Volumen 41(2007)2, p´aginas 303-323
Localization with change of the base space in uniform bundles and sheaves
Localizaci´on con cambio del espacio base en campos uniformes y haces
Clara Marina Neira
1, Januario Varela
11
Universidad Nacional de Colombia, Bogot´ a, 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 spacesT and S, a continuous function ϕ : T −→ S, a surjective function p : E −→ T, a directed family (di)i∈I of bounded pseudometrics forpgenerating a Hausdorff uniformity and a family Σ of global selections forp. In terms of these data, a uniform bundle is constructed over the base spaceS, 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. Uniform bundle, sheaf of sets, localization, colimit.
2000 Mathematics Subject Classification. 55R65, 54E15, 54B40.
Resumen. En este art´ıculo se presenta un proceso de localizaci´on (germinaci´on) con cambio del espacio base. El conjunto de datos consta de dos espacios topol´ogicosT yS, una funci´on continuaϕ:T −→S, una funci´on sobreyectiva p:E−→T, una familia dirigida (di)i∈I de seudom´etricas acotadas parapque genera una uniformidad de Hausdorff y una familia Σ de selecciones globales para p. En t´erminos de estos datos, se construye un campo uniforme sobre el espacio baseS, cuyas fibras son col´ımites en una categor´ıa de espacios uni- formes. Como aplicaci´on inmediata, se obtienen resultados similares para el caso particular de los haces de conjuntos. Este proceso de localizaci´on da lugar a una flecha universal en un contexto apropiado que es descrito en t´erminos de una categor´ıa de campos uniformes.
Palabras y frases clave. Campo uniforme, haz de conjuntos, localizaci´on, col´ımite.
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 spaceC from a uniform bundle (E, π, B), whereBis a subset ofC, and from a full set of global selections forπ.
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 spaceT, their procedure led to the construction of a bundle of uniform spaces overT, 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 functionϕ: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 spaceS, in place of the given spaceT, 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 spacesT andS, a continuous function ϕfrom T toS (non necessarily surjective), a surjective functionp from a set E onto T, a directed family of bounded pseudometrics for p and a family of global selections forp. 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 overT and construct a sheaf overS via the continuous functionϕ, 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 sec- tions, all of them are treated in detail in the references [7], [10], [11] and [12].
2.1. Uniform bundles. LetEandTbe topological spaces andp:E−→Tbe a surjective function. For eacht∈T, the setEt=p−1(t) ={a∈E:p(a) =t}
is called the fiber above t. Observe that E is the disjoint union of the family (Et)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 idQ 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 ΓQ(p) :={σ |σ :Q−→E is continuous andp◦σ=idQ}.IfQ=T, one writes Γ(p) instead of ΓT(p).A set Σ of local sections is said to befull, if for eachx∈E, there existsσ∈Σ such thatσ(p(x)) =x.
LetE×T E:={(u, v)∈E×E:p(u) =p(v)}.The functiond:E×TE−→R is called a pseudometric for p provided that the restriction of d to Et ×Et
is a pseudometric on Et, for each t ∈ T. A family of pseudometrics for p, (di)i∈I, is directed if for each pair i1, i2 ∈ I, there exists i ∈ I such that di1(u, v)6di(u, v) anddi2(u, v)6di(u, v), for each (u, v)∈E×T E.
Definition 1. Let(di)i∈I be a directed family of pseudometrics for p, σ be a local selection, i∈I and >0, the setTi(σ) ={u∈E:di(u, σ(p(u)))< }is called the-tube around σ with respect todi.
Definition 2. Let E and T be topological spaces, p : E −→ T be a sur- jective function and (di)i∈I a family of pseudometrics for p. The quadruplet (E, p, T,(di)i∈I)is said to be a bundle of uniform spacesor simply a uniform bundleprovided that:
(1) For each u∈E, each >0and each i∈I, there exists a local section σ such that u∈ Ti(σ).
(2) The collection of all tubes around local sections for p form a base for the topology ofE.
The spaceT is called the base space and the spaceEis called the fiber space or the bundle space.
Observe that if (E, p, T,(di)i∈I) is a uniform bundle, then the functionpis continuous and open.
Definition 2 secures the upper semicontinuity of the distance functionss7−→
di(σ(s), τ(s)) :Q−→R+∪ {0}, where σ andτ are arbitrary local sections for p,Qis any open subset of Dom σ∩Dom τ andi∈I.
2.2. Existence Theorem of Uniform 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 setΣof local selections forpand a directed family(di)i∈I
of pseudometrics forp. Assume the following conditions:
a) For each u∈ E, each i ∈I and each > 0, there existsα ∈Σ such thatu∈ Ti(α).
b) The function s 7−→ di(α(s), β(s)) : Dom α ∩ Dom β −→ R is upper semicontinuous, for each i∈I and each (α, β)∈Σ×Σ.
Then there exists a topologyS over E such that:
1) The topology S has a base consisting of the sets of the form Ti(αQ), where i ∈ I, > 0 and αQ is the restriction to an open subset Q⊂ Dom α of a local selectionα∈Σ.
2) Each α∈Σis a local section.
3) (E, p, T,(di)i∈I)is a uniform bundle.
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 (di)i∈I a directed family of bounded pseudometrics forp. Suppose that Σ is a full set of global selections forp. A uniform bundle overSis to be constructed in terms of these data by means of the mapϕ.
For each s ∈S, define the relation Rs in the set Σ as follows: for each pair σ, τ ∈Σ,
σRsτ, if and only if, inf
V∈V(s) sup
r∈ϕ−1(V)
di(σ(r), τ(r)) = 0, for eachi∈I. (i) Note that ifs∈(Srϕ(T))◦, there exists a neighbourhoodV ofssuch that ϕ−1(V) =∅. If this is the case, by convention, one takes supr∈∅di(σ(r), τ(r)) = 0, for eachi∈I.
For eachs∈S,Rsis an equivalence relation on Σ. Sincedi is a pseudometric, thenRs is reflexive and symmetric. The transitivity is also straightforward.
Denote by [σ]s the equivalence class of σ module Rs. To be noted that, for s∈(Srϕ(T))◦, the set Σ/Rsreduces to a single point.
LetEb be the disjoint union of the family{Σ/Rs:s∈S}andbpbe the function b
p:Eb−→S
[σ]s7−→s. (ii)
For eachi∈I, define
dbi :Eb×SEb−→R ([σ]s,[τ]s)7−→ inf
V∈V(s) sup
r∈ϕ−1(V)
di(σ(r), τ(r)). (iii) All thedbi,i∈I, are bounded pseudometrics forp, indeed:b
to show that the functiondbi is well defined, letσ1Rsσ,τ1Rsτ, m:= inf
V∈V(s) sup
r∈ϕ−1(V)
di(σ(r), τ(r)) and
l:= inf
V∈V(s) sup
r∈ϕ−1(V)
di(σ1(r), τ1(r)).
For a given >0, there exists a neighbourhoodV ofssuch that sup
r∈ϕ−1(V)
di(σ(r), σ1(r))<
3, sup
r∈ϕ−1(V)
di(τ(r), τ1(r))<
3 and
sup
r∈ϕ−1(V)
di(σ(r), τ(r))< m+ 3.
Hence, for eachr∈ϕ−1(V),
di(σ1(r), τ1(r))6di(σ1(r), τ(r)) +di(τ(r), τ1(r))
6di(σ1(r), σ(r)) +di(σ(r), τ(r)) +di(τ(r), τ1(r))
<
3+m+ 3+
3
=m+.
It follows that, for each >0, there exists a neighbourhoodV ofssuch that sup
r∈ϕ−1(V)
di(σ1(r), τ1(r))6m+, thus
l= inf
V∈V(s) sup
r∈ϕ−1(V)
di(σ1(r), τ1(r))6m.
A similar argument shows that conversely m= inf
V∈V(s) sup
r∈ϕ−1(V)
di(σ(r), τ(r))6l.
The family dbi
i∈I of pseudometrics forpbis directed, in fact, let i1, i2 ∈I, since by hypothesis the family (di)i∈I is directed, there exists aj∈I such that di1(a, b), di2(a, b) 6dj(a, b), for each (a, b) ∈E×T E. It is immediate that each pair ([σ]s,[τ]s)∈Eb×SEb satisfies
dbi1([σ]s,[τ]s), dbi2([σ]s,[τ]s)6dbj([σ]s,[τ]s).
Define now a set of selections for p: for eachb σ ∈ Σ, letσb : S −→ Eb be the function defined bybσ(s) = [σ]s. The setΣ =b {σb:σ∈Σ}is a full set of global selections for p. It remains to be seen thatb bp, the family Σ and the familyb dbi
i∈I satisfy the hypotheses of the Existence Theorem of Uniform Bundles.
a) Let [σ]s∈E,b i∈I and >0. It is immediate that [σ]s=σb(s)∈ Ti(σ).b b) Let i∈I,
b α,βb
∈Σb×Σ andb >0. Ifdbi
α(sb 0),β(sb 0)
< then we have that infV∈V(s0)supt∈ϕ−1(V)di(α(t), β(t))< , hence there exists an open neighbourhoodV ofs0such that supt∈ϕ−1(V)di(α(t), β(t))< , thus one has that for eachs∈V,dbi
α(s),b βb(s)
< .
It follows that the sets of the form Ti(σbQ), where i ∈I, > 0, bσ ∈ Σ,b Qis an open subset of the domain ofσb andbσQ is the restriction ofbσ toQ, form a base for a topology onEbsuch that
E,b p, S,b dbi
i∈I
is a uniform bundle and each element ofΣ is a section.b
On the other hand, for each i ∈ I, the functions Di : Σ×Σ −→ R and Dbi :Σb×Σb −→R defined byDi(σ, τ) = supt∈Tdi(σ(t), τ(t)) and Dbi(σ,b τb) = sups∈Sdbi(bσ(s),τb(s)) are pseudometrics on Σ and Σ respectively.b
The function
ψ: Σ−→Σ, σb 7−→bσ (iv)
is an isometry with respect to Di and Dbi. In fact, for each t ∈ T and each neighbourhoodV ofϕ(t) inS, one has that
di(σ(t), τ(t))6 sup
r∈ϕ−1(V)
di(σ(r), τ(r)), then
di(σ(t), τ(t))6 inf
V∈V(ϕ(t)) sup
r∈ϕ−1(V)
di(σ(r), τ(r))
=dbi(bσ(ϕ(t)),bτ(ϕ(t))) 6sup
s∈S
dbi(σb(s),bτ(s)),
therefore supt∈Tdi(σ(t), τ(t))6sups∈Sdbi(σ(s),b τb(s)),thusDi(σ, τ)6Dbi(bσ,bτ).
On the other hand, since S is a neighbourhood of s for each s ∈ S and ϕ−1(S) =T, it follows that
V∈V(s)inf sup
r∈ϕ−1(V)
di(σ(r), τ(r))6sup
r∈T
di(σ(r), τ(r)) =Di(σ, τ), for each s∈S, thus sups∈S
infV∈V(s)supr∈ϕ−1(V)di(σ(r), τ(r))
6Di(σ, τ).
HenceDbi(σ,b τb)6Di(σ, τ).Then the function ψsatisfies the identity Dbi(ψ(σ), ψ(τ)) =Di(σ, τ), for eachi∈I.
Ifψ(σ) =ψ(τ) thenDbi(bσ,bτ) = 0, for eachi∈I, hence Di(σ, τ) = sup
t∈T
di(σ(t), τ(t)) = 0,
for eachi ∈ I, then di(σ(t), τ(t)) = 0, for each t ∈ T. It follows that, if the family of pseudometrics (di)i∈I is Hausdorff, that is, a = b if di(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 pdetermined by the family (di)i∈I is Hausdorff (equivalently if each fiber is a Hausdorff space), one can identify Σ withΣ.b
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,(di)i∈I be a directed family of bounded pseudometrics for pgenerating a Hausdorff uniformity and Σbe a full set of global selections forp.
(1) For eachs∈S, the relationRsis an equivalence relation.
(2) dbi
i∈I is a directed family of bounded pseudometrics forp.b
(3) The quadruplet
E,b p, S,b dbi
i∈I
is a uniform bundle whose set of global sections contains an isomorphic copy Σb ofΣ.
For the particular caseϕ=idT, 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 elementdthat when restricted to each fiber is the discrete metric, can be considered as a uniform bundle.
Suppose thatϕ :T −→ S is a continuous function and that Σ, the set of all global sections of the sheaf, is full and let
E,b p, S,b db
be the uniform bundle obtained by localization by means ofϕ.
To show that
E,b p, Sb
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 ∈ Eb then the restriction ofpbto T1
2(σ) =b {bσ(s) :s∈Dombσ}is a homeomorphism ofT1
2(σ)b ontoS.
Letsbe any element ofS and consider two different elements [σ]s and [τ]sof the fiberEbs. Since
db([σ]s,[τ]s) = inf
V∈V(s) sup
r∈ϕ−1(V)
d(σ(r), τ(r)), and [σ]s6= [τ]s, then infV∈V(s)supr∈ϕ−1(V)d(σ(r), τ(r))>0,hence
sup
r∈ϕ−1(V)
d(σ(r), τ(r))>0,
for each neighbourhoodV ofs, thus ifV is a neighbourhood ofs, there exists r ∈ ϕ−1(V) such that d(σ(r), τ(r)) = 1. Then supr∈ϕ−1(V)d(σ(r), τ(r)) = 1 and it follows that
d([σ]b s,[τ]s) = inf
V∈V(s) sup
r∈ϕ−1(V)
d(σ(r), τ(r)) = 1.
This shows that the triplet
E,b p, Sb
, obtained by localization by means ofϕ is also a sheaf of sets.
Example 2. The case of the Stone- ˇCech compactification, cf. [10][p. 47].
Two uniform bundles over the Stone- ˇCech compactification of a spaceT 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, β(T) its Stone- ˇCech compactification and e : T −→ β(T) its canonical function. Consider a Hausdorff uniform space Y whose uniformity is defined by a directed family of bounded pseudometrics (di)i∈I,E =T×Y and let p: E −→T be the first projection. The set Λ of
all global selections forpcan be identified withYT, by means of the bijection f 7−→ σf : YT −→ Λ where σf(t) = (t, f(t)), whose inverse is the function σ 7−→fσ : Λ−→YT, where fσ(t) =π2(σ(t)). Consider a full set Σ of global selections forpcontaining only elements of the formσf, such that f :T −→Y is continuous and Im f is compact. Ifσ ∈Σ there exists a unique continuous function ˘fσ fromβ(T) toY such that diagram
T β(T)
T×Y Y
-
e
QσQQs + f˘σ QQs
π2
commutes.
There are two ways leading to the same uniform bundle with base spaceβ(T):
first by localization with change of the base space by means ofe:T −→β(T), consider Σ in the set of data and define for eachs∈β(T) the equivalence rela- tionRson Σ byσfRsσg, if and only if, infV∈V(s)supt∈e−1(V)di(σf(t), σg(t)) = 0, for each i ∈ I. As a second alternative one can obtain the same bun- dle directly, without resorting to the expedient of change of the base space as follows: define, for each s ∈ β(T), an equivalence relation ˘Rs on the set ˘Σ = n
f˘σ:σ∈Σo
by ˘fσR˘sf˘τ, if and only if, for each i ∈ I, we have infV∈V(s)supr∈V di
f˘σ(r),f˘τ(r)
= 0.
For eachs∈β(T) the corresponding fibers in each one of the resulting bundles are isomorphic, indeed, ifs∈β(T) the function
ψ: Σ/Rs−→Σ/˘ R˘s
[σf]s7−→h f˘σ
i
s
is an isometry for the families of pseudometrics defined on those quotients.
Indeed, the functionψis surjective and for eachi∈I and eachσf,τf in Σ we have that
V∈V(s)inf sup
t∈e−1(V)
di(σf(t), τf(t)) = inf
V∈V(s)sup
r∈V
di
f˘σ(r),f˘τ(r) . In fact, let
m= inf
V∈V(s) sup
t∈e−1(V)
di(σf(t), τf(t)), l= inf
V∈V(s)sup
r∈V
di
f˘σ(r),f˘τ(r)
and let >0. There existsV ∈ V(s) such that supr∈V di
f˘σ(r),f˘τ(r)
< l+.
If t ∈ e−1(V) then e(t) ∈ V, thus di
f˘σ(e(t)),f˘τ(e(t))
< l+, therefore
di(σf(t), τf(t))< l+, hence supt∈e−1(V)di(σf(t), τf(t))6l+.Then inf
V∈V(s) sup
t∈e−1(V)
di(σf(t), τf(t))≤l+. This implies thatm6l.
On the other hand, let again >0. There existV ∈ V(s) andδ >0 such that sup
t∈e−1(V)
di(σf(t), τf(t))< δ < m+. For an indirect argument, suppose that for ak ∈V, γ =di
f˘σ(k),f˘τ(k)
≥ m+ > δ.Letα= γ−δ
2 .Since ˘fσand ˘fτ are continuous functions, there exists a neighbourhoodW ofkwithW ⊂V such that ifq∈Wthendi
f˘σ(q),f˘σ(k)
<
αanddi
f˘τ(q),f˘τ(k)
< α.
Again by contradiction, if for someq∈W,di
f˘σ(q),f˘τ(q)
6δ, we have that di
f˘σ(q),f˘σ(k) +di
f˘τ(q),f˘τ(k) +di
f˘σ(q),f˘τ(q)
< α+α+δ=γ, thusdi
f˘σ(q),f˘σ(k) +di
f˘τ(k),f˘σ(q)
< γ,thendi
f˘σ(k),f˘τ(k)
< γwhich is a contradiction. Then for each q ∈W we have that di
f˘σ(q),f˘τ(q)
> δ.
Since e(T) = β(T) there exists t ∈ T such that e(t) ∈ W ⊂ V. Then di
f˘σ(e(t)),f˘τ(e(t))
> δ,thereforedi(σf(t), τf(t))> δ, which is not possible, then supr∈V di
f˘σ(r),f˘τ(r)
6m+, thus infV∈V(s)supr∈V di
f˘σ(r),f˘τ(r) 6 m+and we havel6m.
Note that infV∈V(s)supr∈V di
f˘σ(r),f˘τ(r)
= 0, for eachi∈I, implies that for >0 there exists a neighbourhoodV ofssuch that supr∈V di
f˘σ(r),f˘τ(r)
<
. Therefore for eachi∈I, we havedi
f˘σ(s),f˘τ(s)
= 0 then ˘fσ(s) = ˘fτ(s), since the family (di)i∈I is Hausdorff. On the other hand, if ˘fσ(s) = ˘fτ(s), i ∈ I and > 0, then, since ˘fσ and ˘fτ are continuous functions, there ex- ists a neighbourhoodV ofssuch that ifr∈V thendi
f˘σ(s),f˘σ(r)
<
2 and di
f˘τ(s),f˘τ(r)
<
2. Therefore for eachr∈V we have thatdi
f˘σ(r),f˘τ(r)
<
. We conclude that supr∈V di
f˘σ(r),f˘τ(r)
6and that
V∈V(s)inf sup
r∈V
di
f˘σ(r),f˘τ(r)
= 0.
It follows that infV∈V(s)supr∈V di
f˘σ(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/Rs, coincides
with the spaceY, for eachs∈β(T). Therefore these two bundles over the base spaceβ(T) are the same.
Next, we want to look at the sections corresponding to functions having a singularity at a pointa, 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.
LetS be a topological space,a∈S be any point ofS,T=Sr{a},E =T×R andp:E−→T the map defined byp(t, x) =t. Letdbe the pseudometric for pdefined by d((t, x1),(t, x2)) =|x1−x2|.
For each continuous function f : T −→ R denote by σf the selection for p, defined byσf(t) = (t, f(x)), let Σ ={σf |f :T −→Ris continuous}and denote byϕthe inclusion map fromT intoS.
Consider the bundle obtained by localization over S from the given data.
For eachz∈S define the equivalence relationRz in Σ by σf Rz σg if and only if inf
V∈V(z) sup
t∈ϕ−1(V)
d(σf(t), σg(t)) = 0;
that is, if and only if infV∈V(z)supt∈ϕ−1(V)|f(t)−g(t)|= 0. Denote by [σf]z
the equivalent class ofσf moduleRz.
Takez∈T, that isz6=a. It is apparent that ifσf Rzσg, thenf(z) =g(z).
Conversely, if f, g : T −→ R are continuous, if f(z) = g(z) and if > 0, by the continuity of f −g there exists an open neighbourhood V of z in T, such that |f(t)−g(t)| <
2, for each t ∈ V. Taking into account that V is also a neighbourhood of z in S, it follows that supt∈ϕ−1(V)|f(t)−g(t)| < , then infV∈V(z)supt∈ϕ−1(V)|f(t)−g(t)|< and sincewas chosen arbitrarily, one concludes that infV∈V(z)supt∈ϕ−1(V)|f(t)−g(t)|= 0. That is,σf Rz σg. Thus, ifz6=a, it turns out thatσf Rzσg if and only iff(z) =g(z). Moreover, by using similar arguments, it follows that if f, g : T −→ R are continuous functions, then db([σf]z,[σg]z) =|f(z)−g(z)|. Indeed, if >0, there exists an open neighbourhoodV of z in T such that |f(t)−g(t)| <|f(z)−g(z)|+ 2 for each t ∈ V. Then supt∈ϕ−1(V)|f(t)−g(t)| < |f(z)−g(z)|+, hence infV∈V(z)supt∈ϕ−1(V)|f(t)−g(t)|<|f(z)−g(z)|+. thereforedb([σf]z,[σg]z) = infV∈V(z)supt∈ϕ−1(V)|f(t)−g(t)|=|f(z)−g(z)|. One can now assert that the mapψ:Ebz = Σ/Rz−→Rdefined byψ([σf]z) =f(z) is an isometry. In other words, ifz∈T, then the fiber overzin the bundle
E,b p,bR,db isR.
Since each continuous functionf :Sr{a} −→Rcan be identified with the element σf of Σ and sinceσf is identified with the section cσf of the bundle E,b bp,R,db
obtained by localization, it follows that each continuous function defined fromSr{a}inRcan be extended to a (continuous) section fromS to E.b
In particular ifS=Randa= 0, the functions sinzand 1/zcan be extended,
despite their non removable discontinuity ata, to a (continuous) sections to the domainS that does include the pointa. This is an indication of the awesome intricacy of the fiber above the pointa.
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 byΛin a categoryX is a covariant functor F, from Λ toX, that sends each object α of Λ into an object Xα of X, and such that, forα6β, there exists a morphismfαβ:Xα−→XβofX satisfying:
(1) For eachα∈Λ, fαα= 1Xα. (2) If α6β6γ thenfβγ◦fαβ=fαγ.
Denote by ((Xα)α∈Λ,(fαβ)α6β)the direct system that corresponds toF. Definition 4. Let((Xα)α∈Λ,(fαβ)α6β)be a direct system in a categoryX.
(1) An inductive cone for this direct system is a pair(X,(Fα)α∈Λ)consist- ing of an objectX of X and a family of morphisms Fα:Xα−→X of X such that for each α, β∈Λ, withα6β, Fβ◦fαβ=Fα.
(2) A direct limit or colimit for the direct system is an inductive cone (X,(Fα)α∈Λ) satisfying the following universal property: given any other inductive cone (Y,(Gα)α∈Λ) in X, for this direct system, there exists a unique morphism ϕ : X −→ Y, such that for every α ∈ Λ, ϕ◦ Fα=Gα.
Following S. Bautista [1], consider the category U metH. Its objects are pairs consisting of a set X and a family of bounded pseudometrics (di)i∈I defined onX, such that x1 =x2 if and only ifdi(x1, x2) = 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,(di)i∈I) and (Y,(mj)j∈J), is a pair of functions (f, l), where f :X −→Y andl:J −→I satisfymj(f(x1), f(x2))6 dl(j)(x1, x2), for eachx1, x2∈X and eachj ∈J. If (X,(di)i∈I) is an object of U metH, the identity morphismid(X,(di)i∈I)is the pair (idX, idI), and if (f, l) : (X,(di)i∈I) −→ (Y,(mj)j∈J) and (f0, l0) : (Y,(mj)j∈J) −→ (Z,(nk)k∈K) are morphisms of U metH, then the composition of (f, l) and (f0, l0) is given by (f0, l0)◦(f, l) = (f0f, ll0).
LetT be a topological space,p:E−→T be a surjective function, (di)i∈I be a directed family of bounded pseudometrics forp, such that the fiberEt ={a∈ E : p(a) = t} is a Hausdorff space for each t ∈ T, Σ be a full set of global selection forpandϕ:T −→S be a continuous function.
Takes∈S and for eachV ∈ V(s) let ΣV =
σ ϕ−1(V):σ∈Σ and dVi
i∈I be the family of pseudometrics over ΣV, where for eachi∈I,dVi : ΣV×ΣV −→R
is defined by dVi σϕ−1(V), τ ϕ−1(V)
= supt∈ϕ−1(V)di(σ(t), τ(t)), then the pair ΣV,(dVi )i∈I
is an object ofU metH.
Now consider the order relation 6 over the neighbourhood filter of s, given by V 6 W, if and only if, W ⊂ V. For each V, W ∈ V(s) such that V 6 W, let fV W : ΣV −→ ΣW and lV W : I −→ I be the functions defined by fV W σϕ−1(V)
=σϕ−1(W) andlV W(i) =i. For each i∈I, one has that dWi σϕ−1(W), τ ϕ−1(W)
= sup
t∈ϕ−1(W)
di(σ(t), τ(t)) 6 sup
t∈ϕ−1(V)
di(σ(t), τ(t))
=dVlV W(i) σϕ−1(V), τ ϕ−1(V)
,
thus the pair (fV W, lV W) is a morphism ofU metH. Then the pair
ΣV, dVi
i∈I
V∈V(s),(fV W, lV W)W⊂V
is a direct system in U metH.
Let
E,b p, S,b dbi
i∈I
be the uniform bundle over the topological space S, constructed by localization from the topological spaceT, the surjective function p:E−→T, the directed family of bounded pseudometrics forp, (di)i∈I such thatEt ={a∈E:p(a) =t}is a Hausdorff space, for eacht∈T, the set Σ of global selections forpand the continuous functionϕ:T −→S.
For s ∈S, the equivalence relationRs in Σ is defined by, σRsτ, if and only if, infV∈V(s)supt∈ϕ−1(V)di(σ(t), τ(t)) = 0, for each i ∈ I, and the fiber Ebs
is the set Σ/Rs. Consider the family of pseudometrics dbi
i∈I onEbs, where dbi([σ]s,[τ]s) = infV∈V(s)supt∈ϕ−1(V)di(σ(t), τ(t)), for each i ∈ I. The pair
Ebs, dbi
i∈I
is then an object ofU metH.
For eachV ∈ V(s), let ~V : ΣV −→Ebs andξV :I −→I, the functions defined by~V(σ ϕ−1(V)) = [σ]s (class ofσ moduleRs) and ξV(i) =irespectively. It follows that
dbi ~V σϕ−1(V)
,~V τ ϕ−1(V)
=dbi([σ]s,[τ]s)
= inf
U∈V(s) sup
t∈ϕ−1(U)
di(σ(t), τ(t)) 6 sup
t∈ϕ−1(V)
di(σ(t), τ(t))
=dVi σ ϕ−1(V), τ ϕ−1(V)
=dVξV(i) σϕ−1(V), τ ϕ−1(V) ,
then (~V, ξV) is a morphism ofU metH.
Note that ifV, W ∈ V(s) andV 6W, then for eachσ ϕ−1(V)∈ΣV,
~W fV W σ ϕ−1(V)
=~W σϕ−1(W)
= [σ]s=~V σ ϕ−1(V) , thus~WfV W =~V, therefore
Ebs, dbi
i∈I
,(~V, ξV)V∈V(s)
is an inductive cone for the direct system
ΣV,(dVi )i∈I
V∈V(s),(fV W, lV W)W⊂V
.
Now suppose that
Y,(mj)j∈J
,(ðV, χV)V∈V(s)
is an other inductive cone for this direct system.
For each V ∈ V(s), the functions ðV : ΣV −→ Y and χV : J −→ I satisfy mj ðV σϕ−1(V)
,ðV τ ϕ−1(V)
6 dVχ
V(j) σϕ−1(V), τ ϕ−1(V)
, for each σ ϕ−1(V), τ ϕ−1(V)∈ΣV and each j∈J, and ifV, W ∈ V(s) and V 6W, thenðWfV W =ðV andlV WχW =χV. Therefore, for eachV ∈ V(s) and each σ ∈ Σ, ðV(fSV(σ)) = ðS(σ), that is, ðV σϕ−1(V)
= ðS(σ). Furthermore, for eachj ∈J, lSV(χV(j)) =χS(j), hence χV(j) =χS(j).
Defineψ:Ebs−→Y byψ([σ]s) =ðS(σ), suppose that [σ]s= [τ]sand letj∈J and >0. There exists aV ∈ V(s) such that supt∈ϕ−1(V)dχS(j)(σ(t), τ(t))< . Then
mj(ψ([σ]s), ψ([τ]s)) =mj(ðS(σ),ðS(τ))
=mj ðV σϕ−1(V)
,ðV τ ϕ−1(V) 6dVχ(j) σϕ−1(V), τ ϕ−1(V)
< ,
This means thatmj(ψ([σ]s), ψ([τ]s)) = 0, for eachj ∈J, thus ðS(σ) =ðS(τ) sinceY is a Hausdorff space. It follows thatψis well defined. Defineζ :J −→I byζ(j) =χS(j). It is immediate that the pair (ψ, ζ) is a morphism ofU metH
and is the only morphism such that for each V ∈ V(s), (ψ, ζ)◦(~V, ξV) = (ðV, χV).
It follows that
Ebs, dbi
i∈I
,(~V, ξV)V∈V(s)
is the colimit of the direct system
ΣV,(dVi )i∈I
V∈V(s),(fV W, lV W)W⊂V
inU metH.
In terms of the definitions given above the following proposition summarizes the preceding considerations.
Proposition 2. LetT be a topological space, p:E−→T a surjective function, (di)i∈I a directed family of bounded pseudometrics for p, such that the fiber Et = {a ∈ E : p(a) = t} is a Hausdorff space for each t ∈ T, Σ a full set of global selection for pandϕ:T −→S a continuous function. Consider the
uniform bundle
E,b p, S,b dbi
i∈I
constructed by localization from the data above and lets∈S. Then:
(1) The pair
ΣV, dVi
i∈I
is an object of U metH.
(2) The pair(fV W, lV W)is a morphism ofU metH and the pair
ΣV, dVi
i∈I
V∈V(s),(fV W, lV W)W⊂V
is a direct system inU metH.
(3) The pair (~V, ξV) is a morphism of U metH and the inductive cone Ebs,
dbi
i∈I
,(~V, ξV)V∈V(s)
is the colimit of the direct system ΣV, dVi
i∈I
V∈V(s),(fV W, lV W)W⊂V
inU metH.
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(di)i∈I
of bounded pseudometrics for psuch that the fiber Et ={a∈E:p(a) =t}is a Hausdorff space for each t ∈ T, a full setΣ of global selection for pand a continuous functionϕ: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 presheafMover the spaceS. IfHdenotes the functor that to each uniform bundle overS associates the presheaf of its local sections, it will be shown that the uniform bundle
E,b bp, S, dbi
i∈I
obtained by localization from the data, together with an obvious morphism of presheavesφbetweenMandH
E,b p, S,b dbi
i∈I
, establish a universal arrow fromMto the functorH.
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 U metH as follows:
(1) To each open setU ∈τ it is assigned a setF(U)and a family (di)i∈IU
of bounded pseudometrics generating a Hausdorff uniformity.
(2) To each pair of open setsU, V ∈τ, with V ⊂U, it is assigned a pair (fU V, lU V)of functionsfU V :F(U)−→F(V),lU V :IV −→IU in such a way that di(fU V(x), fU V(y)) ≤dlU V(i)(x, y), for each x, y ∈ F(U) and each i∈IV.
Notice that a presheaf is a direct system inU metH.
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 U of S, let M(U) = Σϕ−1(U) and let dUi
i∈I the family of pseudometrics overM(U), where for eachi∈I it holds that dUi σϕ−1(U), τϕ−1(U)
= sup
t∈ϕ−1(U)
di
σϕ−1 (U)(t), τϕ−1 (U)(t) . (2) IfU andV are open subsets ofS andV ⊂U, define the functions
mU V : Σϕ−1(U)−→Σϕ−1(V)
σϕ−1(U)7−→σϕ−1(V)
and
lMU V :I −→I i7−→i.
It follows that
dVi (mU V(σϕ−1(U)), mU V(τϕ−1(U))) =dVi σϕ−1(V), τϕ−1(V)
= sup
t∈ϕ−1(V)
di σϕ−1(V), τϕ−1(V)
≤ sup
t∈ϕ−1(U)
di σϕ−1(U), τϕ−1(U)
=dUi σϕ−1(U), τϕ−1(U)
. Example 5. Let
F, q, S,(mj)j∈J
be a uniform bundle where (mj)j∈J is a directed family of bounded pseudometrics. The presheafFof local sections for qis defined as follows:
(1) Given an open set U ofS letF(U) = ΓU the set of all local sections for q with domainU and let dUj
j∈J be the family of pseudometrics overF(U), where for eachj∈J,mUj(σ, τ) = sups∈Umj(σ(s), τ(s)).
(2) IfU andV are open subsets ofS and ifV ⊂U, define fU V : ΓU −→ΓV
σ7−→σV
and
lFU V :J −→J j7−→j.
In the particular case of the bundle
E,b p, S,b dbi
i∈I
obtained by localization, denote by Eb the presheaf of local sections for pband by eU V andlU VEb the functionsfU V andlU VF respectively.
5.1. The category Preh. Extending the ideas of S. Bautista [1], consider as objects of the categoryPrehall presheaves of Hausdorff uniform spaces over the topological space (S, τ).
Let F : τ −→ U metH and G : τ −→ U metH be presheaves over S. For U, V ∈τ, with V ⊂U, denote by fU V and lU V the functions defining Fand bygU V andhU V those definingG.
A morphism fromFto Gis given by a natural transformationφfromF to G sending each object U ∈ τ into the morphism (φU, bU) : F(U)×JU −→
G(U)×IU in U metH in such a way that if U, V ∈ τ and V ⊂U, then the diagrams
F(U) G(U)
F(V) G(V)
-
φU
fU V ?
?
gU V
-
φV
and
IV JV
IU JU
-
bV
lU V ?
?
hU V
-
bU
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 byUthe category of Hausdorff uniform bundles. The objects ofUare bundles of uniform spaces with base spaceS, with uniformities determined by directed families of bounded pseudometrics, having a full set of global sections and whose fibers are Hausdorff spaces.
Let
F, q, S,(mj)j∈J and
F0, q0, S, m0j0
j0∈J0
be two objects ofU. A mor- phism (∆, `) :
F, q, S,(mj)j∈J
−→
F0, q0, S, m0j0
j0∈J0
in the categoryU is a pair of maps ∆ : F −→ F0, ` : J0 −→ J such that ∆ is continuous, m0j0(∆(a),∆(b)) ≤ m`(j0)(a, b), for each a, b ∈ F and each j0 ∈ J0 and the diagram
F F0
S -
∆
@@qR q0
commutes (that is, ∆ preserves fibers).
6.1. The functor H. Define the functorH from the category Uto Prehas follows:
(1) To each element
F, q, S,(mj)j∈J
of U, H assigns the presheafF of local sections forq.
(2) To each morphism (∆, `) :
F, q, S,(mj)j∈J
−→
F0, q0, S, m0j0
j0∈J0
inUassigns the morphism of presheaves given by the natural transfor- mationφ(∆,`) from F to F0 that sends each open setU of S into the morphism ofU metH
φ(∆,`)U :
ΓU, mUj
j∈J
−→
Γ0U,
m0Uj0
j0∈J0
defined by the functions
fU(∆,`): ΓU −→Γ0U σ7−→σ0, whereσ0(s) = ∆(σ(s)), for eachs∈S and
l(∆,`)U :J0 −→J j07−→`(j0). Notice that
m0Uj0
fU(∆,`)(σ), fU(∆,`)(τ)
=m0Uj0(σ0, τ0)
= sup
s∈U
m0j0(σ0(s), τ0(s))
= sup
s∈U
m0j0(∆ (σ(s)),∆ (τ(s)))
≤sup
s∈U
m`(j0)(σ(s), τ(s))
=mU
l(∆,`)U (σ, τ).
Furthermore, if U andV are open sets ofS andV ⊂U, then the diagram
ΓU Γ0U
ΓV Γ0V
fU(∆,`) -
fU V ?
?f
0 U V
-
fV(∆,`)
commutes. In fact, ifσ∈ΓU, then fU V0
fU(∆,`)(σ)
=fU V0 (σ0)
=σV0
= ∆σV
=fV(∆,`)(σV)
=fV(∆,`)(fU V (σ)).
6.2. A universal arrow. Consider the morphism of presheavesφ:M−→Eb assigning to each open setU in Sthe morphism ofU metH,
φU :
Σϕ−1(U), dUi
i∈I
−→
ΓbU, dbUi
i∈I
defined by the functions
fU : Σϕ−1(U)−→ΓbU
σϕ−1(U)7−→σbU
and
lU :I −→I i7−→i.
Notice that for eachσ, τ ∈Σ, dbUi fU σϕ−1(U)
, fU τϕ−1(U)
=dbUi (bσU,bτU)
= sup
s∈U
dbi(bσU(s),bτU(s))
= sup
s∈U
V∈V(s)inf sup
t∈ϕ−1(V)
di(σ(t), τ(t))
!
≤sup
s∈U
sup
t∈ϕ−1(U)
di(σ(t), τ(t))
!
= sup
t∈ϕ−1(U)
di(σ(t), τ(t))
=dUi σϕ−1(U), τϕ−1(U) , it follows thatφU is a morphism inU metH.