Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

PUSHOUT STABILITY OF EMBEDDINGS, INJECTIVITY AND CATEGORIES OF ALGEBRAS

LURDES SOUSA

Abstract. In several familiar subcategories of the category Top of topological spaces and continuous maps, embeddings are not pushoutstable. But, an interesting feature, capturable in many categories, namely in categories B of topological spaces, is the following: For M the class of all embeddings, the subclass of all pushout-stable M-morphisms (that is, of those M-morphisms whose pushout along an arbitrary morphism always belongs to M) is of the form A^{Inj} for some space A, where A^{Inj} consists of all morphisms m : X → Y such that the map Hom(m, A) : Hom(Y, A) → Hom(X, A) is surjective. We study this phenomenon. We show that, under mild assumptions, the reflective hull of such a space A is the smallest M-reflective subcategory of B; furthermore, the opposite category of this reflective hull is equivalent to a reflective subcategory of the Eilenberg-Moore category Set^{𝕋}, where 𝕋 is the monad induced by the right adjoint Hom(−, A) : Top^{op} → Set. We also find conditions on a category B under which the pushout-stable M-morphisms are of the form A^{Inj} for some category A.

0. Introduction

It is well-known that in the category Top of topological spaces and continuous maps, the class of all embeddings is pushout-stable, that is, if we

2000 Mathematics Subject Classification. 18A20, 18A40, 18B30, 18G05, 54B30, 54C10, 54C25.

Key words and phrases. embeddings, injectivity, pushout-stability,(epi)reflective subcategories of Top, closure operator, Eilenberg-Moore categories.

Partial financial support by Centro de Matemática da Universidade de Coimbra and by Escola Superior de Tecnologia de Viseu is acknowledged.

This article will be revised and submitted for publication elsewhere. Lurdes Sousa, Pushout stability of embeddings, injectivity and categories of algebras,

Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 295–308, Topology Atlas, Toronto, 2002; arXiv:math.CT/0204140.

have a pushout in Top,

[figure: •; m; •; •; f; \overline{m}; •; \overline{f}]

with m an embedding, then \overline{m} is also an embedding. But this is not true for several subcategories of Top, namely, it is not true for various epireflective subcategories of Top.

Let B be a subcategory^{∗} of Top, let M denote the class of embeddings in B. By P_{B}(M) we denote the class of all morphisms m ∈ M such that every pushout in B of m along any morphism belongs to M. In this paper, this class of morphisms is shown to have an important rôle concerning some nice behaviours of the reflective hull of a topological space.

In several reflective subcategories of Top, it is possible to select a subcategory A such that P_{B}(M) = A^{Inj^{B}}, where A^{Inj^{B}} denotes the class of all morphisms m of B such that each A ∈ A is m-injective, that is, the map Hom(m, A) : Hom(Y, A) → Hom(X, A) is surjective. Under smooth conditions, this fact implies that the reflective hull of A is just the smallest M-reflective subcategory of the epireflective hull of A. We recall that an M-reflective subcategory is a reflective subcategory whose corresponding reflections belong to M. The most interesting case, which is rather common, is when A is chosen as one-object subcategory. In fact, if A is a topological space for which P_{B}(M) consists of all morphisms m such that A is m-injective, then, it often holds that the dual category of the reflective hull of A in Top is a reflective subcategory of Set^{𝕋} for 𝕋 the monad induced by Hom(−, A). Furthermore, under convenient requisites, it is equivalent to Set^{𝕋}.

As very helpful tools, two Dikranjan-Giuli closure operators will be used: the regular closure operator (introduced in [8, 2]) and the orthogonal closure operator (introduced in [10]). We shall see that the equality P_{B}(M) = A^{Inj^{B}}, combined with mild conditions, implies that the two closure operators coincide in P_{B}(M), and this plays an important rôle in the main result of this paper, Theorem 3.1.

1. Injectivity and pushout-stability

Let A be a topological space and let f : X → Y be a morphism in Top. We say that A is f-injective (respectively, orthogonal to f) whenever the map Hom(f, A) : Hom(Y, A) → Hom(X, A) is surjective (respectively, bijective). For A a class of spaces, that is, a full subcategory of Top, we denote by A^{Inj} the class of those morphisms f such that all objects of A are f-injective. In case A has just one object A, we write A^{Inj}. Analogously, A^{⊥} is the class of morphisms f such that any object of A is orthogonal to f. Given a class N of morphisms, the subcategory of n-injective objects (respectively, objects orthogonal to n), for all n ∈ N, is named N_{Inj} (respectively, N_{⊥}).

∗Along this paper we assume that all subcategories are full.

Let B be a subcategory of Top. We denote by A^{Inj^{B}} the class A^{Inj} restricted to B-morphisms. P_{B}(M) designates the class of all M-morphisms whose pushout in B along any morphism exists and belongs to M.

Throughout this paper, unless something is said on the contrary, M denotes the class of embeddings of Top.

An interesting feature, capturable in various reflective subcategories B of Top, is that the class P_{B}(M) coincides with A^{Inj^{B}} for some subcategory A, in many cases this subcategory A having just one object. Next, some examples of this occurrence are given.

Examples 1.1.

(1) Since in Top embeddings are pushout-stable, we have the equality P_{Top}(M) = M. Moreover, M = A^{Inj} for A the topological space {0, 1, 2} whose only non trivial open is {0}.

(2) For the subcategory Top_{0} of T_{0}-spaces, again P_{Top_{0}}(M) = M. In this case, it holds M = S^{Inj^{Top_{0}}} , where S is the Sierpiński space.

(3) For B, the subcategory of 0-dimensional spaces, M ≠ P_{B}(M), but again P_{B}(M) = A^{Inj^{B}}, where A is the space {0, 1, 2} whose topology has as only non trivial opens {0} and {1, 2}. The morphisms of P_{B}(M) are just those embeddings m : X → Y such that for each clopen set G of X there is some clopen H in Y such that G = m^{−1}(H) (cf. [11]).

(4) If B is the subcategory of 0-dimensional Hausdorff spaces, M ≠ P_{B}(M) = D^{Inj^{B}}, where D is the space {0, 1} with the discrete topology. Analogously to the above example, the morphisms of P_{B}(M) are those embeddings m : X → Y in B such that each clopen in X is the inverse image by m of some clopen in Y (cf. [11]).

(5) For T ych the subcategory of Tychonoff spaces, M ≠ P_{T ych}(M) = I^{Inj^{T ych}}, where I is the unit interval. The P_{T ych}(M)-morphisms are just the C^{∗}-embeddings (cf. [11]).

(6) In the category Ind of indiscrete spaces embeddings are pushoutstable, and we have that P_{Ind}(M) = M = {0, 1}^{Inj^{Ind}}.

Examples 1.2. This phenomenon arises in several other categories, not just reflective subcategories of Top. Some examples are given in the following.

(1) For Ab the category of abelian groups and homomorphisms of group, and M the class of all monomorphisms, P (M) = M = A^{Inj}, where A is the subcategory of divisible abelian groups. Similarly, in the category of torsion-free abelian groups and homomorphisms of group, and M the class of all monomorphisms, P (M) = M = A^{Inj}, where A is the subcategory of divisible groups.

(2) For Vec_{𝕂} the category of linear spaces and linear maps over 𝕂, and M the class of all monomorphisms, P (M) = M = 𝕂^{Inj}.

(3) Recall that a separated quasi-metric space is a pair (X, d), where X is a set and d is a function with domain X × X which satisfies the same axioms as a metric in X, but d may take the value ∞. A separated quasi-metric space is said to be complete if all its Cauchy sequences converge. A function f between two separated quasi-metric spaces (X, d) and (Y, e) is non-expansive provided that e(f(x), f(y)) ≤ d(x, y) for all (x, y) ∈ X × X. Let SMet denote the category of separated quasi-metric spaces and non-expansive maps and let CSMet denote its subcategory of complete separated quasimetric spaces. Then, for M the class of all non-expansive monomorphisms, M is pushout-stable, so it coincides with P (M), and it holds M = CSMet^{Inj}.

(4) For the category Met of metric spaces and non-expansive maps and M defined similarly as in 3, we have that

P (M) = {m ∈ M| the pushout of m along any morphism exists}

and it coincides with CMet^{Inj}, where CMet denotes the subcategory of complete metric spaces. An analogous situation holds for the category of normed spaces and non-expansive maps and the subcategory of Banach spaces.

One first question which comes from these examples is the following: When is P_{B}(M) of the form A^{Inj^{B}} for some subcategory A of a given category, in particular, for A a subcategory of Top? The next proposition is a partial answer for this.

Let us recall that, if N is a class of morphisms in a category B, containing all isomorphisms and being closed under composition with isomorphisms, we say that B has enough N -injectives provided that, for each B ∈ B, there is an N -morphism n : B → A with the codomain A in N_{Inj}. A morphism n : B → A of N is said to be N -essential whenever any composition k · n belongs to N only if k ∈ N. The category B is said to have N -injective hulls if, for each B ∈ B, there is some N -essential morphism n : B → A with A N -injective.

The following easy lemma to prove will be useful in this paper.

Lemma 1.1. For any epireflective subcategory B of Top, the class P_{B}(M) has the following properties:

(i) It is closed under composition. (ii) If m · n ∈ P_{B}(M) then n ∈ P_{B}(M).

Proposition 1.1. Let B be an epireflective subcategory of Top.

(1) If B has enough P_{B}(M)-injectives, then P_{B}(M) = A^{Inj^{B}} for some subcategory A of B.

(2) If P_{B}(M) = A^{Inj^{B}} for some M-reflective subcategory A of B, then B has P_{B}(M)-injective hulls, and [P_{B}(M)]_{Inj_{B}} = A.

Proof. The fact that B is an epireflective subcategory of Top ensures that B has pushouts and an (Epi, Embedding)-factorization system for morphisms.

(1). Let A = [P_{B}(M)]_{Inj_{B}}; it is clear that P_{B}(M) ⊆ A^{Inj^{B}}. In order to show the converse inclusion, let f : X → Y belong to A^{Inj^{B}} and let f = me be the (Epi, Embedding)-factorization of f. By hypothesis, there is a P_{B}(M)- morphism n : X → A with A ∈ [P_{B}(M)]_{Inj_{B}}. Let \overline{n} : Y → A be such that \overline{n}f = n; then the equality n1_{X} = (\overline{n}m)e guarantees the existensce of a unique morphism t such that te = 1_{X} and, consequently, e is an isomorphism and f ∈ M. Now, to conclude that f : X → Y belongs to P_{B}(M), let

[figure: X; f; Y; Z; g; \overline{f}; W; \overline{g}]

be a pushout in B. Since B has enough P_{B}(M)-injectives, there is some P_{B}(M)-morphism p : Z → A, with A ∈ B being P_{B}(M)-injective. Let h : Y → A be such that hf = pg; then, there is t such that t\overline{f} = p and so the fact that p ∈ M implies that also \overline{f} ∈ M.

(2). Let us first show that each reflection r_{B} : B → RB of B ∈ B in A belongs to P_{B}(M). Let (\overline{f}, \overline{r}) be the pushout of (r_{B}, f) for some morphism f : B → C. Since Rf · r_{B} = r_{C} · f, there is a unique t such that t · \overline{r} = r_{c} and t · \overline{f} = Rf. By hypothesis, r_{C} ∈ M, and this implies that \overline{r} ∈ M. Then r_{B} ∈ P_{B}(M).

To show that r_{B} is P_{B}(M)-essential, let k : RB → K be a morphism such that k · r_{B} belongs to P_{B}(M). It is easy to see that the fact that A is M-reflective in B implies that each reflection in A is an epimorphism in B. Consequently, for any morphism f : RB → D, (1_{D}, f) is a pushout of (r_{B}, f · r_{B}). Now, let f : RB → D be a morphism and let (\overline{k}, \overline{f}) be the pushout of (k, f). Then, since (1_{D}, f) is a pushout of (r_{B}, f · r_{B}), \overline{k} is the pushout of k · r_{B} along f · r_{B}. Thus, \overline{k} belongs to M and, therefore, k ∈ P_{B}(M).

In order to conclude that [P_{B}(M)]_{Inj_{B}} = A, since the inclusion A ⊆ [P_{B}(M)]_{Inj_{B}} is clear, it remains to show the reverse inclusion. Let B ∈ [P_{B}(M)]_{Inj_{B}}; the fact that the reflection r_{B} : B → RB belongs to P_{B}(M) implies that there is some t : RB → B such that t·r_{B} = 1_{B}, and this implies that r_{B} is an isomorphism. □

Given a subcategory A of Top, let R(A) denote the reflective hull of A in Top, provided it exists. It is well known that, when A is small, R(A) exists and coincides with the limit closure of A.

The subcategories A such that P_{B}(M) = A^{Inj^{B}}, for B its epireflective hull, have a very special property, as established next.

Proposition 1.2. Let B be the epireflective hull of A in Top. If P_{B}(M) = A^{Inj^{B}}, then R(A) is the smallest M-reflective subcategory of B (provided R(A) exists).

Proof. Since B is the epireflective hull of A, it is clear that R(A) is Mreflective in B (cf. [1]). Let C be another M-reflective subcategory of B. Then, as shown in the proof of 2 of Proposition 1.1, each reflection s_{B} : B → SB of B ∈ B into C belongs to P_{B}(M). Therefore, given B ∈ A, since A ⊆ [P_{B}(M)]_{Inj_{B}}, there is some t : SB → B such that t · s_{B} = 1_{B}, and so s_{B} is an isomorphism. Consequently, A ⊆ C and, thus, R(A) ⊆ C. □

2. On closure operators

In this section we recall some notions concerning closure operators, in the sense of Dikranjan-Giuli ([2, 3, 4]), and obtain some results, which are going to be useful in the next section.

Let N be a class of monomorphisms in a category X, which contains all isomorphisms, is closed under composition and is left-cancellable, (i.e., if m · n, m ∈ N then n ∈ N). Considering N as a subcategory of the category X^{2} (of all morphisms of X), let u : N → X be the codomain functor, that is, the functor which, to each morphism (r, s) : (m : X → Y) ⟶ (n : Z → W) of N, assigns the morphism s : Y → W of X.

A closure operator in X with respect to N consists of a functor c : N → N such that u · c = u and of a natural transformation δ : Id_{N} → c such that u · δ = Id_{u}. Therefore, a closure operator determines, for each m : X → Y in N, morphisms c(m) and d(m) and a commutative diagram

[figure: X; d(m); \overline{X}; Y; m; 1_{Y}; Y; c(m)]

where δ_{m} = (d(m), 1_{Y}) : m → c(m).

We recall that, if N is part of an (E, N)-factorization system for morphisms, then a closure operator c : N → N may be equivalently described by a family of functions

(c_{X} : N_{X} → N_{X})_{X∈X} ,

where c_{X}(m) = c(m) for each m, and N_{X} denotes the class of all N - morphisms with codomain in X, fulfilling the conditions:

(1) m ≤ c_{X}(m), m ∈ N_{X};

(2) if m ≤ n, then c_{X}(m) ≤ c_{X}(n), m, n ∈ N_{X}; and

(3) c_{X}(f^{−1}(m)) ≤ f^{−1}(c_{Y} (m)), for each morphism f : X → Y and m ∈ N_{Y}.

If c is a closure operator in X with respect to N, then a morphism (n : X → Y) ∈ N is said to be c-dense if c(n) ≅ 1_{Y}. It is said to be c-closed when c(n) ≅ n. We say that c is weakly hereditary, if d(n) is c-dense for any n ∈ N, and idempotent if c(n) is c-closed for any n ∈ N. An object X ∈ X is said to be absolutely c-closed whenever each N -morphism with domain in X is c-closed.

The regular closure operator. Let A be a reflective subcategory of Top. The regular closure operator induced by A, which will be denoted by c_{A}^{reg} : M → M (just c^{reg}, without the index A, when there is no confusion about the subcategory A which is involved), assigns, to each m : X → Y in M, the equalizer eq(u · r_{W} , v · r_{W}) where u, v : Y → W form the cokernel pair of m and r_{W} is a reflection in A. The regular closure operator, which is an idempotent closure operator, in the sense described above, has been widely studied, not only in the topological context, but in a more general setting (see [2]).

It is easy to conclude that if B is the epireflective hull of A in Top (or, equivalently, A is M-reflective in B), then c_{A}^{reg} = c_{B}^{reg}.

From now on, given a subcategory A of Top, E(A) always denotes the epireflective hull of A in Top.

The orthogonal closure operator. Let A be a reflective subcategory in its epireflective hull E(A) in Top. Let r_{X} : X → RX be the reflection of X in A. To each m : X → Y in P_{E(A)}(M), we assign the M-morphism c_{Y}^{ort}(m) : \overline{X} → Y which is the pullback of the pushout, in E(A), of m along r_{X}, according to the next diagram.

X  \overline{m} Y

[figure: c_{Y}^{ort}(m)]

\overline{X}

[figure]

[figure: RX; r_{X}; \overline{m}; •; \overline{r}]

Using results of [10], we get the next theorem.

Theorem 2.1. For every reflective subcategory A of Top, if c_{Y}^{ort}(m) : \overline{X} → Y belongs to P_{E(A)}(M) for all m ∈ P_{E(A)}(M), then

c_{A}^{ort} : P_{E(A)}(M) ⟶ P_{E(A)}(M)

defines a weakly hereditary idempotent closure operator in E(A), and A is just the subcategory of all absolutely c_{A}^{ort}-closed objects.

As expressed for c_{A}^{reg} before, we remove A from c_{A}^{ort} if there is no reason for ambiguity.

We are going to make use of the following definition.

Definition 2.1. We say that epimorphisms in a category B are left-cancell-able with respect to M provided that q ∈ Epi(B) whenever p · q ∈ Epi(B) with p, q ∈ M.

Corollary 2.1. If A is a reflective subcategory of Top such that, in E(A), epimorphisms are left-cancellable with respect to M, then

c_{A}^{ort} : P_{E(A)}(M) ⟶ P_{E(A)}(M)

defines a weakly hereditary idempotent closure operator and A is just the subcategory of all absolutely c_{A}^{ort}-closed objects.

Proof. It follows from the above proposition and the fact that the leftcancellability of epimorphisms in E(A) with respect to M implies that c_{Y}^{ort}(m) : \overline{X} → Y belongs to P_{E(A)}(M) for all m ∈ P_{E(A)}(M), as it is shown in Remark 4.4 of [10]. □

Lemma 2.1. In any reflective subcategory of Top where embeddings are pushout-stable, epimorphisms are left-cancellable with respect to embeddings.

Proof. First, we point out the following fact: In any category with pushouts, a morphism f is an epimorphism iff (f, f) has a pushout of the form (f′, f′) for some morphism f′; in this case, f′ is an isomorphism.

Let B be a reflective subcategory of Top and let m : X → Y and n : Y → Z be M-morphisms of B such that nm is an epimorphism. Consider the following diagram, where the smallest four squares are pushouts in B.

[figure: X; m; Y; n; Z]

[figure: Y; m; v; •; u; r; •; a]

[figure: Z; n; c; •; s; d; •; b]

Then b · a = d · c and we have that

(br)u = ban = dcn = dsv = (br)v.

Since br is the composition of two embeddings, it is an embedding, and thus u = v; consequently, m is an epimorphism. □

Remark 2.1. The above lemma provides several examples of subcategories of Top where epimorphisms are left-cancellable with respect to embeddings. This occurrence is far from to confine to subcategories under the condition of Lemma 2.1. In fact, it is a very common property in epireflective subcategories of Top. This is the case, for instance, for T_{0}-spaces, T_{1}-spaces, Hausdorff spaces, completely regular Hausdorff spaces, compact Hausdorff spaces, compact 0-dimensional Hausdorff spaces. The left-cancellability of epimorphisms with respect to embeddings, in these categories, is easy to conclude, taking into account that in each one of them epimorphisms are the surjective continuous maps or the dense continuous maps (cf. [4, 7]).

We are going to make use of the following lemma.

Lemma 2.2 (cf. [10]). Let A be a subcategory of Top. Then, in E(A),

A^{⊥} = ((A^{⊥})_{⊥})^{⊥}
= {c_{A}^{ort}-dense P_{E(A)}-morphisms}
⊆ P_{E(A)}(M) ∩ Epi(E(A)).

We point out that, although the existence of the reflective hull of any subcategory of Top cannot be guaranteed ([12]), we can state the following:

Proposition 2.1. Let A be a subcategory of Top. If R(A) exists, then it coincides with (A^{⊥})_{⊥}.

Proof. From Proposition 3.1.2 and Theorem 4.1.3 of [5], one derives that if X is a category fulfilling the following conditions:

• it is complete, cocomplete and co-wellpowered;

• it has an (E, M)-factorization system for morphisms with E = Epi(X);

• it has a generator;

• for every countable family (m_{i} : C_{i} → B)_{i∈ω} of M-subobjects of B ∈ X, and any morphism g with codomain in B, it holds the equality ⋁_{i∈ω} g^{−1}(m_{i}) = g^{−1}(⋁_{i∈ω} m_{i});

then, for each morphism f of X, the subcategory {f}_{⊥} is reflective. The category Top is under these conditions for M the class of all embeddings. Consequently, given a subcategory A of Top which has a reflective hull, we have the inclusions:

R(A) ⊆ ⋂_{f∈A^{⊥}}{f}_{⊥}
= (⋃_{f∈A^{⊥}}{f})_{⊥}
= (A^{⊥})_{⊥} .

Therefore, since the inclusion (A^{⊥})_{⊥} ⊆ R(A) always occurs, we conclude that R(A) = (A^{⊥})_{⊥}. □

The next proposition establishes conditions under which the closure operators c_{A}^{ort} and c_{A}^{reg} coincide in P_{E(A)}(M). This result is going to be very useful in the proof of the theorem of the next section.

Proposition 2.2. Let A be a subcategory of Top with reflective hull. If epimorphisms in E(A) are left-cancellable with respect to M, c_{A}^{reg} is weakly hereditary in E(A) and P_{E(A)}(M) = A^{Inj^{E(A)}}, then

c_{R(A)}^{reg} = c_{R(A)}^{ort} in P_{E(A)}(M).

Proof. In order to simplify the writing, let A^{Inj} and A^{⊥} refer to just morphisms in E(A). It is easy to conclude that the fact that E(A) is the epireflective hull of A implies that

A^{⊥} ⊆ P_{E(A)}(M) ⋂ Epi(E(A))

(see Lemma 2.2). On the other hand, it is clear that

A^{Inj} ⋂ Epi(E(A)) ⊆ A^{⊥}.

Therefore, we have that

A^{⊥} ⊆ P_{E(A)}(M) ⋂ Epi(E(A))
= A^{Inj} ⋂ Epi(E(A))
⊆ A^{⊥},

and so,

(1)  A^{⊥} = P_{E(A)}(M) ⋂ Epi(E(A)).

Furthermore, from Lemma 2.2 and Proposition 2.1,

(2)

A^{⊥} = ((A^{⊥})_{⊥})^{⊥}
= (R(A))^{⊥}
= {c_{R(A)}^{ort}-dense P_{E(A)}-morphisms}.

It is well-known that the c_{R(A)}^{reg}-dense morphisms are just the epimorphisms of E(A) (see, for instance, [4]). Consequently, using (1) and (2), we get that a morphism of P_{E(A)}(M) is c_{R(A)}^{reg}-dense iff it is c_{R(A)}^{ort}-dense.

Let now m be a morphism in P_{E(A)}(M). Since the orthogonal closure operator is always smaller than the regular one (cf. [10]), there is some morphism d such that c_{R(A)}^{ort}(m) = c_{R(A)}^{reg}(m) · d. Let d′ be the morphism such that m = c_{R(A)}^{ort}(m)·d′. Since c_{R(A)}^{reg} is weakly hereditary, the morphism d · d′ is c_{R(A)}^{reg}-dense, and, thus, also c_{R(A)}^{ort}-dense. Since c_{R(A)}^{ort} is a weakly hereditary idempotent closure operator, it follows that it has a (c_{R(A)}^{reg}-dense, c_{R(A)}^{reg}-closed)-factorization system with respect to P_{E(A)}(M) ([3]) and so, the equality

c_{R(A)}^{reg}(m) · (d · d′) = c_{R(A)}^{ort}(m) · d′ · 1_{X}

implies the existence of a morphism t such that t · d · d′ = d′. Consequently, d is an isomorphism and c_{R(A)}^{reg}(m) ≅ c_{R(A)}^{ort}(m). □

3. Injectivity and categories of algebras

Let A, R(A) and E(A) be a topological space, the reflective hull of A and the epireflective hull of A in Top, respectively. It is well-known that the functor U = Hom(−, A) : Top^{op} → Set is a right adjoint whose left adjoint is given by the power functor A^{−}. Let K′ be the corresponding comparison functor and let R be the reflection functor from Top to R(A). It is known that the restriction of U to R(A) is also a right adjoint and induces the same monad as U. Furthermore, if K is the comparison functor concerning to the last adjunction, then, up to isomorphism, K′ = K ·R^{op}, as illustrated by the following diagram.

[figure: Topop; Rop; R(A)op; K; Set^{𝕋}]

[figure: U = Hom(−, A); U^{𝕋}]

Set

We are going to show that, under a convenient injectivity assumption on A, R(A)^{op} is a reflective subcategory of Set^{𝕋}. We also give conditions under which K is an equivalence.

We are going to make use of the following result:

Split Monadicity Theorem under presence of coequalizers (G. Janelidze, private communication). Let A and X be categories with coequalizers. A functor U : A → X is monadic if the following conditions hold:

(a) U has a left adjoint F;

(b) U reflects isomorphisms;

(c) the counit F U → 1_{A} is a split epimorphism.

Theorem 3.1. Let A be a topological space such that epimorphisms in E(A) are left-cancellable with respect to M and c_{R(A)}^{reg} is weakly hereditary in E(A).

Then, if P_{E(A)}(M) = A^{Inj^{E(A)}}, the comparison functor K is full and faithful and R(A)^{op} is equivalent to a reflective subcategory of the Eilenberg-Moore category Set^{𝕋}.

Furthermore, if P_{E(A)}(M) = (R(A))^{Inj^{E(A)}}, then K is an equivalence and so R(A)^{op} is monadic.

Proof. Under the stated conditions, Proposition 2.2 ensures that the closure operators c_{R(A)}^{reg} and c_{R(A)}^{ort} coincide in P_{E(A)}.

Since Hom(−, A) is a right adjoint and R(A)^{op} has coequalizers (because it is a full reflective subcategory of Top^{op}), we know that K is a right adjoint. In order to show that K is full and faithful, it suffices to prove that the counits of the adjunction Hom(−, A) are regular monomorphisms in R(A). For each B ∈ R(A), the co-unit is given by ε_{B} : B → A^{Hom(B,A)}, defined as the unique morphism such that π_{g} · ε_{B} = g for each g ∈ Hom(B, A), where π_{g} is the corresponding projection. Since ε_{B} clearly belongs to A^{Inj^{E(A)}}, it belongs to P_{E(A)}(M). But, by Theorem 2.1, B is an absolutely c_{R(A)}^{ort}-closed object. Consequently, ϵ_{B} is c_{R(A)}^{ort}-closed and, since c_{R(A)}^{reg} and c_{R(A)}^{ort} coincide in P_{E(A)}, ε_{B} is c_{R(A)}^{reg}-closed, so it is a regular monomorphism.

In order to prove the second part of the theorem, let

P_{E(A)}(M) = R(A)^{Inj^{E(A)}}.

Then each P_{E(A)}(M)-morphism with domain in R(A) is a split monomorphism. In particular, for each B ∈ R(A), the morphism ε_{B} : B → A^{Hom(B,A)} is a split monomorphism. Thus, using the Split Monadicity Theorem, to conclude that Hom(−, A) : R(A)^{op} → Set is monadic, it remains to show that it reflects isomorphisms. Let f : B → C be such that Hom(f, A) : Hom(C, A) → Hom(B, A) is a bijective function. This means that f ∈ A^{⊥}. But

A^{⊥} = (R(A))^{⊥} ⊆ R(A)^{Inj_{E}(A)} ∩ Epi(R(A)).

Thus f is a split monomorphism, because it is a morphism of R(A)^{Inj_{E}(A)} with domain in R(A), and it is an epimorphism in R(A); therefore it is an isomorphism. □

Remark 3.1. We point out that the equality P_{E(A)}(M) = R(A)^{Inj^{E(A)}} implies that the reflective hull of A is just the subcategory of all retracts of powers of A. Furthermore, the converse implication is true, whenever P_{E(A)}(M) = A^{Inj^{E(A)}}.

As an immediate consequence of Theorem 3.1, we get Corollary below, which was proved by M. Sobral in [9]. We need to make use of the following lemma.

Lemma 3.1. If A is a subcategory of Top such that A^{Inj^{E(A)}} coincides with the class of embeddings in E(A), then, in E(A), embeddings are pushoutstable and c_{R(A)}^{reg} is weakly hereditary.

Proof. It is well-known that each object in E(A) is the domain of some initial monosource with codomain in A (see [1]). This fact, combined with the assumption of the lemma on A, is easily seen to imply that embeddings are pushout-stable in E(A).

In order to show that c_{R(A)}^{reg} (which coincides with c_{E(A)}^{reg}) is weakly hereditary in E(A), let m : X → Y be an embedding in E(A) and let (u, v) be the cokernel pair of m in E(A). Then \overline{m} = c_{R(A)}^{reg}(m) = eq(u, v). Let d be the morphism such that m = \overline{m} · d. We want to show that d is an epimorphism. Let f and g be morphisms such that fd = gd. As E(A) is the epireflective hull of A, we may assume, without loss of generality, that the codomain of f and g belongs to A. Since \overline{m} ∈ A^{Inj(E(A))}, there are morphisms \overline{f} and \overline{g} such that \overline{f} ·\overline{m} = f and \overline{g} ·\overline{m} = g. Then, we get \overline{f} ·m = \overline{f} ·\overline{m}·d = \overline{g} ·\overline{m}·d = \overline{g} ·m, what implies the existence of a morphism t such that \overline{f} = t · u and \overline{g} = t · v. Consequently, f = \overline{f} · \overline{m} = t · u · \overline{m} = t · v · \overline{m} = \overline{g} · \overline{m} = g. □

Corollary 3.1 ([9]). Let A be an M-injective topological space. Then the comparison functor K is full and faithful and R(A)^{op} is equivalent to a reflective subcategory of the Eilenberg-Moore category Set^{𝕋}.

Proof. The fact that A is M-injective implies that M = A^{Inj^{E(A)}}. Thus, by Lemma 3.1, P_{E(A)}(M) is just the class of all embeddings in E(A), and, from Lemma 2.1, this implies that epimorphisms in E(A) are left-cancellable with respect to M. Furthermore, Lemma 3.1 assures that c_{R(A)}^{reg} is weakly hereditary. Therefore, we are under the conditions of Theorem 3.1. □

Examples 3.1.

(1) The reflective hull of the topological space

A = ({0, 1, 2}, {∅, {0}, {1, 2}, {0, 1, 2}})

is Top (cf. [6]). Since in Top embeddings are pushout-stable and A is M-injective, it follows, from Theorem 3.1, that the dual of Top is a full reflective subcategory of the Eilenberg-Moore category induced by the right adjoint Hom(−, A) : Top^{op} → Set.

(2) Analogously for Top_{0}, we have that embeddings are pushout-stable and the Sierpiński space S is M-injective, so that, since the category Sob of sober spaces is the reflective hull of S, the dual category of Sob is a reflective subcategory of the Eilenberg-Moore category induced by the right adjoint Hom(−, S).

(3) In the category 0-Top_{2} of 0-dimensional Hausdorff spaces, epimorphisms are just the dense continuous maps, that is, the continuous maps f : X → Y such that, for each y and each clopen C in Y, it holds that y ∈ C ⇒ C ∩ f(X) ≠ ∅. Thus, it is clear that, in 0-Top_{2}, epimorphisms are left-cancellable with respect to M. On the other hand, given a subspace X of Y in 0-Top_{2}, being m : X → Y the corresponding embedding, it is easily seen that the regular closure of X in Y is just the intersection of all clopens which contain X, and that it is just the Kuratowski closure of X in Y. Therefore the morphism, which composed with c^{reg}(m) is equal to m, is dense, so an epimorphism; that is, c^{reg} is weakly hereditary. As seen in 1.1.4, P = D^{Inj}. The reflective hull of D is the category of 0dimensional compact Hausdorff spaces (cf. [6]). Therefore, applying the corollary of Theorem 3.1, we conclude that the opposite category of 0-dimensional compact Hausdorff spaces is a full reflective subcategory of an Eilenberg-Moore category.

(4) Another topological space which fulfils the conditions of Theorem 3.1 is the unit interval I. The reflective hull of I is the category of compact Hausdorff spaces, and its epireflective hull is the category T ych of Tychonoff spaces. As mentioned in 1.1.5, P_{T ych}(M) = I^{Inj^{T ych}}. The regular closure operator c_{R(I)}^{reg} in T ych coincides with the Kuratowski closure (cf. [4]); consequently it is weakly hereditary, and epimorphisms are dense morphisms, which are left-cancellabble with respect to M.

(5) The category Ind of indiscrete spaces is the epireflective hull and the reflective hull of the indiscrete space {0, 1} (cf. [6]). It holds P_{Ind}(M) = M = {0, 1}^{Inj_{I}nd} = R({0,)^{Inj_{I}nd}. Then Ind^{op} is equivalent to the Eilenberg-Moore category induced by the two-points indiscrete space.

References

[1] Jiří Adámek, Horst Herrlich, and George E. Strecker, Abstract and concrete categories, John Wiley & Sons Inc., New York, 1990. MR 91h:18001

[2] Dikran Dikranjan and Eraldo Giuli, Closure operators. I, Topology Appl. 27 (1987), no. 2, 129–143. MR 89e:18002

[3] Dikran Dikranjan, Eraldo Giuli, and Walter Tholen, Closure operators. II, Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988), World Sci. Publishing, Teaneck, NJ, 1989, pp. 297–335. MR 91e:18002

[4] Dikran Dikranjan and Walter Tholen, Categorical structure of closure operators, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 97i:18004

[5] P. J. Freyd and G. M. Kelly, Categories of continuous functors. I, J. Pure Appl. Algebra 2 (1972), 169–191. MR 48 #369

[6] Rudolf-E. Hoffmann, Reflective hulls of finite topological spaces, Arch. Math. (Basel) 33 (1979/80), no. 3, 258–262. MR 82f:54015

[7] E. W. Kiss, L. Márki, P. Pröhle, and W. Tholen, Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity, Studia Sci. Math. Hungar. 18 (1982), no. 1, 79–140. MR 85k:18003

[8] Sergio Salbany, Reflective subcategories and closure operators, Categorical topology (Proc. Conf., Mannheim, 1975), Springer, Berlin, 1976, pp. 548–565. Lecture Notes in Math., Vol. 540. MR 56 #9473

[9] Manuela Sobral, Absolutely closed spaces and categories of algebras, Portugal. Math. 47 (1990), no. 4, 341–351. MR 91m:18009

[10] Lurdes Sousa, Orthogonality and closure operators, Cahiers Topologie Géom. Différentielle Catég. 36 (1995), no. 4, 323–343. MR 97c:18004

[11] , Reflective and orthogonal hulls, Ph.D. thesis, Universidade de Coimbra, 1997.

[12] Věra Trnková, Jiří Adámek, and Jiří Rosický, Topological reflections revisited, Proc. Amer. Math. Soc. 108 (1990), no. 3, 605–612. MR 90e:18004

Departamento de Matemática, Escola Superior de Tecnologia, Instituto Politécnico de Viseu, Repeses 3500 Viseu, Portugal

E-mail address: sousa@mat.estv.ipv.pt