] E^{∗}X⟩ is given by a pair (f , γ) with f : A [figure] B a morphism in ℰ and γ : Y [figure] Z a morphism in Loc over X, such that

\xymatrix{E^{∗}Y ×_{E^{∗}X} U \ar[r]^{θ} \ar[d]^{E^{∗}γ×U} & A × U \ar[d]^{f×U} \\ E^{∗}Z ×_{E^{∗}X} U \ar[r]_{κ} & B × U}

commutes.

3.2. Proposition. Let φ : ℱ [figure] ℰ be a local homeomorphism in Top_{𝒮} . Let 𝒰 = ⟨U [figure] 1, X, η : U [figure] E^{∗}U⟩ be a cover in ℰ. Then

(i) φ^{∗}𝒰 = ⟨φ^{∗}U [figure] φ^{∗}1 ≅ 1, X, φ^{∗}η : (φ^{∗} · E^{∗})X ≅ F^{∗}X⟩ is a cover in ℱ.

(ii) If A is V-covering projection object in ℰ split by 𝒰 , then φ^{∗}A is a V-covering projection object in ℱ, split by φ^{∗}𝒰 .

Proof. The conclusion is easily checked using Definition 3.1, and the fact that the 2-cell φ^{∗}E^{∗} [figure] F^{∗} is an isomorphism when φ is a local homeomorphism (see Remark 1.8). □

The following statement is similar to one shown in [11] for locally constant coverings (or covering projections). Due to the restricted nature of the BCC in the V-case the proof given therein is not directly applicable. We give one which relies directly on the V-reflection universal property.

3.3. Proposition. If e : ℰ [figure] 𝒮 is V-determined, A an object of ℰ, and U [figure] 1 (a ‘cover’) in ℰ, the following are equivalent:

(i) A is a V-covering projection object split by U.

(ii) There is an etale morphism Y [figure: α] E_{!}U in Loc(𝒮), and a pullback

\xymatrix{A × U \ar[r]^{π_{2}} \ar[d] & U \ar[d]^{η_{U}} \\ E^{∗}Y \ar[r]_{E^{∗}α} & E^{∗}E_{!}U}

where η_{U} is the unit of adjointness E_{!} ⊣ E^{∗} evaluated at U.

(iii) There is an etale morphism Y [figure: β] X in Loc(𝒮) with X ∈ V(𝒮), a morphism η : U [figure] E^{∗}X, and a pullback

\xymatrix{A × U \ar[r]^{π_{2}} \ar[d] & U \ar[d]^{η} \\ E^{∗}Y \ar[r]_{E^{∗}β} & E^{∗}X}

Proof. The items (i) and (iii) are equivalent. Trivially, (ii) ⇒ (iii) As for (iii) ⇒ (ii), suppose that A is U-split via β : Y [figure] X, ζ : U [figure] E^{∗}X and an iso θ : E^{∗}Y ×_{E^{∗}X}U [figure] A×U over U. Then the composite rectangle

\xymatrix{A × U \ar[r]^{π_{2}} \ar[d] & U \ar[d]^{η_{U}} \\ E^{∗}Y \ar[r]^{E^{∗}α} \ar[d]_{=} & E^{∗}E_{!}U \ar[d]^{E^{∗}ζ′} \\ E^{∗}Y \ar[r]_{E^{∗}β} & E^{∗}X}

is a pullback and, in the lower ‘square’, ζ′ : E_{!}U [figure] X exists and is the unique such that ζ′ · α = β is obtained from the universal property of the V-localic reflection. it follows easily that the top square is a pullback. □

Let ℰ be an V-determined topos. Denote by 𝒞_{V}(ℰ)(𝒰 ) the category of the V-covering projections objects split by 𝒰 = ⟨U [figure] 1, η : U [figure] E^{∗}E_{!}U⟩ and their morphisms. There is a forgetful functor Ψ_{U} : 𝒞_{V}(𝒰 ) [figure] ℰ defined by ⟨A, α, θ⟩ ↦ A and (f, γ) ↦ f. This functor is in general neither full nor faithful.

3.4. Proposition. Let 𝒢_{U}(ℰ) be the topos in the fundamental pushout applied to a V-determined topos ℰ and a cover U. Then there exists an equivalence functor Φ_{U} : 𝒢_{U}(ℰ) [figure] 𝒞_{V}(ℰ)(U). Under this equivalence, the functor σ_{U}^{∗} : 𝒢_{U}(ℰ) [figure] ℰ corresponds to the forgetful functor Ψ_{U} : 𝒞_{V}(ℰ)(U) [figure] ℰ, defined by ⟨A, α, ζ⟩ ↦ A and ⟨a, γ⟩ ↦ a. This functor is in general neither full nor faithful.

Proof. An object of 𝒢_{U}(ℰ) in the fundamental pushout of toposes is precisely a locally V-trivial object split by U in the sense of Definition 3.3. A similar observation applies to morphisms in the category 𝒢_{U}(ℰ). This is a consequence of the well-known fact that pushouts in Top_{𝒮} are calculated as pullbacks in CAT of the inverse image parts. □

3.5. Remark. An alternative construction of the fundamental progroupoid of a general Grothendieck topos ℰ which also uses pushout toposes is that of [14]. However, the locally constant pushouts employed therein (albeit as a first approximation) just give the toposes 𝒫_{𝒰} (ℰ) for a cover 𝒰 = ⟨U [figure] 1, I, U [figure] e^{∗}I⟩. That this is not adequate in the non locally connected case is indicated by the fact that the canonical ‘point’ p_{𝒰} : 𝒮 /I [figure] 𝒫_{𝒰} (ℰ) need not be of effective descent . A correction is subsequently made in [14] by means of categories 𝒟_{𝒰} (ℰ) ⊆ 𝒫_{𝒰} (ℰ), with 𝒟_{𝒰} (ℰ) defined as the full subcategory of 𝒫_{𝒰} (ℰ) whose objects are sums of 𝒰 -split ‘covering projections’, a notion that is a generalization to Grothendieck toposes of the ‘covering projections’ off [18] for localic toposes, and which is itself inspired by the overlays of [17].

In pictures, the locally split pushout (top) can be completed to a commutative diagram

[figure: 𝒮 /I; 𝒫_{𝒰} (ℰ); p_{𝒰}; ℰ /U; 𝒮 /I; λ_{𝒰}; ℰ /U; ℰ; φ_{U}; ℰ; 𝒫_{𝒰} (ℰ); τ_{𝒰}; 𝒮 /I; 𝒟_{𝒰} (ℰ); r_{𝒰}; 𝒮 /I; id; 𝒟_{𝒰} (ℰ); ν_{𝒰}; μ_{𝒰}]  (12)

The category 𝒟_{𝒰} (ℰ) ⊆ 𝒫_{𝒰} (ℰ) is then shown to be a topos, that it is atomic, and that there exists an etale complete localic groupoid 𝔻_{𝒰} such that 𝒟_{𝒰} (ℰ) ≅ ℬ(𝔻_{𝒰} ), and such that it classifies 𝒰 -split K-torsors in ℰ for discrete groups K.

The construction of the 𝒟_{𝒰} (ℰ) is different from our construction of the 𝒢_{U}(ℰ). For instance, in our case (first exposed in [4]), the use of the fundamental pushouts requires no correction.

We also point out that (without further information) we cannot conclude that there is an equivalence between the toposes 𝒟_{𝒰} (ℰ) and 𝒢_{𝒰} (ℰ). Since both constructions generalize the locally connected case, these toposes are certainly equivalent when ℰ is locally connected. For a general ℰ, a property that 𝒟_{𝒰} (ℰ) and 𝒢_{𝒰} (ℰ) both share is the classification of torsors in ℰ for discrete groups. However, the localic groupoids that these toposes respectively classify are not in general discrete, so no conclusion about their possible equivalence can be drawn from this observation alone.

4. The comprehensive V-factorization

We now review and update the V-comprehensive factorization theorem [9], modeled on the pure, complete spread factorization in case of geometric morphisms with a locally connected domain [7], and on the hyperpure, complete spread factorization of [8] for geometric morphisms with a definable dominance domain.

The following is a generalization of the notion of a Lawvere distribution on a topos [23].

4.1. Definition. A V–distribution on a topos ℰ is an 𝒮 -indexed functor μ : ℰ [figure] V with an 𝒮 -indexed right adjoint μ ⊣ μ_{∗}. Denote by E_{V}(ℰ) the category of V-distributions on ℰ.

Let μ be a V-distribution on ℰ ≃ Sh(ℂ, J). Let 𝕄 be the category in 𝒮 with objects (C, U) with U ∈ 𝒪(μ(C)), and morphisms (C, U) [figure] (D, V) given by C [figure: m] D in ℂ such that U ≤ μ(m)^{∗}(V). For U ∈ 𝒪(μ(C)), denote by U [figure] μ(C) the corresponding open sublocale.

Let 𝒵 be the topos of sheaves for the topology on 𝕄 generated by the following families, which we call weak μ-covers: a family

{(C, U_{a}) [figure: 1_{C}] (C, U) | a^{∈} A}

is a weak μ-cover if ⋁ U_{a} = U in 𝒪(μ(C)). As usual there is a functor 𝕄 [figure] ℂ that

induces a geometric morphism P(𝕄)  [figure] P(ℂ), and hence one 𝒵  [figure] P(ℂ).

4.2. Definition. Let μ be a V-distribution on ℰ. The geometric morphism φ in the topos pullback

[figure: ℰ; P(ℂ); 𝒳; ℰ; φ; 𝒳; 𝒵; 𝒵; P(ℂ)]

is said to be the support of μ, denoted {μ}_{ℰ}. (A priori, this construction depends on a site presentation of ℰ.)

4.3. Proposition. Consider a topos ℰ ≃ Sh(ℂ, J). Suppose that ℱ is a V-determined topos and that ψ : ℱ [figure] ℰ is a geometric morphism. Let μ = F_{!}ψ^{∗}. As in Def. 4.2, we may consider the category 𝕄, and the support φ : 𝒳 [figure] ℰ associated with μ as in Def 4.2. Then, 𝒳 is V-determined.

4.4. Definition. A geometric morphism φ : 𝒳 [figure] ℰ is said to be a V-fibration if its domain ℱ is a V-determined topos and furthermore φ equals the support of the V-distribution μ = F_{!}ψ^{∗} .

The following is the analogue of the comprehensive factorization [22, 29, 7] in the V-setting. It was established in [9].

4.5. Theorem. Any geometric morphism ψ : ℱ [figure] ℰ with a V-determined domain ℱ admits a factorization

[figure: ℱ; ℰ; ψ; ℱ; 𝒳; η; 𝒳; ℰ; φ]  (13)

into a first factor η that is V-initial followed by a V-fibration φ. This factorization, said to be comprehensive, is unique up to equivalence.

In the rest of this section we obtain several new results about the factors in the Vcomprehensive factorization, needed in the rest of the paper.

4.6. Lemma. Let π : ℱ [figure] ℰ be a local homeomorphism. If ℰ is V-determined then also ℱ is V-determined.

Proof. We observed (Remark 1.8) that since π is a local homeomorphism, the 2-cell α in (4) is an isomorphism. Hence, a left adjoint F_{!} to F^{∗} ≅ (π^{∗} ·E^{∗}) is given by the composite (E_{!} · π_{!}). The BCC for etales holds for F_{!} ⊣ F^{∗} since it does for E_{!} ⊣ E^{∗} and since π is in particular locally connected, π_{!} ⊣ π^{∗} is an ℰ -indexed adjointness.

□

4.7. Proposition. The pullback of a V-initial geometric morphism along a local homeomorphism is V-initial.

Proof. This is a straightforward diagram chase, using the fact that (4) is an isomorphism when ψ is a local homeomorphism. □

4.8. Lemma. Consider a triangle of geometric morphisms

[figure: 𝒳; 𝒵; τ; ℱ; 𝒳; η; ℱ; 𝒵; p]

in which τ is an inclusion.

1. If p and τ are both V-initial, then so is η.

2. If p is V-initial and τ^{∗}Z^{∗} [figure] X^{∗} is an isomorphism, then η is V-initial.

Proof. 1. Consider

[figure: τ_{∗}X^{∗}; τ_{∗}η_{∗}F^{∗}; Z^{∗}; τ_{∗}X^{∗}; Z^{∗}; τ_{∗}η_{∗}F^{∗}]

If p = τη is V-initial, then the hypotenuse is an isomorphism. If τ is V-initial, then the vertical is an isomorphism. Therefore, the horizontal is an isomorphism, and therefore η is V-initial since τ is an inclusion.

2. Applying τ^{∗} to the isomorphism Z^{∗} ≅ p_{∗}F^{∗} ≅ τ_{∗}η_{∗}F^{∗} gives the top horizontal in the following diagram, which is an isomorphism.

[figure: X^{∗}; η_{∗}F^{∗}; τ^{∗}Z^{∗}; X^{∗}; τ^{∗}Z^{∗}; τ^{∗}τ_{∗}η_{∗}F^{∗}; τ^{∗}τ_{∗}η_{∗}F^{∗}; η_{∗}F^{∗}]

The right vertical is the counit of τ^{∗} ⊣ τ_{∗}, which is an isomorphism since τ is an inclusion. The left vertical is an isomorphism by assumption. We conclude the bottom horizontal is an isomorphism, which says that η is V-initial. □

4.9. Proposition. Consider a commutative triangle

[figure: 𝒢; 𝒵; η; 𝒳; 𝒢; ρ; 𝒳; 𝒵; δ]

where 𝒳 , 𝒢, and 𝒵 are V-determined toposes. If ρ and η are V-initial, so is the composite δ.

Proof. From Remark 1.8, we have the canonical isomorphisms X_{!} · ρ^{∗} ≅ G_{!} and G_{!} · η^{∗} ≅ Z_{!}. It follows that

X_{!} · δ^{∗} ≅ X_{!} · (ρ^{∗} · η^{∗}) ≅ (X_{!} · ρ^{∗}) · η^{∗} ≅ G_{!} · η^{∗} ≅ Z_{!}

and this is also given by a canonical (iso) 2-cell. Hence the result. □

4.10. Proposition. Consider a bipullback in Top_{𝒮} , in which ψ : 𝒴 [figure] ℰ is a Vfibration and ξ is a local homeomorphism:

[figure: ℱ; ℰ; ξ; 𝒳; ℱ; φ; 𝒳; 𝒴; π; 𝒴; ℰ; ψ]

Then φ is a V-fibration.

Proof. First factor the V-determined φ into its V-initial and V-fibration parts: φ ≅ τ ·ρ. Then factor the V-determined ξ · τ into its V-initial and V-fibration parts.

We now have a diagram as follows.

[figure: ℱ; ℰ; ξ; 𝒳; ℱ; φ; 𝒳; 𝒴; π; 𝒴; ℰ; ψ; 𝒵; 𝒵; ζ; 𝒲; τ; 𝒲; ρ; η]

There is such a ζ because ψ is a V-fibration and since η · ρ is the composite of two V-initial hence a V-initial (by Proposition 4.9).

Since the outer square is a pullback, there is 𝒲 [figure: θ] 𝒳 such that π · θ ≅ ζ · η and φ · θ ≅ τ. The universal property of the pullback implies that θ · ρ ≅ id_{𝒳} . Hence, ρ · θ · ρ ≅ ρ. Therefore, the two geometric morphisms ρ · θ and id_{𝒲} from the V-fibration τ to itself agree when precomposed with the V-initial ρ. They must therefore agree: ρ · θ ≅ id_{𝒲} . We have shown that ρ is an equivalence, so that φ is a V-fibration. □

We have already considered the transpose α̂ : F^{∗} [figure] ρ_{∗}G^{∗} of α : ρ^{∗}F^{∗} [figure] G^{∗} under ρ^{∗} ⊣ ρ_{∗}.

[figure: ρ_{∗}ρ^{∗}F^{∗}; ρ_{∗}G^{∗}; ρ_{∗}α; F^{∗}; ρ_{∗}ρ^{∗}F^{∗}; ηF^{∗}; F^{∗}; ρ_{∗}G^{∗}; α̂]

Recall that ageometric morphism 𝒢 [figure: ρ] ℱ is V-initial if α̂ is an isomorphism .

4.11. Definition. A geometric morphism 𝒢 [figure: ρ] ℱ is such that its direct image part ρ_{∗} preserves V-coproducts if ηF^{∗} is an isomorphism.

4.12. Proposition. Assume that the 2-cell α of (4) is an isomorphism. Then, the following are equivalent:

1. ρ is V-initial.

2. ρ_{∗} preserves V-coproducts.

Proof.

Consider

[figure: ρ_{∗}ρ^{∗}E^{∗}; ρ_{∗}F^{∗}; E^{∗}; ρ_{∗}ρ^{∗}E^{∗}; E^{∗}; ρ_{∗}F^{∗}]

where the bottom 2-cell is ρ_{∗}(α), hence an isomorphism by assumption. Then, the vertical 2-cell ηE^{∗} is an isomorphism iff the hypothenuse α̂ is an isomorphism. □

5. Branched V-coverings

The following ‘pullback lemma’ follows an argument similar to that employed in [13] (Proposition 8.2), shown therein in the locally connected case.

5.1. Lemma. Let ℰ /S [figure] ℰ be a local homeomorphism which is also a V-initial subtopos, where ℰ is V-determined. Let ψ : ℰ /T [figure] ℰ /S be induced by a morphism p : T [figure] S in ℰ. Assume that ψ is a V-fibration. Then the associated V-fibration φ of i · ψ (exists and) forms a topos pullback square.

[figure: ℰ /S; ℰ; i; ℰ /T; ℰ /S; ψ; ℰ /T; 𝒴; ρ; 𝒴; ℰ; φ]

Consequently, the V-initial factor ρ is an inclusion.

Proof.

We claim that in the diagram below, where the bottom square is a bipullback, the induced geometric morphism τ is an equivalence.

[figure: ℰ /S; ℰ; i; 𝒢; ℰ /S; ζ; 𝒢; 𝒴; j; 𝒴; ℰ; φ; ℰ /T; 𝒴; ρ; ℰ /T; 𝒢; τ; ℰ /T; ℰ /S; ψ]

First, observe that, since ℰ is V-determined, so is the slice topos ℰ /T, by Lemma 4.6. Hence the comprehensive V-factorization of the composite

ℰ /T [figure: ψ] ℰ /S [figure: i] ℰ

exists as indicated, with ρ a V-initial geometric morphism, and φ a V-fibration.

That τ is a V-fibration follows from the equation ψ ≅ ζ · τ and the facts that ψ is a V-fibration by assumption, and that ζ is a V-fibration as it is obtained from the V-fibration φ by pullback along a local homeomorphism geometric morphism. For this, quote Proposition 4.10.

That τ is V-initial follows from the fact that j · τ ≅ ρ, ρ V-initial, and j an inclusion, using Lemma 4.8. Therefore, τ is both V-initial and a V-fibration hence an equivalence. This completes the proof. □

5.2. Definition. Let ℰ be a V-determined topos. An object A of ℰ is said to be a locally V-trivial object split by a cover 𝒰 = ⟨U [figure] 1, X, η : U [figure] E^{∗}X⟩ if A is part of a 3-tuple ⟨A, α, θ⟩ where α is an etale morphism Y [figure] X in ℰ, with X ∈ V and Y ∈ V, and θ is an isomorphism

E^{∗}Y ×_{E^{∗}X} U [figure: θ] A × U

over U. We call the corresponding geometric morphism ℰ /A [figure] ℰ a locally V-trivial covering split by 𝒰 .

A morphism ⟨A, α, θ⟩ [figure] ⟨B, β, κ⟩ between locally V-trivial objects split by 𝒰 = ⟨U [figure] 1, X, η : U [figure] E^{∗}X⟩ is given by a pair (f , γ) with f : A [figure] B a morphism in ℰ and γ : Y [figure] Z a morphism in Loc over X, such that

\xymatrix{E^{∗}Y ×_{E^{∗}X} U \ar[r]^{θ} \ar[d]^{E^{∗}γ×U} & A × U \ar[d]^{f×U} \\ E^{∗}Z ×_{E^{∗}X} U \ar[r]_{κ} & B × U}

commutes.

5.3. Remark. The analogue of Proposition 3.3 holds for locally V-trivial objects in a V-determined topos. Moreover, in this case, the proof is entirely analogous to the corresponding one in [11] for locally constant objects. This is so on account of the BCC holding for etale maps Y [figure] X where both X and Y are in V.

Denote by 𝒵_{V}(ℰ)(U) the category of locally V-trivial objects of ℰ split by U = ⟨U [figure] 1, η : U [figure] E^{∗}E_{!}U⟩ and their morphisms.

5.4. Remark. There is an inclusion

𝒵_{V}(ℰ)(U) ⊆ 𝒞_{V}(ℰ)(U).

It follows from Example 1.13 that this inclusion is in general proper. For instance, not every Z-covering projection is locally Z-trivial. On the other hand, every D-covering projection is locally D-trivial (or locally constant).

5.5. Definition. A geometric morphism ψ : 𝒲 [figure] ℰ of a V-determined topos ℰ is said to be an unramified V-covering if it is both a local homeomorphism and a V-fibration.

5.6. Proposition. A locally V-trivial covering is an unramified V-covering.

Proof. Let ⟨A, α, θ⟩ be a locally V-trivial object of a V-determined topos ℰ, trivialized by U [figure] 1, in the sense of Definition 3.1. Recall that α : Y [figure] X, η : U [figure] E^{∗}U, and that

E^{∗}Y ×_{E^{∗}X} U [figure: θ] A × U

is an isomorphism over U.

For any etale morphism p : Y [figure] X and corresponding geometric morphism Sh(Y) [figure] Sh(X), with X in V, we have that for each open inclusion p_{a} : W_{a} [figure] X, where lim{W_{a}} ≅ Y, there is induced an unramified V-covering Sh(W_{a}) ≅ Sh(X)/(p_{a}) [figure] Sh(X). For the local homeomorphism Sh(Y) [figure] Sh(X) to be a Vfibration, hence an unramified V-covering, we must have that Y ∈ V, by Proposition 3.3. □

5.7. Remark. A V-covering projection of ℰ is not necessarily an unramified V-covering. This follows from the observation, already employed in the proof of Proposition 5.6, that for a local homeomorphism that Sh(Y) [figure] Sh(X) to be a V-fibration, hence an unramified V-covering, we must have, by Proposition 3.3, that Y ∈ V. By Example 1.15 this need not be the case, for instance for V = Z.

5.8. Definition. Let ℰ be a V-determined topos. A branched V-covering of ℰ (with ‘non-singular part’ i : ℰ /S/ [figure] ℰ) is given by the following data:

(i) A V-fibration ψ : ℱ [figure] ℰ.

(ii) A V-initial inclusion i : ℰ /S [figure] ℰ.

(iii) An locally trivial V-covering φ : ℰ /T [figure] ℰ /S induced by a morphism p : T [figure] S.

(iv) A V-initial geometric morphism π : ℰ /T [figure] ℱ for which the square

[figure: ℰ /S; ℰ; i; ℰ /T; ℰ /S; φ; ℰ /T; ℱ; π; ℱ; ℰ; ψ]  (14)

commutes.

5.9. Corollary. The square (14) is a pullback.

Proof. This is a consequence of Lemma 5.1 and Proposition 5.6. □

In view of Remark 5.4, Proposition 5.6, and Corollary 5.9, the definition we have given of a branched V-covering not only generalizes that of [7] in the locally connected case, but it is also the only possible such. An alternative is to require directly unramified Vcoverings in both the discrete and the general V-case, but unless special assumptions are made on ℰ (locally paths simply connected) there is no connection with the fundamental groupoid.

Let ℰ be a V-determined topos in Top_{𝒮} . Denote by ℬ_{V,S}(ℰ) the full sub 2-category of Top_{𝒮} whose objects are the branched V-coverings of ℰ with non-singular part i : ℰ /S [figure] ℰ.

5.10. Theorem. The category ℬ_{V,S}(ℰ) is canonically equivalent to the category 𝒵_{V}(ℰ /S).

Proof. We pass from a branched V-covering ψ of ℰ as given in Definition 5.8 to a locally V-trivial covering of ℰ /S by pullback.

Conversely, we pass from a locally V-trivial covering φ of ℰ /S to a branched covering of ℰ with non-singular part i : ℰ /S [figure] ℰ by the comprehensive V-factorization of the composite

𝒳 [figure: φ] ℰ /S [figure] ℰ .

By Lemma 5.9, the square

[figure: ℰ /S; ℰ; i; 𝒳; ℰ /S; φ; 𝒳; 𝒴; ρ; 𝒴; ℰ; ψ]  (15)

is a pullback so that the V-initial ρ is an inclusion. The passages indicated give an equivalence of categories.

□

5.11. Corollary. Let ℰ be a V-determined topos. Then, for each cover U in ℰ, there is an inclusion

ℬ_{V,S}(ℰ)(U) ⊆ 𝒞_{V}(ℰ /S)(U).

This inclusion is in general proper.

5.12. Remark. In the locally connected case, the results above reduce to the known ones [7]. Moreover, If V is the category D, of discrete locales, then there is an equivalence ℬ_{V,S}(ℰ)(U) ≅ 𝒞_{V}(ℰ /S)(U) so that the branched fundamental groupoid topos with respect to a given ‘non-singular part’ i : ℰ /S [figure] ℰ may be identified with the fundamental groupoid topos Π_{1}^{(c)}(ℰ /S) = lim{𝒢_{U} (ℰ /S) | U ∈ Cov(ℰ)}, for the pushout toposes 𝒢_{U}(ℰ /S).

We may always consider the largest topology in any Grothendieck topos ℰ for which certain objects of the topos are sheaves [27]. A monomorphism m : A [figure] B is dense for this largest topology iff B^{∗}A [figure] B^{∗}A^{m} (transpose of the projection) is an isomorphism.

We apply this idea, as in [7], to the class of objects of the form E^{∗}X for X ∈ V. Clearly, a monomorphism m : S [figure] T in ℰ is dense for such a topology iff the induced geometric morphism σ : ℰ /S [figure] ℰ /T is such that σ_{∗} preserves V-coproducts. Let us call this the V-topology. Consider the topos of sheaves for the V-topology on ℰ, with inclusion i : ℰ_{V} [figure] ℰ.

5.13. Theorem. ℰ_{V} is the smallest subtopos of ℰ whose inclusion preserves Vcoproducts.

Proof.

The unit E^{∗}(Ω_{𝒮} ) [figure] i_{∗}i^{∗}E^{∗}(Ω_{𝒮} ) for the inclusion i : ℰ_{V} [figure] ℰ is an isomorphism because E^{∗}(Ω_{𝒮} ) is a V-initial-sheaf. But this says that i_{∗} preserves V-coproducts.

We now show that it is the smallest such subtopos. Let Sh_{j}(ℰ) [figure] ℰ be an inclusion for which j_{∗} preserves V-coproducts. Every E^{∗}(Ω_{𝒮} ) is a j-sheaf. Hence, every j-dense monomorphism is dense for the V-topology. Therefore, ℰ_{V} is included in Sh_{j}(ℰ). □

Since for each local homeomorphism i : ℰ /S [figure] ℰ is V-initial iff i_{∗} preserves Vcoproducts, ℰ_{V} [figure] ℰ factors through every V-initial local homeomorphism i : ℰ /S [figure] ℰ. However, ℰ_{V} [figure] ℰ itself can ‘never’ be a local homeomorphism.

It follows that the smallest V-initial subtopos of ℰ does not exist in general and that therefore the intrinsic characterization of branched coverings [7] does not hold in general either. In particular, the ‘ideal knot’ does not exist in the general case. We conclude that the locally connected assumption leads to miracles which we cannot expect without it.

6. Acknowledgements

This paper is partially based on lectures given by the author at the International Category Theory Meeting 2007, and at the Session on “Mathematical Applications of Category Theory” of the Winter Meeting of the Canadian Mathematical Society 2007. I thank Eduardo Dubuc and Gábor Lukács for interesting discussions. I also thank the anonymous referee for useful remarks. Research partially supported from an NSERC individual grant. Diagrams typeset with Michael Barr’s diagxy package for xy-pic.

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Department of Mathematics and Statistics, McGill University 805 Sherbrooke St. W, Montreal, QC, Canada H3A 2K6

Email: marta.bunge@mcgill.ca

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