Theory and Applications of Categories, Vol. 30, No. 34, 2015, pp. 1121–1131.

A NOTE ON TRANSPORT OF ALGEBRAIC STRUCTURES

HENRIK HOLM

Abstract. We study transport of algebraic structures and prove a theorem which subsumes results of Comfort and Ross on topological group structures on Stone-Čech compactifications, of Chevalley and of Gil de Lamadrid and Jans on topological group and ring structures on universal covering spaces, and of Gleason on topological group structures on universal locally connected refinements.

1. Introduction

Transport of algebraic structures—a concept that will be made precise in Section 3—is a well-known phenomenon. To illustrate what we have in mind, we mention some results from the literature, which have motivated this note.

(a) Let M be a metric space with completion M → M̂. Algebraic structures on M tend to be inherited by M̂. For example, let M = ℚ be equipped with the metric induced by the Euclidian norm | · | or the p-adic norm | · |_{p}. Then one has M̂ = ℝ (the real numbers) or M̂ = ℚ_{p} (the p-adic numbers). In either case, M is a ring^{1} in which addition and multiplication are continuous functions. As it is well-known, the completion M̂ can, in both of these cases, also be made into a ring with continuous addition and multiplication in such a way that M → M̂ becomes a ring homomorphism. This exemplifies that the algebraic structure of type “ring” ascends^{2} along the map M → M̂.

(b) Let X be a topological space with Stone-Čech compactification X → βX. [Comfort and Ross (1966), Thm. 4.1] showed that if X is a pseudocompact topological group, then βX admits a

We thank Prof. Anders Kock for useful comments and suggestions. Received by the editors 2015-04-28 and, in revised form, 2015-08-17. Transmitted by Anders Kock. Published on 2015-08-20. 2010 Mathematics Subject Classification: 03C05; 18C10. Key words and phrases: Algebraic structure; algebraic theory; completion of metric space; equational

class; Stone-Čech compactification; universal covering space; universal locally connected refinement. \begin{matrix}c \\ ◯ \end{matrix} Henrik Holm, 2015. Permission to copy for private use granted. 1 Of course, ℚ, ℝ and ℚ_{p} are even fields (p is a prime number), but a field is not an “algebraic

structure” in the sense discussed in this paper; see 2.1.

2 In this paper, we say, loosely speaking, that an algebraic structure of a given type (such as “group” or “ring”) ascends, respectively, descends, along a map f: X → Y (more precisely, along an arrow in some category) if every algebraic structure of that type on X, respectively, on Y, can be “transported” to Y (that is, in the direction of the arrow), respectively, to X (that is, against the direction of the arrow), in such a way that f: X → Y becomes a homomorphism of algebraic structures of the type in question. The precise definitions can be found in Section 3.

structure of a topological group in such a way that X → βX becomes a group homomorphism (and a homeomorphism onto its image). This illustrates that the algebraic structure of type “group” ascends along the map X → βX for certain types of spaces X.

(c) Let X be a topological space which has a universal covering space X̃ → X. [Chevalley (1946), Chap. II§8 Prop. 5] proved that if X is a topological group, then X̃ can be made into a topological group in such a way that X̃ → X becomes a group homomorphism. By similar methods, [Gil de Lamadrid and Jans (1957), Thm. 1] showed that if X is a topological ring, then X̃ can be equipped with the structure of a topological ring such that X̃ → X becomes a ring homomorphism. This illustrates that the algebraic structures of type “group” and “ring” descend along the map X̃ → X.

(d) Let X be a topological space and let X^{∗} → X be its universal locally connected refinement in the sense of [Gleason (1963), Thm. A]. In Thm. D in loc. cit. it is proved that if X is a topological group, then X^{∗} can be made into a topological group in such a way that the map X^{∗} → X becomes a group homomorphism. This illustrates that the algebraic structure of type “group” descends along the map X^{∗} → X.

The purpose of this note is to describe circumstances under which algebraic structures ascend or descend along certain types of morphisms and to give useful applications. Our main result, Theorem 3.4, is certainly not profound; it deals with an adjoint situation and its proof is completely formal. However, despite its simplicity, the result has several useful applications; some of them are collected in Theorem 1.1 below, which subsumes the classic results (a)–(d) mentioned above. We consider these applications to be the main content of this note, and they are our justification for presenting the details that lead to Theorem 3.4.

1.1. Theorem. The following assertions hold.

(a) Let M be any metric space with completion M̂. Every algebraic structure on M ascends uniquely along the canonical map M → M̂.

(b) Let X be any pseudocompact and locally compact topological space with Stone-Čech compactification βX. Every algebraic structure on X ascends uniquely along the canonical map X → βX.

(c) Let X be any pointed topological space which has a pointed universal covering space X̃. Every algebraic structure on Xdescends uniquely along the canonical map X̃ → X.

(d) Let X be any topological space with universal locally connected refinement X^{∗}. Every algebraic structure on X descends uniquely along the canonical map X^{∗} → X.

The paper is organized as follows: Section 2 contains a few preliminaries on universal algebra, algebraic structures, and algebraic theories. In Section 3 we prove our main result and in Section 4 we apply this result in various settings and thereby give a proof of Theorem 1.1 above.

2. Algebraic structures and algebraic theories

Algebraic structures are objects like groups and rings. In general, they are sets equipped with operations subject to identities. The language of universal algebra makes this precise, and we refer to e.g. [Burris and Sankappanavar (1981), Chap. II§10,11] for the relevant notions.

2.1. Let Σ be an algebraic signature and let 𝕀 be a set of identities of type Σ. An algebraic structure of type (Σ, 𝕀) (called a (Σ, 𝕀)-algebra in [Mac Lane (1971), Chap. V§6]) is a Σ-algebra that satisfies every identity in 𝕀. A morphism of such structures is a morphism of the underlying Σ-algebras. We write Struc(Σ, 𝕀) for the category of all algebraic structures of type (Σ, 𝕀) and U^{Σ,𝕀}: Struc(Σ, 𝕀) → Set for the forgetful functor. By an algebraic structure we just mean an algebraic structure of some type (Σ, 𝕀).

Note that a field, for example, is not an algebraic structure in the sense above. In the literature, the category Struc(Σ, 𝕀) is often referred to as a variety or an equational class.

By definition, an algebraic structure of type (Σ, 𝕀) is, in particular, a set (and the definition of what is means for a Σ-algebra to satisfy an identity refers specifically to elements). In other words, what is defined above is algebraic structures in the category Set. The standard way to deal with algebraic structures in more general categories goes through algebraic theories. The book [Adámek, Rosický, and Vitale (2011)] is an excellent account on algebraic theories, and we shall refer to this for relevant notions and results.

2.2. Following [Adámek, Rosický, and Vitale (2011), Chap. 1] an algebraic theory is a small category with finite products. If T is an algebraic theory, then a T -algebra is a functor T → Set that preserves finite products. A morphism of T -algebras is a natural transformation. The category of all T -algebras and their morphisms is denoted by Alg(T ).

2.3. Example. Let \underline{ℕ}_{0} be the category whose objects are natural numbers and zero and in which the hom-set \underline{ℕ}_{0}(m, n) consists of all functions \{0, …, m − 1\} → \{0, …, n − 1\}. This category has finite coproducts; indeed, the coproduct of objects m, n ∈ \underline{ℕ}_{0} is the sum m + n. Thus the opposite category \underline{ℕ}_{0}^{op} is an algebraic theory; we denote it by N as in [Adámek, Rosický, and Vitale (2011), Exa. 1.9]. In this category, every object n ∈ N is the n-fold product n ≅ 1 × ⋯ × 1 of the object 1 (this also holds for n = 0, as the empty product in a category yields the terminal object). In the cited example, it is also proved that the functor e: Alg(N) → Set given by A ↦ A(1) is an equivalence of categories. It is customary to suppress this functor.

2.4. Let C be a fixed category. Recall that a concrete category over C is a pair (U, U) where U is a category and U: U → C is a faithful functor. If (U, U) and (V, V) are concrete categories over C, then a concrete functor (U, U) → (V, V) is a functor F: U → V with V F = U. A concrete equivalence of concrete categories (U, U) and (V, V) over C is a pair of quasi-inverse concrete functors (U, U) ⇄ (V, V).

2.5. Following [Adámek, Rosický, and Vitale (2011), Def. 11.3] a one-sorted algebraic theory is a pair (T , T), where T is an algebraic theory whose objects are natural numbers and zero and T: N → T is a product preserving functor which is the identity of objects. This definition goes back to [Lawvere (1963)].

Let (T , T) be a one-sorted algebraic theory. As noted above, one has n ≅ 1 × ⋯ × 1 (n copies) in the category N. By applying the functor T (which is the identity on objects and preserves products) to this isomorphism, one gets that n ≅ 1×⋯×1 also holds in T .

2.6. For every one-sorted algebraic theory (T , T), the functor \xymatrix{Alg(T ) \ar[r]^{Alg(T)} & Alg(N) ≃ Set} is faithful by [Adámek, Rosický, and Vitale (2011), Prop. 11.8], and hence (Alg(T ), Alg(T)) is a concrete category over Set. The pair (Struc(Σ, 𝕀), U^{Σ,𝕀}) from 2.1 is also a concrete category over Set. It is proved in [Adámek, Rosický, and Vitale (2011), Thm. 13.11] that for every algebraic signature Σ and every set 𝕀 of identities of type Σ, there exists a one-sorted algebraic theory (T_{Σ,𝕀}, T_{Σ,𝕀}) and a concrete equivalence,

(1)  (Alg(T_{Σ,𝕀}), Alg(T_{Σ,𝕀})) ≃ (Struc(Σ, 𝕀), U^{Σ,𝕀}) .

Moreover, by [Adámek, Rosický, and Vitale (2011), Thm. 11.39] (which is called “onesorted algebraic duality”) such a one-sorted algebraic theory (T_{Σ,𝕀}, T_{Σ,𝕀}) is unique up to isomorphism of one-sorted algebraic theories.

As shown in [Adámek, Rosický, and Vitale (2011), Rmk. 11.24 and Chap. 13], the algebraic theory T_{Σ,𝕀} satisfying (1) can be constructed directly from the left adjoint of the forgetful functor U^{Σ,𝕀}.

Now, the standard way to define algebraic structures of type (Σ, 𝕀) in a general category (with finite products) is as follows.

2.7. Definition. Let T be an algebraic theory and let C be any category with finite products. A T -algebra in C is a functor T → C that preserves finite products. A morphism of T -algebras in C is a natural transformation. The category of all T -algebras in C and their morphisms is denoted by Alg_{C}(T ). Thus, Alg_{Set}(T ) is nothing but Alg(T ) from 2.2.

The next lemma is straightforward to prove (the proof is the same as for C = Set).

2.8. Lemma. Let C be a category with finite products. There is an equivalence of categories e_{C}: Alg_{C}(N) → C given by C̃ ↦ C̃(1).

2.9. Definition. Let Σ be an algebraic signature, let 𝕀 be a set of identities of type Σ, and let C be a category with finite products. Let (T_{Σ,𝕀}, T_{Σ,𝕀}) be the unique one-sorted algebraic theory for which there is a concrete equivalence (1). Define the category

Struc_{C}(Σ, 𝕀) := Alg_{C}(T_{Σ,𝕀})

and the forgetful functor U_{C}^{Σ,𝕀} as the composition \xymatrix{Alg_{C}(T_{Σ,𝕀}) \ar[r]^{Alg_{C}(T_{Σ,𝕀})} & Alg_{C}(N) \ar[r]^{e_{C}}_{≃} & C} .

We refer to an object in Struc_{C}(Σ, 𝕀) as an algebraic structure of type (Σ, 𝕀) in C (even though it is not actually an object in C, but a product preserving functor T_{Σ,𝕀} → C).

2.10. Remark. The action of U_{C}^{Σ,𝕀} on an object X̃ (i.e. a functor T_{Σ,𝕀} → C) is X̃(1) ∈ C, and the action of U_{C}^{Σ,𝕀} on a morphism σ̃: X̃ → Ỹ (i.e. a natural transformation) is σ̃_{1}.

As noted in 2.5 one has n ≅ 1 × ⋯ × 1 in T_{Σ,𝕀}. Thus, for any morphism σ̃: X̃ → Ỹ of T_{Σ,𝕀}-algebras in C one has σ̃_{n} ≅ σ̃_{1} × ⋯ × σ̃_{1}. This has two immediate consequences:

(a) The functor U_{C}^{Σ,𝕀} is faithful; thus (Struc_{C}(Σ, 𝕀), U_{C}^{Σ,𝕀}) is a concrete category over C.

(b) If U_{C}^{Σ,𝕀}(σ̃) is an isomorphism, then so is σ̃.

2.11. A priori an algebraic structure of type (Σ, 𝕀) in a category C is not actually an object in C, but instead a product preserving functor T_{Σ,𝕀} → C. However, it is possible to— and well-known that one can—interpret such a functor as an actual object in C equipped with some additional structure. For example, if Σ_{Grp} is the algebraic signature and 𝕀_{Grp} is the set of identities for groups, then the concrete category Struc_{C}(Σ_{Grp}, 𝕀_{Grp}) of algebraic structures of type (Σ_{Grp}, 𝕀_{Grp}) in C is concretely equivalent to the concrete category Grp(C) of group objects in C. We remind the reader that a group object in C is a quadruple (C, m, u, i) where C is an (actual) object in C and m, u, i are morphisms (where T is the terminal object in C):

C × C ⟶^{m} C (called multiplication)
T ⟶^{u} C (called unit)
C ⟶^{i} C (called inverse)

that make the expected diagrams commutative. For example, if C = Top is the category of topological spaces, then Grp(C) = Grp(Top) is the category of topological groups.

The category Struc_{C}(Σ_{Grp}, 𝕀_{Grp}) is convenient for working with group structures in C from a theoretical point of view, however, in specific examples (see Section 4) it is more natural to have the equivalent category Grp(C) in mind.

3. The main result

Throughout this section, we fix an algebraic signature Σ and a set 𝕀 of identities of type Σ. For a category C with finite products we consider the category Struc_{C}(Σ, 𝕀) of algebraic structures of type (Σ, 𝕀) in C and its forgetful functor U_{C}: Struc_{C}(Σ, 𝕀) → C from Def. 2.9.

3.1. Definition. By an algebraic structure of type (Σ, 𝕀) on an object C ∈ C we mean an object C̃ ∈ Struc_{C}(Σ, 𝕀) such that U_{C}(C̃) = C.

This is the definition we shall formally use. However, as illustrated in 2.11, one can think of an algebraic structure C̃ on an object C ∈ C as a pair C̃ = (C, {f_{σ}}_{σ∈Σ}) where {f_{σ}}_{σ∈Σ} is a collection of morphisms in C, determined by the signature Σ, that make certain diagrams, determined by the identities 𝕀, commutative.

Now suppose that F: C → D is a product preserving functor between categories with finite products. There is a commutative diagram,

\xymatrix{Alg_{C}(T_{Σ,𝕀}) \ar[d]_{Alg_{F} (T_{Σ,𝕀})} \ar[r]^{Alg_{C}(T_{Σ,𝕀})} & Alg_{C}(N) \ar[d]^{Alg_{F} (N)} \ar[r]^{e_{C}} & C \ar[d]^{F} \\ Alg_{D}(T_{Σ,𝕀}) \ar[r]^{Alg_{D}(T_{Σ,𝕀})} & Alg_{D}(N) \ar[r]^{e_{D}} & D,}

where the horizontal functors Alg_{C}(T_{Σ,𝕀}) and Alg_{D}(T_{Σ,𝕀}) map a functor X̃ to X̃ ∘ T_{Σ,𝕀}, and the vertical functors Alg_{F} (T_{Σ,𝕀}) and Alg_{F} (N) map a functor X̃ to F ∘ X̃. If we write F̃ for the functor Alg_{F} (T_{Σ,𝕀}), i.e. F̃(X̃) = F ∘ X̃, then the commutative diagram above is:

(2)  \xymatrix{Struc_{C}(Σ, 𝕀) \ar[r]^{U_{C}} \ar[d]_{F̃} & C \ar[d]^{F} \\ Struc_{D}(Σ, 𝕀) \ar[r]^{U_{D}} & D.}

Suppose that D̃ is an algebraic structure of type (Σ, 𝕀) on a given object D ∈ D. The commutative diagram (2) induces two functors between comma categories ([Mac Lane (1971), Chap. II§6]):

U D̃ ; F: (D̃ ↓ F̃) ⟶ (D ↓ F) given by (C̃, φ̃) ⟼ (U_{C}(C̃), U_{D}(φ̃)) , and
U F ; D̃: (F̃ ↓ D̃) ⟶ (F ↓ D) given by (C̃, ψ̃) ⟼ (U_{C}(C̃), U_{D}(ψ̃)) .

With these functors at hand, we can now make precise what is meant by transport (ascent and descent) of algebraic structures along morphisms.

3.2. Definition. Let D̃ be an algebraic structure of type (Σ, 𝕀) on a given object D ∈ D.

(a) Let C ∈ C be an object and let φ: D → F (C) be a morphism, i.e. (C, φ) ∈ (D ↓ F).

We say that the algebraic structure D̃ ascends along φ if there exists an algebraic structure C̃ of type (Σ, 𝕀) on C and a morphism φ̃: D̃ → F̃(C̃) in Struc_{D}(Σ, 𝕀) such that U_{D}(φ̃) = φ, that is, (C̃, φ̃) is an object in (D̃ ↓ F̃) with U D̃ ; F(C̃, φ̃) = (C, φ).

We say that the algebraic structure D̃ ascends uniquely along φ if there a unique, up to isomorphism, object (C̃, φ̃) in (D̃ ↓ F̃) with U D̃ ; F(C̃, φ̃) = (C, φ).

(b) Let C ∈ C be an object and let ψ: F (C) → D be a morphism, i.e. (C, ψ) ∈ (F ↓ D).

We say that the algebraic structure D̃ descends along ψ if there exists an algebraic structure C̃ of type (Σ, 𝕀) on C and a morphism ψ̃: F̃(C̃) → D̃ in Struc_{D}(Σ, 𝕀) such that U_{D}(ψ̃) = ψ, that is, (C̃, ψ̃) is an object in (F̃ ↓ D̃) with U F ; D̃(C̃, ψ̃) = (C, ψ).

We say that the algebraic structure D̃ descends uniquely along ψ if there a unique, up to isomorphism, object (C̃, ψ̃) in (F̃ ↓ D̃) with U F ; D̃(C̃, ψ̃) = (C, ψ).

3.3. Lemma. Let ⟨F, G, η, ε⟩: C → D be an adjunction of product preserving functors between categories with finite products^{3}. The induced functors F̃,G̃ (see the diagram (2)) are part of an adjunction

⟨F̃, G̃, η̃, ε̃⟩: Struc_{C}(Σ, 𝕀) ⟶ Struc_{D}(Σ, 𝕀)

with U_{C}(η̃C̃) = η_{U_{C}(C̃)} for C̃ ∈ Struc_{C}(Σ, 𝕀) and U_{D}(ε̃D̃) = ε_{U_{D}(D̃)} for D̃ ∈ Struc_{D}(Σ, 𝕀).

Proof. We use the shorthand notation T = T_{Σ,𝕀} and T = T_{Σ,𝕀}.

By definition, Struc_{X} (Σ, 𝕀) is a full subcategory of the functor category X^{T} (X = C, D). Thus, for all product preserving functors C̃: T → C and D̃: T → D we must construct a natural bijection (where we have written F C̃ = F ∘ C̃ and GD̃ = G ∘ D̃):

\xymatrix{D^{T}(F C̃, D̃) = D^{T}(F̃(C̃), D̃) \ar[r]^{α} & C^{T}(C̃, G̃(D̃)) = C^{T}(C̃, GD̃) \ar[l]^{β}} (where β = α^{−1}).

For a natural transformation σ: F C̃ → D̃ we set α(σ) = Gσ ∘ ηC̃, and for a natural transformation τ: C̃ → GD̃ we set β(τ) = εD̃ ∘ F τ. We then have

βα(σ) = β(Gσ ∘ ηC̃) = εD̃ ∘ F Gσ ∘ F ηC̃ = σ ∘ εF C̃ ∘ F ηC̃ = σ ∘ id_{F C̃} = σ ,

where the first and second equalities are by definition, the third equality follows as ε is a natural transformation, and the fourth equality follows as εF ∘ F η = id_{F}. Thus βα is the identity on D^{T}(F C̃, D̃), and a similar argument shows that αβ is the identity on C^{T}(C̃, GD̃).

The unit of this adjunction is η̃C̃ = α(id_{F̃(C̃)}) = ηC̃: C̃ → G̃F̃(C̃) = GF C̃. Thus, U_{C}(η̃C̃) is the morphism η_{C̃(1)} : C̃(1) → GF C̃(1), which is η_{U_{C}(C̃)} : U_{C}(C̃) → GF (U_{C}(C̃)); see Remark 2.10. A similar argument shows that U_{D}(ε̃D̃) = ε_{U_{D}(D̃)}.

3.4. Theorem. Let ⟨F, G, η, ε⟩: C → D be an adjunction of product preserving functors between categories with finite products. Let Σ be any algebraic signature and let 𝕀 be a set of identities of type Σ. The following conclusions hold.

(a) Let C ∈ C, set D = F (C) ∈ D, and consider the unit η_{C}: C → G(D). Every algebraic structure of type (Σ, 𝕀) on C ascends uniquely along η_{C}.

(b) Let D ∈ D, set C = G(D) ∈ C, and consider the counit ε_{D}: F (C) → D. Every algebraic structure of type (Σ, 𝕀) on D descends uniquely along ε_{D}.

3 Of course, G always preserves products since it is a right adjoint, so the assumption on the functors is really just that F preserves finite products.

Proof. We only prove part (a) since the proof of (b) is similar. In the proof, we use the notation of Lemma 3.3. Let C̃ be an algebraic structure of type (Σ, 𝕀) on C. Then D̃ = F̃(C̃) is an algebraic structure of type (Σ, 𝕀) on D as U_{D}(D̃) = U_{D}F̃(C̃) = F U_{C}(C̃) = F (C) = D. By Lemma 3.3, the unit η̃C̃: C̃ → G̃(D̃) has the property that U_{C}(η̃C̃) = η_{C}. This proves that the given algebraic structure C̃ ascends along η_{C}. To prove that it ascends uniquely, let D̃_{0} be any algebraic structure of type (Σ, 𝕀) on D and let φ̃: C̃ → G̃(D̃_{0}) be any morphism with U_{C}(φ̃) = η_{C}. As η̃C̃: C̃ → G̃(D̃) is a universal arrow from C̃ to G̃, that is, (D̃, η̃C̃) is the initial object in the comma category (C̃ ↓ G̃), there is a (unique) morphism δ̃: D̃ → D̃_{0} which makes the left diagram in the following display commutative:

\xymatrix{C̃ \ar[r]^{η̃C̃} \ar[d]_{φ̃} & G̃(D̃) \ar[dl]^{G̃(δ̃)} & C \ar[r]^{η_{C}} \ar[d]_{η_{C}} & G(D) \ar[dl]^{G(U_{D}(δ̃))} \\ G̃(D̃_{0}) &  & G(D)}

The right diagram above is obtained from the left one by applying the functor U_{C}. As the unit η_{C}: C → G(D) is a universal arrow from C to G, that is, (D, η_{C}) is the initial object in the comma category (C ↓ G), then the morphism U_{D}(δ̃) must be the identity id_{D} on D. It follows from Remark 2.10(b) that δ̃ is an isomorphism.

4. Applications

In this final section, we apply Theorem 3.4 in some specific examples and thereby give a proof of Theorem 1.1 from the Introduction.

4.1. Example. Let C = Met be the category whose objects are all metric spaces and whose morphisms are all continuous functions. This category has finite products, indeed, the product of metric spaces (M, d_{M}), (N, d_{N}) is (M × N, d_{M×N}) where d_{M×N} is given by

d_{M×N}((x_{1}, y_{1}), (x_{2}, y_{2})) = max{d_{M}(x_{1}, x_{2}), d_{N}(y_{1}, y_{2})} .

Let D = CompMet be the full subcategory of Met consisting of all complete metric spaces. Note that CompMet is closed under finite products in Met and write G: CompMet → Met for the inclusion functor. The functor G has a left adjoint, namely the functor F that maps a metric space M to its completion F (M) = M̂ (i.e. CompMet is a reflective subcategory of Met). It is not hard to see that F preserves finite products. The unit of the adjunction is the canonical isometry η_{M}: M → M̂ whose image is dense in M̂.

Theorem 3.4(a) applies to this setting and shows that every algebraic structure on a metric space M ascends uniquely along η_{M}: M → M̂, as asserted in Theorem 1.1(a).

4.2. Example. Let Top be the category of topological spaces and let D = CompHaus be the full subcategory hereof whose objects are all compact Hausdorff spaces. We note that CompHaus is closed under finite products in Top and write G: CompHaus → Top for the inclusion functor. The functor G has a left adjoint, namely the functor F that maps a topological space X to its Stone-Čech compactification F (X) = βX. This is a classic application of the Special Adjoint Functor Theorem; see [Mac Lane (1971), Chap. V§8]. However, the functor β does not preserve finite products, for example, β(ℝ × ℝ) is not βℝ × βℝ; see [Walker (1974), 1.67].

Consider therefore the full subcategory C = PsLocComp of Top whose objects are pseudocompact^{4} and locally compact^{5} spaces. A finite product of locally compact spaces is locally compact. A product of two pseudocompact spaces need not be pseudocompact^{6} but if, in addition, one of the factors is locally compact, then the product is in fact pseudocompact by [Glicksberg (1959), Thm. 3] or [Walker (1974), 8.21]. Hence C is closed under finite products in Top. Every compact space is also pseudocompact, and every compact Hausdorff space is locally compact. Thus there is an inclusion D ⊂ C. Although this inclusion is “close” to being an equality, it is strict^{7}. The point is now that the restriction of F = β to C does preserve finite products; this is part of proof of [Glicksberg (1959), Thm. 3]. In conclusion, Theorem 3.4(a) applies to the situation

\xymatrix{C = PsLocComp \ar[r]^{F =β} & CompHaus = D , \ar[l]^{G}}

and shows that every algebraic structure on a pseudocompact and locally compact topological space X ascends uniquely along the canonical map η_{X}: X → βX (which is the unit of the adjunction), as asserted in Theorem 1.1(b).

4.3. Example. It is proved in [Munkres (2000), Cor. 82.2] (or [May (1999), Chap. 3§8]) that a topological space has a universal covering space if and only if it is path connected, locally path connected, and semi-locally simply connected^{8}. Write D for the category of all such spaces, and C = SConn for the subcategory of simply connected topological spaces. Note that C and D are closed under finite products in Top; see e.g. [Chevalley (1946), II§7 Prop. 1 and II§8 Prop. 4].

Let Top_{∗} be the category of pointed topological spaces and denote by C_{∗} and D_{∗} the full subcategories of Top_{∗} whose objects are the ones in C and D, respectively. As noted above, the inclusion functor F: C_{∗} → D_{∗} preserves finite products. Since we work with pointed spaces, the universal covering space C_{∗} ∋ X̃ → X of a space X ∈ D_{∗} has the unique mapping property, see e.g. [Chevalley (1946), II§8 Prop. 1], in other words, F (X̃) = X̃ → X is a universal arrow from the inclusion functor F to the object X. By [Mac Lane (1971), IV§1 Thm. 2] this means that there is a well-defined functor G: D_{∗} → C_{∗}, which assigns to each X ∈ D_{∗} its universal covering space G(X) = X̃, and that this functor G is right adjoint to F.

4 A topological space X is pseudocompact if every continuous function X → ℝ is bounded.

5 A topological space is locally compact if every point has a local base consisting of compact neighbourhoods.

6 [Gillman and Jerison (1960), 9.15] present an example, due to Novák and Terasaka, of a pseudocompact space X for which X × X is not pseudocompact.

7 For a pathological example of a topological space X which is both pseudocompact and locally compact, but neither compact nor Hausdorff, let X be any countable set with the particular point topology.

8 As (path connected) ⇒ (connected) and since (connected) + (locally path connected) ⇒ (path connected), the conditions (path connected) + (locally path connected) and (connected) + (locally path connected) are the same.

Thus Theorem 3.4(b) applies to this setting and shows that every algebraic structure on a space X ∈ D_{∗} descends uniquely along the canonical map ε_{X}: X̃ → X (which is the counit of the adjunction), as asserted in Theorem 1.1(c).

4.4. Example. Let D = Top be the category of all topological spaces and let C = LocConn be the full subcategory hereof whose objects are all locally connected spaces. As C is closed under finite products in D, the inclusion functor F: LocConn → Top preserves finite products. The main result in [Gleason (1963)] is that F has a right adjoint G, which to every space X assigns its so-called universal locally connected refinement G(X) = X^{∗}.

Theorem 3.4(b) applies to this setting and shows that every algebraic structure on a space X descends uniquely along the map ε_{X}: X^{∗} → X, as asserted in Theorem 1.1(d).

References

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Department of Mathematical Sciences, University of Copenhagen Universitetsparken 5, 2100 Copenhagen Ø, Denmark.

Email: holm@math.ku.dk
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