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Universidad Polit´c ecnica de Valencia Volume 9, No. 1, 2008pp. 77-108
Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics
S. E. Rodabaugh∗
Abstract. This paper studies various functors between (lattice- valued) topology and (lattice-valued) bitopology, including the ex- pected “doubling” functor Ed : L-Top → L-BiTop and the “cross”
functor E× : L-BiTop → L2-Top introduced in this paper, both of which are extremely well-behaved strict, concrete, full embeddings.
Given the greater simplicity of lattice-valued topology vis-a-vis lattice- valued bitopology and the fact that the class of L2-Top’s is strictly smaller than the class of L-Top’s encompassing fixed-basis topology, the class of E×’s makes the case that lattice-valued bitopology is cat- egorically redundant. As a special application, traditional bitopology as represented by BiTop is (isomorphic in an extremely well-behaved way to) a strict subcategory of 4-Top, where 4 is the four element Boolean algebra; this makes the case that traditional bitopology is a special case of a much simpler fixed-basis topology.
2000 AMS Classification: Primary: 03E72, 54A10, 54A40, 54E55.
Secondary: 06F07, 18A22, 18A30, 18A35, 18B30.
Keywords: unital-semi-quantale, unital quantale, (fixed-basis) topology, (fixed-basis) bitopology, order-isomorphism, categorical (functorial) embed- ding, redundancy.
1. Introduction and preliminaries
1.1. Motivation. Bitopology has a long and distinguished history spanning five decades and a literature of some 700 papers [29] with traditional bitopology playing a wide range of roles in Baire spaces, homotopy and algebraic topology, generalizations of metric spaces, biframes, programming semantics, etc.
∗Support of Youngstown State University via a sabbatical for the 2005–2006 academic year is gratefully acknowledged.
First defined and used in [31, 32, 3, 4], a bitopological space was originally defined as a triple ((X, T) , (X, S) , e) with (X, T) , (X, S) topological spaces and e : (X, T) → (X, S) a continuous bijection—cf. [3]. But if we set
T′ = {e→(U ) : U ∈ T} ,
then T′ is a topology on X and the continuity of e insures that idX : (X, T′) → (X, S) is continuous, i.e., that T′ ⊃ S. It is therefore not surprising that almost immediately [4] the original definition was replaced by the simpler, equivalent definition that a bitopological space is a triple (X, T, S) with T, S topologies on X with T ⊃ S, T being called the strong topology and S the weak topology. Even in the broader lattice-valued topology setting, this definition plays a categorical role (Proposition 3.5 below).
Since a quasi-pseudo-metric p on a set X determines its conjugate quasi- pseudo-metric q, namely by q (x, y) = p (y, x) , quasi-pseudo-metrics necessarily occur in conjugate pairs which generate pairs of topologies that need not be related. Thus the definition of a traditional bitopological space was generalized in [22] to its modern form to be an ordered triple (X, T, S) with T, S topologies on X (and no relationship assumed between T and S). Further, a bicontinuous mapping f : (X, T1, T2) → (Y, S1, S2) is a mapping f : X → Y satisfying
T1⊃ (f←)→(S1) , T2⊃ (f←)→(S2) ,
i.e., f : (X, T1) → (Y, S1) and f : (X, T2) → (Y, S2) are both continuous. With the composition and identities of Set, one has the category BiTop, which is a topological construct and hence strongly complete and strongly cocomplete along with many other properties.
There is a voluminous literature for BiTop concerning separation, com- pactness, connectedness, completion, connections to uniform and quasi-uniform spaces, homotopy groups and algebraic topology, relationships to bilocales [2], a recently emerging role in programming semantics [25], etc. A significant part of the recent literature on bitopology is in lattice-valued mathematics [30, 27, 50, 51]. Letting L be a us-quantale (Subsection 1.2 below) and X a set, the triple (X, τ, σ) is an L-bitopological space if τ, σ are L-topologies on X (Subsection 1.5); and such spaces with L-bicontinuous mappings comprise the category L-BiTop. This category is a topological construct, strongly complete, strongly cocomplete, and so on. The schemum {L-BiTop : L ∈ |USQuant|}
essentially includes BiTop via its functorial isomorph 2-BiTop.
This paper studies functorial relationships between (lattice-valued) bitopol- ogy and (lattice-valued) topology in Sections 2–3. The expected functor Ed
strictly embeds L-Top into L-BiTop, a functor we dub the “doubling” func- tor; and to fully study Ed, it is necessary to construct several functors from L-BiTop to L-Top whose relationships with Ed lead us to conclude that Ed
is extremely well-behaved. But on the other hand, for each L ∈ |USQuant| , the direct product L2 ∈ |USQuant| and there is an embedding E× of L- BiTopinto L2-Top (3.4.1) which is extremely well-behaved (Subsubsections 3.4.2, 3.4.3) if L is a u-quantale (Subsection 1.2) and a strict embedding if L
is consistent (Subsubsection 3.4.1). Given that this embedding is strict (for consistent L) and that the L2’s form a proper subclass of USQuant—which means (lattice-valued) bitopology is properly “contained” in the proper sub- classL2-BiTop: L ∈ |USQuant| of {L-Top : L ∈ |USQuant|} , it follows (lattice-valued) topology (twice) strictly generalizes bitopology. In Section 4 we summarize some metamathematical facts: given that lattice-valued topol- ogy is fundamentally simpler than lattice-valued bitopology—a membership lattice and one topology vis-a-vis a membership lattice and two topologies, it follows that topology and the class of embeddings E×’s make lattice-valued bitopology categorically redundant; and as a special application, traditional bitopology BiTop strictly embeds in an extremely well behaved way into 4- Top, the latter being lattice-valued topology based on the four-element Boolean algebra 4, so that traditional bitopology both is a strictly special case of the simpler lattice-valued topology and demonstrates the necessity of lattice-valued topology. On the other hand, this last fact points the way for bringing over into lattice-valued topology successful ideas from the extensive literature of traditional bitopology; in particular, traditional bicompactness mandates, via the embedding of BiTop into 4-Top, the compactness of [5] for lattice-valued topology (Corollary 4.7).
1.2. Lattice theoretics. A semi-quantale (L, ≤, ⊗) (s-quantale) is a com- plete lattice (L, ≤) equipped with a binary operation ⊗ : L × L → L, with no additional assumptions, called a tensor product; an ordered semi-quantale (os-quantale) is an s-quantale in which ⊗ is isotone in both variables; a com- plete quasi-monoidal lattice(cqml) [20, 41] is an os-quantale for which ⊤ is an idempotent element for ⊗; a unital semi-quantale (us-quantale) is an s-quantale in which ⊗ has an identity element e ∈ L called the unit [33]—units are unique; a quantale is an s-quantale with ⊗ associative and distributing across arbitraryW from both sides (implying ⊥ is a two-sided zero) [20, 33, 49];
and a unital quantale (u-quantale) is a us-quantale which is a quantale; and a strictly two-sided quantale (st-quantale) is a u-quantale for which e = ⊤ [20]. All quantales are os-quantales. The notions of s-quantales, os-quantales, and us-quantales are from [45, 46].
SQuant comprises all semi-quantales together with mappings preserving
⊗ and arbitrary W; OSQuant is the full subcategory of SQuant of all os- quantales; USQuant is a subcategory of SQuant comprising all us-quantales together with all mappings preserving arbitraryW, ⊗, and e; Quant is the full subcategory of OSQuant of all quantales; and UQuant is the full subcategory of UOSQuant of all unital quantales. Note uos-quantales for which ⊗ = ∧ (binary) are semiframes and SFrm is the full subcategory of UOSQuant of all semiframes; and u-quantales for which ⊗ = ∧ (binary) are frames—
in which case e = ⊤—and Frm is the full subcategory of UQuant of all frames. Semiframes equipped with an order-reversing involution are com- plete DeMorgan algebras; and s-quantales equipped with a semi-polarity
[16] (∀α, β ∈ L, α ≤ β ⇒ β′ ≤ α′ and α ≤ (α′)′) are complete semi- DeMorgan s-quantales.
Throughout this paper, the requirement of us-quantale [u-quantale] can be relaxed to s-quantale [quantale, resp.] if one wishes to consider the relationships between q-topology and q-bitopology ([46] and Subsection 1.5 below).
Justifying the above lattice-theoretic notions is a wealth of examples (see [17, 20, 21, 23, 33, 35, 39, 40, 41, 44, 46] and their references). The lattice 2 = {⊥, ⊤} with ⊥ 6= ⊤; and a lattice is consistent if it contains 2 and inconsistentif it is singleton (with ⊥ = ⊤).
1.3. Powerset operators. Let X ∈ |Set| and L ∈ |SQuant|. Then LXis the L-powerset of X of all L-subsets of X. The constant L-subset member of LXhaving value α is denoted α. All order-theoretic operations (e.g.,W, V) and algebraic operations (e.g., ⊗) on L lift point-wise to LX and are denoted by the same symbols. In the case L ∈ |USQuant| , the unit e lifts to the constant map e, which is the unit of ⊗ as lifted to LX.
The operator ℘∅ : |Set| → |Set| is useful in this paper, where ℘∅(X) denotes the poset of all the nonempty subsets of X.
Let L ∈ |SQuant| , X, Y ∈ |Set| , and f : X → Y be in Set. Then the standard (traditional) image and preimage operators f→ : ℘ (X) → ℘ (Y ) , f←: ℘ (X) ← ℘ (Y ) are
f→(A) = {f (x) ∈ Y : x ∈ A} , f←(B) = {x ∈ X : f (x) ∈ B} , and the Zadeh image and preimage operators fL→: LX→ LY, fL←: LX ← LY [53] are
fL→(a) (y) =_
{a (x) : x ∈ f←({y})} , fL←(b) = b ◦ f.
If L is understood, it may be dropped providing the context distinguishes these operators from the traditional operators. It is observed that f→ and f← are naturally isomorphic to f2→ and f2←, resp.
It is well-known [36, 37, 39, 40, 46] that each fL← preserves arbitrary W, arbitraryV, ⊗, and all constant maps, as well as the unit e if L ∈ |USQuant|;
each fL→preserves arbitraryW;
f→⊣ f←, fL→⊣ fL←;
f→ and fL→ are left-inverses [right-inverses] of f← and fL←, resp., if f is sur- jective [injective, resp.]; and f→, f←, fL→, fL←are all order-isomorphisms if and only if f is a bijection.
Powerset operators and the powerset theories underlying lattice-valued math- ematics are studied extensively in [6, 14, 7, 8, 15, 10, 11, 36, 37, 39, 40, 46].
1.4. Category theoretics. The main reference for categorical notions is [1], to which we refer the reader for various properties of functors as well as various versions of the Adjoint Functor Theorem and related notions.
The proving of functorial adjunctions is done via lifting (or major) and naturality (or minor) diagrams in the manner of [28, 36, 37, 41].
1.5. Topology and bitopology. Given L ∈ |USQuant|, the category L-Top comprises objects of the form (X, τ ), where τ ⊂ LX is closed under arbitrary W and binary ⊗ and contains e—so that τ is a sub-us-quantale of LX, together with morphisms f : (X, τ ) → (Y, σ), where f : X → Y is a function and
τ ⊃ (fL←)→(σ) ,
namely fL←(v) ∈ τ for each v ∈ σ. The objects (X, τ ) are called L-topological spaces and τ is an L-topology on X comprising L-subsets of X; and the morphisms f are called L-continuous. Cf. [20, 41, 46].
Similarly, the category L-BiTop comprises objects of the form (X, τ, σ) , where τ, σ are L-topologies on X, together with morphisms f : (X, τ1, τ2) → (Y, σ1, σ2), where f : X → Y is a function and
τ1⊃ (fL←)→(σ1) , τ2⊃ (fL←)→(σ2) .
The objects (X, τ, σ) are called L-bitopological spaces and (τ, σ) is an L- bitopologyon X; and the morphisms f are called L-bicontinuous. If the L is clear in context, it may be dropped from the labels.
As noted in Subsection 1.1, the traditional category BiTop is isomorphic to 2-BiTop (cf. 3.25 below) and embeds into each L-BiTop, and similarly Top is isomorphic to 2-Top and embeds into each L-Top.
Each of L-Top and L-BiTop has the base L of the category fixed and so is part of fixed-basis (lattice-valued) topology and fixed-basis (lattice-valued) bitopology, resp. The disciplines of fixed-basis topology and fixed-basis bitopol- ogy are encompassed by the respective classes
{L-Top : L ∈ |USQuant|} , {L-BiTop : L ∈ |USQuant|} .
Both L-Top and L-BiTop are topological over Set and have small fi- bres, hence are [co]complete and [co]well-powered, and hence are strongly [co]complete with many other nice properties (see 3.36 and its proof below).
The categorical product for L-Top is given in, or adapted from, [12, 52] (cf.
[20, 41]) and for {(Xγ, τγ) : γ ∈ Γ} denoted by
Y
γ∈ Γ
(Xγ, τγ) , {πγ}γ∈ Γ
, Y
γ∈ Γ
(Xγ, τγ) ≡ (×γ∈ ΓXγ, Πγ∈ Γτγ) , where {πγ : γ ∈ Γ} are the projections. The binary L-topological product for two spaces (X, τ ) , (Y, σ) is denoted (X, τ ) Π (Y, σ) or (X × Y, τ Π σ) with pro- jections {π1, π2}. The categorical product for L-BiTop for {(Xγ, τγ) : γ ∈ Γ}
is
Y
γ∈ Γ
(Xγ, τγ, σγ) , {πγ}γ∈ Γ
, Y
γ∈ Γ
(Xγ, τγ, σγ) ≡ (×γ∈ ΓXγ, Πγ∈ Γτγ, Πγ∈ Γσγ) ,
where Πγ∈ Γτγ, Πγ∈ Γσγ are the L-topological product topologies in each slot and the projections are as above.
An L-topology τ is weakly stratified [20] if {α : α ∈ L} ⊂ τ , non- stratifiedif it is not weakly stratified, and anti-stratified [9, 35] if
{α : α ∈ L, α ∈ τ } = {⊥, e} ;
so a weakly stratified topology contains all constant L-subsets, while an anti- stratified topology contains precisely the constant L-subsets ⊥ and e (which are the same if L is inconsistent with ⊥ = ⊤). An L-topological space is weakly stratified [anti-stratified] if its topology is weakly stratified [anti-stratified], and an L-bitopological space is weakly stratified [anti-stratified] if both topologies are weakly stratified [anti-stratified]. The inclusionist position that the axioms of a fixed-basis topology must allow for all types of stratification has recently received additional, emphatic confirmations from both lattice-valued frames [35] and topological systems in domain theory [9].
The following definition and proposition are needed in this paper.
Definition 1.1. Let X be a set and let L be a us-quantale. Then the L- topological fibre, respectively, L-bitopological fibre on X is
L-T (X) ≡ τ ⊂ LX: (X, τ ) ∈ |L-Top| , L-BT (X) ≡ {(τ, σ) : (X, τ, σ) ∈ |L-BiTop|} .
Proposition 1.2. Let X be a set, let L be a us-quantale, and recall ℘∅ from Subsection 1.3.
(1) L-T (X) is a complete meet subsemilattice of ℘ LX; and since each L-topology is nonempty, L-T (X) ⊂ ℘∅ LX .
(2) L-BT (X), ordered coordinate-wise by inclusion, is a complete meet subsemilattice of ℘ LX×℘ LX; and further, L-BT (X) ⊂ ℘∅ LX×
℘∅ LX.
Proof. The first part of (1) is well-known, and the second part of (1) is trivial.
Now (2) follows from (1) since
L-BT (X) = L-T (X) × L-T (X) ⊂ ℘∅ LX × ℘∅ LX .
Finally, we need the notion of a subbase of an L-topology τ on X [41]. We say σ ⊂ LX is a subbase of τ , written τ = hhσii , if
τ =\
{τ′ ∈ L-T (X) : σ ⊂ τ′} ,
the right-hand side always existing by Proposition 1.2(1), and we say β ⊂ LX is a base of τ , written τ = hβi , if
∀u ∈ τ, ∃ Bu⊂ β, u =_ Bu.
One can always pass from a subbase σ to a topology τ through a base β in the traditional way, written
τ = hβi = hhσii ,
if and only if ⊗ is associative and distributes across arbitrary W, i.e., if and only if L is a u-quantale.
2. Functorial interpretations of topology as bitopology For each us-quantale L, this section records a simple (and expected) “dou- bling” embedding Ed : L-Top → L-BiTop. The behavior of Ed w.r.t. limits and colimits—it preserves, reflects, detects both—is examined completely in Subsections 3.1–3.2 below. It emerges that Ed is an extremely well-behaved embedding.
Proposition 2.1. Let L be a us-quantale. Define Ed: L-Top → L-BiTop by the following correspondences:
Ed(X, τ ) = (X, τ, τ ) , Ed(f ) = f.
Then Ed is a concrete, full, strict embedding; and so L-Top is isomorphic to a full subcategory of L-BiTop.
Proof. All details are straightforward.
3. Functorial interpretations of bitopology as topology This section records several interpretations of bitopology as topology, the most important of which would seem to be the extremely well-behaved embed- ding E× of Subsection 3.4.
3.1. Fl, Fr, F∧ : L-BiTop → L-Top and behavior of Ed : L-Top → L- BiTop w.r.t. limits. This subsection constructs the concrete, faithful, full forgetful functors—the “left-forgetful” functor Fl : L-BiTop → L-Top and the “right-forgetful” Fr: L-BiTop → L-Top—as well as the concrete, faithful
“meet” functor F∧ : L-BiTop → L-Top and shows F∧ is the left-adjoint of Ed of the previous section and that each of Fl, Fris a left-adjoint of Ed under certain restrictions.
Proposition 3.1. Let L be a us-quantale and define Fl, Fr : L-BiTop → L- Topas follows:
Fl(X, τ, σ) = (X, τ ) , Fl(f ) = f, Fr(X, τ, σ) = (X, σ) , Fr(f ) = f.
Then each of Fl, Fr is a concrete, faithful, full, object-surjective functor, but need not be an embedding.
Proof. We comment only on Fl. Trivially, Fl is a concrete, faithful, object- surjective functor. As for fullness, let f : (X, τ ) → (Y, σ) in L-Top; then f : (X, τ, τ ) → (Y, σ, σ) is L-bicontinuous, so is in L-BiTop, and maps to f : (X, τ ) → (Y, σ). Now suppose that either [|X| ≥ 1 and |L| ≥ 3] or [|X| ≥ 2 and |L| ≥ 2]; then ∃ τ, σ ∈ L-T (X) with τ 6= σ, so that Fl(X, τ, σ) = (X, τ ) = Fl(X, τ, τ ), and hence Fldoes not inject objects and is not an embedding. Proposition 3.2. Let L be a us-quantale and define F∧: L-BiTop → L-Top as follows:
F∧(X, τ, σ) = (X, τ ∩ σ) , F∧(f ) = f.
Then F∧ is a concrete, faithful, object-surjective functor, but need not be full nor an embedding.
Proof. Since L-Top has complete fibres, F∧is well-defined on objects. Now let f : (X, τ1, τ2) → (Y, σ1, σ2) be L-bicontinuous. Since the image operator of the Zadeh preimage operator preserves ⊂, it follows
τ1 ⊃ (fL←)→(σ1) , τ2⊃ (fL←)→(σ2) ⇒
τ1∩ τ2 ⊃ (fL←)→(σ1) ∩ (fL←)→(σ2) ⊃ (fL←)→(σ1∩ σ2) ,
so that f : (X, τ1∩ τ2) → (Y, σ1∩σ2) is L-continuous. Immediately, F∧ is a concrete, faithful functor which surjects objects.
Now let L be a complete DeMorgan algebra (with ⊗ = ∧ (binary)) and consider each of the L-bitopological spaces (R (L) , τl(L) , τl(L)) and (R (L) , τl(L) , τ (L)) , where R (L) is the L-fuzzy real line, τl(L) is the left-hand L-topology on R (L) determined by the Lt operators and τ (L) is the standard L-topology on R (L) [43]. Then (R (L) , τl(L) , τl(L)) 6= (R (L) , τl(L) , τ (L)) and
F∧(R (L) , τl(L) , τl(L)) = (R (L) , τl(L))
= (R (L) , τl(L) ∩ τ (L))
= F∧(R (L) , τl(L) , τ (L)) ,
showing that F∧ does not inject objects, so is not an embedding. Now letting f : R (L) → R (L) be idR(L), we have
f : F∧(R (L) , τl(L) , τ (L)) → F∧(R (L) , τ (L) , τl(L)) is L-continuous, but
f : (R (L) , τl(L) , τ (L)) → (R (L) , τ (L) , τl(L))
cannot be L-bicontinuous (because of the first slot). The concreteness of F∧
implies there exists no g ∈ L-BiTop with F∧(g) = f , so F∧ is not full. Theorem 3.3. Let L be a us-quantale. Then F∧ ⊣ Ed, this adjunction is a monocoreflection, and F∧ takes L-BiTop to a monocoreflective subcategory of L-Top. On the other hand, Ed⊣/ F∧.
Proof. Let (X, τ1, τ2) ∈ |L-BiTop| , choose
η = id : (X, τ1, τ2) → EdF∧(X, τ1, τ2) = (X, τ1∩ τ2, τ1∩ τ2) ,
and note η is an L-continuous injection. Now let (Y, σ) ∈ |L-Top| , suppose f : (X, τ1, τ2) → Ed(Y, σ) = (Y, σ, σ) is L-bicontinuous, and note
τ1⊃ (fL←)→(σ) , τ2⊃ (fL←)→(σ) ⇒ τ1∩ τ2⊃ (fL←)→(σ) ,
making f : F∧(X, τ1, τ2) = (X, τ1∩ τ2) → (Y, σ) L-continuous. Then f = f is the unique choice making f = f ◦ η. The naturality diagram now follows by concreteness as do the other claims concerning F∧ ⊣ Ed. Finally, given F∨
of Subsection 3.2 and Ed ⊣ F∨ of 3.9 below, Ed⊣/ F∧ since F∧ ≇ F∨ and
right-adjoints are essentially unique.
Definition 3.4. L-BiTop (⊂) [L-BiTop (⊃)] is the full subcategory of L- BiTopof all spaces (X, τ, σ) in which τ ⊂ σ [τ ⊃ σ].
Note BiTop (⊂) and BiTop (⊃) (essentially setting L = 2) express the original sense of traditional bitopology [3, 4].
Proposition 3.5. Let L be a us-quantale. Then
Fl| L-BiTop(⊂)= F∧ | L-BiTop(⊂), Fr| L-BiTop(⊃)= F∧ | L-BiTop(⊃). Hence Fl| L-BiTop(⊂) ⊣ Ed and Fr| L-BiTop(⊃) ⊣ Ed, but Ed⊣/ Fl| L-BiTop(⊂)
and Ed⊣/ Fr| L-BiTop(⊃).
Proof. The restricted forgetful functors obviously coincide with the meet func- tor. Observing that Edmaps into each of L-BiTop (⊂) and L-BiTop (⊃), the claimed adjunctions are then immediate from 3.3. The claimed non-adjunctions
follow from 3.10 below.
Corollary 3.6. Let L be a us-quantale. The following hold:
(1) Ed preserves all strong limits and F∧ preserves all strong colimits.
(2) Flpreserves the strong colimits of L-BiTop (⊂), Frpreserves the strong colimits of L-BiTop (⊃), and Ed preserves strong limits into each of L-BiTop (⊂) and L-BiTop (⊃).
Proposition 3.7. For each us-quantale L, Ed : L-Top → L-BiTop reflects and detects all limits and hence lifts all limits and is transportable.
Proof. The details are straightforward using 3.6 and Proposition 13.34 [1]. 3.2. F∨ : L-BiTop → L-Top and behavior of Ed : L-Top → L-BiTop w.r.t. colimits. This subsection constructs the concrete, faithful “join” func- tor F∨ : L-BiTop → L-Top and shows it is the right-adjoint of Ed of the previous section.
Proposition 3.8. Let L be a us-quantale and define F∨: L-BiTop → L-Top as follows:
F∨(X, τ, σ) = (X, τ ∨ σ) , F∨(f ) = f, where
τ ∨ σ = hhτ ∪ σii
Then F∨ is a concrete, faithful, object-surjective functor, but need not be full nor an embedding.
Proof. Since L-Top has complete fibres, F∨is well-defined on objects. Now let f : (X, τ1, τ2) → (Y, σ1, σ2) be L-bicontinuous. Since the image operator of the Zadeh preimage operator preserves unions, then
τ1⊃ (fL←)→(σ1) , τ2⊃ (fL←)→(σ2) ⇒
τ1∨ τ2⊃ τ1∪ τ2⊃ (fL←)→(σ1) ∪ (fL←)→(σ2) = (fL←)→(σ1∪ σ2) ,
so that f : (X, τ1∨ τ2) → (Y, σ1∨ σ2) is L-subbasic continuous. By Theorem 3.2.6 of [41] as restricted to the fixed-basis case and then adapted to the us- quantalic case, f : (X, τ1∨ τ2) → (Y, σ1∨ σ2) is L-continuous. Immediately, F∨is a concrete, faithful functor which surjects objects.
Now let L be a complete DeMorgan algebra (with ⊗ = ∧ (binary)) and con- sider each of the L-bitopological spaces (R (L) , τl(L) , τr(L)) and (R (L) , τ (L) , τ (L)) , where R (L) is the L-fuzzy real line, τl(L) is the left-hand L-topology on R (L) determined by the Ltoperators, τr(L) is the left-hand L- topology on R (L) determined by the Rt operators, and τ (L) is the standard L-topology on R (L) [43]. Then (R (L) , τl(L) , τr(L)) 6= (R (L) , τ (L) , τ (L)) and
F∨(R (L) , τl(L) , τr(L)) = (R (L) , τl(L) ∨ τr(L))
= (R (L) , τ (L))
= F∨(R (L) , τ (L) , τ (L)) ,
showing that F∨ does not inject objects, so is not an embedding. Now letting f : R (L) → R (L) be idR(L), we have
f : F∨(R (L) , τl(L) , τr(L)) → F∨(R (L) , τr(L) , τl(L)) is L-continuous, but
f : (R (L) , τl(L) , τr(L)) → (R (L) , τr(L) , τl(L))
cannot be L-bicontinuous. The concreteness of F∨ implies there exists no g ∈ L-BiTop with F∨(g) = f , so F∨ is not full. Theorem 3.9. Let L be a us-quantale. Then Ed ⊣ F∨, this adjunction is an isoreflection, and F∨ takes L-BiTop to an isoreflective subcategory of L-Top.
On the other hand, F∨⊣/ Ed. Proof. Let (X, τ ) ∈ |L-Top| , choose
η = id : (X, τ ) → F∨Ed(X, τ ) = (X, τ ∨ τ ) = (X, τ ) ,
and note η is an L-homeomorphism. Now let (Y, σ1, σ2) ∈ |L-BiTop| , suppose f : (X, τ ) → F∨(Y, σ1, σ2) = (Y, σ1∨ σ2) is L-continuous, and note
τ ⊃ (fL←)→(σ1∨ σ2) ⊃ (fL←)→(σ1∪ σ2) ⊃ (fL←)→(σ1) , (fL←)→(σ2) , making f : Ed(X, τ ) = (X, τ, τ ) → (Y, σ1, σ2) L-bicontinuous. Then f = f is the unique choice making f = f ◦ η. The naturality diagram now follows by concreteness, as do the other claims concerning Ed ⊣ F∨. Finally, given F∧ of Subsection 3.1 and F∧ ⊣ Ed of 3.3 above, F∨⊣/ Ed since F∧ ≇F∨ and
left-adjoints are essentially unique.
Corollary 3.10. Let L be a us-quantale. The following hold:
(1) Ed preserves all strong colimits and F∨ preserves all strong limits.
(2) Ed⊣/ Fl| L-BiTop(⊂) and Ed⊣/ Fr| L-BiTop(⊃), and hence Ed⊣/ Fl and Ed⊣/ Fr.
Proof. (1) is immediate. As for (2), it is clear that Fl| L-BiTop(⊂), Fr| L-BiTop(⊃)
≇ F∨ | L-BiTop(⊂), F∨ | L-BiTop(⊃), resp., implying Ed⊣/ Fl| L-BiTop(⊂) and Ed⊣/ Fr| L-BiTop(⊃)by the essential uniqueness of the right-adjoint in 3.9; and
hence Ed ⊣/ Fl and Ed⊣/ Fr.
Proposition 3.11. For each us-quantale L, Ed : L-Top → L-BiTop reflects and detects all colimits.
Proof. The details are straightforward.
3.3. FΠ : L-BiTop → L-Top. This subsection constructs the non-concrete, faithful “product” functor FΠ: L-BiTop → L-Top which, when appropriately restricted, is an embedding. It need not preserve finite products and hence lacks a left-adjoint.
Proposition 3.12. Let L be a us-quantale and define FΠ: L-BiTop → L-Top as follows:
FΠ(X, τ, σ) = (X × X, τ Π σ) , FΠ(f ) = f × f,
where τ Π σ is the L-product topology on X × X (Subsection 1.5). Then FΠ is a non-concrete, faithful functor which need not be full nor object-surjective nor an embedding.
Proof. Immediately FΠ is well-defined on objects. Let f : (X, τ1, τ2) → (Y, σ1, σ2) be L-bicontinuous and let v ∈ σ1Π σ2 be a subbasic open set of the form (π1)←L (s1) with s1∈ σ1. Then given (x1, x2) ∈ X × X,
(f × f )←L (v) (x1, x2) = (π1)←L (s1) (f (x1) , f (x2))
= s1(π1(f (x1) , f (x2)))
= s1(f (x1))
= fL←(s1) (x1)
= fL←(s1) (π1(x1, x2))
= (π1)←L (fL←(s1)) (x1, x2) ,
so that (f × f )←L (v) = (π1)←L (fL←(s1)) ∈ τ1Π τ2; and similarly, if v is a sub- basic open set of the form (π2)←L (s2) with s2 ∈ σ2, (f × f )←L (v) ∈ τ1Π τ2. So FΠ(f ) : FΠ(X, τ1, τ2) → FΠ(Y, σ1, σ2) is L-subbasic continuous and hence L-continuous (cf. Theorem 3.2.6 of [41]). It is easy to show FΠ preserves com- position and identities—and so is a functor—and is faithful and need not be full nor object-surjective.
To see that FΠ need not inject objects, let L = {⊥, α, β, ⊤} be a chain with ⊗ = ∧ (binary), X = {x} , τ1 = {⊥, α, ⊤} , and τ2 =⊥, β, ⊤ . Then (X, τ1, τ2) 6= (X, τ2, τ1) , yet FΠ(X, τ1, τ2) = FΠ(X, τ2, τ1). Proposition 3.13. FΠ does not preserve binary products and hence has no left-adjoint.
Proof. Let (X, τ1, τ2) , (Y, σ1, σ2) be given with X 6= Y . Then the carrier set of FΠ[(X, τ1, τ2) Π (Y, σ1, σ2)] is (X × Y ) × (X × Y ) and the carrier set of FΠ(X, τ1, τ2) Π FΠ(Y, σ1, σ2) is (X × X) × (Y × Y ), clearly not the same.
Definition 3.14. Letting L be a us-quantale, L-NBiTop is the full subcategory of all spaces (X, τ, σ) satisfying the condition that each open L-subset u 6= ⊥ in each of τ, σ is L-normalized, i.e., has the property that
_
x∈ X
u (x) = e.
If L is an st-quantale, then the notion of L-normalized subsets coincides with the usual notion, namely W
x∈ X u (x) = ⊤.
Theorem 3.15. Let L be a u-quantale. Then FΠ | L-NBiTop: L-NBiTop → L- Top is an embedding. This embedding does not preserve binary products and hence has no left-adjoint.
Proof. Because of 3.12, it suffices to show FΠas restricted injects objects. For two distinct objects, let us consider
(X, τ1, σ) 6= (X, τ2, σ)
with τ1 6= τ2; all other cases are similar and left to the reader. Suppose W.L.O.G. there is u ∈ τ1− τ2 and assume τ1Π σ = τ2Π σ on X × X. Then setting ⊠ ≡ ⊗ ◦ ×,
∃ {uγ⊠vγ}γ∈ Γ ⊂ τ2Π σ such that
(π1)←L (u) = _
γ∈ Γ
(uγ⊠vγ) .
Applying the surjectivity of π1 and properties of Zadeh image operators (Sub- section 1.3), we obtain the contradiction
u = (π1)→L ((π1)←L (u))
= (π1)→L
_
γ∈ Γ
(uγ⊠vγ)
= _
γ∈ Γ
((π1)→L (uγ⊠vγ))
= _
γ∈ Γ
uγ∈ τ2,
where we have used the fact, for each γ ∈ Γ and each x ∈ X, that (π1)→L (uγ⊠vγ) (x) = _
y∈ X
(uγ⊠vγ) (x, y)
= _
y∈ X
(uγ(x) ⊗ vγ(y))
= uγ(x) ⊗ _
y∈ X
vγ(y)
= uγ(x) ⊗ e
= uγ(x) .
The non-preservation of products follow for the restricted functor as in the
proof of 3.13.
Corollary 3.16. FΠ◦ Gχ : BiTop → Top is an embedding. This embedding does not preserve binary products and hence has no left-adjoint.
Proof. The first statement is a corollary of 3.15 as follows: given any non-empty subset A of set X, χA : X → 2 is normalized; 2-NBiTop = 2-BiTop; and Gχ : BiTop → 2-BiTop is a categorical isomorphism. The non-preservation of products follows for the composite functor as in the proof of 3.13. Remark 3.17. Corollary 3.16 furnishes an embedding of BiTop into Top;
but this is not enough to say that Top may be categorically regarded as a generalization of BiTop since FΠ◦ Gχ is not sufficiently well-behaved. This motivates the search for a better behaved embedding of bitopology into topol- ogy conducted in the next subsection.
3.4. E× : L-BiTop → L2-Top. This subsection constructs the concrete, full, strict “cross” embedding E× : L-BiTop → L2-Top, establishes its behavior w.r.t. limits and colimits—for appropriate L, E× preserves both and detects and reflects the former, and shows that E× is essentially neutral w.r.t. strati- fication issues. It follows that E× is an extremely well-behaved embedding.
3.4.1. Construction of E×: L-BiTop → L2-Top.
Proposition 3.18(cf. [16]). Let X be a set.
(1) For each set L the mapping ϕX : LX× LX → L2X
given by ϕX(a1, a2) = a1× a2, i.e., ϕX(a1, a2) (x) = (a1(x) , a2(x)) is a bijection with inverse mapping ϕ−1X : LX× LX ← L2X
given by ϕ−1X (a) = (π1◦ a, π2◦ a) ,
where π1, π2 are the projections from L2 to L.
(2) If L is a poset, then ϕX is an order-isomorphism.
(3) If L is a semi-DeMorgan s-quantale, then ϕXpreserves semi-complements.
(4) If L is an [u]s-quantale, then ϕXis an [u]s-quantalic isomorphism (i.e., ϕX also preserves tensor products [and the unit]).
Proof. The details of (1)−(3) are the same as, or analogous to, those of Lemma 4.4.1 of [16]. The details of (4) are straightforward. Corollary 3.19. ϕ→X : ℘ LX× LX → ℘
L2X
is an order-isomorphism.
Proof. This is immediate from 3.18(1) using Subsection 1.3. Proposition 3.20. Let A, B be nonempty sets. Then ζ : ℘∅(A) × ℘∅(B) →
℘∅(A × B) given by
ζ (C, D) = C × D
is an order-isomorphism onto its image, i.e., an order-embedding.
Proof. Clearly ζ is well-defined. As for injectivity, let (C1, D1) 6= (C2, D2).
Then there are several cases, and a typical case is C1 6= C2, D1= D2. Then W.L.O.G. there is x ∈ C1− C2. Since D1 = D2 6= ∅, there is y ∈ D1 = D2. So (x, y) ∈ (C1× D1) − (C2× D2) ; hence ζ (C1, D1) 6= ζ (C2, D2). Since all orderings in question are coordinate-wise, it follows that both ζ and ζ−1 (on
Im (ζ)) are isotone.
Proposition 3.21. Let X be a set, L be a us-quantale, and ζ denote any restriction of the ζ of 3.20.
(1) ζ : ℘∅ LX ×℘∅ LX → ℘∅ LX× LXis an order-isomorphism onto its image.
(2) ζ : L-BT (X) → ℘∅ LX× LX is an order-isomorphism onto its im- age.
Proof. Conjoin Proposition 1.2 and 3.20.
Lemma 3.22. Let X be a set and L be a us-quantale, and put E×: L-BT (X) → L2-T (X) by
E× = ϕ→X ◦ ζ.
Then E× is an order-isomorphism onto its image.
Proof. It must be first verified that E× actually maps into L2-T (X). Let (τ1, τ2) ∈ L-BT (X). Then τ1, τ2 are L-topologies on X and hence sub-us- quantales of LX. It is straightforward to check that as direct products,
ζ (τ1, τ2) = τ1× τ2 ⊂ LX× LX and τ1× τ2 is a sub-us-quantale of LX× LX. It follows
E×(τ1, τ2) = ϕ→X (ζ (τ1, τ2)) = ϕ→X(τ1× τ2) ⊂ ℘
L2X and that E×(τ1, τ2) is a sub-us-quantale of L2X
, namely an L2-topology on X. Hence E×(τ1, τ2) ∈ L2-T (X).
The remaining claims concerning E× follow from 3.19 and 3.21. Theorem 3.23. Let L be a us-quantale, let
f ∈ L-BiTop ((X, τ1, τ2) , (Y, σ1, σ2)) , and put
E×(X, τ1, τ2) = (X, E×(τ1, τ2)) , E×(f ) = f.
Then E× : L-BiTop → L2-Top is a concrete, full embedding; and hence L-BiTop is concretely isomorphic to a full subcategory of L2-Top. Further, if L is consistent, E× is a strict embedding (not a functorial isomorphism).
Proof. It is immediate from 3.22 that E×is well-defined at the object-level into L2-Top. It must be now checked that E×is well-defined at the morphism-level, i.e., that f : (X, τ1, τ2) → (Y, σ1, σ2) is L-bicontinuous implies f : (X, E×(τ1, τ2)) → (Y, E×(σ1, σ2)) is L2-continuous. To that end, let
v ∈ E×(σ1, σ2) = ϕ→Y (σ1× σ2) .
Then ∃ (v1, v2) ∈ σ1× σ2 with v = ϕY (v1, v2) . Now let x ∈ X. Then fL←(v) (x) = v (f (x))
= ϕY (v1, v2) (f (x))
= (v1(f (x)) , v2(f (x)))
= (fL←(v1) (x) , fL←(v2) (x)) . Since f is L-bicontinuous,
u1≡ fL←(v1) ∈ τ1, u2≡ fL←(v2) ∈ τ2; and so choosing
u = ϕX(u1, u2) ∈ E×(τ1, τ2) , we have
fL←(v) = u, finishing the proof that f is L2-continuous.
Since E× is concrete (with respect to the usual forgetful functors), it is im- mediate that E× is a functor and that E×injects hom-sets. To verify that E×
is full, we show that f : (X, E×(τ1, τ2)) → (Y, E×(σ1, σ2)) is L2-continuous im- plies f : (X, τ1, τ2) → (Y, σ1, σ2) is L-bicontinuous. Let v ∈ σ1and note ⊥ ∈ σ2. Then (v, ⊥) ∈ σ1×σ2, so that ϕY (v, ⊥) ∈ E×(σ1, σ2). Hence fL←(ϕY (v, ⊥)) ∈ E×(τ1, τ2) by the L2-continuity of f . It follows ∃ u ∈ E×(τ1, τ2) , and hence
∃ (u1, u2) ∈ τ1× τ2, such that
fL←(ϕY (v, ⊥)) = u = ϕX(u1, u2) . Now let x ∈ X. Then
(u1(x) , u2(x)) = ϕX(u1, u2) (x)
= fL←(ϕY (v, ⊥)) (x)
= (v (f (x)) , ⊥)
= (fL←(v) (x) , ⊥) ,
so that u1(x) = fL←(v) (x) . It follows fL←(v) = u1 ∈ τ1. Similarly, it can be shown that if v ∈ σ2, then fL←(v) ∈ τ2. Hence f is L-bicontinuous.
For E× to be an embedding, it remains to show that E× injects objects. To that end let (X, τ1, τ2) 6= (Y, σ1, σ2) . If X 6= Y, we are done. So suppose that X = Y and that (τ1, τ2) 6= (σ1, σ2). Then immediately by 3.22,
E×(τ1, τ2) 6= E×(σ1, σ2) . It follows that
E×(X, τ1, τ2) 6= E×(Y, σ1, σ2) .
Finally, the strictness of E×, when L is consistent, follows from 3.24 below.
Many more properties of E× are developed in the next three subsections which show that it is an extremely well-behaved embedding.
Counterexample 3.24. If E× were to surject objects, then E× would be a functorial isomorphism. This however is usually not the case. Let L be any consistent us-quantale, note L ⊃ 2 = {⊥, e}, and consider the L2-topological space (X, τ ) with X nonempty and τ the indiscrete L2-topology
τ =n
(⊥, ⊥), (e, e)o .
Suppose a space (X, E×(τ1, τ2)) from the image of E× is (X, τ ). This forces ϕ→X (τ1× τ2) = E×(τ1, τ2) = τ.
Noting
{⊥, e} ⊂ τ1, {⊥, e} ⊂ τ2, it follows
ϕX(⊥, e) ∈ ϕ→X(τ1× τ2) , ϕX(⊥, e) = (⊥, e) /∈ τ,
a contradiction. Hence the space (X, τ ) is not in the image of E×. Hence, for consistent L it is the case that E× is not a functorial isomorphism, but a strict embedding. This justifies examining E×’s behavior w.r.t. limits and colimits in the next Subsubsections 3.4.2–3.4.3 as well as characterizing
E×→(L-BiTop) in 3.28.
Corollary 3.25. Let 4 be the 4-element Boolean algebra {⊥, α, β, ⊤}with
⊗ = ∧ (binary). Then the traditional category BiTop of bitopological spaces and bicontinuous maps concretely, fully, strictly embeds into 4-Top as a full monocoreflective subcategory that is closed under all limits and colimits.
Proof. Consider the bitopological version Gχ : BiTop → 2-BiTop of the cha- racteristic functor given by
Gχ(T) = {χU : U ∈ T} , Gχ(S) = {χV : V ∈ S} , Gχ(X, T, S) = (X, Gχ(T) , Gχ(S)) , Gχ(f ) = f.
Then this bitopological Gχ is a concrete functorial isomorphism. Now clearly by the direct product of us-quantales, 22 ∼= 4, so by 3.23 and 3.24, 2-BiTop concretely, fully, strictly embeds into 4-Top. Hence via the composition
E×◦ Gχ: BiTop ֒→ 4-Top,
BiTopconcretely, fully, strictly embeds into 4-Top. For the monoreflectivity claim, see 3.26 below; and the claim regarding limits and colimits follows from Subsubsections 3.4.2–3.4.3 below, the limit claim needing the observation that
4is a u-quantale with ⊗ = ∧.
3.4.2. Behavior of E× : L-BiTop → L2-Top w.r.t. colimits. Since for all consistent L the full concrete embedding E× is not a functorial isomorphism, but only a strict embedding, it is worthwhile to investigate its behavior w.r.t.
limits and colimits. This subsection shows for any us-quantale L that the embedding E× has a right-adjoint—and hence preserves colimits. The next subsection then shows step by step for L a u-quantale that the Special Adjoint Functor Theorem constructs for E× a left-adjoint—and hence E× preserves limits; and further the next subsection shows for any us-quantale L that the embedding E× reflects and detects all limits and is transportable. Therefore, this subsection—in concert with the preceding and subsequent subsections—
shows that E× is an extremely well-behaved embedding.
Theorem 3.26 (E× ⊣ Fπ). Let L be a us-quantale and put the “projection”
functor Fπ: L-BiTop ← L2-Top as follows:
Fπ(X, τ ) = (X, Fπ(τ )) , Fπ(f ) = f, where the fibre level of Fπ
Fπ(τ ) = (π1◦ τ ≡ {π1◦ u : u ∈ τ } , π2◦ τ ≡ {π2◦ u : u ∈ τ })
uses the projections π1, π2 : L × L → L for the us-quantalic (direct) product.
Then the following hold:
(1) Fπ is a concrete embedding which is not full and does not lift limits.
(2) E× ⊣ Fπ, so E× preserves all strong colimits and Fπ preserves all strong limits.
(3) Fπ need not detect limits nor be transportable.
(4) Fπ◦ E×= IdL-BiTop.
(5) L-BiTop is isomorphic (via E×) to a full monocoreflective subcategory of L2-Top.
(6) L2-Top is isomorphic (via Fπ) to an isoreflective subcategory of L-BiTop.
Proof. Ad(1). Since us-quantalic projections preserve arbitrary joins, the ten- sor, and the unit, it follows that Fπ(τ ) ∈ L-BT (X); and hence (X, Fπ(τ )) ∈
|L-BiTop| and Fπ is well-defined at the object level. As for morphisms, let f : (X, τ ) → (Y, σ) be L2-continuous in L2-Top. Then, given v ∈ σ, the identities
fL←(π1◦ v) = π1◦ fL←(v) , fL←(π2◦ v) = π2◦ fL←(v)
are easily checked and immediately imply that f : (X, Fπ(τ )) → (Y, Fπ(σ)) is L-bicontinuous in L-BiTop. Now by the concreteness of Fπ, it is immediately a concrete and faithful functor. To show that Fπ is an embedding, it remains to check that Fπ injects objects: but if u, v : X → L2 are distinct, there exists x ∈ X such that W.L.O.G.
π1(u (x)) 6= π1(v (x)) ;
which implies that if τ 6= σ as L2-topologies on X, then Fπ(τ ) 6= Fπ(σ) as L-bitopologies on X, showing that Fπ injects objects.
To see that Fπneed not be full, let L = 2, write the Boolean algebra L2= 4 as {(⊥, ⊥) , (⊥, ⊤) , (⊤, ⊥) , (⊤, ⊤)}, let X = {x} , and choose
τ =n
(⊥, ⊥), (⊤, ⊤)o
, σ =n
(⊥, ⊥), (⊥, ⊤), (⊤, ⊥), (⊤, ⊤)o .
Then it follows idX : (X, τ ) → (X, σ) is not L2-continuous (since σ is not a subset of τ ). Now
π1◦ (⊥, ⊥) = π1◦ (⊥, ⊤) = ⊥, π1◦ (⊤, ⊥) = π1◦ (⊤, ⊤) = ⊤, π2◦ (⊥, ⊥) = π2◦ (⊤, ⊥) = ⊥, π2◦ (⊥, ⊤) = π2◦ (⊤, ⊤) = ⊤, so that
Fπ(τ ) = (π1◦ τ, π2◦ τ )
= ({⊥, ⊤} , {⊥, ⊤})
= (π1◦ σ, π2◦ σ)
= Fπ(σ) ,
implying idX : (X, Fπ(τ )) → (X, Fπ(σ)) is L-bicontinuous. The concreteness of Fπ implies there exists no g ∈ L-Top with Fπ(g) = idX, so Fπ is not full.
To see that Fπ need not lift limits, let the diagram in L2-Top be the space (X, σ) of the preceding paragraph. Then the image of this diagram is the space (X, Fπ(σ)) in L-BiTop. Now the space (X, Fπ(τ )), together with the arrow idX : (X, Fπ(τ )) → (X, Fπ(σ)), is a limit of the diagram (X, Fπ(σ)): any L-bicontinuous f : (Z, υ1, υ2) → (X, Fπ(σ)) trivially factors uniquely through idX. But as seen in the preceding paragraph, there is no g ∈ L-Top with Fπ(g) = idX, which means there is no limiting cone of (X, σ) in L2-Top which Fπcarries over to the limit idX: (X, Fπ(τ )) → (X, Fπ(σ)) in L-BiTop. Hence Fπ need not lift limits.
Ad(2). Let (X, τ1, τ2) ∈ |L-BiTop| be given. Then
FπE×(X, τ1, τ2) = Fπ(X, ϕ→X (τ1× τ2)) ≡ (X, ˆτ1, ˆτ2) , where it follows that
ˆ
τ1 = {π1◦ ϕX(u, v) : u ∈ τ1, v ∈ τ2} , ˆ
τ2 = {π1◦ ϕX(u, v) : u ∈ τ1, v ∈ τ2} .
We choose the right unit η to be the identity mapping id : X → X. Then for each x ∈ X,
(π1◦ ϕX(u, v)) (x) = π1(u (x) , v (x)) = u (x) , (π2◦ ϕX(u, v)) (x) = π2(u (x) , v (x)) = v (x) , which immediately gives the L-bicontinuity of η.
For universality of the lifting, let (X, τ ) ∈
L2-Top
be given, along with an L-bicontinuous map f : (X, τ1, τ2) → (X, Fπ(τ )). Choosing ¯f = f , we now check ¯f : E×(X, τ1, τ2) → (X, τ ) is an L-continuous map from E×(X, τ1, τ2) to (X, τ ) by letting v ∈ τ and x ∈ X. Then the L-bicontinuity of f implies
fL←(π1◦ u) ∈ τ1, fL←(π2◦ u) ∈ τ2,
from which it follows
ϕX(fL←(π1◦ u) ∈ τ1, fL←(π2◦ u) ∈ τ2) ∈ ϕ→X (τ1× τ2) . Further, we note
fL←(u) (x) = u (f (x)) = (π1(u (x)) , π2(u (x)))
= (fL←(π1◦ u) (x) , fL←(π2◦ u) (x))
= ϕX(fL←(π1◦ u) ∈ τ1, fL←(π2◦ u) ∈ τ2) (x) .
Finally, it is immediate that ¯f is the unique L-continuous map from E×(X, τ1, τ2) to (X, τ ) such that
f = ¯f ◦ η,
completing the universality of the lifting. The naturality diagram now follows by concreteness.
Ad(3). This is an immediate consequence of (1), (2), and Proposition 13.34 [1].
Ad(4). Since ˆτ1 = τ1, ˆτ2 = τ2 in the proof of (2), it is immediate that Fπ◦ E×= IdL-BiTop.
Ad(5). Using Fπ◦ E×= IdL-BiTop, the components of the left unit (counit) of E×⊣ Fπfurnish the needed monocoreflection arrows to L2-topological spaces from the E× image of L-BiTop.
Ad(6). Using Fπ◦ E× = IdL-BiTop, the components of the right unit of E× ⊣ Fπ furnish the needed isocoreflection arrows to L-bitopological spaces
from the Fπ image of L2-Top.
Remark 3.27. We collect some facts concerning E×, Fπ, and their fibre- dependent constructions, where L is a us-quantale:
(1) Fπ ⊣/ E× if L is consistent. This is a consequence of 3.24.
(2) E×⊣ Fπ need not be a categorical equivalence. This follows from (1).
(3) For each (X, τ1, τ2) ∈ |L-BiTop| ,
FπE×(τ1, τ2) = Fπ(ϕ→X(τ1× τ2))
= (π1◦ ϕ→X (τ1× τ2) , π2◦ ϕ→X(τ1× τ2))
= (τ1, τ2) . (4) For each (X, τ ) ∈
L2-Top ,
“E×(Fπ(τ )) = ϕ→X((π1◦ τ ) × (π2◦ τ )) ⊃ τ ” always holds; but for L consistent,
“E×(Fπ(τ )) = ϕ→X((π1◦ τ ) × (π2◦ τ )) ⊂ τ ”
need not hold. The latter statement is another version of (1).
Theorem 3.28(characterization of
E×→(L-BiTop)
). Let L be a us-quantale and (X, τ ) ∈
L2-Top
. Then (X, τ ) ∈
E×→(L-BiTop)
if and only if E×(Fπ(τ )) = τ , i.e., both inequalities of 3.27(4) hold.
3.4.3. Behavior of E× : L-BiTop → L2-Top w.r.t. limits. The question of a left-adjoint for E×is open for general us-quantales L; and it is our conjecture is that for general us-quantales L, E×would not preserve products or intersections and hence would not have a left-adjoint. But on the other hand, this section shows E× has a left-adjoint (and therefore preserves all limits) for L any u- quantale. We point out that our proof of this left-adjoint is existential (via the Special Adjoint Functor Theorem) and not constructive; and it is an additional open question whether there is a direct construction of this left adjoint not essentially factoring through our proof. It is further proved that E× reflects and detects limits and is transportable.
Lemma 3.29 (preservation of products). For each u-quantale L, E× : L- BiTop→ L2-Top preserves arbitrary (small) products.
Sublemma 3.30. Let L be a u-quantale and suppose X is a set and τ1, τ2 are L-topologies on X with respective subbases σ1, σ2, namely
τ1= hhσ1ii , τ2= hhσ2ii , such that {⊥, e} ⊂ σ1∩ σ2. Then
(*) ϕ→X(τ1× τ2) = hhϕ→X(σ1× σ2)ii . Proof. To see that “⊃” holds in (*), note that
σ1× σ2⊂ τ1× τ2, ϕ→X (σ1× σ2) ⊂ ϕ→X(τ1× τ2) , hhϕ→X (σ1× σ2)ii ⊂ ϕ→X(τ1× τ2) .
For “⊂” in (*), we first invoke the associativity of ⊗ and its infinite distribu- tivity overW to write members of τ1, τ2as joins of tensor products of members of σ1, σ2, respectively. More precisely, consider these typical members
_
α∈ A1
O
β∈ B1
uαβ
, _
α∈ A2
O
β∈ B2
vαβ
of τ1, τ2, respectively, where A1, A2 are arbitrary indexing sets, B1, B2 are ar- bitrary finite indexing sets, each uαβ∈ σ1, each vαβ∈ σ2, and where W.L.O.G.
we assume
A1∩ A2= ∅ = B1∩ B2.
Next, we augment the uαβ’s and vαβ’s as follows, using the assumption that {⊥, e} ⊂ σ1∩ σ2:
α ∈ A1, β ∈ B2, uαβ≡ e, α ∈ A2, β ∈ B1∪ B2, uαβ≡ ⊥,
α ∈ A2, β ∈ B1, vαβ≡ e, α ∈ A1, β ∈ B1∪ B2, vαβ≡ ⊥.
