Localization with change of the base space in uniform bundles and sheaves

Transcription

1 Revista Colombiana de Matemáticas Volumen , páginas Localization with change of the base space in uniform bundles and sheaves Localización con cambio del espacio base en campos uniformes y haces Clara Marina Neira 1, Januario Varela 1 1 niversidad Nacional de Colombia, Bogotá, Colombia To Professor Jairo A. Charris in memoriam Abstract. In this paper a localization germination process with change of the base space is presented. The data consist of two topological spaces T and S, a continuous function ϕ : T S, a surjective function p : E T, a directed family d i of bounded pseudometrics for p generating a Hausdorff uniformity and a family Σ of global selections for p. In terms of these data, a uniform bundle is constructed over the base space S, whose fibers are colimits in a category of uniform spaces. Similar results follow for the case of sheaves of sets. This localization process leads to a universal arrow in a context described in terms of a category of uniform bundles. Key words and phrases. niform bundle, sheaf of sets, localization, colimit Mathematics Subject Classification. 55R65, 54E15, 54B40. Resumen. En este artículo se presenta un proceso de localización germinación con cambio del espacio base. El conjunto de datos consta de dos espacios topológicos T y S, una función continua ϕ : T S, una función sobreyectiva p : E T, una familia dirigida d i de seudométricas acotadas para p que genera una uniformidad de Hausdorff y una familia Σ de selecciones globales para p. En términos de estos datos, se construye un campo uniforme sobre el espacio base S, cuyas fibras son colímites en una categoría de espacios uniformes. Como aplicación inmediata, se obtienen resultados similares para el caso particular de los haces de conjuntos. Este proceso de localización da lugar a una flecha universal en un contexto apropiado que es descrito en términos de una categoría de campos uniformes. Palabras y frases clave. Campo uniforme, haz de conjuntos, localización, colímite. 303

2 304 CLARA MARINA NEIRA & JANARIO VARELA 1. Introduction In the theories of sheaves of sets and sheaves of groups one resorts to the classical germination process to construct fibers of a sheaf from data provided by a given presheaf. In [7] K. H. Hofmann presented a localization process to obtain a uniform bundle over a topological space C from a uniform bundle E, π, B, where B is a subset of C, and from a full set of global selections for π. This method was just slightly different from the process presented in [5], where J. Dauns and K. H. Hofmann considered a surjective function ϕ : B C instead of the inclusion map. In [2] S. Bautista et al, gave a localization process in the context of bundles of Hausdorff uniform spaces over a fixed topological space T, their procedure led to the construction of a bundle of uniform spaces over T, in terms of a given presheaf of local selections defined on open subsets of T. The fibers of these bundles are colimits in the category of Hausdorff uniform spaces. In this paper, the function ϕ : T S, introduced as a new ingredient in the above localization process, allows to have at our disposal an alternative and eventually better behaved and more interesting base space S, in place of the given space T, that could enjoy some desirable topological properties like compactness. In the present article, the data for the generalized localization process consists of a pair of topological spaces T and S, a continuous function ϕ from T to S non necessarily surjective, a surjective function p from a set E onto T, a directed family of bounded pseudometrics for p and a family of global selections for p. In terms of these data a uniform bundle over the base space S is contructed, its fibers are colimits in a category consisting of sets equipped with a family of pseudometrics generating a Hausdorff uniformity and appropriate morphisms between them. In particular, one can start with a sheaf over T and construct a sheaf over S via the continuous function ϕ, as described in Example 1 below. This localization process leads to a universal arrow in a context described in terms of the category of uniform bundles. 2. Preliminaries Definitions and results that are required in what follows are given in this sections, all of them are treated in detail in the references [7], [10], [11] and [12] niform bundles. Let E and T be topological spaces and p : E T be a surjective function. For each t T, the set E t = p 1 t = {a E : pa = t} is called the fiber above t. Observe that E is the disjoint union of the family E t t T. A local selection for p is a function σ : Q E such that Q T is an open subset and p σ is the identity map id Q of Q. If Q = T, σ is a global selection. A local section for p is a continuous local selection and a global section is a continuous global selection. For each open subset Q of T, let Γ Q p := {σ σ : Q E is continuous and p σ = id Q }. If Q = T, one writes Γp instead of Γ T p. A set Σ of local sections is said to be full, if for each x E, there exists σ Σ such that σpx = x. Volumen 41, Número 2, Año 2007

3 LOCALIZATION WITH CHANGE OF THE BASE SPACE 305 Let E T E := {u, v E E : pu = pv}. The function d : E T E R is called a pseudometric for p provided that the restriction of d to E t E t is a pseudometric on E t, for each t T. A family of pseudometrics for p, d i, is directed if for each pair i 1, i 2 I, there exists i I such that d i1 u, v d i u, v and d i2 u, v d i u, v, for each u, v E T E. Definition 1. Let d i be a directed family of pseudometrics for p, σ be a local selection, i I and ɛ > 0, the set Tɛ i σ = {u E : d i u, σpu < ɛ} is called the ɛ-tube around σ with respect to d i. Definition 2. Let E and T be topological spaces, p : E T be a surjective function and d i a family of pseudometrics for p. The quadruplet E, p, T, d i is said to be a bundle of uniform spaces or simply a uniform bundle provided that: 1 For each u E, each ɛ > 0 and each i I, there exists a local section σ such that u Tɛ i σ. 2 The collection of all tubes around local sections for p form a base for the topology of E. The space T is called the base space and the space E is called the fiber space or the bundle space. Observe that if E, p, T, d i is a uniform bundle, then the function p is continuous and open. Definition 2 secures the upper semicontinuity of the distance functions s d i σs, τs : Q R + {0}, where σ and τ are arbitrary local sections for p, Q is any open subset of Dom σ Dom τ and i I Existence Theorem of niform Bundles. The following result, cf. [12], is an indispensable tool in the constructions that follow. Theorem 1. Let T be a topological space and p : E T be a surjective function. Consider a set Σ of local selections for p and a directed family d i of pseudometrics for p. Assume the following conditions: a For each u E, each i I and each ɛ > 0, there exists α Σ such that u Tɛ iα. b The function s d i αs, βs : Dom α Dom β R is upper semicontinuous, for each i I and each α, β Σ Σ. Then there exists a topology S over E such that: 1 The topology S has a base consisting of the sets of the form T i ɛ α Q, where i I, ɛ > 0 and α Q is the restriction to an open subset Q Dom α of a local selection α Σ. 2 Each α Σ is a local section. 3 E, p, T, d i is a uniform bundle. Revista Colombiana de Matemáticas

4 306 CLARA MARINA NEIRA & JANARIO VARELA 3. Localization with change of the base space Let T and S be topological spaces, ϕ : T S a continuous function not necessarily surjective, p : E T a surjective function and d i a directed family of bounded pseudometrics for p. Suppose that Σ is a full set of global selections for p. A uniform bundle over S is to be constructed in terms of these data by means of the map ϕ. For each s S, define the relation R s in the set Σ as follows: for each pair σ, τ Σ, σr s τ, if and only if, V Vs r ϕ 1 V d i σr, τr = 0, for each i I. Note that if s S ϕt, there exists a neighbourhood V of s such that ϕ 1 V =. If this is the case, by convention, one takes r d i σr, τr = 0, for each i I. For each s S, R s is an equivalence relation on Σ. Since d i is a pseudometric, then R s is reflexive and symmetric. The transitivity is also straightforward. Denote by [σ] s the equivalence class of σ module R s. To be noted that, for s S ϕt, the set Σ/R s reduces to a single point. Let Ê be the disjoint union of the family {Σ/R s : s S} and p be the function For each i I, define d i : Ê S Ê R [σ] s, [τ] s p : Ê S [σ] s s. V Vs r ϕ 1 V d i σr, τr. All the d i, i I, are bounded pseudometrics for p, indeed: to show that the function d i is well defined, let σ 1 R s σ, τ 1 R s τ, and m := l := V Vs r ϕ 1 V V Vs r ϕ 1 V d i σr, τr d i σ 1 r, τ 1 r. For a given ɛ > 0, there exists a neighbourhood V of s such that d i σr, σ 1 r < ɛ r ϕ 1 V 3, i ii iii and r ϕ 1 V d i τr, τ 1 r < ɛ 3 d i σr, τr < m + ɛ r ϕ 1 V 3. Volumen 41, Número 2, Año 2007

5 LOCALIZATION WITH CHANGE OF THE BASE SPACE 307 Hence, for each r ϕ 1 V, d i σ 1 r, τ 1 r d i σ 1 r, τr + d i τr, τ 1 r d i σ 1 r, σr + d i σr, τr + d i τr, τ 1 r < ɛ 3 + m + ɛ 3 + ɛ 3 = m + ɛ. It follows that, for each ɛ > 0, there exists a neighbourhood V of s such that thus l = d i σ 1 r, τ 1 r m + ɛ, r ϕ 1 V V Vs r ϕ 1 V A similar argument shows that conversely m = V Vs r ϕ 1 V d i σ 1 r, τ 1 r m. d i σr, τr l. The family di of pseudometrics for p is directed, in fact, let i 1, i 2 I, since by hypothesis the family d i is directed, there exists a j I such that d i1 a, b, d i2 a, b d j a, b, for each a, b E T E. It is immediate that each pair [σ] s, [τ] s Ê S Ê satisfies d i1 [σ] s, [τ] s, di2 [σ] s, [τ] s d j [σ] s, [τ] s. Define now a set of selections for p: for each σ Σ, let σ : S Ê be the function defined by σs = [σ] s. The set Σ = { σ : σ Σ} is a full set of global selections for p. It remains to be seen that p, the family Σ and the family di satisfy the hypotheses of the Existence Theorem of niform Bundles. a Let [σ] s Ê, i I and ɛ > 0. It is immediate that [σ] s = σs Tɛ i σ. b Let i I, α, β Σ Σ and ɛ > 0. If d i αs 0, βs 0 < ɛ then we have that V Vs0 t ϕ 1 V d i αt, βt < ɛ, hence there exists an open neighbourhood V of s 0 such that t ϕ 1 V d i αt, βt < ɛ, thus one has that for each s V, d i αs, βs < ɛ. It follows that the sets of the form Tɛ i σ Q, where i I, ɛ > 0, σ Σ, Q is an open subset of the domain of σ and σ Q is the restriction of σ to Q, form a base for a topology on Ê, Ê such that p, S, di is a uniform bundle and each element of Σ is a section. On the other hand, for each i I, the functions D i : Σ Σ R and D i : Σ Σ R defined by D i σ, τ = t T d i σt, τt and D i σ, τ = s S di σs, τs are pseudometrics on Σ and Σ respectively. Revista Colombiana de Matemáticas

6 308 CLARA MARINA NEIRA & JANARIO VARELA The function ψ : Σ Σ, σ σ is an isometry with respect to D i and D i. In fact, for each t T and each neighbourhood V of ϕt in S, one has that then d i σt, τt d i σt, τt d i σr, τr, r ϕ 1 V V Vϕt r ϕ 1 V = d i σϕt, τ ϕt d i σs, τs, s S d i σr, τr therefore t T d i σt, τt s S di σs, τs, thus D i σ, τ D i σ, τ. On the other hand, since S is a neighbourhood of s for each s S and ϕ 1 S = T, it follows that V Vs r ϕ 1 V d i σr, τr d i σr, τr = D i σ, τ, r T for each s S, thus s S V Vs r ϕ 1 V d i σr, τr Hence D i σ, τ D i σ, τ. Then the function ψ satisfies the identity D i ψσ, ψτ = D i σ, τ, for each i I. iv D i σ, τ. If ψσ = ψτ then D i σ, τ = 0, for each i I, hence D i σ, τ = d i σt, τt = 0, t T for each i I, then d i σt, τt = 0, for each t T. It follows that, if the family of pseudometrics d i is Hausdorff, that is, a = b if d i a, b = 0, for each i I, then σ = τ. If that is the case, the function ψ is one to one. Summing up, if the uniformity for p determined by the family d i is Hausdorff equivalently if each fiber is a Hausdorff space, one can identify Σ with Σ. In terms of the definitions given by i, ii, iii and iv above, the following proposition makes precise the preceding arguments, cf. [10][p. 44]. Proposition 1. Let T and S be two topological spaces, ϕ : T S be a continuous function, p : E T be a surjective function, d i be a directed family of bounded pseudometrics for p generating a Hausdorff uniformity and Σ be a full set of global selections for p. 1 For each s S, the relation R s is an equivalence relation. 2 di is a directed family of bounded pseudometrics for p. Volumen 41, Número 2, Año 2007

7 LOCALIZATION WITH CHANGE OF THE BASE SPACE The quadruplet Ê, p, S, di is a uniform bundle whose set of global sections contains an isomorphic copy Σ of Σ. For the particular case ϕ = id T, cf. [10][p. 34]. Example 1. The case of a sheaf of sets, cf. [10][p. 48]. Consider a sheaf of sets E, p, T, here p : E T is a local homeomorphism. The sheaf with the family of pseudometrics reduced to a single element d that when restricted to each fiber is the discrete metric, can be considered as a uniform bundle. Suppose that ϕ : T S is a continuous function and that Σ, the set of all global sections of the sheaf, is full and let Ê, p, S, d be the uniform bundle obtained by localization by means of ϕ. To show that Ê, p, S is a sheaf of sets it suffices to verify that each fiber is endowed with the discrete metric since in this case, if [σ] s Ê then the restriction of p to T 1 σ = { σs : s Dom σ} is a homeomorphism of T 1 σ 2 2 onto S. Let s be any element of S and consider two different elements [σ] s and [τ] s of the fiber Ês. Since d [σ] s, [τ] s = V Vs r ϕ 1 V dσr, τr, and [σ] s [τ] s, then V Vs r ϕ 1 V dσr, τr > 0, hence dσr, τr > 0, r ϕ 1 V for each neighbourhood V of s, thus if V is a neighbourhood of s, there exists r ϕ 1 V such that dσr, τr = 1. Then r ϕ 1 V dσr, τr = 1 and it follows that d[σ] s, [τ] s = dσr, τr = 1. This shows that the triplet is also a sheaf of sets. V Vs r ϕ 1 V Ê, p, S, obtained by localization by means of ϕ Example 2. The case of the Stone-Čech compactification, cf. [10][p. 47]. Two uniform bundles over the Stone-Čech compactification of a space T that turn out to be essentially the same can be constructed, one resorting to change of the base space and the other one directly. Let T be a topological space, βt its Stone-Čech compactification and e : T βt its canonical function. Consider a Hausdorff uniform space Y whose uniformity is defined by a directed family of bounded pseudometrics d i, E = T Y and let p : E T be the first projection. The set Λ of Revista Colombiana de Matemáticas

8 310 CLARA MARINA NEIRA & JANARIO VARELA all global selections for p can be identified with Y T, by means of the bijection f σ f : Y T Λ where σ f t = t, ft, whose inverse is the function σ f σ : Λ Y T, where f σ t = π 2 σt. Consider a full set Σ of global selections for p containing only elements of the form σ f, such that f : T Y is continuous and Im f is compact. If σ Σ there exists a unique continuous function f σ from βt to Y such that diagram e T βt σ T Y f σ π2 Y commutes. There are two ways leading to the same uniform bundle with base space βt : first by localization with change of the base space by means of e : T βt, consider Σ in the set of data and define for each s βt the equivalence relation R s on Σ by σ f R s σ g, if and only if, V Vs t e 1 V d i σ f t, σ g t = 0, for each i I. As a second alternative one can obtain the same bundle directly, without resorting to the expedient of change of the base space as follows: define, for each s βt, an equivalence relation R s on the set Σ { = fσ : σ Σ} by f σ Rs fτ, if and only if, for each i I, we have V Vs r V d i fσ r, f τ r = 0. For each s βt the corresponding fibers in each one of the resulting bundles are isomorphic, indeed, if s βt the function ψ : Σ/R s Σ/ R s [ ] [σ f ] s fσ s is an isometry for the families of pseudometrics defined on those quotients. Indeed, the function ψ is surjective and for each i I and each σ f, τ f in Σ we have that d i σ f t, τ f t = d i fσ r, f τ r. In fact, let V Vs t e 1 V m = V Vs t e 1 V V Vs r V d i σ f t, τ f t, l = d i fσ r, f τ r V Vs r V and let ɛ > 0. There exists V Vs such that r V d i fσ r, f τ r < l +ɛ. If t e 1 V then et V, thus d i fσ et, f τ et < l + ɛ, therefore Volumen 41, Número 2, Año 2007

9 LOCALIZATION WITH CHANGE OF THE BASE SPACE 311 d i σ f t, τ f t < l + ɛ, hence t e 1 V d i σ f t, τ f t l + ɛ. Then V Vs t e 1 V d i σ f t, τ f t l + ɛ. This implies that m l. On the other hand, let again ɛ > 0. There exist V Vs and δ > 0 such that d i σ f t, τ f t < δ < m + ɛ. t e 1 V For an indirect argument, pose that for a k V, γ = d i fσ k, f τ k m+ɛ > δ. Let α = γ δ 2. Since f σ and f τ are continuous functions, there exists a neighbourhood W of k with W V such that if q W then d i fσ q, f σ k < α and d i fτ q, f τ k < α. Again by contradiction, if for some q W, d i fσ q, f τ q δ, we have that d i fσ q, f σ k + d i fτ q, f τ k + d i fσ q, f τ q < α + α + δ = γ, thus d i fσ q, f σ k +d i fτ k, f σ q < γ, then d i fσ k, f τ k < γ which is a contradiction. Then for each q W we have that d i fσ q, f τ q > δ. Since et = βt there exists t T such that et W V. Then d i fσ et, f τ et > δ, therefore d i σ f t, τ f t > δ, which is not possible, then r V d i fσ r, f τ r m+ɛ, thus V Vs r V d i fσ r, f τ r m + ɛ and we have l m. Note that V Vs r V d i fσ r, f τ r = 0, for each i I, implies that for ɛ > 0 there exists a neighbourhood V of s such that r V d i fσ r, f τ r < ɛ. Therefore for each i I, we have d i fσ s, f τ s = 0 then f σ s = f τ s, since the family d i is Hausdorff. On the other hand, if fσ s = f τ s, i I and ɛ > 0, then, since f σ and f τ are continuous functions, there exists a neighbourhood V of s such that if r V then d i fσ s, f σ r < ɛ 2 and d i fτ s, f τ r < ɛ 2. Therefore for each r V we have that d i fσ r, f τ r < ɛ. We conclude that r V d i fσ r, f τ r ɛ and that d i fσ r, f τ r = 0. V Vs r V It follows that V Vs r V d i fσ r, f τ r = 0 if and only if fσ s = f τ s. This shows that the quotient Σ/ R s, and hence also Σ/R s, coincides Revista Colombiana de Matemáticas

10 312 CLARA MARINA NEIRA & JANARIO VARELA with the space Y, for each s βt. Therefore these two bundles over the base space βt are the same. Next, we want to look at the sections corresponding to functions having a singularity at a point a, and their values in the fiber over that point, in a bundle obtained by localization by change of the base space. Example 3. The bundle of continuous functions around a singular point. Let S be a topological space, a S be any point of S, T = S {a}, E = T R and p : E T the map defined by pt, x = t. Let d be the pseudometric for p defined by dt, x 1, t, x 2 = x 1 x 2. For each continuous function f : T R denote by σ f the selection for p, defined by σ f t = t, fx, let Σ = {σ f f : T R is continuous} and denote by ϕ the inclusion map from T into S. Consider the bundle obtained by localization over S from the given data. For each z S define the equivalence relation R z in Σ by σ f R z σ g if and only if V Vz t ϕ 1 V dσ f t, σ g t = 0; that is, if and only if V Vz t ϕ 1 V ft gt = 0. Denote by [σ f ] z the equivalent class of σ f module R z. Take z T, that is z a. It is apparent that if σ f R z σ g, then fz = gz. Conversely, if f, g : T R are continuous, if fz = gz and if ɛ > 0, by the continuity of f g there exists an open neighbourhood V of z in T, such that ft gt < ɛ, for each t V. Taking into account that V is 2 also a neighbourhood of z in S, it follows that t ϕ 1 V ft gt < ɛ, then V Vz t ϕ 1 V ft gt < ɛ and since ɛ was chosen arbitrarily, one concludes that V Vz t ϕ 1 V ft gt = 0. That is, σ f R z σ g. Thus, if z a, it turns out that σ f R z σ g if and only if fz = gz. Moreover, by using similar arguments, it follows that if f, g : T R are continuous functions, then d [σ f ] z, [σ g ] z = fz gz. Indeed, if ɛ > 0, there exists an open neighbourhood V of z in T such that ft gt < fz gz + ɛ 2 for each t V. Then t ϕ 1 V ft gt < fz gz + ɛ, hence V Vz t ϕ 1 V ft gt < fz gz +ɛ. therefore d [σ f ] z, [σ g ] z = V Vz t ϕ 1 V ft gt = fz gz. One can now assert that the map ψ : Êz = Σ/R z R defined by ψ [σ f ] z = fz is an isometry. In other words, if z T, then the fiber over z in the bundle Ê, p, R, d is R. Since each continuous function f : S {a} R can be identified with the element σ f of Σ and since σ f is identified with the section σ f of the bundle Ê, p, R, d obtained by localization, it follows that each continuous function defined from S {a} in R can be extended to a continuous section from S to Ê. In particular if S = R and a = 0, the functions sin z and 1/z can be extended, Volumen 41, Número 2, Año 2007

11 LOCALIZATION WITH CHANGE OF THE BASE SPACE 313 despite their non removable discontinuity at a, to a continuous sections to the domain S that does include the point a. This is an indication of the awesome intricacy of the fiber above the point a. 4. An observation regarding the categorical setting In this section it is shown that the fibers of the uniform bundle constructed by localization with change of the base space are colimits of direct systems in a certain category of uniform spaces. The following definitions are required. Definition 3. Let Λ be a directed set viewed as a category by means of its preorder. A directed system indexed by Λ in a category X is a covariant functor F, from Λ to X, that sends each object α of Λ into an object X α of X, and such that, for α β, there exists a morphism f αβ : X α X β of X satisfying: 1 For each α Λ, f αα = 1 Xα. 2 If α β γ then f βγ f αβ = f αγ. Denote by X α α Λ, f αβ α β the direct system that corresponds to F. Definition 4. Let X α α Λ, f αβ α β be a direct system in a category X. 1 An inductive cone for this direct system is a pair X, F α α Λ consisting of an object X of X and a family of morphisms F α : X α X of X such that for each α, β Λ, with α β, F β f αβ = F α. 2 A direct limit or colimit for the direct system is an inductive cone X, F α α Λ satisfying the following universal property: given any other inductive cone Y, G α α Λ in X, for this direct system, there exists a unique morphism ϕ : X Y, such that for every α Λ, ϕ F α = G α. Following S. Bautista [1], consider the category met H. Its objects are pairs consisting of a set X and a family of bounded pseudometrics d i defined on X, such that x 1 = x 2 if and only if d i x 1, x 2 = 0, for each i I, that is, the uniformity associated with the family of pseudometrics is Hausdorff, and a morphism between two given objects X, d i and Y, m j j J, is a pair of functions f, l, where f : X Y and l : J I satisfy m j fx 1, fx 2 d lj x 1, x 2, for each x 1, x 2 X and each j J. If X, d i is an object of met H, the identity morphism id X,di is the pair id X, id I, and if f, l : X, d i Y, m j j J and f, l : Y, m j j J Z, n k k K are morphisms of met H, then the composition of f, l and f, l is given by f, l f, l = f f, ll. Let T be a topological space, p : E T be a surjective function, d i be a directed family of bounded pseudometrics for p, such that the fiber E t = {a E : pa = t} is a Hausdorff space for each t T, Σ be a full set of global selection for p and ϕ : T S be a continuous function. Take s S and for each V Vs let Σ V = { σ ϕ 1 V : σ Σ } and d V i be the family of pseudometrics over Σ V, where for each i I, d V i : Σ V Σ V R Revista Colombiana de Matemáticas

12 314 CLARA MARINA NEIRA & JANARIO VARELA is defined by d V i σ ϕ 1 V, τ ϕ 1 V = t ϕ 1 V d i σt, τt, then the pair Σ V, d V i is an object of meth. Now consider the order relation over the neighbourhood filter of s, given by V W, if and only if, W V. For each V, W Vs such that V W, let f V W : Σ V Σ W and l V W : I I be the functions defined by f V W σ ϕ 1 V = σ ϕ 1 W and l V W i = i. For each i I, one has that d W i σ ϕ 1 W, τ ϕ 1 W = d i σt, τt t ϕ 1 W d i σt, τt t ϕ 1 V = d V l V W i σ ϕ 1 V, τ ϕ 1 V, thus the pair f V W, l V W is a morphism of met H. Then the pair Σ V, d V i, f V W, l V W W V is a direct system in V Vs met H. Let Ê, p, S, di be the uniform bundle over the topological space S, constructed by localization from the topological space T, the surjective function p : E T, the directed family of bounded pseudometrics for p, d i such that E t = {a E : pa = t} is a Hausdorff space, for each t T, the set Σ of global selections for p and the continuous function ϕ : T S. For s S, the equivalence relation R s in Σ is defined by, σr s τ, if and only if, V Vs t ϕ 1 V d i σt, τt = 0, for each i I, and the fiber Ês is the set Σ/R s. Consider the family of pseudometrics di on Ês, where d i [σ] s, [τ] s = V Vs t ϕ 1 V d i σt, τt, for each i I. Ê s, di is then an object of met H. The pair For each V Vs, let V : Σ V Ês and ξ V : I I, the functions defined by V σ ϕ 1 V = [σ] s class of σ module R s and ξ V i = i respectively. It follows that d i V σ ϕ 1 V, V τ ϕ 1 V = di [σ] s, [τ] s = Vs t ϕ 1 = d V i t ϕ 1 V d i σt, τt d i σt, τt σ ϕ 1 V, τ ϕ 1 V = d V ξ V i σ ϕ 1 V, τ ϕ 1 V, Volumen 41, Número 2, Año 2007

13 LOCALIZATION WITH CHANGE OF THE BASE SPACE 315 then V, ξ V is a morphism of met H. Note that if V, W Vs and V W, then for each σ ϕ 1 V Σ V, W fv W σ ϕ 1 V = W σ ϕ 1 W = [σ]s = V σ ϕ 1 V, thus W f V W = V, therefore Ê s, di, V, ξ V V Vs is an inductive cone for the direct system ΣV, d V i V Vs, f V W, l V W W V. Now pose that Y, m j j J, ð V, χ V V Vs is an other inductive cone for this direct system. For each V Vs, the functions ð V : Σ V Y and χ V : J I satisfy m j ðv σ ϕ 1 V, ðv τ ϕ 1 V d V χv j σ ϕ 1 V, τ ϕ 1 V, for each σ ϕ 1 V, τ ϕ 1 V Σ V and each j J, and if V, W Vs and V W, then ð W f V W = ð V and l V W χ W = χ V. Therefore, for each V Vs and each σ Σ, ð V f SV σ = ð S σ, that is, ð V σ ϕ 1 V = ðs σ. Furthermore, for each j J, l SV χ V j = χ S j, hence χ V j = χ S j. Define ψ : Ês Y by ψ [σ] s = ð S σ, pose that [σ] s = [τ] s and let j J and ɛ > 0. There exists a V Vs such that t ϕ 1 V d χsjσt, τt < ɛ. Then m j ψ [σ] s, ψ [τ] s = m j ð S σ, ð S τ = m j ðv σ ϕ 1 V, ðv τ ϕ 1 V d V χj σ ϕ 1 V, τ ϕ 1 V < ɛ, This means that m j ψ[σ] s, ψ[τ] s = 0, for each j J, thus ð S σ = ð S τ since Y is a Hausdorff space. It follows that ψ is well defined. Define ζ : J I by ζj = χ S j. It is immediate that the pair ψ, ζ is a morphism of met H and is the only morphism such that for each V Vs, ψ, ζ V, ξ V = ð V, χ V. It follows that Ê s, di, V, ξ V V Vs is the colimit of the direct ΣV system, d V i V, f Vs V W, l V W W V in met H. In terms of the definitions given above the following proposition summarizes the preceding considerations. Proposition 2. Let T be a topological space, p : E T a surjective function, d i a directed family of bounded pseudometrics for p, such that the fiber E t = {a E : pa = t} is a Hausdorff space for each t T, Σ a full set of global selection for p and ϕ : T S a continuous function. Consider the Revista Colombiana de Matemáticas

14 316 CLARA MARINA NEIRA & JANARIO VARELA uniform bundle Ê, p, S, di constructed by localization from the data above and let s S. Then: 1 The pair Σ V, d V i is an object of met H. 2 The pair f V W, l V W is a morphism of met H and the pair Σ V, d V i, f V W, l V W W V V Vs is a direct system in met H. 3 The pair V, ξ V is a morphism of met H and the inductive cone Ê s, di, V, ξ V V Vs is the colimit of the direct system in met H. Σ V, d V i, f V W, l V W W V V Vs Remark 1. From now on, the following set of data is to be considered: a topological space T, a surjective function p : E T, a directed family d i of bounded pseudometrics for p such that the fiber E t = {a E : pa = t} is a Hausdorff space for each t T, a full set Σ of global selection for p and a continuous function ϕ : T S. 5. A category of presheaves In order to present a universal arrow associated with the process of localization with change of the base space, the data provided to construct the uniform bundle ought to be presented as a presheaf M over the space S. If H denotes the functor that to each uniform bundle over S associates the presheaf of its local sections, it will be shown that the uniform bundle Ê, p, S, di obtained by localization from the data, together with an obvious morphism of presheaves φ between M and H Ê, p, S, di, establish a universal arrow from M to the functor H. This and the following sections outline the categorical context in which we can state without ambiguity the aforementioned universal arrow. Definition 5. Let X, τ be a topological space. A presheaf F of Hausdorff uniform spaces over X is a contravariant functor defined on τ with values in the category met H as follows: 1 To each open set τ it is assigned a set F and a family d i of bounded pseudometrics generating a Hausdorff uniformity. Volumen 41, Número 2, Año 2007

15 LOCALIZATION WITH CHANGE OF THE BASE SPACE To each pair of open sets, V τ, with V, it is assigned a pair f V, l V of functions f V : F F V, l V : I V I in such a way that d i f V x, f V y d lv ix, y, for each x, y F and each i I V. Notice that a presheaf is a direct system in met H. Example 4. The data defined in Remark 1 give rise to a presheaf M of Hausdorff uniform spaces in the following way: 1 Given an open set of S, let M = Σ ϕ 1 and let d i the family of pseudometrics over M, where for each i I it holds that d i σ ϕ 1, τ ϕ 1 = d i σ ϕ t ϕ 1 1 t, τ ϕ 1 t. 2 If and V are open subsets of S and V, define the functions and m V : Σ ϕ 1 Σ ϕ 1 V σ ϕ 1 σ ϕ 1 V l M V : I I i i. It follows that d V i m V σ ϕ 1, m V τ ϕ 1 = d V i σ ϕ 1 V, τ ϕ 1 V = d i σ ϕ 1 V, τ ϕ 1 V t ϕ 1 V d i σ ϕ 1, τ ϕ 1 t ϕ 1 = d i σ ϕ 1, τ ϕ 1. Example 5. Let F, q, S, m j j J be a uniform bundle where m j j J is a directed family of bounded pseudometrics. The presheaf F of local sections for q is defined as follows: 1 Given an open set of S let F = Γ the set of all local sections for q with domain and let d j be the family of pseudometrics j J over F, where for each j J, m j σ, τ = s m jσs, τs. 2 If and V are open subsets of S and if V, define f V : Γ Γ V and σ σ V l F V : J J j j. Revista Colombiana de Matemáticas

16 318 CLARA MARINA NEIRA & JANARIO VARELA In the particular case of the bundle Ê, p, S, di obtained by localization, denote by Ê the presheaf of local sections for p and by e V and lêv the functions f V and lv F respectively The category Preh. Extending the ideas of S. Bautista [1], consider as objects of the category Preh all presheaves of Hausdorff uniform spaces over the topological space S, τ. Let F : τ met H and G : τ met H be presheaves over S. For, V τ, with V, denote by f V and l V the functions defining F and by g V and h V those defining G. A morphism from F to G is given by a natural transformation φ from F to G sending each object τ into the morphism φ, b : F J G I in met H in such a way that if, V τ and V, then the diagrams F φ G f V φ V g V F V GV and I V b V J V l V I b J h V commute. 6. The category of Hausdorff uniform bundles In this section it is shown that the uniform bundle obtained by localization with change of the base space satisfies a universal property in a category of uniform bundles as it was announced in the first paragraph of Section 5. The localization process, studied in the previous sections, does not require any hypothesis on the base spaces T and S, but now for the purpose of the present section, consider Hausdorff uniform bundles with Hausdorff base space. Denote by the category of Hausdorff uniform bundles. The objects of are bundles of uniform spaces with base space S, with uniformities determined by directed families of bounded pseudometrics, having a full set of global sections and whose fibers are Hausdorff spaces. Volumen 41, Número 2, Año 2007

17 LOCALIZATION WITH CHANGE OF THE BASE SPACE 319 Let F, q, S, m j j J and F, q, S, m j j J be two objects of. A morphism, l : F, q, S, m j j J F, q, S, m j j J in the category is a pair of maps : F F, l : J J such that is continuous, m j a, b m lj a, b, for each a, b F and each j J and the diagram commutes that is, preserves fibers. F F q q S 6.1. The functor H. Define the functor H from the category to Preh as follows: 1 To each element F, q, S, m j j J of, H assigns the presheaf F of local sections for q. 2 To each morphism, l : F, q, S, m j j J F, q, S, m j j J in assigns the morphism of presheaves given by the natural transformation φ,l from F to F that sends each open set of S into the morphism of met H φ,l : Γ, m j Γ j J, m j j J defined by the functions f,l : Γ Γ σ σ, where σ s = σs, for each s S and Notice that m j f,l l,l : J J j l j. σ, f,l τ = m j σ, τ = m j σ s, τ s s = m j σ s, τ s s m lj σ s, τ s s = m σ, τ. l,l Furthermore, if and V are open sets of S and V, then the diagram Revista Colombiana de Matemáticas

18 320 CLARA MARINA NEIRA & JANARIO VARELA Γ f,l Γ f V Γ V f,l V commutes. In fact, if σ Γ, then σ = f V σ f V f,l = σ V = σ V Γ V f V = f,l V σ V = f,l V f V σ A universal arrow. Consider the morphism of presheaves φ : M Ê assigning to each open set in S the morphism of met H, φ : Σ ϕ 1, d i Γ, d i defined by the functions and f : Σ ϕ 1 Γ σ ϕ 1 σ l : I I i i. Notice that for each σ, τ Σ, f σ ϕ 1, f τ ϕ 1 = d i σ, τ d i = d i σ s, τ s s = s s = = d i t ϕ 1 it follows that φ is a morphism in met H. V Vs t ϕ 1 V d i σ t, τ t d i σ t, τ t t ϕ 1 d i σ t, τ t σ ϕ 1, τ ϕ 1, Volumen 41, Número 2, Año 2007

19 LOCALIZATION WITH CHANGE OF THE BASE SPACE 321 It remains to be seen that < Ê, p, S, d i, φ > is a universal arrow from M to the functor H. Suppose that F, q, S, m j j J is an object of and for each open set of S denote by Γ the set of all local sections for q with domain. Let γ : M F be a morphism of presheaves assigning to each open set of S the morphism γ : Σ ϕ 1, d i Γ, m j in met j J H determined by the functions g : Σ ϕ 1 Γ and h : J I. Notice that m j g σ ϕ 1, g τ ϕ 1 d h j σ ϕ 1, τ ϕ 1 for each σ, τ Σ and each j J. Furthermore, since γ is a morphism of presheaves, if and V are open subsets of S and V, then h V j = h j, for each j J and f V g σ ϕ 1 = gv mv σ ϕ 1, for each σ Σ. That is, g σ ϕ 1 = g V V σ ϕ 1 V, for each σ Σ. In particular, g S σ = g σ ϕ 1, for each σ Σ and each open subset of S. Consider the morphism, l : Ê, p, S, di F, q, S, m j j J defined by [σ] s = g S σs and lj = h S j. The function is well defined, in fact: if [σ] s = [τ] s then V Vs t ϕ 1 V d i σt, τt = 0, for each i I. Let j J, for each open neighbourhood of s one can assert that: m j g σ ϕ 1, g τ ϕ 1 d h j σ ϕ 1, τ ϕ 1, then m j g σ ϕ 1 r, g τ ϕ 1 r d h j σ t, τ t. r t ϕ 1 In particular, m j g σ ϕ 1 s, g τ ϕ 1 s d h j σ t, τ t t ϕ 1 and since S, that is, then m j g S σ s, g S τ s m j g S σ s, g S τ s m j g S σ s, g S τ s d h j σ t, τ t, t ϕ 1 d hsj σ t, τ t, t ϕ 1 Vs t ϕ 1 d hsj σ t, τ t = 0. This shows that g S σ s = g S τ s, hence is well defined. Revista Colombiana de Matemáticas

20 322 CLARA MARINA NEIRA & JANARIO VARELA The commutativity of the diagram M γ φ F Ê φ,l is secured. Indeed, if is an open subset of S, σ Σ and s, then This means that the diagram σ s = [σ] s Σ ϕ 1 g commutes. The commutativity of the diagram J l l = g S σ u s = g σ ϕ 1 s. Γ I f h Γ,l f and the uniqueness of γ are immediate. It follows that Ê, p, S, di, φ is a universal arrow from M to the functor H. Acknowledgments. The authors are indebted to the referee for his very pertinent suggestions that led to a revision of this paper. l I References [1] Bautista, S. Co-completez de categorías de espacios uniformes y procesos de localización. Master tesis, Departamento de Matemáticas, niversidad Nacional de Colombia, Bogotá, [2] Bautista, S., and Varela, J. Localización en campos de espacios uniformes separados. Rev. Colombiana Mat , [3] Bourbaki, N. General Topology. Addison Wesley, París, [4] De Castro, R., and Varela, J. Localization in bundles of uniform spaces. Rev. Colombiana Mat. 24, , [5] Dauns, J., and Hofmann, K. H. Representation of rings by sections. Mem. Amer. Math. Soc., Providence, R.I., [6] García, M., Roig, J. M., Olano, C., Domínguez, E. O., and Pinilla, J. L. Topología. Alhambra, Madrid, [7] Hoffman, K. H. Representation of algebras by continuous sections. Bull. Amer. Math. Soc , Volumen 41, Número 2, Año 2007

21 LOCALIZATION WITH CHANGE OF THE BASE SPACE 323 [8] Kelley, J. L. General Topology. D. Van Nostrand Company Inc., Canada, [9] MacLane, S. Categories for the working mathematician. Springer-Verlag, New York, [10] Neira, C. M. Sobre campos de espacios uniforme. Doctoral dissertation, Departamento de Matemáticas, niversidad Nacional de Colombia, Bogotá, [11] Varela., J. Existence of uniform bundles. Rev. Colombiana Mat. 18, , 1 8. [12] Varela, J. On the existence of uniform bundles. Rev. Colombiana Mat , [13] Willard, S. General Topology. Addison Wesley, New York, Recibido en agosto de Aceptado en agosto de 2007 Departamento de Matemáticas niversidad Nacional de Colombia Carrera 30, calle 45 Bogotá, Colombia cmneirau@unal.edu.co Departamento de Matemáticas niversidad Nacional de Colombia Carrera 30, calle 45 Bogotá, Colombia jvarelab13@yahoo.com Revista Colombiana de Matemáticas