THE ENVELOPE OF A SUBCATEGORY IN TOPOLOGY AND GROUP THEORY

A collection of results are presented which are loosely centered around the notion of reflective subcategory. For example, it is shown that reflective subcategories are orthogonality classes, that the morphisms orthogonal to a reflective subcategory are precisely the morphisms inverted under the reflector, and that each subcategory has a largest “envelope” in the ambient category in which it is reflective. Moreover, known results concerning the envelopes of the category of sober spaces, spectral spaces, and jacspectral spaces, respectively, are summarized and reproved. Finally, attention is focused on the envelopes of one-object subcategories, and examples are considered in the category of groups.

A morphism f : A → B and an object X in a category C are called orthogonal [15] if the mapping hom C ( f ,X) : hom C (B,X) → hom C (A,X) which takes g to g f is bijective. For a class of morphisms Σ (resp., a class of objects D), we denote by Σ ⊥ the class of objects orthogonal to every f in Σ (resp., denote by D ⊥ the class of morphisms orthogonal to all X in D) [15].
Over the years, reflective subcategories have been studied extensively. It is worth noting that in a remarkable paper [6], Cassidy et al. have given an important study of reflective subcategories.
The present paper is a contribution to the study of reflective subcategories. We state the following three natural questions. Let D be a reflective subcategory in C. The following question is credited to [15].

Question 1.2. Is D ⊥⊥ = D?
Let C be a category and D a subcategory of C closed under isomorphisms. It is easily seen that there is a largest full subcategory of C (the envelope of D) in which D is a reflective subcategory.

Question 1.3. How envelopes of various categories can be described in concrete cases?
Note that in [19], Herrlich has considered the envelope question in some generality. Question 1.1 has been answered by Casacuberta et al. in [5]. The second section of this paper is devoted to collect some information about Questions 1.1 and 1.2.
The third section deals with an example of orthogonal class of morphisms. The fourth, fifth, and sixth sections deal with Question 1.3. Note that this question has been tackled long ago (in the setting of HAUS, where HAUS is the category of Hausdorff spaces with continuous maps as morphisms) by Porter in [28].
The last two sections treat Question 1.3 in the particular case when D is the subcategory of C whose objects are those isomorphic to a given object X. Particular study of Question 1.3 is given in the category of groups.

Reflective subcategories and orthogonality
Let us first fix some notations which will be used throughout this section. The symbol C will always denote a category, D will denote a full reflective subcategory of C which is isomorphism-closed. We also denote by F a left adjoint functor of the inclusion functor I : D → C and µ the unit of the adjunction (for precise definitions, see [25]).

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Thus, if two of the three maps hom C (θ 1 ,X), hom C (θ 2 ,X), hom C (θ 3 ,X) are bijective, then so is the third one. This leads to the following proposition. → C be two morphisms in C and θ 3 = θ 2 • θ 1 . If two of the morphisms θ 1 , θ 2 , and θ 3 are in D ⊥ , then so is the third one.
Proposition 2.2. For each object A in C, the following statements are equivalent: The following result is an easy observation from [6].
Then the following statements are equivalent: Let F : C → C be a functor. The class of morphisms of C rendered invertible by F is sometimes denoted by Σ F [6] or (F) [4]. Hence, Proposition 2.2 says exactly that D ⊥ = (F).
Using Propositions 2.2 and 2.3, one may check easily the following.
Then the following properties hold.
(2) If A and B are objects in D, then f is an isomorphism. Remark 2.5. It is possible to have an arrow f : A → B in D ⊥ such that B is in D, but f has neither section nor retraction (see Example 3.4).
An affirmative answer to Question 1.2 is already in [19]; this yields the following. Proof. Clearly, D is contained in D ⊥⊥ .
Conversely, let C be an object of C lying in D ⊥⊥ . Since µ C is in D ⊥ , there exists a unique morphism g : is commutative.
3390 The envelope of a subcategory in topology and group theory Thus, the diagram commutes.
Since, in addition, µ C ∈ D ⊥ , we get µ C • g = 1 F(C) . Therefore, µ C is an isomorphism, and consequently C is a D-object by Proposition 2.2.
Proposition 2.7. Let D 1 and D 2 be two full reflective isomorphism-closed subcategories of a category C. For i ∈ {1, 2}, denote by F i a left adjoint functor of the inclusion functor from D i into C.
Then the following statements are equivalent: Proof. (i)⇒(ii). It is well known that any two left adjoint functors for a given functor are naturally isomorphic [25].
(ii)⇒(iii). Let η be a natural isomorphism from F 1 to F 2 and f : A → B a morphism in C. Hence, the diagram commutes. On the other hand, η A and η B are isomorphisms, and it follows that F 1 ( f ) is an isomorphism if and only if so is F 2 ( f ). Therefore, hom C (A,B) ∩ D 1 ⊥ = hom C (A,B) ∩ D 2 ⊥ , for each objects A, B in C by Proposition 2.3. (iii)⇒(i). The proof follows immediately from Proposition 2.6.
The following result gives more information about reflective subcategories. hold: Proof. (1)⇒(2). Statement (1) is equivalent to D being a reflective subcategory of C. Now, since µ A is in D ⊥ , we conclude that F(µ A ) is an isomorphism, by Proposition 2.3. If A is in D, then, according to Proposition 2.2, µ A is an isomorphism.
(2)⇒(1). We are aiming to prove that (F(A),µ A ) is a universal to the inclusion functor I : D → C from A.
Let C be an object of D and f : A → C a morphism in C. We must prove that there is a unique morphism f : Suppose that such a morphism f exists. Then we have On the other hand, the diagram commutes. Consequently, This implies the uniqueness of f , if it exists. Now, it suffices to verify that does the job. Indeed, the following diagrams are commutative. Hence, (2.8)

An example of orthogonality class of morphisms
Let X be a topological space, we denote by O(X) the set of all open subsets of X. Recall that a continuous map g : Recall that a subset of a topological space X is said to be locally closed if it is the intersection of an open subset and a closed subset of X. A subset S of a topological space X is said to be strongly dense in X if S meets every nonempty locally closed subset of X. Thus, a subset S of X is strongly dense if and only if the canonical injection S X is a quasihomeomorphism. It is well known that a continuous map q : X → Y is a quasihomeomorphism if and only if the topology of X is the inverse image by q of that of Y and the subset q(X) is strongly dense in Y [16].
The notion of quasihomeomorphism is used in algebraic geometry. It has been recently shown that this notion arises naturally in the theory of some foliations associated to closed connected manifolds (see [2,3]). It is worth noting that quasihomeomorphisms are also linked with sober spaces. A Recall that a topological space X is said to be sober if any nonempty irreducible closed subset of X has a unique generic point. Let X be a topological space and S(X) the set of all irreducible closed subsets of X [16].
provides a topology on S(X) and the following properties hold.
(i) The map η X : The topological space S(X) is called the soberification of X, and the assignment X → S(X) defines a functor from the category of topological spaces TOP to TOP [16]. The soberification serves, sometime, to give topological characterization of particular spaces (see, e.g., [11]).
We denote by SOB the full subcategory of TOP whose objects are sober spaces. It is clear that η is a natural transformation from the functor 1 TOP to the functor I • S, where I : SOB → TOP is the inclusion functor. Let X and Y be two topological spaces, we will denote QH(X,Y ) the set of all quasihomeomorphisms from X to Y .
It is well known that SOB is a reflective full subcategory of TOP [16]. We are aiming to determine the orthogonality class of morphisms SOB ⊥ .
Notice that all the material of this section may be derived from [1,12]; and for the sake of completeness, we will give all the details.
First, notice that Proposition 3.1 follows from well-established results in frame (or locale) theory. The adjunction Ω Σ : FRAME −→ TOP, (3.1) where Ω is the open set functor, describes the soberification reflector by Σ • Ω. The A. Ayache and O. Echi 3393 quasihomeomorphisms are exactly those maps inverted by the reflector (see, e.g., Johnstone [23]).
Proposition 3.1. The orthogonality class SOB ⊥ of TOP is the class of all quasihomeomorphisms.
We need a technical lemma.
Lemma 3.2. Let q : X → Y be a quasihomeomorphism. Then the following properties hold.
(2) (i) We start with the obvious observation that if S is a closed subset of Y , then S is irreducible if and only if so is q −1 (S). (ii) Let us prove that q is surjective. For this end, let y ∈ Y , according to the above observation, q −1 ({y}) is a nonempty irreducible closed subset of X. Hence, q −1 ({y}) has a generic point x. Thus, we have the containments This proves that q is a surjective map, and thus q is bijective. One may easily see that bijective quasihomeomorphisms are homeomorphisms.
Proof of Proposition 3.1. According to Proposition 2.3, one is brought back to prove that for each topological spaces X, Y and each continuous map q : X → Y , the following statements are equivalent: (a) q is a quasihomeomorphism; is commutative.
(b)⇒(a). Since η X = ((S(q)) −1 • η Y ) • q and (S(q)) −1 • η Y are quasihomeomorphisms, it is easily seen that q is a quasihomeomorphism. Example 3.4. Let X be a T 0 -space which is not sober. Then η X : X → S(X) is a quasihomeomorphism which is not a homeomorphism.
The mapping η X is in SOB ⊥ and S(X) is an object of the reflective subcategory SOB of TOP, but η X has neither section nor retraction. Indeed, suppose that the following hold.
(i) Suppose that there exists a continuous map g : S(X) → X such that g • η X = 1 X . Then (η X • g) • η X = η X . This is to say that the diagram commutes.
Since S(X) is sober and η X is orthogonal to SOB, we must have η X • g = 1 S(X) . It follows that η X is a homeomorphism, a contradiction.
(ii) Suppose that there exists a continuous map g : S(X) → X such that η X • g = 1 S(X) . Then η X is a surjective quasihomeomorphism. Thus η X is a bijective quasihomeomorphism. Hence η X is a homeomorphism, a contradiction.

The envelope of a subcategory
This section is devoted to answer Question 1.3.
Let C be a category and let D be a subcategory (no longer reflective and not assumed to be isomorphism closed).

Definitions and remarks.
Let C be a category and X an object of C. By a D-ification of X, we mean a morphism p : X → X such that X is an object of D and p is orthogonal to D. The object X is said to be D-ifiable if it has a D-ification. We denote by Env C (D) the full subcategory of C whose objects are the D-ifiable objects of C, this subcategory will be called the envelope of D in C.

Remarks 4.1.
(1) D is a subcategory of Env C (D) and Env C (D) is isomorphism-closed.
(2) D is a reflective subcategory of Env C (D).
(3) Env C (D) is the largest subcategory of C in which D is a reflective subcategory.
(4) Env C (Env C (D)) = Env C (D). (5) If D is a subcategory of C 1 and C 1 is a subcategory of C, then Env C (Env C1 (D)) = Env C (D).
The following result is equivalent to Remark 4.1 (5). This result facilitates verification of the fact that some objects are D-ifiable.
Proposition 4.2. Let D be a subcategory of C 1 and C 1 a subcategory of C. Suppose that X is a C 1 -ification of the object X of C. Then the following statements are equivalent: The semispectral spaces and spectral maps form a category U. It is easily seen that is a full subcategory of U.
In the same paper [21], Hochster has introduced the notion of spectralifiable space. By a spectralification of a semispectral space X, we mean a spectral embedding g of X into a spectral space X such that for every spectral space Y and spectral map f from X to Y , there is a unique spectral map f from X to Y such that f = f • g. The space X is said to be spectralifiable if it has a spectralification [21]. When a semispectral space is spectralifible, we will say that it is H-spectralifiable.
A complete characterization of H-spectralifiable spaces is given by Hochster in the following.
Theorem 4.4 (see Hochster [21]). Let X be a semispectral space. Then the following conditions are equivalent:

Jacspectralifiable spaces.
Recall that a topological space X is said to be a Jacobson space if the set Ꮿ(X) of all closed points of X is strongly dense in X [16]. Obviously, when X is a topological space, Jac(X) = {x ∈ X | {x} = {x} ∩ Ꮿ(X)} is a Jacobson space; we call it the Jacobson subspace of X. Clearly, Jac(X) is the largest subset of X in which Ꮿ(X) is strongly dense. Hence, the canonical injection Ꮿ(X) Jac(X) is a quasihomeomorphism.
Let R be a ring, we denote by Jac(R) the Jacobson subspace of Spec(R). It is easily seen that a prime ideal p of R is in Jac(R) if and only if p is the intersection of all maximal ideals m of R such that p ⊆ m. A jacspectral space is defined to be a topological space homeomorphic to the Jacobson space of Spec(R) for some ring R.
In [1], Bouacida et al. have given a nice topological characterization of jacspectral spaces. For the sake of completeness, we will prove this result but with some changes in the proof.
We need a lemma, its proof is obvious, and therefore it is omitted.
Lemma 4.6. Let q : X → Y be a quasihomeomorphism.
(1) If U is an open subset of Y , then the following statements are equivalent: (3) Let X be a T 0 -space. Then q(Ꮿ(X)) = Ꮿ(S(X)), where q : X → S(X) is the injection of X onto its soberification S(X). (4) Let S be a subset of X. Then the following statements are equivalent: (i) S is strongly dense in X;

then X is Jacobson if and only if so is S(X).
We now head towards an important result which completely characterizes jacspectral spaces.

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Theorem 4.7. Let X be a topological space. The following statements are equivalent: (i) X is a jacspectral space; (ii) X is a compact Jacobson sober space.
Proof. (i)⇒(ii). Let R be a ring and X = Jac(R) = Jac(Spec(R)) the Jacobson space of Spec(R); then X is a Jacobson space. One may check easily that Jac(R) is the soberification of Max(R) (the set of all maximal ideals of R). Thus Jac(R) is a sober space. On the other hand, Max(R) is compact by Hochster [21]. Moreover, the canonical injection Max(R) Jac(R) is a quasihomeomorphism. Hence X is compact by Lemma 4.6.
(ii)⇒(i). Suppose that X is a compact Jacobson sober space. We know that the canonical injection Ꮿ(X) X is a quasihomeomorphism, whence Ꮿ(X) is compact by Lemma 4.6. It follows that Ꮿ(X) is a compact T 1 -space. Therefore, there exists some ring R such that Ꮿ(X) is homeomorphic to Max(R) (see Hochster [21,Proposition 11]). Let ϕ : Ꮿ(X) → Max(R) be a homeomorphism and i : Max(R) → Jac(R) the canonical injection; then f = i • ϕ : Ꮿ(X) → Jac(R) is a quasihomeomorphism. In view of Corollary 3.3, there exists a continuous extensionf : X → Jac(R). This extension is also a quasihomeomorphism. Now, since X and Jac(R) are sober,f is a homeomorphism by Proposition 2.4 (2).
Let be the full subcategory of TOP whose objects are jacspectral spaces. By a jacspectralifiable space, we mean a -ifiable topological space. Next, we give some examples of jacspectralifiable spaces.
Proposition 4.8. Let X be a topological space. If the T 0 -identification T 0 (X) of X is a Jacobson compact space, then X is jacspectralifiable.
Proof. Following Proposition 4.2, it suffices to prove that each T 0 compact Jacobson space is jacspectralifiable (since each topological space is TOP 0 -ifiable).
Lemma 4.6 assures that the soberification S(X) is a compact Jacobson sober space. Hence, S(X) is a jacspectral space, by Theorem 4.7. Now, Corollary 3.3 tells us that the canonical injection of X into its soberification S(X) is a -ification of X.
We state a similar problem to Problem 4.5.

COMP-ifiable spaces.
Let COMP be the full subcategory of TOP consisting of compact topological spaces. It is well known that a completely regular space is COMP-ifiable.
The following is a more general result.
Let TOP 3.5 be the full subcategory of TOP consisting of completely regular spaces. It is known that every space has a TOP 3.5 -ification (see [26,27]). Now, if F : A → B and G : B → C are reflectors, then so is the composition G • F. This yields the following proposition.
Let f : X → Y be a continuous map between completely regular spaces. A natural question is asked; when is the extension β( f ) : β(X) → β(Y ) a homeomorphism?
According to the paper of Holgate [22], we get the following. Proof. First, suppose that β( f ) is a homeomorphism. Then, the restriction of β( f ) to X is an embedding, which implies that f : X → Y is an embedding. Clearly, f (X) ⊆ Y ⊆ β(Y ) and (4.1) Now, we have the following commutative diagram:

Definitions and remarks.
Let X, Y be two objects of a category C. By an X-ification of Y , we mean a morphism p : Y → X such that p is orthogonal to X. We say that Y is X-ifiable if there is an X-ification of Y . This is a particular case when D is the subcategory of C whose objects are those isomorphic to a given object X.
The following result will be needed in order to discuss the notion of X-ifiable objects.
Proposition 5.1. Let C be a category and let D be a subcategory of C. If X is in D and Z is a D-ification of Y , then the following statements are equivalent: Proof. (i) First, suppose that d = 1. According to Proposition 6.2, Z/nZ and Z/mZ are relatively terminal. Hence, hom(Z/nZ,Z/mZ) is a cyclic group of order 1.
(ii) Now, suppose that d = 1. We denote byx the equivalence class of x modulo nZ anḋ x the equivalence class of x modulo mZ. Let p ∈ hom(Z/nZ,Z/mZ) be a nonzero morphism. To define p, it suffices to know that p(1) =k, where k ∈ {1, 2,...,m − 1}. Since n1=0, we have nk =0. Thus nk ≡ 0 (mod m). Hence, by Remark 6.3, k ≡ 0 (mod (m/d)), where d = gcd(m,n) so that there exists t ∈ Z such that k = t(m/d). Let ρ : Z/nZ → Z/mZ be the morphism of groups defined by ρ(x) = (m/d)ẋ. Then p = tρ. It follows that hom(Z/ nZ,Z/mZ) is a cyclic group generated by the morphism ρ. It is clear that ρ is of order d.
Let us derive two important consequences of Proposition 6.4.  Proof. It is well known that a finite Abelian group is in a unique manner a direct sum of primary cyclic groups (fundamental theorem of finite Abelian groups). Now, Remark 5.6 and Corollary 6.5 permit to check easily the equivalence (i)⇔(ii). (2) Let G be a group not isomorphic to Z. Is there a group H(G) such that G, H(G) are relatively terminal?
The following observation will be useful in the next theorem.
Remark 6.8. Let (G,+) be a finite cyclic group generated by ρ ∈ G and let d be the order of ρ. Let α ∈ Z and p = αρ ∈ G. Then, p generates G if and only if gcd(α,d) = 1.
We are now in a position to state the main result of this section. Theorem 6.9. Let (m,n) ∈ N * × N such that gcd(m,n) = 1. Then the following statements are equivalent: (i) Z/nZ is Z/mZ-ifiable; (ii) m divides n.
The following corollary gives examples of non-T-groups of arbitrary order and proves that the quotient of a T-group need not be a T-group. Proof. Let p be a prime factor of n. By Theorem 6.9, Z/ pnZ is Z/nZ-ifiable. If we suppose that Z/nZ is a T-group, then there exists a group H such that Z/nZ is H-terminal and Z/ pnZ is isomorphic to Z/nZ × H. For a reason of cardinality, H is a group of order p.

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Hence, H is isomorphic to Z/ pZ. It follows that Z/ pnZ is isomorphic to Z/nZ × Z/ pZ. Thus Z/nZ × Z/ pZ is a cyclic group. But it is a part of the folklore of algebra that the direct product of two cyclic groups is cyclic if and only if their order are relatively prime. Therefore, Z/nZ is not a T-group in Ᏼ.