Representation of H-closed monoreflections in archimedean `-groups with weak unit

The category of the title is called W. This has all free objects F (I) (I a set). For an object class A, HA consists of all homomorphic images of A-objects. This note continues the study of the H-closed monoreflections (R, r) (meaning HR = R), about which we show (inter alia): A ∈ A if and only if A is a countably up-directed union from H{rF (ω)}. The meaning of this is then analyzed for two important cases: the maximum essential monoreflection r = c, where cF (ω) = C(R), and C ∈ H{c(R)} means C = C(T ), for T a closed subspace of R; the epicomplete, and maximum, monoreflection, r = β, where βF (ω) = B(R), the Baire functions, and E ∈ H{B(R)} means E is an epicompletion (not “the”) of such a C(T ).


Introduction
W is the category of archimedean -groups G with distinguished weak order unit e G , and morphisms G ϕ − → H the -group homomorphisms with ϕ(e G ) = e H .We compress the discussion in §1 of [11], which see for more detail."A ≤ B" means A is a W-subobject of B.
The forgetful functor W → Sets has the left adjoint F .An F (I) is the free object on the set I, and this is the W-subobject of R R I generated by the constant function 1, and all projections π i : R I → R (i → π i is the "insertion of generators" I → F (I)).
A full subcategory rA, rA ∈ R, with the property: We usually write A ≤ rA for the r A .We abuse language and notation by saying as convenient (R, r) or R, or r, is a monoreflection.
The class of monoreflections is ordered by: r ≤ s means ∀A ∃ monic f with "ω" stands for the natural numbers, or any countable set, or the ordinal or cardinal.
Theorem 1.1 is one of the main results of [11] and is the cornerstone of that paper.It devolves from categorical generalities, and many special features of W, some of which we describe below, and some later when needed.
Another main result of [11] is the characterization of the rF (ω) in Theorem 1.1.Namely, 3.6 there says these are exactly the S with F (ω) σ ≤ S ≤ B(R ω ) (B the Baire functions), with σ epic and S • S ω = S (that is, ∀s and countable {s n } from S, the function R ω sn −−→ R ω s − → R lies in S).The cases for c 3 and β are mentioned in the Abstract, and will be deployed below. Let From the form of the F (I), and the fact that any f ∈ C(R I ) factors through a countable subproduct, we have A crucial ingredient to what we have said so far, and necessary later, is the Yosida representation of W-objects: R is the real numbers, and R = R ∪ {±∞} under the obvious topology and order.For X a topological space, D(X) = {f ∈ C(X, R) | f −1 R dense in X}.This is a lattice containing C(X), but has only partly defined +.
, where A is closed under the partly defined data required to make A ∈ W.
The Yosida representation of A ∈ W (see [12]) says: (1) A ≤ D(YA) for a unique compact Hausdorff YA for which A separates the points. (

Main Theorem
We expand on Theorem 1.1.
Theorem 2.1.Suppose (R, r) is an H-closed monoreflection in W. For A ∈ W, the following are equivalent: (1) A ∈ R.

c (Closed under countable composition)
"c 3 " stands for "closed under countable composition", originally studied in [13].The definition goes as follows.
Each A ∈ W has its Yosida representation A ≤ D(YA).A sequence a 1 , a 2 , . . .from A has a −1 R dense in YA (Baire Category Theorem) and c 3 will denote either the object class, or reflections A ≤ c 3 A. We assemble known facts.Theorem 3.1.(Each item without specific reference can be located in [11] §1, with reference to original sources.) The class c 3 is H-closed.(d) c 3 is the largest essential monoreflection (with the smallest class of objects).
Proof.First note: For any Tychonoff space X and T ⊆ X, the restriction For the converse, we shall use details of the Yosida representation; see §1.Any A ϕ − B has the quasi-dual embedding YA ← YB for which ϕ(a) = a|YB ∀a ∈ A. This entails a −1 R ∩ YB dense in YB, and thus and closed in the normal X (thus C-embedded) so B = C(T ).
Summing up, we interpret Theorem 2.1 for c 3 through Theorem 3.2 and some of Theorem 3.1.
(2) There is I with a surjection C(R Proof.This is all quite immediate.We just note: (4) is just Theorem 2.1(5), using Theorem 3.2 for X = R ω .
(4) is the statement that in Theorem 2.1(4) the i B are one-to-one.This follows solely from the essentiality of the reflection maps B ≤ c 3 B.
(c) We note [7], p. 74: T is (≈) a closed subspace of R ω if and only if T is completely metrizable and separable.
(d) In Corollary 3.3 (5), the A = ω ↑ C(T )'s is a countably directed direct limit, A = lim − →ω C(T )'s.The Yosida functor converts this to an inverse limit YA = lim ← −ω βT 's.Using A = C(X) with X real compact, and a little fiddling yields X = lim ← −ω T 's, and if X is compact, so are the T 's.This is more or less a result of Pasynkov [15].See also [7], p. 220.
(e) An essential reflection (R, r) has r ≤ c 3 (Theorem 3.1 (d)), and if R = HR, Corollary 3.3 holds mutatis mutandis.For rF (ω) = S (see the second paragraph after Theorem 1.1), we have F (ω) σ ≤ S ≤ C(R ω ), and "σ epic" is automatic.Examples of this are: R = "rings" (W-objects A with a compatible f -ring multiplication with identity the W-unit e A ), vector lattices, algebras, . . . .For example: for rings, rF (ω) is the sub-f -ring of C(R ω ) generated by F (ω).In Corollary 3.3 (4), each C(T ) is to be replaced by the set of restrictions rF (ω)|T .An additional feature of any essential r is that rF (ω)|T = r(F (ω)|T ).
(f) The present paper began with an analysis of a version of Corollary 3.3 and some related matters, in the view of a c 3 -object as the f -ring of real-valued continuous functions on a frame.As such, it was reported in [6]: where c 3 was taken as condition 3.3(3), thus avoiding a reference to the Yosida representation and the reflection is then given an explicit frametheoretic form.See [4] for details.

β (Epicomplete)
E is called epicomplete if E ϕ − → • monic and epic implies ϕ an isomorphism.The class of epicomplete objects is denoted EC.
Recall that, for a Tychonoff space X, B(X) denotes the W-object of real-valued Baire functions on X.
We summarize known features of EC, prior to the interpretation of Theorem 2.1 for R = EC.Theorem 4.1.(Each item without specific reference can be located in [11], with reference to original sources.)(a) E ∈ EC if and only if E is σ-complete both conditionally, and laterally if and only if E ≈ D(X) with X basically disconnected (the X is YE).Thus, any B(X) ∈ EC.
(b) ( [3]).E ≈ C(P) with P a P -locale.(Such a P is the localic intersection of {S | S is dense cozero in YE}.) (c) EC is monoreflective, thus the maximum monoreflection.The reflection of A is βA = B(YA)/N , for a certain σ-ideal N .
(e) For every set I, βF We now interpret Theorem 2.1.Most of this is the routine writing-down of items in Theorem 2.1 using information in Theorem 4.1.An exception is Theorem 2.1 (5), which says A ∈ H{B(R ω )}."An" epicompletion of A ∈ W is an epic A ≤ E, with E EC.These are exactly the quotients over A of βA.Theorem 4.2.Suppose X is LČ (as is R ω ).
(a) E ∈ H{B(X)} if and only if there is (b) (Note that an F in (a) is again LČ.) C(X) has a unique epicompletion if and only if X is discrete and countable (and thus X ≈ N, C(X) ≈ C(N), is already EC).
(c) If X is not countable discrete, there are many epicompletions of C(X).
Proof.(a) Suppose E ∈ H{B(X)}, as B(X) where ϕ 0 is the restriction of ϕ, e labels the inclusion, and ϕβ C = eϕ 0 (obviously), so e is epic (as a second factor of the epic ϕβ C ).By Theorem 3.2, ϕ(C(X)) is the desired C(F ).
Suppose F is closed in X and C(F ) e ≤ E is an epicompletion.We then have E where ρ is the restriction map described at the beginning of the proof of Theorem 3.2, and then ∃ρ with ρβ C = eρ by the universal mapping property of β.
(b) If C(X) ≈ C(N), already C(X) ∈ EC, so is its only epicompletion.If C(X) has a unique epicompletion, it must be C(X) ≤ B(X) (Theorem 4.1 (d)), and this must be an essential embedding (because any A ∈ W has a (unique) essential epicompletion ( [2], §9)).If X has a non-void nowhere dense zero-set Z, then the characteristic function χ(Z) ∈ B(X), and there is no 0 < a ∈ C(X) with a ≤ χ(Z): C(X) ≤ B(X) is not essential.Thus there is no such Z, so X is what is called an almost P -space.But the only almost P -space which is LČ is (≈) N.
Referring to Theorem 4.2, let ECS(R ω ) stand for the family of epicompletions of objects of the form C(T ), for T closed in R ω .
Summing up, we write down Theorem 2.1 for W β − → EC using Theorem 4.2 and some of Theorem 4.1.
(2) There is I with a surjection B(R The comparison of Corollary 4.3 (4) and ( 5) with Corollary 3.3 (4) and ( 5), shows a huge difference between c 3 (or any essential reflection) with β and identifies some special classes of EC objects which might deserve further study.(It is quite rare that any A ≤ βA is essential; see [2], §9.) We consider the analogue of Corollary 3.3 (4) for β.Recall that for B ≤ A, βB ≤ A means that the i B in 2.8 (4) is one-to-one.
( * * ) All we have to say is: sometimes this happens, sometimes not.We leave the subject for now.

Theorem 4 . 4 .
Suppose A ∈ W. For every countable B ≤ A, βB ≤ A if and only if A ≈ R n for some n ∈ N. ( * ) with the inclusion B j B ≤ A being j B = ki B and with i

Remark 4 . 5 .
(a) There are A satisfying ( * * ): Obviously, any B(T ); less trivially, ([11]) for uncountable I, B(R I ) = ω ↑ {B(R J ) | J ∈ P 0 (I)}.(b) There are many A failing ( * * ).The countable chain condition, ccc, of a space or W-object is relevant here.X (resp., A) has ccc if there is no uncountable pairwise disjoint family of non-void open sets in X (respectively, non-zero positive elements in A).A has ccc if and only if YA does (because cozA is a base in YA).If A has ccc and satisfies ( * * ), then in the Yosida representation A ≈ D(YA), each a ∈ A is locally constant on a dense open subset of YA. (If A = ω ↑ B(T α ), then each B(T α ) has ccc, and it follows that T α is a copy N α of N. For each α, C(N α ) = βN α c − YA).If a ∈ C(N α ), then a" = a • τ is locally constant on τ −1 (N α ).)Consider the absolute (projective cover) [0, 1] π − a[0, 1].Using irreducibility of π: Since [0, 1] has ccc, so do a[0, 1], and also A = D(a[0, 1]).Here C([0, 1]) ≤ A, as f → f • π.No continuous nonconstant f has f • π locally constant on a dense subset of a[0, 1].Thus A fails ( * * ).(c) The class EC consists exactly of the D(X), X compact and basically disconnected.The class σBA of σ-complete Boolean algebras consists exactly of the clopen algebras clopX for the same X [16].So, the various properties of EC's considered here have direct translations to σBA.For example, corresponding to 4.6 are the σBA's of the form A ∈ ω ↑ {B(T ) | T closed in R ω }, B denoting the σ-field of Baire sets.