Theory and Applications of Categories, Vol. 21, No. 10, 2008, pp. 182–190.

DESCENT FOR COMPACT 0-DIMENSIONAL SPACES

Dedicated to Walter Tholen on the occasion of his 60th birthday

GEORGE JANELIDZE^{1} AND MANUELA SOBRAL^{2}

Abstract. Using the reflection of the category 𝒞 of compact 0-dimensional topological spaces into the category of Stone spaces we introduce a concept of a fibration in 𝒞. We show that: (i) effective descent morphisms in 𝒞 are the same as the surjective fibrations; (ii) effective descent morphisms in 𝒞 with respect to the fibrations are all surjections.

Introduction

Our original intention was to describe effective descent morphisms in the category 𝒞 of compact 0-dimensional topological spaces by combining the following well-known facts:

• A compact 0-dimensional space is nothing but a set equipped with a surjection into a Stone space (see Theorem 2.1 for the precise formulation).

• The effective descent morphisms in the categories of sets and of Stone spaces are just surjections.

It is still the main purpose of the paper, although it turned out that:

• Not all pullbacks exist in 𝒞. Therefore the definition of an effective descent morphism p in 𝒞 should include the requirement: all pullbacks along p must exist (see Definition 3.2).

• When p is surjective, that requirement holds if and only if p is a fibration in a suitable sense (see Definition 2.2), which is very different from what is happening in the situations studied by H. Herrlich [1], and which makes the descent problem much easier. In a somewhat different situation, this is made clear in [3].

• The surjectivity requirement does not create any problem since it is independently forced by the reflection of isomorphisms by the pullback functor along an effective descent morphism.

1. Partially supported by South African NRF.

2. Partially supported by FCT/Centro de Matemática da Universidade de Coimbra.

Received by the editors 2008-07-10 and, in revised form, 2008-11-12.

Published on 2008-11-15 in the Tholen Festschrift.

2000 Mathematics Subject Classification: 18B30, 18A25, 18A20.

Key words and phrases: comma categories, effective descent, effective 𝔽-descent.

\begin{matrix}c \\ ◯ \end{matrix} George Janelidze^{1} and Manuela Sobral^{2}, 2008. Permission to copy for private use granted.

• Therefore the problem of describing effective descent morphisms in 𝒞 has an easy solution: Theorem 3.3 says that they are the same as the surjective fibrations.

• However, this suggests a new question, namely, what are the effective descent morphisms with respect to fibrations? Fortunately there is a complete answer again: they are all surjections (Theorem 3.1).

• In particular, even though the spaces we consider are not necessarily Hausdorff spaces, which prevents their convergence relations to be maps, our characterization of their effective descent morphisms avoids using the Reiterman-Tholen characterization of effective descent morphisms in the category of all topological spaces [4].

Accordingly, the paper is organized as follows:

Section 1 contains preliminary categorical observations with no topology involved. The ground category 𝒞 there is constructed as a full subcategory in the comma category (𝒮 ↓ U), where U : 𝒳 → 𝒮 is a pullback preserving functor between categories with pullbacks, using also a distinguished class 𝔼 of morphisms in 𝒮. This class is also used to define what we call fibrations in 𝒞. The sufficient conditions for a morphism to be an effective descent morphism (globally or with respect to the class of fibrations) given in Section 1 will become also necessary in the topological context of Sections 2 and 3.

Section 2 begins by recalling relevant topological concepts, presents the category of compact 0-dimensional spaces as a special case of 𝒞 above, introduces fibrations of 0dimensional spaces accordingly, and ends by proving that a surjective morphism in 𝒞 admits all pullbacks along morphisms with the same codomain if and only if it is a fibration.

The purpose of Section 3 is to formulate and prove the two main results, namely the above mentioned Theorems 3.1 and 3.3.

1. Categorical framework

We fix the following data: categories 𝒮 and 𝒳 with pullbacks, a pullback preserving functor U : 𝒳 → 𝒮 and a class 𝔼 of morphisms in 𝒮 that has the following properties:

• contains all isomorphisms;

• is pullback stable;

• is closed under composition;

• forms a stack (=coincides with its localization), which means that if

\xymatrix{· \ar[r] \ar[d]_{u} & · \ar[d]^{v} \\ · \ar[r]_{w} & ·}

184  GEORGE JANELIDZE^{1} AND MANUELA SOBRAL^{2}

is a pullback diagram with w being an effective descent morphism, then u ∈ 𝔼 ⇒ v ∈ 𝔼.

Let 𝒞 = 𝒞[𝒳, 𝒮, U, 𝔼] be the full subcategory in the comma category (𝒮 ↓ U) with objects all triples A = (A_{1}, e_{A}, A_{0}), in which e_{A} : A_{1} → U(A_{0}) is in 𝔼; accordingly, a morphism A → B in 𝒞 is a pair f = (f_{1}, f_{0}), in which f_{1} : A_{1} → B_{1} and f_{0} : A_{0} → B_{0} are morphisms in 𝒮 and 𝒳 respectively,

\xymatrix{A_{1} \ar[r]^{e_{A}} \ar[d]_{f_{1}} & U(A_{0}) \ar[d]^{U(f_{0})} \\ B_{1} \ar[r]_{e_{B}} & U(B_{0})}

such that U(f_{0})e_{A} = e_{B}f_{1}.

1.1. Definition. A morphism f : A → B in (𝒮 ↓ U) is said to be a fibration if the morphism

< f_{1}, e_{A} >: A_{1} → B_{1} ×_{U(B_{0})} U(A_{0})

is in 𝔼.

1.2. Observation. If f : A → B is a fibration, and B is in 𝒞, then, since the class 𝔼 is pullback stable, A also is in 𝒞.

1.3. Proposition. Let

(1.1)  \xymatrix{D \ar[r]^{q} \ar[d]_{g} & A \ar[d]^{f} \\ E \ar[r]_{p} & B}

be a pullback diagram in (𝒮 ↓ U) with p : E → B in 𝒞. Then:

(a) If f is a fibration, then so is g.

(b) If g is a fibration, and p_{1} is an effective descent morphism in 𝒮, then f also is a fibration.

(c) If p is a fibration and A is in 𝒞, then D is in 𝒞.

(d) If 𝔼 has the (weak left) cancellation property (e′, e · e′ ∈ 𝔼 ⇒ e ∈ 𝔼) and p_{1} and U(p_{0}) are in 𝔼 and D is in 𝒞, then A is in 𝒞.

Proof. Consider the diagram

\xymatrix{D_{1} \ar[rrr]^{q_{1}} \ar[ddd]_{g_{1}} \ar[dr]_{d} \ar[ddrr]^{e_{D}} &  &  & A_{1} \ar[ddd]_{f_{1}} \ar[dr]_{a} \ar[ddrr]^{e_{A}} \\  & S \ar[rrr]^{h} \ar[dr]_{s′} \ar[ddl]^{s} &  &  & T \ar[dr]_{t′} \ar[ddl]^{t} \\  &  & U(D_{0}) \ar[rrr]^{U(q_{0})} \ar[dd]^{U(g_{0})} &  &  & U(A_{0}) \ar[dd]^{U(f_{0})} \\ E_{1} \ar[rrr]^{p_{1}} \ar[drr]_{e_{E}} &  &  & B_{1} \ar[drr]^{e_{B}} \\  &  & U(E_{0}) \ar[rrr]^{U(p_{0})} &  &  & U(B_{0})}

in which:

• the enveloping cube represents the diagram (1.1);

• e_{E}s = U(g_{0})s′ and e_{B}t = U(f_{0})t′ are pullbacks;

• d =< g_{1}, e_{D} >, a =< f_{1}, e_{A} >, and h = p_{1} × U(q_{0}) are the suitable induced morphisms.

Since the front square U(p_{0})U(g_{0}) = U(f_{0})U(q_{0}) and the quadrilaterals e_{E}s = U(g_{0})s′ and e_{B}t = U(f_{0})t′ are pullbacks, so is the quadrilateral p_{1}s = th. Next, since p_{1}g_{1} = f_{1}q_{1} and p_{1}s = th are pullbacks, so is hd = aq_{1}. This proves (a).

(b): Since p_{1} is an effective descent morphism and p_{1}s = th is a pullback, h also is an effective descent morphism ([5], Theorem 3.1). Since hd = aq_{1} is a pullback, this proves (b).

For (c) and (d), in order to use the same observations, let us “turn the diagram (1.1) around the diagonal connecting D and B”, i.e. let us reformulate (c) and (d) as follows:

(c′) If f is a fibration and E is in 𝒞, then D is in 𝒞. (d′) If f_{1} and U(f_{0}) are in 𝔼 and D is in 𝒞, then E is in 𝒞.

Proof of (c′):

• Since f is a fibration, a is in 𝔼.

• Since E is in 𝒞 and e_{E}s = U(g_{0})s′ is a pullback, s′ is in 𝔼.

• Since a and s′ are in 𝔼, so is e_{D}, i.e. D is in 𝒞.

Proof of (d′):

• Since f_{1} and U(f_{0}) are in 𝔼, so are g_{1} and U(g_{0}).

• Since g_{1}, U(g_{0}) and e_{D} are in 𝔼, the cancellation property of (d′) implies that e_{E} is in 𝔼, as desired.

From Observation 1.2 and Proposition 1.3(a) we obtain:

1.4. Corollary. The category 𝒞 is closed in (𝒮 ↓ U) under pullbacks along fibrations; that is, if (1.1) is a pullback diagram in (𝒮 ↓ U) with f in 𝒞 and p being a fibration in 𝒞, then it is a pullback diagram in 𝒞.

When 𝒮 has coequalizers of equivalence relations, all effective descent morphisms in 𝒮 are regular epimorphisms. Using this fact it is easy to show that if p : E → B is a morphism in (𝒮 ↓ U), for which p_{0} and p_{1} are effective descent morphisms in 𝒳 and in 𝒮 respectively, then p itself is an effective descent morphism. After that, using Proposition 1.3 and Corollary 1.4, we obtain:

1.5. Proposition. If 𝒮 has coequalizers of equivalence relations and p : E → B is a morphism in 𝒞, for which p_{0} and p_{1} are effective descent morphisms in 𝒳 and in 𝒮 respectively, then

(a) p is an effective 𝔽-descent morphism in 𝒞, where 𝔽 is the class of all fibrations (in 𝒞).

(b) if p is a fibration, then it is an effective descent morphism in 𝒞.

2. The category of compact 0-dimensional spaces

For a topological space A, we shall write Open(A) for the set of open subsets in A and Clopen(A) for the set of those subsets in A that are clopen, i.e. closed and open at the same time. Let us recall the definitions of the following full subcategories of the category 𝒯op of topological spaces:

– 𝒯op_{0}, the category of T_{0}-spaces; a space A is a T_{0}-space if, for every two distinct points a and a′ in A, either there exists U ∈ Open(A) with a ∈ U and a′ ∉ U, or there exists U ∈ Open(A) with a′ ∈ U and a ∉ U. Note that 𝒯op_{0} is a reflective subcategory in 𝒯op, with the reflection given by

(2.1)  A ↦ A_{0} = A/ ∼, where a ∼ a′ ⇔ ∀_{U∈Open(A)}(a ∈ U ⇔ a′ ∈ U).

– 0-𝒟im𝒯op, the category of 0-dimensional spaces; a space is 0-dimensional if it has a basis of clopen subsets, i.e. if every open subset in it can be presented as a union of clopen subsets.

– The category of compact 0-dimensional spaces, which is the category of interest in this paper, will be simply denoted by 𝒞; hence

𝒞 = 𝒞omp𝒯op ∩ 0-𝒟im𝒯op

where 𝒞omp𝒯op is the category of compact spaces.

– 𝒮tone, the category of Stone spaces = spaces that occur as spectra of Boolean algebras = spaces that occur as limits of finite discrete spaces = compact Hausdorff 0dimensional spaces = compact spaces A, such that for every two distinct points a and a′ in A, there exists U ∈ Clopen(A) with a ∈ U and a′ ∉ U. The T_{0}-reflection (2.1) of course induces a reflection

(2.2)  𝒞 ↦ 𝒮tone, A ↦ A_{0}.

The following theorem is a reformulation of well-known results (see also Example 3.3 in [2] for the same result for arbitrary topological spaces, which, together with other similar results was mentioned already in [1]):

2.1. Theorem. The category 𝒞 of compact 0-dimensional spaces is equivalent to the category 𝒞[𝒳, 𝒮, U, 𝔼] (see Section 1), for 𝒳 = 𝒮tone, 𝒮 = 𝒮et, U : 𝒮tone → 𝒮et, U being the usual forgetful functor into the category of sets and 𝔼 being the class of all surjective maps. Under this equivalence a space A corresponds to the triple (A_{1}, e_{A}, A_{0}), in which A_{1} is the underlying set of A, A_{0} is the T_{0}-reflection of A, and e_{A} : A_{1} → U(A_{0}) is the canonical map (and we write again A = (A_{1}, e_{A}, A_{0})). Conversely, the space corresponding to a triple (A_{1}, e_{A}, A_{0}) has A_{1} as its underlying set and the inverse e_{A}-images of open sets in A_{0} as its open subsets.

According to this theorem and Definition 1.1, we introduce:

2.2. Definition. A morphism f : A → B in 𝒞 is said to be a fibration if so is the corresponding morphism in 𝒞[𝒳, 𝒮, U, 𝔼] of Theorem 2.1, i.e. if for every a in A and b in B with f(a) ∼ b there exists a′ in A with a′ ∼ a and f(a′) = b.

After that Proposition 1.3 helps to prove:

2.3. Theorem. Let p : E → B be a morphism in 𝒞. If p is surjective, then the following conditions are equivalent:

(a) every morphism f : A → B in 𝒞 admits pullback along p;

(b) p is a fibration.

Proof. (a)⇒(b): Suppose p is not a fibration. This means that there are e in E and b in B with

(2.3)  p(e) ∼ b and (x ∈ p^{−1}(b) ⇒ ∃_{U_{x}∈Clopen(E)}(x ∈ U_{x} and e ∉ U_{x})).

We choose U_{x} as in (2.3) for each x in p^{−1}(b), and consider two cases:

Case 1. There exists a finite subset Y in X, for which

p^{−1}(b) ⊆ ⋃_{x∈Y} U_{x}.

Case 2. There is no such Y.

In Case 1 we take

V = ⋂_{x∈Y} (E \ U_{x}),

and observe that since Y is finite, V is clopen; and of course V contains e and has empty intersection with p^{−1}(b). After that we take

A = {n^{−1}|n = 1, 2, 3, ⋯ } ∪ {0}

with the topology induced from the real line, and define f : A → B by f(n^{−1}) = b and f(0) = p(e). Suppose the pullback of p and f does exist, and let us write it as the diagram (1.1). Using the universal property of this pullback with respect to maps from a one-point space, we easily conclude that it is preserved by the forgetful functor into the category of sets. In particular, since V contains e and has empty intersection with p^{−1}(b), we have

q(g^{−1}(E \ V)) = {n^{−1}|n = 1, 2, 3, ⋯ }.

This is a contradiction since g^{−1}(E \ V) being clopen in D must be compact in it, while {n^{−1}|n = 1, 2, 3, ⋯ } is not compact in A.

In Case 2 we take A = {a} to be a one-point space, and define f : A → B by f(a) = b. Then, using (1.1) as above, we observe that g(D) = p^{−1}(b) – which is again a contradiction because now p^{−1}(b) is not compact.

That is, whenever p is not a fibration, there exists a morphism f : A → B in 𝒞 that has no pullback along p.

(b)⇒(a) follows from Corollary 1.4 and Theorem 2.1.

3. 𝔽-Descent and global descent

Let 𝒞 be as in Section 2.

3.1. Theorem. The following conditions on a morphism p : E → B in 𝒞 are equivalent: (a) p is an effective 𝔽-descent morphism in 𝒞;

(b) p is a surjective map.

Proof. (a)⇒(b): Suppose p is not surjective, and choose b ∈ B \ p(E). Let A be the equivalence class of b with respect to the equivalence relation ∼ (see (2.1)). We take

A′ = (A \ {b}) ∪ {b} × {1, 2}

equipped with the indiscrete topology, and define α : A′ → A by α(a) = a, for a ∈ A, and α(b, 1) = b = α(b, 2); then α becomes a morphism (A′, αf) → (A, f), where f : A → B is the inclusion map, in the category 𝔽(B) of fibrations over B (in 𝒞). Since the image of this morphism under the pullback functor p^{∗} : F (B) → F (E) is an isomorphism, p cannot be an effective 𝔽-descent morphism in 𝒞.

(b)⇒(a): Let (p_{1}, p_{0}) : (E_{1}, e_{E}, E_{0}) → (B_{1}, e_{B}, B_{0}) be the morphism in 𝒞[𝒳, 𝒮, U, 𝔼] corresponding to p under the category equivalence of Theorem 2.1, where X = 𝒮tone, 𝒮 = 𝒮et, U : 𝒮tone → 𝒮et being the usual forgetful functor into the category of sets, and 𝔼 being the class of all surjective maps. Then p_{1} is surjective and this makes p_{0} surjective too. Since in both 𝒮tone and 𝒮et surjections are effective descent morphisms, this makes p an effective 𝔽-descent morphism by Proposition 1.5(a).

Since 𝒞 does not admit some pullbacks, we define effective (global-)descent morphisms in 𝒞 as follows:

3.2. Definition. A morphism p : E → B in 𝒞 is said to be an effective descent morphism if every morphism f : A → B in 𝒞 admits pullback along p, and the pullback functor

p∗ : (𝒞 ↓ B) → (𝒞 ↓ E)

is monadic.

3.3. Theorem. The following conditions on a morphism p : E → B in 𝒞 are equivalent: (a) p is an effective descent morphism;

(b) p is a surjective fibration.

Proof. (a)⇒(b): Surjectivity can be proved in the same way as in the proof of Theorem 3.1 (or even much simpler by considering the empty and one-point space instead of A′ and A there). The fact that p must be a fibration follows from the implication (a)⇒(b) of Theorem 2.3.

(b)⇒(a) can be deduced from Proposition 1.5(b) and Theorem 2.1 with the same arguments as in the proof of Theorem 3.1(b)⇒(a).

References

[1] H. Herrlich, Topological functors, General Topology and Appl. 4 (1974) 125-142.

[2] G. Janelidze, Light morphisms for generalized T_{0}-reflections, submitted.

[3] G. Janelidze and M. Sobral, Finite preorders and topological descent I, J. Pure Appl. Algebra 175(1-3)(2001) 187-205.

[4] J. Reiterman and W. Tholen, Effective descent maps of topological spaces, Topology Appl. 57 (1994) 53-69.

[5] M. Sobral and W. Tholen, Effective descent morphisms and effective equivalence relations, in: Category Theory 1991, CMS Conference Proceedings (Amer. Math. Soc., Providence, R. I. 1992), 421-432.

Dept. Math. and Appl. Math. University of Cape Town Rondebosh 7701, Cape Town South Africa

Departamento de Matemática Universidade de Coimbra Ap. 3008, 3001-454 Coimbra Portugal

Email: janelidg@maths.uct.ac.za
      sobral@mat.uc.pt

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