Theory and Applications of Categories, Vol. 30, No. 35, 2015, pp. 1132–1162.

GENERALIZED COVERING SPACE THEORIES

JEREMY BRAZAS

Abstract. In this paper, we unify various approaches to generalized covering space theory by introducing a categorical framework in which coverings are defined purely in terms of unique lifting properties. For each category C of path-connected spaces having the unit disk as an object, we construct a category of C-coverings over a given space X that embeds in the category of π_{1}(X, x_{0})-sets via the usual monodromy action on fibers. When C is extended to its coreflective hull ℋ (C), the resulting category of based ℋ (C)-coverings is complete, has an initial object, and often characterizes more of the subgroup lattice of π_{1}(X, x_{0}) than traditional covering spaces.

We apply our results to three special coreflective subcategories: (1) The category of Δ-coverings employs the convenient category of Δ-generated spaces and is universal in the sense that it contains every other generalized covering category as a subcategory.

(2) In the locally path-connected category, we preserve notion of generalized covering due to Fischer and Zastrow and characterize the topology of such coverings using the standard whisker topology. (3) By employing the coreflective hull Fan of the category of all contractible spaces, we characterize the notion of continuous lifting of paths and identify the topology of Fan-coverings as the natural quotient topology inherited from the path space.

1. Introduction

When a topological space X is path-connected, locally path-connected, and semilocally simply-connected, the entire subgroup lattice of π_{1}(X, x_{0}) can be understood in terms of the covering spaces of X [22]. More precisely, the monodromy functor μ : Cov(X) → π_{1}(X, x_{0})Set from the category of coverings over X to the category of π_{1}(X, x_{0})-sets is an equivalence of categories. If X lacks a simply connected covering space, more sophisticated machinery is often needed to understand the combinatorial structure of π_{1}(X, x_{0}); however, significant advancements have been made in the past two decades. In this paper, we develop a categorical framework which unifies various attempts to generalize covering space theory, clarifies their relationship to classical topological constructions, and illuminates the theoretical extent to which such methods provide information about the fundamental group.

Many authors have attempted to extend the covering-theoretic approach to more general spaces, e.g. [1, 3, 7, 20, 21]. The usefulness of one generalization over another depends

Received by the editors 2014-09-28 and, in revised form, 2015-08-20. Transmitted by Ronald Brown. Published on 2015-08-23. 2010 Mathematics Subject Classification: 55R65, 57M10, 55Q52. Key words and phrases: Fundamental group, generalized covering map, coreflective hull, unique path

lifting property.

\begin{matrix}c \\ ◯ \end{matrix} Jeremy Brazas, 2015. Permission to copy for private use granted.

on the intended application. For instance, Fox’s overlays [20] provide no more information about the subgroup lattice of π_{1}(X, x_{0}) than traditional covering maps but admit a much more general classification in terms of the fundamental pro-group. Semicoverings [3] are intimately related to topological group structures on fundamental groups [5, 17] and have natural applications to general topological group theory [4]. In the current paper, we consider maps defined purely in terms of unique lifting properties. Our definitions are inspired by the initial approach of Hanspeter Fischer and Andreas Zastrow in [16] and the subsequent papers [8] and [11].

Generalized covering maps defined in terms of unique lifting properties often exist when standard covering maps do not and provide combinatorial information about fundamental groups of spaces which are not semilocally simply connected [15, 16]. For instance, one-dimensional spaces such as the Hawaiian earring, Menger curve, and Sierpinski carpet admit certain generalized “universal” coverings having the structure of topological ℝ-trees (called universal lpc_{0}-coverings in the current paper) on which π_{1}(X, x_{0}) acts by homeomorphism. These ℝ-trees behave like generalized Caley graphs [18] and have produced an explicit word calculus for the fundamental group of the Menger curve [19] in which the fundamental groups of all other one-dimensional and planar Peano continua embed.

In the current paper, we begin with a category C of path-connected spaces having the unit disk as an object and define C-coverings to have a unique lifting property with respect to maps on the objects of C. In Section 2, we introduce categories of C-coverings and explore their properties. In particular, we show the category Cov_{C}(X) of C-coverings over a given space X canonically embeds into π_{1}(X, x_{0})Set by a fully faithful monodromy functor μ : Cov_{C}(X) → π_{1}(X, x_{0})Set. Thus a C-covering p : X̃ → X is completely characterized up to isomorphism by the conjugacy class of the stabilizer subgroup H = p_{∗}(π_{1}(X̃, x̃_{0})). The category of based C-coverings becomes highly structured - exhibiting many properties that categories of classical covering maps lack - when we take C to be the coreflective hull of a category of simply connected spaces.

The literature on fundamental groups of wild spaces and generalized covering spaces primarily takes the viewpoint of considering fundamental groups at a chosen basepoint and applying the relevant infinite group theoretic concepts. In some situations, it may be preferable to avoid picking a basepoint and work with the fundamental groupoid π_{1}(X). In Section 3, we relate our categorical treatment of generalized coverings to covering morphisms of groupoids [9].

In Section 4, we identify the convenient (in the sense of [2, 23]) category of Δ-generated spaces, used in directed topology [13] and diffeology [10] as the setting for a “universal” theory of generalized covering maps. The category of Δ-coverings is universal in the sense that any other category of C-coverings canonically embeds within it. In a more practical sense, any attempt to characterize the subgroup structure of π_{1}(X, x_{0}) using maps having unique lifting of paths and homotopies of paths is retained as a special case of Δ-coverings.

Since we define C-coverings only in terms of abstract unique lifting properties, we are left with two important questions for given C.

Structure Question: If a C-covering exists, is there a simple characterization of the topology of the C-covering space X̃?

Existence Question: Monodromy μ : Cov_{C}(X) → π_{1}(X, x_{0})Set is fully faithful but need not be essentially surjective. Thus a subgroup H ⊆ π_{1}(X, x_{0}) is said to be a C-covering subgroup if there exists a corresponding C-covering p : X̃ → X and x̃_{0} ∈ X̃ such that H = p_{∗}(π_{1}(X̃, x̃_{0})). Is there a practical characterization of the C-covering subgroups of π_{1}(X, x_{0})?

In general, the Existence Question is challenging and must be taken on a case-by-case basis depending on which category C is being used. Some necessary and sufficient conditions for the existence of C-coverings are known for the locally path-connected category C = lpc_{0} [6, 8, 15, 16]; however, general characterizations remain the subject of ongoing research beyond the scope of the current paper. Nevertheless, it is reasonable to expect that an answer to the Structure Question will provide some help in answering the Existence Question.

In Sections 5 and 6, we answer the Structure Question in two important cases. First, we show the topology of lpc_{0}-coverings must always be the so-called “standard” or “whisker” topology. Moreover, we show that if X is first countable, then the notions of lpc_{0}- covering and Δ-covering over X agree. Second, we consider the coreflective hull Fan of all contractible spaces, which is generated by so-called directed arc-fan spaces. We identify the topology of Fan-coverings, which characterize the notion of continuous lifting of paths [3], as the natural quotient topology inherited from the path space. The Existence Question for Fan-coverings remains open and is likely to have application to topological group theory.

2. Generalized covering theories

2.1. Notational considerations. Throughout this paper, X will denote a pathconnected topological space with basepoint x_{0} ∈ X. We take Top_{0} to be the categories of path-connected topological spaces and maps (i.e. continuous function) and bTop_{0} to be the category of based path-connected spaces and based maps. All subcategories of Top_{0} and bTop_{0} considered in this paper are assumed to be full subcategories. Given a subcategory C ⊂ Top_{0}, the corresponding category of based spaces (X, x) where X ∈ C is denoted bC. If f : (X, x) → (Y, y) is a based map, f_{∗} : π_{1}(X, x) → π_{1}(Y, y) will denote the homomorphism induced by f on fundamental groups.

Let [0, 1] denote the unit interval and D^{2} = {(x, y) ∈ ℝ^{2}|x^{2} + y^{2} ≤ 1} the closed unit disk with basepoint d = (1, 0). A path in a space X is a continuous function α : [0, 1] → X. If α : [0, 1] → X is a path, then α^{−}(t) = α(1 − t) is the reverse path. If α, β : [0, 1] → X are paths such that α(1) = β(0), then α ·β denotes the usual concatenation of paths. The constant path at x ∈ X is denoted by c_{x}.

2.2. Definition. A map f : X → Y has the unique path lifting property if for any two paths α, β : [0, 1] → X, we have α = β whenever f ∘ α = f ∘ β and α(0) = β(0).

Let P (X) denote the space of paths in X with the compact-open topology. The compact-open topology of P (X) is generated by the subbasic sets

⟨K, U⟩ = {α ∈ P (X)|α(K) ⊆ U}

where K ⊂ [0, 1] is compact and U ⊂ X is open. For given x ∈ X, let P (X, x) = {α ∈ P (X)|α(0) = x} denote the subspace of paths which start at x and Ω(X, x) = {α ∈ P (X)|α(0) = x = α(1)} denote the subspace of loops based at x. A map f : (X, x) → (Y, y) induces a continuous function f_{#} : P (X, x) → P (Y, y) given by f_{#}(α) = f ∘ α. Note that f : X → Y has the unique path lifting property if and only if f_{#} : P (X, x) → P (Y, p(x)) is injective for every x ∈ X.

2.3. Disk-coverings. In the attempt to minimize the conditions one might impose on a covering-like map p : E → X, we are led to the following definition due to Dydak [11]. The definition is minimal in the sense that our goal is to retain information about the traditional fundamental group π_{1}(X, x_{0}) and to do this one should require unique lifting of all paths and homotopies of paths.

2.4. Definition. A map p : E → X is a disk-covering if E is non-empty, pathconnected and if for every e ∈ E and map f : (D^{2}, d) → (X, p(e)), there is a unique map f̂: (D^{2}, d) → (E, e) such that p ∘ f̂ = f.

Since the unit interval is a retract of D^{2}, it is clear that if p : E → X is a disk-covering, then every path α : ([0, 1], 0) → (X, p(e)) also has a unique lift α̃_{e} : ([0, 1], 0) → (E, e) such that p ∘ α̃_{e} = α. It follows that p must be surjective. The induced homomorphism p_{∗} : π_{1}(E, e) → π_{1}(X, p(e)) is injective for every e ∈ E and we have [α] ∈ p_{∗}(π_{1}(E, e)) if and only if the unique lift α̃_{e} : ([0, 1], 0) → (E, e) such that p ∘ α̃_{e} = α is a loop.

A morphism of disk-coverings p : E → X and q : E′ → X′ is a pair (f, g) of maps f : E → E′ and g : X → X′ such that g ∘ p = q ∘ f. Let DCov denote the category of disk-coverings and for a given space X, let DCov(X) be the subcategory of disk-coverings over X where morphisms are pairs (f, id), that is, commuting triangles:

\xymatrix{E \ar[dr]_{p} \ar[rr]^{f} &  & E′ \ar[dl]^{q} \\  & X}

If G = π_{1}(X, x_{0}), then traditional arguments from classical covering space theory imply the existence of a canonical “monodromy” functor μ : DCov(X) → GSet to the category GSet of G-Sets (sets A with a group action (g, a) ↦ g · a) and G-equivariant functions (functions f : A → B satisfying f(g · a) = g · f(a)). On objects, μ is defined as the fiber μ(p) = p^{−1}(x_{0}). If q : E′ → X is a disk-covering and f : E → E′ is a map such that q ∘ f = p, then μ(f) is the restriction of f to a G-equivariant function p^{−1}(x_{0}) → q^{−1}(x_{0}).

2.5. Lemma. If (X, x_{0}) ∈ bTop_{0} and G = π_{1}(X, x_{0}), the functor μ : DCov(X) → GSet is faithful.

Proof. Suppose p : E → X and q : E′ → X are disk-coverings over X. Let f, g : E → E′ be maps such that q ∘ f = p = q ∘ f and μ(f) = μ(g) as functions p^{−1}(x_{0}) → q^{−1}(x_{0}). To see that μ is faithful, we check that f = g. Fix e_{0} ∈ p^{−1}(x_{0}), pick a point e ∈ E, and a path γ : [0, 1] → E from e_{0} to e. By assumption, we have f(e_{0}) = g(e_{0}) = e′_{0} for some point e′_{0} ∈ q^{−1}(x_{0}). If β : [0, 1] → E′ is the unique path such that q ∘ β = p ∘ γ and β(0) = e′_{0}, then f ∘ γ = β = g ∘ γ by unique path lifting. In particular, f(e) = β(1) = g(e) and thus f = g.

The functor μ : DCov(X) → GSet will not typically be full since there are nonisomorphic disk-coverings corresponding to isomorphic group actions (see Example 2.23 below). In this sense, disk coverings - despite their great generality - are not an ideal candidate for a generalized covering theory. We will often restrict μ to a subcategory D ⊆ DCov(X). When we do this, we will still use the symbol μ to represent the restriction functor D → GSet.

The subgroups H ⊆ G which arise as the stabilizer subgroups of G-sets in the image of μ are precisely those for which there is a disk covering p : (E, e) → (X, x) with p_{∗}(π_{1}(E, e)) = H. Since some subgroups of π_{1}(X, x_{0}) need not arise as such stabilizers, we give special attention to those that do.

2.6. Definition. A subgroup H ⊆ π_{1}(X, x_{0}) is a disk-covering subgroup if there is a disk covering p : (E, e) → (X, x) with p_{∗}(π_{1}(E, e)) = H.

Another useful property not held by classical covering maps is the following 2-of-3 lemma. We leave the straightforward proof to the reader.

2.7. Lemma. Suppose p : E → X and q : E′ → E are maps. If two of the maps p, q, p ∘ q are disk-coverings, then so is the third.

2.8. C-coverings and their properties. The following definition is based on the definition of generalized covering in [16] but, in the spirit of [11], allows for a wider range of possible lifting criteria.

2.9. Definition. Let C ⊆ Top_{0} be a full subcategory of non-empty, path-connected spaces having the unit disk D^{2} as an object. A C-covering map is a map p : X̃ → X such that

1. X̃ ∈ C,

2. For every space Y ∈ C, point x̃ ∈ X̃, and based map f : (Y, y) → (X, p(x̃)) such that f_{∗}(π_{1}(Y, y)) ⊆ p_{∗}(π_{1}(X̃, x̃)), there is a unique map f̃ : (Y, y) → (X̃, x̃) such that p ∘ f̃ = f.

We call p a universal C-covering if X̃ is simply connected. Furthermore, we call p a weak C-covering map if p only satisfies condition 2.

2.10. Remark. The second condition in the definition of C-covering is reminiscent of the unique lifting criterion used in classical covering space theory. Since D^{2} is an object of C and π_{1}(D^{2}, d) = 1, it is clear that every weak C-covering is a disk-covering and is therefore surjective. In general, the lift f̃ : (Y, y_{0}) → (X̃, x̃_{0}) in condition 2. can be described as follows: Let y ∈ Y and γ : [0, 1] → Y be any path from y_{0} to y. Then f̃(y) is the endpoint of the unique lift { \atop f ∘ γ}x̃_{0} : [0, 1] → X̃ starting at x̃_{0}.

If C is a category of simply connected spaces, the notion of weak C-covering agrees with the maps of study in [11]. For instance, if D is the category whose only object is D^{2}, a weak D-covering is precisely a disk-covering. In general, the “weak” coverings are not unique up to homeomorphism and, as we will show, can always be replaced by some category of genuine C-coverings without losing information about π_{1}(X, x_{0}).

Since we always assume D^{2} is an object of C, we have the following implications for a given map p : X̃ → X:

p is a C-covering ⇒ p is a weak C-covering ⇒ p is a disk-covering

Let Cov_{C} denote the category of C-coverings and Cov_{C}(X) denote the category of Ccoverings over X; we view these as full subcategories of DCov and DCov(X) respectively. Since every C-covering p : X̃ → X is a disk-covering, we may apply the monodromy functor μ to p to obtain the corresponding group action of G = π_{1}(X, x_{0}) on the fiber p^{−1}(x_{0}). The following embedding theorem illustrates that C-coverings, unlike disk-coverings, are characterized up to isomorphism by this group action.

2.11. Theorem. The functor μ : Cov_{C}(X) → GSet is fully faithful.

Proof. Since μ : DCov(X) → GSet is faithful by Lemma 2.5, the restriction is also faithful. To check that μ is full, suppose p : E → X and q : E′ → X are C-coverings and that f : p^{−1}(x_{0}) → q^{−1}(x_{0}) is a G-equivariant function. Being G-equivariant means that f satisfies the equation f(α̃_{e}(1)) = α̃_{f(e)}(1) for every loop α ∈ Ω(X, x_{0}) and point e ∈ p^{−1}(x_{0}). Fix e_{0} ∈ p^{−1}(x_{0}), let e′_{0} = f(e_{0}), and consider a loop β ∈ Ω(E, e_{0}). We have

e′_{0} = f(e_{0}) = f (α̃_{e_{0}}(1)) = { \atop p ∘ β}_{e_{0}}(1)

which implies that p∘β lifts to a loop in E′ based at f(e_{0}) = e′_{0} ∈ q^{−1}(x_{0}). Thus p_{∗}([β]) ∈ q_{∗}(π_{1}((E′, e′_{0}))). Since p_{∗}(π_{1}(E, e_{0})) ⊆ q_{∗}(π_{1}((E′, e′_{0}))), there is a unique morphism p̃ : (E, e_{0}) → (E′, e′_{0}) such that q ∘ p̃ = p. It suffices to check that f is the restriction of p̃ to p^{−1}(x_{0}). Let e ∈ p^{−1}(x_{0}) and γ : [0, 1] → E be a path from e_{0} to e. If α = p ∘ γ, then

f(e) = f(α̃_{e_{0}}(1)) = α̃_{e′_{0}}(1) = p̃(e).

The functor μ in Theorem 2.11 is not necessarily an equivalence of categories since subgroups H ⊆ π_{1}(X, x_{0}) exist for which there may be no C-covering p : (X̃, x̃_{0}) → (X, x_{0}) such that H = p_{∗}(π_{1}(X̃, x̃_{0})): see Examples 5.14 and 5.15.

2.12. Definition. A subgroup H ⊆ π_{1}(X, x_{0}) is a C-covering subgroup if there exists a C-covering map p : (X̃, x̃) → (X, x_{0}) such that p_{∗}(π_{1}(X̃, x̃)) = H.

2.13. Remark. By changing the basepoint of the path-connected space X̃ in the fiber p^{−1}(x_{0}), it is clear that whenever H is a C-covering subgroup, every conjugate of H is also a C-covering subgroup.

The following proposition is straightforward to verify based on arguments from traditional covering space theory. In particular, it confirms that Cov_{C} is closed under the operation of composition, a property not generally held by covering maps in the classical sense.

2.14. Proposition. Suppose p : X̃ → X and q : Ỹ → X̃ are maps.

1. If p and q are C-coverings, then so is p ∘ q,

2. If p and p ∘ q are C-coverings, then so is q.

2.15. Coreflections. Throughout this section, we assume C is a category having D^{2} as an object. We typically want C to have “enough” objects to provide an interesting theory. Certainly, if the only object of C is D^{2}, then the category Cov_{C} is rather uninteresting; we’d even have Cov_{C}([0, 1]) = ∅. In this section, we will show that replacing any C with its coreflective hull in Top_{0} provides a richer theory of coverings without sacrificing any monodromy data.

2.16. Definition. The coreflective hull of C in Top_{0} is the full subcategory ℋ (C) of Top_{0} consisting of all path-connected spaces which are the quotient of a topological sum of objects of C.

Certainly ℋ (C) is closed under quotients and C ⊆ ℋ (C).

2.17. Example. Well-known examples of coreflective hulls taken within Top_{0} include the following:

1. If D is the category whose only object is D^{2}, then ℋ (D) is the category ΔTop_{0} of so-called Δ-generated spaces^{1} The category of Δ-generated spaces has been used in directed topology [13] and diffeology [10]. We treat the Δ-generated category in more detail in Section 4.

1The term “Δ-generated” comes from the fact that ΔTop_{0} is the coreflective hull of the category consisting of the standard n-simplices Δ_{n}.

2. The category lpc_{0} of path-connected, locally path-connected spaces is it’s own coreflection since the quotient of every locally path-connected space is locally pathconnected. More practically, lpc_{0} is the coreflective hull of the category of directed arc-hedgehogs: see Lemma 5.2 below.

3. The category Top_{0} is itself the coreflective hull of the category Top_{1} of all simply connected spaces. The proof is a nice exercise and is left to the reader.

2.18. Proposition. If D^{2} is an object of C, then ΔTop_{0} ⊆ ℋ (C). In particular, [0, 1] is an object of ℋ (C).

The category ℋ (C) is coreflective in the sense that the inclusion functor ℋ (C) → Top_{0} has a right adjoint c : Top_{0} → ℋ (C) where c(X) has the quotient (i.e. final) topology with respect to all maps g : Y → X with Y ∈ C. A set U ⊆ X is C-open in X if for every map f : Z → X where Z ∈ C, f^{−1}(U) is open in Z. A set U is open in c(X) if and only if U is C-open in X. The fact that c : Top_{0} → ℋ (C) is right adjoint is equivalent to the following more practical formulation.

2.19. Proposition. The identity function id : c(X) → X is continuous. Moreover, if g : Y → X is continuous where Y ∈ ℋ (C), then g : Y → c(X) is also continuous.

Since [0, 1] and D^{2} are objects of ℋ (C), every path and homotopy of paths in X is also continuous with respect to the topology of c(X). Thus the continuous identity id : c(X) → X is a disk-covering, which induces an isomorphism π_{1}(c(X), x) → π_{1}(X, x) on fundamental groups for every x ∈ X.

2.20. Corollary. For every X ∈ Top_{0}, the identity function id : c(X) → X is an ℋ (C)-covering.

Proof. By construction, c(X) ∈ ℋ (C). Suppose Y ∈ ℋ (C) and f : (Y, y) → (X, x) is a based map such that f_{∗}(π_{1}(Y, y)) ⊆ (id)_{∗}(π_{1}(c(X), x)) = π_{1}(X, x). Since Y ∈ ℋ (C), f : Y → c(X) is also continuous and certainly satisfies f ∘ id = f.

The next lemma tells us that every weak C-covering induces a ℋ (C)-covering which retains identical monodromy data of π_{1}(X, x_{0}).

2.21. Lemma. If p : X̃ → X is a weak C-covering, then ψ(p) : c(X̃) → X is a ℋ (C)- covering. Moreover, the morphism

\xymatrix{c(X̃) \ar[dr]_{ψ(p)} \ar[rr]^{id} &  & X̃ \ar[dl]^{p} \\  & X}

of disk-coverings over X induces an isomorphism μ(ψ(p) : c(X̃) → X) ≅ μ(p : X̃ → X) of G-sets.

Proof. First, we check that ψ(p) is a ℋ (C)-covering. By construction, c(X̃) ∈ ℋ (C). We check that ψ(p) is a weak ℋ (C)-covering. Suppose x̃ ∈ X̃, p(x̃) = x, and f : (Y, y) → (X, x) is a based map where Y ∈ ℋ (C) and f_{∗}(π_{1}(Y, y)) ⊆ (ψ(p))_{∗}(π_{1}(c(X̃), x̃)). Since the continuous identity function c(X̃) → X̃ induces an isomorphism of fundamental groups, we have (ψ(p))_{∗}(π_{1}(c(X̃), x̃)) = p_{∗}(π_{1}(X̃, x̃)).

Define a function f̃: (Y, y) → (X̃, x̃) as follows: for z ∈ Y, pick a path γ : [0, 1] → Y from y to z. If { \atop f ∘ γ} : [0, 1] → X̃ is the unique lift such that p ∘ { \atop f ∘ γ} = f ∘ γ and { \atop f ∘ γ}(0) = x̃, we let f̃(z) = { \atop f ∘ γ}(1). Since f_{∗}(π_{1}(Y, y)) ⊆ p_{∗}(π_{1}(X̃, x̃)) and p is a diskcovering, it is clear that f̃ is a well-defined function and is unique. It suffices to show f̃: Y → c(X̃) is continuous with respect to the topology of c(X̃). Suppose U ⊆ c(X̃) is open. Since Y ∈ ℋ (C), we need to check that g^{−1}(f̃^{−1}(U)) is open in Z for every map g : Z → Y where Z ∈ C. Suppose we have a point z ∈ g^{−1}(f̃^{−1}(U)). Pick any path γ from y to g(z) so that f̃(g(z)) = { \atop f ∘ γ}(1) = x̃′. Note that

(f ∘ g)_{∗}(π_{1}(Z, z)) ⊆ f_{∗}(π_{1}(Y, g(z)))
= [f ∘ γ]^{−1}f_{∗}(π_{1}(Y, y))[f ∘ γ]
⊆ [f ∘ γ]^{−1}p_{∗}(π_{1}(X̃, x̃))[f ∘ γ]
= p_{∗} (\left[{ \atop f ∘ γ}\right]^{−1} π_{1}(X̃, x̃) \left[{ \atop f ∘ γ}\right])
= p_{∗}(π_{1}(X̃, x̃′))

Since Z ∈ C and p : X̃ → X is a weak C-covering, there is a unique map k : (Z, z) → (X̃, x̃′) such that p ∘ k = f ∘ g. Since Z ∈ ℋ (C), the function k : Z → c(X̃) is also continuous. Since p ∘ f̃∘ g = f ∘ g = p ∘ k and f̃(g(z)) = x̃′, we have f̃∘ g = k by the uniqueness of the function k. Thus g^{−1}(f̃^{−1}(U)) = k^{−1}(U) is open in Z.

The last statement of the lemma follows directly from the fact that id : c(X̃) → X̃ is a disk-covering which induces an isomorphism of fundamental groups.

2.22. Corollary. Every C-covering subgroup of π_{1}(X, x_{0}) is also a ℋ (C)-covering subgroup of π_{1}(X, x_{0}).

2.23. Example. The previous lemma indicates that if X is path-connected but not an object of ℋ (C), then the identity functions id : X → X and ψ(id) : c(X) → X are nonisomorphic disk-coverings (only the later of which is a ℋ (C)-covering) which correspond to isomorphic G-sets under the functor μ. For instance, when ℋ (C) = lpc_{0}, a non-locally path-connected space X provides an example.

2.24. Corollary. The coreflection c : Top_{0} → ℋ (C) induces a fully faithful functor ψ : Cov_{C}(X) → Cov_{ℋ (C)} (X) which is the identity on all underlying sets and functions.

Moreover, the following diagram of functors commutes up to natural isomorphism.

\xymatrix{Cov_{C}(X) \ar[dr]_{μ} \ar[rr]^{ψ} &  & Cov_{ℋ (C)} (X) \ar@{-->}[dl]^{μ} \\  & GSet}

Proof. Lemma 2.21 shows that a C-covering p : X̃ → X gives rise to the ℋ (C)-covering ψ(p) : c(X̃) → X. If q : Ỹ → X is another C-covering and f : X̃ → Ỹ is a map such that q ∘ f = p, then the coreflection c(f) : c(X̃) → c(Ỹ) satisfies ψ(q) ∘ c(f) = ψ(p). Functoriality follows easily from here. Since c : Cov_{C}(X) → Cov_{ℋ (C)} (X) is the identity functor on underlying sets and functions, it is obviously faithful.

The component of the natural isomorphism at a given C-covering p : X̃ → X, is the isomorphism of G-sets (ψ(p))^{−1}(x_{0}) → p^{−1}(x_{0}) induced by the following morphism of disk-coverings over X.

\xymatrix{c(X̃) \ar[dr]_{ψ(p)} \ar[rr]^{id} &  & X̃ \ar[dl]^{p} \\  & X}

Naturality is straightforward to check. Finally, ψ is full since it is faithful and the monodromy functors are fully faithful.

2.25. Proposition. Suppose the objects of C are simply connected and include the unit disk. Let p : X̃ → X and q : Ỹ → X̃ be maps.

1. If p and q are ℋ (C)-coverings, then so is p ∘ q,

2. If p and p ∘ q are ℋ (C)-coverings, then so is q,

3. If q and p ∘ q are ℋ (C)-coverings, then so is ψ(p) : c(X̃) → X.

Proof. 1. and 2. follow directly from Proposition 2.14. For 3. suppose q and r are ℋ (C)- coverings. By Lemma 2.21, it suffices to show that p : X̃ → X is a weak C-covering. Let Y ∈ C and f : (Y, y) → (X, x) be a map and p(x̃) = x. Since q is surjective, we may pick a point ỹ ∈ q^{−1}(x̃). By Lemma 2.7, p is a disk-covering and thus has unique lifting with respect to path-connected spaces. Therefore, we only need to check that a based lift of f to (X̃, x̃) exists. Since Y is simply connected, f_{∗}(π_{1}(Y, y)) = 1 ⊆ (p ∘ q)_{∗}(π_{1}(Ỹ, ỹ)). Thus there is a unique map f̃ỹ : (Y, y) → (Ỹ, ỹ) such that p ∘ q ∘ f̃ỹ = f. The map q ∘ f̃ỹ : (Y, y) → (X̃, x̃) is the desired lift.

2.26. Theorem. For every X ∈ Top_{0}, the continuous identity function c(X) → X induces an equivalence of categories Cov_{ℋ (C)} (X) ≅ Cov_{ℋ (C)} (c(X))

Proof. Define the functor F : Cov_{ℋ (C)} (X) → Cov_{ℋ (C)} (c(X)) simply by applying the coreflection c: If p : X̃ → X is a ℋ (C)-covering, then F (p) : X̃ = c(X̃) → c(X) is continuous by Proposition 2.19. Given a morphism f : X̃ → Ỹ of ℋ (C)-coverings as in the diagram below, we have X̃, Ỹ ∈ ℋ (C) so F is the identity on morphisms.

\xymatrix{X̃ \ar[rr]^{f} \ar[dr]_{p} &  & Ỹ \ar[dl]^{q} & X̃ \ar[rr]^{f} \ar[dr]_{F (p)} &  & Ỹ \ar[dl]^{F (q)} \\  & X &  &  & c(X)}

The inverse G : Cov_{ℋ (C)} (c(X)) → Cov_{ℋ (C)} (X) is defined as follows: If p′ : X̃ → c(X) is a ℋ (C)-covering, then the composition p = G(p′) = X̃ → c(X) → X is a ℋ (C)-covering by Proposition 2.14. On morphisms, G is the identity. A straightforward check shows that F and G are inverse equivalences.

Theorem 2.26 implies that if one wishes to consider the ℋ (C)-coverings of a space X, then for all practical purposes one may assume X is an object of ℋ (C). We now consider subcategories C ⊆ D of Top_{0}. The fact that ℋ (C) ⊂ ℋ (D) allows us to apply the coreflection functor c : Top_{0} → ℋ (C) to every object of ℋ (D).

2.27. Corollary. If p : X̃ → X is a ℋ (D)-covering, then ψ(p) : c(X̃) → X is a ℋ (C)-covering.

Proof. Since C ⊂ ℋ (D), a ℋ (D)-covering p : X̃ → X is a weak C-covering. Now apply Lemma 2.21.

The coreflection c : ℋ (D) → ℋ (C) induces a functor ϕ : Cov_{ℋ (D)} (X) → Cov_{ℋ (C)} (X): on objects ϕ sends a ℋ (D)-covering p : X̃ → X to ϕ(p) = p : c(X̃) → X and if q : Ỹ → X is another ℋ (D)-covering and f : X̃ → Ỹ satisfies q ∘ f = p, then ϕ(f) = c(f) : c(X̃) → c(Ỹ) is the coreflection.

2.28. Theorem. If C ⊂ D ⊂ Top_{0}, the functor ϕ : Cov_{ℋ (D)} (X) → Cov_{ℋ (C)} (X) is fully faithful. Moreover, the following diagram of fully faithful functors commutes up to natural isomorphism.

\xymatrix{Cov_{ℋ (D)} (X) \ar@{-->}[dr]_{μ} \ar[rr]^{ϕ} &  & Cov_{ℋ (C)} (X) \ar@{-->}[dl]^{μ} \\  & GSet}

Proof. By definition, ϕ is the identity on underlying sets and functions, which makes faithfulness clear. Suppose p : X̃ → X and q : X̃ → X are ℋ (D)-coverings and f : c(X̃) → c(Ỹ) is map such that ϕ(q) ∘ f = ϕ(p). Since ℋ (C) ⊂ ℋ (D), we have c(X̃) = X̃ and c(Ỹ) = Y. Thus f : X̃ → Ỹ satisfies q ∘ f = p and c(f) = f. We conclude that ϕ is full.

The component of the natural isomorphism at a given ℋ (D)-covering p : X̃ → X, is the isomorphism of G-sets (ϕ(p))^{−1}(x_{0}) → p^{−1}(x_{0}) induced by the morphism id : c(X̃) → X̃ of disk-coverings ϕ(p) and p over X. Naturality is straightforward to check.

We conclude that using a smaller category C allows us to retain more of the subgroup lattice of π_{1}(X, x_{0}).

2.29. Corollary. If C ⊂ D ⊂ Top_{0} and H is a ℋ (D)-covering subgroup of π_{1}(X, x_{0}), then H is also a ℋ (C)-covering subgroup of π_{1}(X, x_{0}). In particular, if X admits a universal ℋ (D)-covering, then X also admits a universal ℋ (C)-covering.

We summarize the results of this section with a simple diagram: Suppose D^{2} ∈ C ⊂ D ⊂ Top_{0} with coreflections Top_{0} → ℋ (C) and Top_{0} → ℋ (D). Then the following diagram of fully faithful functors commutes up to natural isomorphism (where ↪ denotes a fully faithful functor).

\xymatrix{ & Cov_{ℋ (D)} (X) \ar@{^{(}->}[ddr]_{μ} \ar@{^{(}->}[rr]^{ϕ} &  & Cov_{ℋ (C)} (X) \ar@{^{(}->}[ddl]^{μ} \\ Cov_{D}(X) \ar@{^{(}->}[drr]_{μ} \ar@{^{(}->}[ur]_{ψ} &  &  &  & Cov_{C}(X) \ar@{^{(}->}[dll]^{μ} \ar@{^{(}->}[ul]^{ψ} \\  &  & GSet}

2.30. Categorical constructions of C-coverings. The category Top_{0} whose objects are unbased path-connected spaces is not complete (i.e. closed under all small categorical limits). In order to construct limits and pullbacks of coverings, we restrict ourselves to based spaces and maps. Let bCov_{C} and bCov_{C}(X, x) denote the categories of all based C-coverings p : (X̃, x̃) → (X, x) and based coverings over (X, x) respectively.

2.31. Lemma. The categories bCov_{ℋ (C)} and bCov_{ℋ (C)} (X, x) are complete.

Proof. Let J be a small category and F : J → bCov_{C} be a diagram of ℋ (C)-coverings: F (j) = p_{j} : (X̃_{j}, x̃_{j}) → (X_{j}, x_{j}) for object j ∈ J and for each morphism m : j → k in J, there are maps a_{m} : (X̃_{j}, x̃_{j}) → (X̃_{k}, x̃_{k}) and b_{m} : (X_{j}, x_{j}) → (X_{k}, x_{k}) such that p_{k} ∘ a_{m} = b_{m} ∘ p_{j}. Let lim p_{j} : (lim X̃_{j}, x̃_{0}) → (lim X_{j}, x_{0}) be the standard limit in bTop_{0}. Let X = lim X_{j}, X̃ be the path-component of lim X̃_{j} containing x̃_{0} and p : X̃ → X be the restriction of lim p_{j}. We check that p : X̃ → X is a weak ℋ (C)-covering.

For j ∈ J, let a_{j} : (X̃, x̃_{0}) → (X̃_{j}, x̃_{j}) and b_{j} : (X, x_{0}) → (X_{j}, x_{j}) be the canonical projections satisfying b_{j} ∘ p = p_{j} ∘ a_{j}. Let f : (Y, y) → (X, x_{0}) be a map such that Y ∈ ℋ (C) and f_{∗}(π_{1}(Y, y)) ⊆ p_{∗}(π_{1}(X̃, x̃_{0})). Let f_{j} = b_{j} ∘ f and note that

(f_{j})_{∗}(π_{1}(Y, y)) ⊆ (b_{j} ∘ p)_{∗}(π_{1}(X̃, x̃_{0})) = (p_{j} ∘ a_{j})_{∗}(π_{1}(X̃, x̃_{0})) ⊆ (p_{j})_{∗}(π_{1}(X̃_{j}, x̃_{j}))

Thus there is a unique map f̃_{j} : (Y, y) → (X̃_{j}, x̃_{j}) such that p_{j} ∘f̃_{j} = f_{j}. Given a morphism m : j → k in J, we have

p_{k} ∘ a_{m} ∘ f̃_{j} = b_{m} ∘ p_{j} ∘ f̃_{j} = b_{m} ∘ f_{j} = b_{m} ∘ b_{j} ∘ f = b_{k} ∘ f = f_{k}

so a_{m}∘f̃_{j} = f_{k} by the uniqueness of lifts. It follows that there is a unique map f̃: (Y, y) → (lim X̃_{j}, x̃_{0}) such that a_{j} ∘ f̃ = f̃_{j}. Since Y is path-connected, f̃ has image in X̃. Finally, since b_{j} ∘ p ∘ f̃ = p_{j} ∘ a_{j} ∘ f̃ = p_{j} ∘ f̃_{j} = f_{j} = b_{j} ∘ f, the universal property of X gives p ∘ f̃ = f. This proves that p is a weak ℋ (C)-covering.

\xymatrix{ &  & X̃_{k} \ar[dd]_{p_{k}} \\  & X̃ \ar[dd]_{p} \ar[ur]^{a_{k}} \ar[rr]_{a_{j}} &  & X̃_{j} \ar[ul]_{a_{m}} \ar[dd]^{p_{j}} \\  &  & X_{k} \\ Y \ar[r]_{f} \ar@{-->}[uur]^{f̃} & X \ar[ur]^{b_{k}} \ar[rr]_{b_{j}} &  & X_{j} \ar[ul]_{b_{m}}}

By Lemma 2.21, the coreflection ψ(p) : c(X̃) → X is a ℋ (C)-covering. It is a standard application of coreflections and limits to check that p is the categorical limit in bCov_{ℋ (C)} .

Applying the same argument to based ℋ (C)-coverings over a given based space (X, x), one can prove that bCov_{ℋ (C)} (X, x) is complete.

Since (X, x_{0}) is the limit lim(X_{j}, x_{j}) in the proof of Lemma 2.31, there is a canonical homomorphism h : π_{1}(X, x_{0}) → lim π_{1}(X_{j}, x_{j}), h([α]) = ([b_{j} ∘ α]) where the limit lim π_{1}(X_{j}, x_{j}) of groups is realized canonically as a subgroup of the product ∏_{j∈J} π_{1}(X_{j}, x_{j}) taken over the objects of J. In the following theorem, we utilize this same notation; the result generalizes a key ingredient in the proof that fundamental groups of one-dimensional spaces inject into their first shape group, i.e. are π_{1}-shape injective [12].

2.32. Theorem. Suppose the ℋ (C)-covering p : (X̃, x̃_{0}) → (X, x_{0}) is the limit a diagram F : J → bCov_{ℋ (C)} of universal ℋ (C)-coverings. Then p is a universal ℋ (C)-covering if and only if the canonical homomorphism h : π_{1}(X, x_{0}) → lim π_{1}(X_{j}, x_{j}) is injective.

Proof. Suppose h is injective. Let α̃ : S^{1} → X̃ be a loop based at x̃_{0}. Since X̃_{j} is simply connected by assumption, the loop p_{j} ∘ a_{j} ∘ α̃ = b_{j} ∘ p ∘ α̃ is null-homotopic in X_{j}. Since h([p ∘ α̃]) = ([b_{j} ∘ p ∘ α̃]) is trivial and h is injective, [p ∘ α̃] = 1. Since p_{∗} is injective, [α̃] = 1 proving that X̃ is simply connected. For the converse, suppose α : S^{1} → X is a loop such that α_{j} = b_{j} ∘ α is null-homotopic in X_{j} for all j ∈ J. To prove the injectivity of h it suffices to show α is null-homotopic. Since p_{j} : X̃_{j} → X_{j} is a ℋ (C)-covering, there is a unique lift α̃_{j} : S^{1} → X̃_{j} such that p_{j} ∘ α̃_{j} = α_{j}. Given a morphism m : j → k in J, we have p_{k} ∘ a_{m} ∘ α̃_{j} = b_{m} ∘ p_{j} ∘ α̃_{j} = b_{m} ∘ α_{j} = α_{k}, which by unique lifting proves that a_{m} ∘ α̃_{j} = α̃_{k}. Since together, the loops α̃_{j} and α_{j} define a cone from the identity ℋ (C)- covering id : S^{1} → S^{1} to F, we see that there is a loop α̃ : S^{1} → X̃ such that p ∘ α̃ = α and a_{j} ∘ α̃ = α̃_{j}. But since X̃ is simply connected, both α̃ and α are null-homotopic.

2.33. Corollary. Suppose (X, x_{0}) = lim(X_{j}, x_{j}) is the limit of a diagram F : J → bTop_{0} in the category of based, path-connected spaces where X_{j} admits a universal ℋ (C)- covering for each j ∈ J and b_{j} : X → X_{j} is the projection. If the canonical homomorphism h : π_{1}(X, x_{0}) → lim π_{1}(X_{j}, x_{j}), h([α]) = ([b_{j} ∘ α]) is injective, then X admits a universal ℋ (C)-covering.

Proof. Let p_{j} : (X̃_{j}, x̃_{j}) → (X_{j}, x_{j}) be a universal ℋ (C)-covering. Given a morphism m : j → k in J let b_{m} = F (m) : X_{j} → X_{k}. Since X̃_{j} is simply connected, there is a unique map a_{m} : (X̃_{j}, x̃_{j}) → (X̃_{k}, x̃_{k}) such that p_{k} ∘ a_{m} = b_{m} ∘ p_{j}. Thus we have a diagram J → bCov_{ℋ (C)} of universal ℋ (C)-coverings with limit p : (X̃, x̃_{0}) → (X, x_{0}). By Theorem 2.32, X̃ is simply connected.

Corollary 2.33 provides a categorical proof of the fact that a π_{1}-shape injective Peano continuum admits a universal lpc_{0}-covering. The general case appears in [16]. Interestingly, the converse of Corollary 2.33 is false in the locally-path connected case [15]; there exist Peano continua (X, x_{0}) = \underleftarrow{lim}_{n}(X_{n}, x_{n}), which are the inverse limit of finite polyhedra and which admit a universal lpc_{0}-covering but for which h : π_{1}(X, x_{0}) → \underleftarrow{lim}_{n} π_{1}(X_{n}, x_{n}) fails to be injective.

Another useful construction is the pullback construction: Fix a ℋ (C)-covering p : X̃ → X and any map f : Y → X. We view the pullback X̃×_{X} Y = {(x̃, y) ∈ X̃×Y |p(x̃) = f(y)} as a subspace of the direct product. Similar to the situation above, the space X̃×_{X}Y need not path-connected and the components need to be objects of ℋ (C). This failure has been fully characterized in the locally path-connected case ℋ (C) = lpc_{0} [14]. To overcome this issue, we again choose a single path component and apply the coreflection c.

Fix points y_{0} ∈ Y, x_{0} = f(x_{0}), and x̃_{0} ∈ p^{−1}(x_{0}). Let P be the path component of X̃ ×_{X} Y containing (x̃_{0}, y_{0}) and let f^{#}X̃ = c(P) be the coreflection. We call the projection f^{#}p : f^{#}X̃ → Y, f^{#}p(x̃, y) = y the pullback of X̃ by f and check that it is a ℋ (C)-covering map.

2.34. Lemma. If p : X̃ → X is a ℋ (C)-covering, Y ∈ Top_{0} and f : Y → X is a map, then the pullback f^{#}p : f^{#}X̃ → Y is a ℋ (C)-covering corresponding to the subgroup f_{∗}^{−1}(p_{∗}(π_{1}(X̃, x̃_{0}))) ⊆ π_{1}(Y, y_{0}).

Proof. By construction, f^{#}X̃ is an object of ℋ (C). Let Z ∈ ℋ (C), (x̃, y) ∈ f^{#}X̃ and g : (Z, z) → (Y, y) be a map such that g_{∗}(π_{1}(Z, z)) ⊆ (f^{#}p)_{∗}(π_{1}(f^{#}X̃, (x̃, y))). Let q : f^{#}X̃ → X̃ be the second projection such that p ∘ q = f ∘ f^{#}p. Now

(f ∘ g)_{∗}(π_{1}(Z, z)) ⊆ f_{∗} ((f^{#}p)_{∗}(π_{1}(f^{#}X̃, (x̃, y))))
= p_{∗}(q_{∗}(π_{1}(f^{#}X̃, (x̃, y))))
⊆ p_{∗}(π_{1}(X̃, x̃))

Thus there is a unique map k : (Z, z) → (X̃, x̃) such that p ∘ k = f ∘ g. By the universal property of pullbacks, we have a unique map g̃ : (Z, z) → (X̃ ×_{X} Y, (x̃, y)) satisfying q ∘ g̃ = k and f^{#}p ∘ g̃ = g. Since Z is path-connected g̃ has image in P and since Z ∈ ℋ (C), g̃: Z → f^{#}X̃ is continuous.

\xymatrix{Z \ar@{-->}[dr]^{g̃} \ar[drr]^{k} \ar[ddr]_{g} \\  & f^{#}X̃ \ar[r]^{q} \ar[d]_{f^{#}p} & X̃ \ar[d]^{p} \\  & Y \ar[r]_{f} & X}

Finally, we check that (f^{#}p)_{∗}(π_{1}(f^{#}X̃, (x̃_{0}, y_{0}))) = f_{∗}^{−1}(p_{∗}(π_{1}(X̃, x̃_{0}))). One inclusion is clear from the commutativity of the square. For the other suppose [γ] ∈ f_{∗}^{−1}(p_{∗}(π_{1}(X̃, x̃_{0}))). Since [f ∘ γ] ∈ p_{∗}(π_{1}(X̃, x̃_{0})), there is a unique lift κ : S^{1} → X̃ such that p ∘ κ = f ∘ γ. Thus there is a unique loop γ̃ : S^{1} → f^{#}X̃ such that f^{#}p ∘ γ̃ = γ and q ∘ γ̃ = κ. It follows that [γ] ∈ (f^{#}p)_{∗}(π_{1}(f^{#}X̃, (x̃_{0}, y_{0}))).

In fact, a based map f : (Y, y_{0}) → (X, x_{0}) induces a functor f^{#} : bCov_{ℋ (C)} (X, x_{0}) → bCov_{ℋ (C)} (X, y_{0}). If p′ : (X̃′, x̃′_{0}) → (X, x_{0}) is another ℋ (C)-covering over X and g : (X̃, x̃_{0}) → (X̃′, x̃′_{0}) is a morphism such that p′ ∘ g = p, then f^{#}g : f^{#}X̃ → f^{#}X̃′ is uniquely induced by a straightforward pullback diagram.

2.35. Corollary. Suppose p : X̃ → X is a universal ℋ (C)-covering. Then the pullback f^{#}p : f^{#}X̃ → Y is a universal ℋ (C)-covering if and only if f_{∗} : π_{1}(Y, y_{0}) → π_{1}(X, x_{0}) is injective.

We use the existence of pullbacks to verify the closure of ℋ (C)-covering subgroups under intersection.

2.36. Theorem. If {H_{j}|j ∈ J} is any set of ℋ (C)-covering subgroups of π_{1}(X, x_{0}), then ⋂_{j} H_{j} is a ℋ (C)-covering subgroup.

Proof. Fix a ℋ (C)-covering p_{j} : (X̃_{j}, x̃_{j}) → (X, x_{0}) such that (p_{j})_{∗}(π_{1}(X̃_{j}, x̃_{j})) = H_{j}. Let (X̃, x̃_{0}) = (∏_{j} X̃_{j}, (x̃_{j})). By Lemma 2.31, the product p = ∏ p_{j} : c(X̃) → ∏_{j} X is a ℋ (C)-covering. Note that p_{∗}(π_{1}(X̃, x̃_{0})) is the product of subgroups ∏_{j} H_{j} when we make the identification ∏_{j} π_{1}(X, x_{0}) ≅ π_{1}(∏_{j} X, (x_{0})). Let δ : X → ∏_{j} X be the diagonal map and δ^{#}p : δ^{#}X̃ → X be the pullback of X̃ by δ. By Lemma 2.34, δ^{#}p is a ℋ (C)-covering map such that the image of the homomorphism (δ^{#}p)_{∗} is δ_{∗}^{−1}(∏_{j} H_{j}). It is easy to see that δ_{∗}^{−1}(∏_{j} H_{j}) = ⋂_{j} H_{j} and thus ⋂_{j} H_{j} is a ℋ (C)-covering subgroup.

Theorem 2.36 implies the existence of a minimal ℋ (C)-covering subgroup for every based path-connected space (X, x_{0}): Let

U(X, C) = ⋂{H|H is a ℋ (C)-covering subgroup of π_{1}(X, x_{0})}.

The existence of minimal ℋ (C)-covering subgroups immediately implies the existence of an initial based ℋ (C)-covering p : (X̃, x̃_{0}) → (X, x_{0}) such that p_{∗}(π_{1}(X̃, x̃_{0})) = U(X, C).

2.37. Theorem. For any space X, the category Cov_{ℋ (C)} (X) has an initial object.

2.38. Corollary. A space X admits a universal ℋ (C)-covering if and only if U(X, C) = 1.

Any non-trivial elements of the subgroup U(X, C) may be viewed as those elements of π_{1}(X, x_{0}) which are indistinguishable from the homotopy class of the constant loop with respect to ℋ (C)-coverings. Thus if U(X, C) ≠ 1 there are homotopy classes of loops which are inaccessible to study via ℋ (C)-coverings.

2.39. Proposition. U(X, C) is a normal subgroup of π_{1}(X, x_{0}).

Proof. Recall from Remark 2.13 that if H is a ℋ (C)-covering subgroup of G = π_{1}(X, x_{0}), then gHg^{−1} is also a ℋ (C)-covering subgroup for each g ∈ G. The normal subgroup N_{H} = ⋂_{g∈G} gHg^{−1} (called the core of H in G) is the largest normal subgroup of G which is contained in H. By Theorem 2.36, N_{H} is a ℋ (C)-covering subgroup. Let N be the intersection of the cores of all ℋ (C)-covering subgroups H. Clearly N is normal. Since N_{H} ⊆ H for all H, we have N ⊆ U(X, C) and since N is a ℋ (C)-covering subgroup (by Theorem 2.36), we see that N = U(X, C).

3. The groupoid approach

Recall that if a map f : X → Y has the unique path-lifting property and f ∘ α = f ∘ β for paths α and β, then we cannot conclude that α = β unless we already know they agree at a point. Consequently, the basepoints in the definition of C-covering cannot be disregarded. Nevertheless, some of the above results have nice analogues involving the fundamental groupoid, which avoid reference to subgroup conjugacy classes. We briefly mention a few of them here.

Given a small groupoid G, let G(x, −) denote the set of morphisms in G whose source is the object x. A covering morphism over G is a functor F : H → G between groupoids such that for each object y ∈ H the induced function H(y, −) → G(F (y), −) is bijective [9]. The category of covering morphisms over G is denoted CovMor(G). It is a standard fact that CovMor(G) is equivalent to the functor category Set^{G} of operations of G on sets. Moreover, if G = G(x, x) is the vertex group viewed as a one-object subcategory and G is connected, then the inclusion G → G is an equivalence of categories which in turn induces an equivalence Set^{G} → GSet. We are interested in the case where G = π_{1}(X) is the fundamental groupoid of X and G = π_{1}(X, x_{0}).

Since a disk-covering p : E → X has unique lifting of all paths and homotopies of paths, it is clear that the induced functor π_{1}(p) : π_{1}(E) → π_{1}(X) is a covering morphism. Thus we obtain a monodromy functor M : DCov(X) → CovMor(π_{1}(X)) by applying π_{1} to disk-coverings and their morphisms. If E : CovMor(π_{1}(X)) → π_{1}(X, x_{0})Set is the equivalence of categories referenced in the previous paragraph, then E ∘ M ≃ μ is the faithful monodromy functor of Lemma 2.5. Consequently, M is faithful.

For the rest of this section, suppose C ⊂ Top_{0} is a category containing the unit disk as an object.

3.1. Theorem. For any space X, there is a canonical fully faithful monodromy functor M : Cov_{C}(X) → CovMor (π_{1}(X)).

Proof. Since Cov_{C}(X) ⊂ DCov(X), we define the functor

M : Cov_{C}(X) → CovMor(π_{1}(X))

simply to be the restriction of M : DCov(X) → CovMor(π_{1}(X)). It follows from Theorem 2.11 that M : Cov_{C}(X) → CovMor(π_{1}(X)) is fully faithful.

Since the following diagram commutes up to natural isomorphism and the bottom functor is an equivalence, it is possible to consistently replace μ with M in all of the previous diagrams involving monodromy.

\xymatrix{ & Cov_{C}(X) \ar@{^{(}->}[dl]_{M} \ar@{^{(}->}[dr]^{μ} \\ CovMor(π_{1}(X)) \ar[rr]^{≅}_{E} &  & π_{1}(X, x_{0})Set}

The results on categorical limits and pullbacks as applied to based ℋ (C)-coverings may be summarized as follows.

3.2. Theorem. If CovMor is the category of all covering morphisms, then for any C the image of M : bCov_{ℋ (C)} → CovMor is complete.

3.3. Theorem. For any (X, x_{0}) ∈ bTop_{0}, the image of M : bCov_{ℋ (C)} (X, x_{0}) → CovMor (π_{1}(X)) is complete and has an initial object.

4. The universal covering theory: Δ-coverings

In this section, we consider in more detail the category in which all other C-covering categories embed, that is, when D^{2} is the only object of C.

4.1. Definition. A subset U ⊆ X is Δ-open in X if f^{−1}(U) is open in D^{2} for every map f : D^{2} → X. A space X is Δ-generated if a set U is open in X if and only if U is Δ-open in X.

Let ΔTop_{0} denote the category of path-connected, Δ-generated spaces. We prefer to use the following characterization of Δ-generated topologies which follows directly from the existence of space filling curves [0, 1] → D^{2}: A set U ⊆ X is Δ-open if and only if α^{−1}(U) is open in [0, 1] for every path α : [0, 1] → X.

Recall a space X is sequential if U ⊂ X is open if and only if for every convergent sequence x_{n} → x in X with x ∈ U, there exists N ≥ 1 such that x_{n} ∈ U for all n ≥ N. The following lemma follows from well-known topological facts.

4.2. Lemma. [10] Every first countable and locally path-connected space is Δ-generated and every Δ-generated space is sequential and locally path-connected.

If D denotes the full subcategory of Top_{0} whose only object is D^{2}, then ℋ (D) = ΔTop_{0}. We denote the coreflection functor by Δ : Top_{0} → ΔTop_{0}. Thus Δ(X) has the same underlying set as X and has the topology consisting of all Δ-open sets. For the sake of convenience we call a ΔTop_{0}-covering p : X̃ → X simply a Δ-covering. Definition 2.9 translates as follows.

4.3. Definition. A map p : X̃ → X is a Δ-covering map if

1. X̃ ∈ ΔTop_{0},

2. for every space Y ∈ ΔTop_{0}, point x̃ ∈ X̃, and based map f : (Y, y) → (X, p(x̃)) such that f_{∗}(π_{1}(Y, y)) ⊂ p_{∗}(π_{1}(X̃, x̃)), there is a unique map f̃ : (Y, y) → (X̃, x̃) such that p ∘ f̃ = f.

The following theorem is the relevant combination of Theorem 2.28 and Corollary 2.24; it implies that among all C-coverings, Δ-coverings retain the most information about the subgroup structure of π_{1}(X, x_{0}).

4.4. Theorem. If C has D^{2} as an object, then there is a canonical fully faithful functor Cov_{C}(X) → ΔCov(X).

4.5. Corollary. Suppose C ⊂ Top_{0} has D^{2} as an object. Then every C-covering subgroup of π_{1}(X, x_{0}) is also a Δ-covering subgroup of π_{1}(X, x_{0}).

We now observe that the images of the functors μ : ΔCov(X) → GSet and μ : DCov(X) → GSet agree, or equivalently that every disk-covering subgroup of π_{1}(X, x_{0}) is a Δ-covering subgroup of π_{1}(X, x_{0}). Recall that a disk-covering is precisely a weak D-covering where the only object of D is D^{2}. The following lemma is a special case of Lemma 2.21.

4.6. Lemma. If X is path-connected and p : E → X is a disk-covering, then p : Δ(E) → X is a Δ-covering.

We can now extend the coreflection Δ : Top_{0} → ΔTop_{0} to a functor DCov(X) → ΔCov(X). If f : E → E′ satisfies p′ ∘ f = p for disk-coverings p : E → X and p : E′ → X, then Δ(f) : Δ(E) → Δ(E′) is a morphism of the Δ-coverings p : Δ(E) → X, p : Δ(E′) → X.

4.7. Corollary. The functor Δ : DCov(X) → ΔCov(X) taking disk-covering p : E → X to the Δ-covering p : Δ(E) → X is right adjoint to the inclusion ΔCov(X) → DCov(X).

4.8. Theorem. Suppose X is path-connected and G = π_{1}(X, x_{0}). The follow diagram of functors commutes up to natural isomorphism.

\xymatrix{DCov(X) \ar[dr]_{μ} \ar[rr]^{Δ} &  & ΔCov(X) \ar@{^{(}->}[dl]^{μ} \\  & GSet}

Proof. To define the natural isomorphism η : μ ∘ Δ → μ, we take the component η_{p} of the disk-covering p : E → X to be the isomorphism of group actions induced by the morphism

\xymatrix{Δ(E) \ar[dr]_{p} \ar[rr]^{id} &  & E \ar[dl]^{p} \\  & X}

of disk-coverings under μ. The naturality of η is straightforward to verify.

4.9. Remark. We do not wish to give much attention to categories of weak C-coverings; however, Theorem 4.8 can be generalized as follows: If wCov_{C}(X) is the category of weak C-coverings over X, then the follow diagram of functors commutes up to natural isomorphism where ψ is right adjoint to the inclusion Cov_{ℋ (C)} (X) ⊂ wCov_{C}(X).

\xymatrix{wCov_{C}(X) \ar[dr]_{μ} \ar[rr]^{ψ} &  & Cov_{ℋ (C)} (X) \ar@{^{(}->}[dl]^{μ} \\  & GSet}

One should interpret the conclusion of Theorem 4.8 as follows: Any attempt to study the combinatorial structure of the traditional fundamental group π_{1}(X, x_{0}) using maps that uniquely lift paths and homotopies of paths can be replaced or generalized by Δcoverings without losing any information about π_{1}(X, x_{0}). Additionally, by Theorem 2.26, we have ΔCov(X) ≅ ΔCov(Δ(X)) and thus there is also no loss of information by assuming from the start that X is Δ-generated. The subgroup U(X, ΔTop_{0}) ⊆ π_{1}(X, x_{0}) is, in some sense, the group of “phantom” homotopy class, i.e. those which are invisible to any generalized covering-theoretic approach.

Given a Δ-covering p : X̃ → X, it is natural to search for explicit descriptions of the topology of X̃. In the following section, we employ the locally path-connected category to provide a simple description of X̃ in the case that X is first countable. For a non-first countable, Δ-generated space X, it seems that we can only describe the topology of X̃ as a final topology with respect to a collection of paths [0, 1] → X̃.

5. Coverings for locally path-connected spaces: lpc_{0}-coverings

The second case we consider is when ℋ (C) is the category lpc_{0} of path-connected, locally path-connected spaces. As discussed in the introduction, the notion of lpc_{0}-covering is a direct generalization of the “generalized regular coverings” in [16]. lpc_{0}-coverings p, which are non-regular in the sense that the image of p_{∗} is a non-normal subgroup, are known to exist [17].

5.1. Definition. [11] Let (J, ≤) be an infinite directed set. The directed wedge (A, a_{0}) = ⋁_{J}(X_{j}, x_{j}) of spaces (X_{j}, x_{j}) indexed by J is the based wedge sum with the following topology: U ⊂ A\{a_{0}} is open if and only if U ∩ X_{j} is open in X_{j} for each j ∈ J and U is an open neighborhood of a_{0} if and only if there is a k ∈ J such that X_{j} ⊂ U for all j > k and U ∩ X_{j} is open in X_{j} for each j ∈ J. The arc-hedgehog over J is the directed wedge (H(J), a_{0}) = ⋁_{J}([0, 1]_{j}, 0).

If t ∈ [0, 1], we let t_{j} denote the image of t ∈ [0, 1]_{j} in H(J). Observe that maps (H(J), a_{0}) → (X, x) are in bijective correspondence with convergent nets α_{j} → c_{x}, j ∈ J (which converge to the constant path) in the path space P (X). Also note that H(J) is path-connected, locally path-connected, and simply-connected. Let H be the category whose objects include H(J) for every directed set J.

5.2. Lemma. ℋ (H) = lpc_{0}.

Proof. Since every quotient of a topological sum of locally path-connected spaces is locally path-connected, it is clear that ℋ (H) ⊆ lpc_{0}. Suppose X is path-connected and locally path-connected and suppose there is a set U ⊆ X such that ϵ^{−1}(U) is open for every map ϵ : H(J) → X from some arc-hedgehog. We check that U is open in X. If U is not open, then there exists x ∈ U such that no neighborhood of x is contained in U. Let {V_{j}|j ∈ J} be a base of path-connected connected neighborhoods at x, where J is a directed set. For every j ∈ J, there is a point x_{j} ∈ V_{j}\U. Pick a path ϵ_{j} : [0, 1] → V_{j} from x to x_{j}. Together these paths induce a unique map ϵ : (H(J), a_{0}) → (X, x) such that the restriction to [0, 1]_{j} is ϵ_{j}. Since ϵ is continuous ϵ^{−1}(U) is an open neighborhood of a_{0} in H(J). Thus there is a k ∈ J such that [0, 1]_{j} ⊆ ϵ^{−1}(U) for all j > k. Thus ϵ_{j}(1) = x_{j} ∈ U for all j > k which is a contradiction. Thus U must be open in X.

The coreflection functor Top_{0} → ℋ (H) is well-known [8, 11, 16]; we denote it by lpc : Top_{0} → lpc_{0}. A convenient description of the coreflection is the following: lpc(X) has the same underlying set as X and has topology generated by the path-components of the open sets of X. Since D^{2} is not an object of H, apparently, we cannot apply much of the above theory to H itself. However, it is certainly true that D^{2} ∈ lpc_{0}. Since ℋ (lpc_{0}) = lpc_{0}, the results in Section 2 apply to lpc_{0} when we view it as the coreflective hull of H with D^{2} added as an object. Definition 2.9 translates as follows.

5.3. Definition. A map p : X̃ → X is a lpc_{0}-covering map if

1. X̃ is path-connected and locally path-connected,

2. for every path-connected, locally path-connected space Y, point x̃ ∈ X̃, and based map f : (Y, y) → (X, p(x̃)) such that f_{∗}(π_{1}(Y, y)) ⊂ p_{∗}(π_{1}(X̃, x̃)), there is a unique map f̃: (Y, y) → (X̃, x̃) such that p ∘ f̃ = f.

5.4. Example. Suppose X is semilocally simply connected and locally path-connected. Certainly every covering (in the classical sense) over X is a lpc_{0}-covering. Thus the traditional classification of covering spaces guarantees that every subgroup of X is a lpc_{0}-covering subgroup. As a consequence, the category Cov_{lpc_{0}}(X) of lpc_{0}-coverings is precisely the usual category of connected coverings over X.

5.5. Example. It is known that if X is a metric space with Lebesgue dimension equal to 1 or if X embeds in ℝ^{2}, then X admits a universal lpc_{0}-covering, or equivalently U(X, lpc_{0}) = 1 [16]. Thus spaces such as the Hawaiian earring, Menger curve, and Sierpinski triangle all admit a universal lpc_{0}-covering which, for many practical purposes, is a suitable replacement for a traditional universal covering.

We now seek to identify the topology of lpc_{0}-covering spaces. The authors of [16] show that the so-called “whisker topology” construction, used in classical covering theory [22], provides a candidate one might use to construct lpc_{0}-coverings. We confirm that all lpc_{0}-coverings arise from the whisker topology.

Fix (X, x_{0}) ∈ bTop_{0} and a subgroup H ⊆ π_{1}(X, x_{0}). Let X̃_{H} = P (X, x_{0})/ ∼ where α ∼ β if and only if α(1) = β(1) and [α · β^{−}] ∈ H. The equivalence class of α is denoted [α]_{H}. We give X̃_{H} the whisker topology, which is generated by the sets

B([α]_{H}, U) = {[α · ϵ]_{H}|ϵ([0, 1]) ⊆ U}

where U is an open neighborhood of α(1). Note that if [β]_{H} ∈ B([α]_{H}, U), then B([α]_{H}, U) = B([β]_{H}, U). In the case that H = 1 is the trivial subgroup, we just write X̃. The basepoint of X̃_{H} is taken to be class x̃_{H} = [c_{x_{0}}]_{H} of the constant path c_{x_{0}} at x_{0}. The map p_{H} : X̃_{H} → X is defined to be the endpoint projection p_{H}([α]_{H}) = α(1).

When X is locally path-connected and semilocally simply-connected, p_{H} : X̃_{H} → X, p_{H}(x̃_{H}) = x_{0} is the classical covering space of X which corresponds to the subgroup H ⊆ π_{1}(X, x_{0}) under the usual classification of coverings [22]. When X is not semilocally simply-connected, p_{H} : X̃_{H} → X may not be a covering map but still has the potential to be a lpc_{0}-covering map. The general properties of p_{H} are studied in great detail in [16].

5.6. Lemma. [16] The space X̃_{H} is path-connected and locally path-connected. The endpoint projection p_{H} : X̃_{H} → X is a continuous surjection. If X is locally path-connected, then p_{H} : X̃_{H} → X, p_{H}([α]_{H}) = α(1) is an open map.

An important feature of the whisker topology is that many maps f : (Y, y) → (X, x) are guaranteed to have at least one lift to X̃_{H}. The following result is extracted from classical covering space theory.

5.7. Lemma. [16, Lemma 2.4] Suppose Y is path-connected and locally path-connected, y_{0} ∈ Y, α ∈ P (X, x_{0}), and f : (Y, y_{0}) → (X, α(1)) is a map such that f_{∗}(π_{1}(Y, y_{0})) ⊆ [α^{−}]H[α]. Then there is a continuous map f̃: (Y, y_{0}) → (X̃_{H}, [α]_{H}) such that p_{H} ∘ f̃ = f defined by f̃(y) = [α · (f ∘ τ)]_{H} where τ : [0, 1] → Y is any path from y_{0} to y.

The lift f̃ : (Y, y_{0}) → (X̃_{H}, x̃_{H}) of a map f : (Y, y_{0}) → (X, x_{0}) defined in Lemma 5.7 is called the standard lift of f. In particular, every path α : ([0, 1], 0) → (X, x_{0}) has a standard lift α̃_{𝒮} : [0, 1] → X̃_{H} starting at x̃_{H} defined as α̃_{𝒮} (t) = [α_{t}]_{H} where α_{t}(s) = α(st).

5.8. Corollary. H ⊆ (p_{H})_{∗} (π_{1}(X̃_{H}, x̃_{H})).

Proof. If [α] ∈ H, then the standard lift α̃_{𝒮} is a loop based at x̃_{H} since α̃_{𝒮} (1) = [α_{1}]_{H} =

[α]_{H} = x̃_{H}. Thus (p_{H})_{∗} (\left[{ \atop α_{𝒮}}\right]) = [α].

5.9. Lemma. For any subgroup H ⊆ π_{1}(X, x_{0}), the following are equivalent:

1. p_{H} : X̃_{H} → X is a lpc_{0}-covering,

2. p_{H} : X̃_{H} → X has the unique path lifting property,

3. (p_{H})_{∗} (π_{1}(X̃_{H}, x̃_{H})) = H.

Proof. (1) ⇒ (2) is obvious. (2) ⇒ (3) Suppose p_{H} : X̃_{H} → X has the unique path lifting property. By Corollary 5.8, one inclusion is clear. Suppose γ : [0, 1] → X̃_{H} is a loop based at x̃_{H}. Let α = p_{H} ∘ γ. By uniqueness of path lifting, γ = α̃_{𝒮} . In particular, α̃_{𝒮} (1) = [α]_{H} = x̃_{H}, which implies (p_{H})_{∗}([γ]) = [α] ∈ H.

(3) ⇒ (2) Suppose (p_{H})_{∗} (π_{1}(X̃_{H}, x̃_{H})) = H. Suppose α ∈ P (X, x_{0}) is a path such

that there is a lift β : [0, 1] → X̃_{H}, β(0) such that β(1) ≠ α̃_{𝒮} (1) = [α]_{H}. Let β(t) = [β_{t}]_{H} and γ = β_{1} and note β · \binom{}{γ_{𝒮}}^{−} is a well-defined loop based at x̃_{H}. By assumption, [α · β_{1}^{−}] = (p_{H})_{∗} ([β · \binom{}{γ_{𝒮}}^{−}]) ∈ H. Thus β(1) = [β_{1}]_{H} = [α]_{H} = α(1) which is a contradiction.

(2) & (3) ⇒ (1). Let Y ∈ lpc_{0}, y ∈ Y, α ∈ P (X, x_{0}), and f : (Y, y) → (X, x) such that α(1) = x and f_{∗}(π_{1}(Y, y)) ⊆ (p_{H})_{∗}(π_{1}(X̃_{H}, [α]_{H})). We have

(p_{H})_{∗}(π_{1}(X̃_{H}, [α]_{H})) = [α^{−}](p_{H})_{∗}(π_{1}(X̃_{H}, x̃_{H}))[α] = [α^{−}]H[α].

By Lemma 5.7, there is a map f̃: (Y, y) → (X̃_{H}, [α]_{H}) such that p_{H} ∘ f̃ = f. Since p_{H} has the unique path lifting property and Y is path-connected, f̃ is unique.

We now verify that these are the only candidates for lpc_{0}-covering maps.

5.10. Lemma. Suppose p : (X̂, x̂) → (X, x_{0}) is a based lpc_{0}-covering and p_{∗}(π_{1}(X̂, x̂)) = H. Then there is a homeomorphism p̃: (X̂, x̂) → (X̃_{H}, x̃_{H}) such that p_{H} ∘ p̃ = p.

Proof. Since p_{∗}(π_{1}(X̂, x̂)) = H ⊆ (p_{H})_{∗}(π_{1}(X̃_{H}, x̃_{H})) (recall Corollary 5.8), there is by Lemma 5.7 a continuous standard lift p̃ : (X̂, x̂) → (X̃_{H}, x̃_{H}) such that p_{H} ∘ p̃ = p. Suppose [α]_{H} ∈ X̃_{H} and α̃ : ([0, 1], 0) → (X̂, x̂) is the unique lift of α such that p∘α̃x̂ = α. By definition of the standard lift, p̃(α̃\overline{x}(1)) = [α]_{H}. Thus p̃ is surjective.

Suppose τ_{1}, τ_{2} : ([0, 1], 0) → (X̂, x̂) are paths such that p̃(τ_{1}(1)) = [p∘τ_{1}]_{H} = [p∘τ_{2}]_{H} = p̃(τ_{2}(1)). Then

p_{∗}([τ_{1} · τ_{2}^{−}]) = [(p ∘ τ_{1}) · (p ∘ τ_{2})^{−}] ∈ H = p_{∗}(π_{1}(X̂, x̂)).

By unique path lifting of p, it follows that τ_{1}(1) = τ_{2}(1) and thus p̃ is injective.

It suffices to check that p̃ is an open map. To obtain a contradiction, suppose V is an open neighborhood in X̂ such that p̃(V) is not open in X̃_{H}. Pick a path β : ([0, 1], 0) → (X̂, x̂) such that β(1) ∈ V and no neighborhood of p̃(β(1)) = [p∘β]_{H} is contained in p̃(V). Let {U_{j}|j ∈ J} be a neighborhood base at x = p(β(1)), which is directed by inclusion (j ≤ k if and only if U_{k} ⊆ U_{j}). Since {B([p ∘ β]_{H}, U_{j})|j ∈ J} is a neighborhood base at [p∘β]_{H}, there is a path ϵ_{j} : ([0, 1], 0) → (U_{j}, x) such that [(p∘β)·ϵ_{j}]_{H} ∈ B([p∘β]_{H}, U_{j})\p̃(V) for each j ∈ J. Let x_{j} = ϵ_{j}(1) ∈ U_{j}.

The paths ϵ_{j} combine to give a map ϵ : (H(J), a_{0}) → (X, x) on the arc-hedgehog. Since H(J) is path-connected, locally path-connected and simply-connected, there is a unique lift ϵ̂: (H(J), a_{0}) → (X̂, β(1)) (such that p ∘ ϵ̂ = ϵ). Since V is an open neighborhood of β(1), there is a k ∈ J such that ϵ̂ maps the k-th arc of H(J) in to V, i.e. ϵ̂([0, 1]_{k}) ⊂ V. If ϵ̂_{k} is the path which is the restriction of ϵ̂ to [0, 1]_{k}, then ϵ̂_{k}(1) ∈ V and p ∘ ϵ̂_{k} = ϵ_{k}. But this implies

[(p ∘ β) · ϵ_{k}]_{H} = [(p ∘ β) · (p ∘ ϵ̂_{k})]_{H} = [p ∘ (β · ϵ̂_{k})]_{H} = p̃(β · ϵ̂_{k}(1)) ∈ p̃(V)

which is a contradiction.

Combining Lemmas 5.9 and 5.10, we obtain the following theorem which, for the locally path-connected category, answers the Structure Question posed in the introduction: the topology of any lpc_{0}-covering space is equivalent to the whisker topology.

5.11. Theorem. For any subgroup H ⊆ π_{1}(X, x_{0}), the following are equivalent:

1. H is a lpc_{0}-covering subgroup of π_{1}(X, x_{0}),

2. p_{H} : X̃_{H} → X is a lpc_{0}-covering,

3. p_{H} : X̃_{H} → X has the unique path lifting property,

4. (p_{H})_{∗} (π_{1}(X̃_{H}, x̃_{H})) = H.

It is not true that every Δ-covering map is a lpc_{0}-covering map. For instance, if X is locally path-connected but not Δ-generated, then the identity map X → X is a lpc_{0}-covering map but not a Δ-covering. We confirm that every Δ-covering map is also a lpc_{0}-covering map when X is first countable.

5.12. Lemma. Suppose X is first countable and p : \overline{X} → X is a map. Then p is a Δ-covering map if and only if p is a lpc_{0}-covering map.

Proof. First, suppose p : \overline{X} → X is a lpc_{0}-covering map. By Theorem 5.11, \overline{X} ≅ X̃_{H} where H = p_{∗}(π_{1}(\overline{X}, \overline{x})). But if X is first countable, then so is X̃_{H} (by the definition of the whisker topology). Since X̃_{H} is also locally path-connected, it is Δ-generated by Lemma 4.2. Since every Δ-generated space is locally path-connected, condition 2. of Definition 4.3 is straightforward to verify. Therefore, p is a Δ-covering map.

For the converse, suppose p : \overline{X} → X is a Δ-covering such that p_{∗}(π_{1}(\overline{X}, \overline{x})) = H. Since \overline{X} is locally path-connected, p : \overline{X} → X is continuous, and p_{∗}(π_{1}(\overline{X}, \overline{x})) = H, there is by Lemma 5.7, a map p̃: \overline{X} → X̃_{H} such that p_{H} ∘ p̃ = p. By definition, if γ : [0, 1] → \overline{X} is a path from \overline{x} to z and α = p ∘ γ, then p̃(γ(t)) = α̃_{𝒮} (t) = [α_{t}]_{H} where α_{t}(s) = α(st) is the path given by restricting α to [0, t]. We show in the next paragraph that p̃ is a homeomorphism. Once this is done, it is clear that p_{H} has the unique path lifting property and is therefore a lpc_{0}-covering (by Theorem 5.11) which is equivalent to p.

Our proof that p̃ is a homeomorphism is quite similar to the proof of Lemma 5.10. Verifying bijectivity is identical so we leave it to the reader. We check that p̃ is open: To obtain a contradiction, suppose V is open in \overline{X} and p̃(V) is not open in X̃_{H}. Pick a path β : ([0, 1], 0) → (\overline{X}, \overline{x}) such that β(1) ∈ V and no neighborhood of p̃(β(1)) = [p ∘ β]_{H} is contained in p̃(V). Let {U_{n}|n ≥ 1} be a countable neighborhood base at x = p(β(1)). Since {B([p ∘ β]_{H}, U_{n})|n ≥ 1} is a neighborhood base at [p ∘ β]_{H} and p̃(V) is not open, there is a path ϵ_{n} : ([0, 1], 0) → (U_{n}, x) such that [(p ∘ β) · ϵ_{n}]_{H} ∈ B([p ∘ β]_{H}, U_{n})\p̃(V) for each n ≥ 1. Let x_{n} = ϵ_{n}(1) ∈ U_{n}.

The paths ϵ_{n} combine to give a map ϵ : (H(ω), a_{0}) → (X, x) on the ω-arc-hedgehog. Since H(ω) is Δ-generated and simply-connected, there is a unique lift \overline{ϵ} : (H(ω), a_{0}) → (\overline{X}, β(1)) (such that p ∘ \overline{ϵ} = ϵ). Since V is an open neighborhood of β(1), there is a N such that \overline{ϵ} maps the n-th arc of H(ω) in to V for all n ≥ N. If \overline{ϵ}_{n} is the path which is the restriction of \overline{ϵ} to [0, 1]_{n}, then \overline{ϵ}_{n}(1) ∈ V and p ∘ \overline{ϵ}_{n} = ϵ_{n}. But this implies

[(p ∘ β) · ϵ_{n}]_{H} = [(p ∘ β) · (p ∘ \overline{ϵ}_{n})]_{H} = [p ∘ (β · \overline{ϵ}_{n})]_{H} = p̃(β · \overline{ϵ}_{n}(1)) ∈ p̃(V)

for some n, which is a contradiction.

Combining the previous lemma with Theorem 5.11, we obtain the following comparison for first countable spaces. This comparison states implies that for a first countable space X, the subgroup structure of π_{1}(X, x_{0}) retained by all disk-coverings, Δ-coverings, and lpc_{0}-coverings is exactly the same.

5.13. Theorem. If X is first countable, then Cov_{lpc_{0}}(X) = ΔCov(X). Moreover, the following diagram of functors commutes up to natural isomorphism.

[figure: DCov(X); μ; Δ; ΔCov(X); μ; Cov_{lpc_{0}}(X); μ; GSet]

The study of wild fundamental groups is typically focused on the fundamental groups of metric spaces. The previous theorem implies that any covering-theoretic attempt to characterize the subgroup structure of the fundamental group of a metric space is retained or improved upon by the lpc_{0}-coverings introduced in [16]. In particular, there can be no mysterious disk-covering subgroup H ⊆ π_{1}(X, x_{0}) for which the classical endpoint projection p_{H} : X̃_{H} → X fails to have unique path lifting. The author does not know of an example of a non-first countable, Δ-generated space X and subgroup H ⊆ π_{1}(X, x_{0}) such that H is a Δ-covering subgroup but not a lpc_{0}-covering subgroup.

5.14. Example. Let ℍ = ⋃_{n≥1}{(x, y) ∈ ℝ^{2}|(x − 1/n)^{2} + y^{2} = 1/n^{2}} be the standard Hawaiian earring space and ℓ_{n} : S^{1} → ℍ be a simple closed curve traversing the n-th circle. We consider the subgroup F_{∞} ⊆ π_{1}(ℍ, (0, 0)) freely generated by the homotopy classes of the loops ℓ_{n}. There does not exist any subcategory C ⊆ Top_{0} such that F_{∞} is a C-covering subgroup. If such a subcategory were to exist, then F_{∞} would be a Δ-covering subgroup and therefore a lpc_{0}-subgroup since ℍ is first countable. But by Theorem 5.11, this would imply that p_{F_{∞}} : X̃_{F_{∞}} → X has unique path-lifting, which is false: see Example 6.2 of [16]

5.15. Example. For an even more extreme example consider any non-simply connected, first countable space X such every non-trivial homotopy class has a representative in an arbitrary neighborhood of the basepoint x_{0}. Such spaces are considered in detail in [24, 25] and include the famous harmonic archipelago and Griffiths twin cone spaces. It is wellknown that the only subgroup H ⊆ π_{1}(X, x_{0}) for which p_{H} : X̃_{H} → X has the unique path lifting property is when H = π_{1}(X, x_{0}). Using the same reasoning as in the previous example, we can conclude that given any C, the only C-covering subgroup of π_{1}(X, x_{0}) is π_{1}(X, x_{0}) itself. Thus one cannot possibly distinguish any elements of π_{1}(X, x_{0}) using any theory of generalized coverings.

6. Continuous lifting: Fan-coverings

6.1. Definition. Let (J, ≤) be a directed set and K = J ∪ {∞} be obtained by adding one maximal point. Give K the topology generated by basis sets {k} and V_{k} = {∞}∪{j ∈ J|j > k} for k < ∞. The directed arc-fan over J is the cone over K, i.e. the quotient space F (J) = K × [0, 1]/K × {0} with basepoint b_{0}, the image of K × {0} in the quotient.

We denote the k-th arc of F (J) (i.e. the image of {k} × [0, 1]) by [0, 1]_{k} and take the point t_{k} to be the image of {(k, t)} in [0, 1]_{k}. The restriction of a map f : F (J) → X to [0, 1]_{k} is a path f_{k} : [0, 1]_{k} → X. Observe that maps f : (F (J), b_{0}) → (X, x) are in bijective correspondence with convergent nets f_{j} → f_{∞} in P (X, x) indexed by J.

Let F be the category whose objects are D^{2} and the directed fans F (J) for all directed sets J. Let Fan = ℋ (F) be the coreflective hull of F in Top_{0}. Note that the archedgehog H(J) = F (J)/({∞} × [0, 1]) is a quotient of F (J) and thus lpc_{0} ⊂ Fan. The directed fan F (ω) is a non-locally path-connected continuum. Thus lpc_{0} is a proper subcategory of Fan.

6.2. Lemma. A path-connected space X is an object of Fan if and only if for some (equivalently any) x ∈ X, the map ev : P (X, x) → X, ev(α) = α(1) is quotient.

Proof. First note that if y ∈ X and β : [0, 1] → X is any path from y to x, then ev : P (X, x) → X factors as the composition of the left concatenation map λ_{β} : P (X, x) → P (X, y), λ_{β}(α) = β · α and evaluation ev : P (X, y) → X. Thus if ev : P (X, x) → X is quotient, then ev : P (X, y) → X is quotient for all y ∈ X.

Suppose ev : P (X, x) → X is quotient for all x ∈ X. We show that X ∈ Fan. Suppose U ⊆ X is a set such that for every directed set J and map f : F (J) → X, f^{−1}(U) is open in F (J). Fix x ∈ X. We check that ev^{−1}(U) is open in P (X, x). If ev^{−1}(U) is not open, then there is a convergent net f_{j} → f_{∞}, j ∈ J in P (X, x) such that f_{∞} ∈ ev^{−1}(U) and f_{j} ∉ ev^{−1}(U) for every j ∈ J. This net uniquely induces a map f : (F (J), b_{0}) → (X, x) for which f^{−1}(U) is open by assumption. But 1_{∞} ∈ [0, 1]_{∞} ∩ f^{−1}(U) which implies the existence of j_{0} such that 1_{j} ∈ f^{−1}(U) for all j > j_{0}. This contradicts the assumption that f_{j}(1) = f(1_{j}) ∉ U for all j ∈ J. Therefore, ev^{−1}(U) must be open in P (X, x). Since ev is quotient, U is open in X. This proves X ∈ Fan.

For the converse, suppose X ∈ Fan, x ∈ X, and U ⊆ X is such that ev^{−1}(U) is open in P (X, x). To see that ev is quotient, we check that U is open. If U is not open, then there is a map f : (F (J), b_{0}) → (X, y) on a directed arc-fan such that f^{−1}(U) is not open in F (J). Pick a point z ∈ f^{−1}(U) such that no neighborhood of z is contained in f^{−1}(U).

Case I: Suppose z = b_{0}. Since F (J) is locally path-connected at b_{0}, there is a directed set K and a map h : (H(K), a_{0}) → (F (J), b_{0}) such that h(1_{k}) ∉ f^{−1}(U) for every k ∈ K. But since H(K) ∈ Fan and f ∘ h : (H(K), a_{0}) → (X, y) is continuous, h^{−1}(f^{−1}(U)) is an open neighborhood of a_{0} in H(K). Thus there is a k_{0} ∈ K such that 1_{k} ∈ h^{−1}(f^{−1}(U)) for all k > k_{0}. This contradicts the fact that h(1_{k}) ∉ f^{−1}(U) for every k ∈ K. Thus U must be open.

Case II: Suppose z ≠ b_{0}. There is a directed set L and a net z_{ℓ} → z_{∞} = z, ℓ ∈ L in F (J) such that z_{ℓ} ∉ f^{−1}(U) for every ℓ ∈ L. Since z ≠ b_{0}, we may assume z_{ℓ} has image in the open set F (J)\{b_{0}} ≅ (J ∪ {∞}) × (0, 1]. Thus for each ℓ ∈ L ∪ {∞}, we have z_{ℓ} = t_{j(ℓ)} ∈ (0, 1]_{j(ℓ)} for some j(ℓ) ∈ J. Define a function h : (F (L), b_{0}) → (F (J), b_{0}) so that for all ℓ ∈ L ∪ {∞}, the restriction of h to [0, 1]_{ℓ} is the linear path h(s_{ℓ}) = (st)_{j(ℓ)}. Since [0, 1] is locally compact and L ∪ {∞} is Hausdorff, the continuity of h follows directly from exponential laws for maps on pairs of spaces. Note that h(1_{ℓ}) = z_{ℓ} for all ℓ ∈ L ∪ {∞}. Since f and h are continuous, (f ∘ h)_{ℓ} → (f ∘ h)_{∞} is a convergent net of paths in P (X, y). Now if γ is any path in X from x to y, then γ · (f ∘ h)_{ℓ} → γ · (f ∘ h)_{∞} in P (X, x). Since γ ·(f ∘h)_{∞}(1) = f(z) ∈ U, we have γ ·(f ∘h)_{∞} ∈ ev^{−1}(U). By assumption, ev^{−1}(U) is open so there is an ℓ_{0} such that γ · (f ∘ h)_{ℓ} ∈ ev^{−1}(U) for all ℓ > ℓ_{0}. However, this contradicts the fact that γ · (f ∘ h)_{ℓ}(1) = f(z_{ℓ}) ∉ U for every ℓ ∈ L. Thus U must be open.

6.3. Corollary. Suppose fan : Top_{0} → Fan is the coreflection functor. The topology of fan(X) is the quotient topology on X with respect to evaluation ev : P (X, x) → X for any x ∈ X.

Proof. Pick any x ∈ X. The continuous identity fan(X) → X is a disk-covering and therefore induces a continuous bijection b : P (fan(X), x) → P (X, x) on path spaces. A convergent net α_{j} → α, j ∈ J in P (X, x) corresponds to a continuous map f : F (J) → X on a directed arc-fan for which f : F (J) → fan(X) is also continuous since F (J) ∈ Fan. Thus α_{j} also converges to α in P (fan(X), x). Consequently, b is a homeomorphism. By Lemma 6.2, ev : P (fan(X), x) → fan(X) is quotient. Composing b^{−1} with this quotient map completes the proof.

6.4. Theorem. Every contractible space is an object of Fan. Consequently, if Cntr is the category of contractible spaces, then ℋ (Cntr) = Fan.

Proof. Since every directed arc-fan is the cone over a convergent net, we have F ⊂ Cntr and thus Fan ⊂ ℋ (Cntr). Recall that every contractible space Z is the retract (and therefore quotient) of the cone CZ = Z × [0, 1]/Z × {0} over Z. Since coreflective hulls are closed under quotients, it suffices to show CZ ∈ Fan for any (possibly non-path-connected) space Z. Let t_{z} denote the image of (z, t) in CZ and v = 0_{z} be the vertex. Using the exponential law for based path spaces it is straightforward to verify that the map f : CZ → P (CZ, v), f(t_{z})(s) = (st)_{z} is continuous. If ev : P (CZ, v) → CZ is evaluation, then ev ∘f = id, showing that ev is a retraction and therefore a quotient map. It follows from Lemma 6.2 that CZ ∈ Fan.

6.5. Definition. [3] A map f : X → Y has continuous lifting of paths if for every x ∈ X, the induced map f_{#} : P (X, x) → P (Y, f(x)) is a homeomorphism.

6.6. Proposition. The following are equivalent for a map p : X̂ → X:

1. p has continuous lifting of paths,

2. p is a weak F-covering map,

3. p is a weak Cntr-covering map.

Proof. (1) ⇔ (2) Suppose x̂ ∈ X̂, and p(x̂) = x. First observe that p_{#} : P (X̂, x̂) → P (X, x) is a homeomorphism if and only if every map f : (F (J), b_{0}) → (X, x) on a directed arc-fan lifts uniquely to a map f̂ : (F (J), b_{0}) → (X̂, x̂) such that p ∘ f̂ = f. Since directed arc-fans are simply connected, it is immediate that every F-covering has continuous lifting of paths. It follows directly from exponential laws of mapping spaces that every map with continuous lifting of paths also is a disk-covering [5, Remark 2.5].

(2) ⇔ (3) We prove the more general fact that if ℋ (C) = ℋ (D), then p : X̂ → X is a weak C-covering if and only if p is a weak D-covering. By symmetry, we only need to prove one direction. Suppose p is a weak C-covering and f : (Y, y) → (X, x) is a map with Y ∈ D and f_{∗}(π_{1}(Y, y)) ⊆ p_{∗}(π_{1}(X̂, x̂)). Since p is a disk-covering it suffices to show the unique function f̂ : (Y, y) → (X̂, x̂) such that p ∘ f̂ = f is continuous. If c : Top_{0} → ℋ (C) is the coreflection, then by Lemma 2.21 ψ(p) : c(X̂) → X is a ℋ (C)- covering. Since Y ∈ ℋ (D) = ℋ (C), the unique lift f̂: (Y, y) → (c(X̂), x̂) is continuous. Composing this map with the continuous identity id : c(X̂) → X̂ shows that f̂: Y → X̂ is continuous.

\xymatrix{c(X̂) \ar[dr]^{ψ(p)} \ar[r]^{id} & X̂ \ar[d]^{p} \\ Y \ar[r]_{f} \ar@{-->}[u]^{f̂} & X}

6.7. Corollary. A map p : X̂ → X is a Fan-covering map if and only if X̂ ∈ Fan and p has continuous lifting of paths.

In the previous section, we identified the topology of lpc_{0}-coverings as the whisker topology. In the same fashion, we identify the topology of Fan-coverings as the natural quotient topology. Let X̃_{H}^{qtop} denote the space with the same underlying set of X̃_{H} (used in the previous section) with the quotient topology with respect to the map q : P (X, x_{0}) → X̃_{H}, q(α) = [α]_{H}.

Using exponential properties of mapping spaces, it is shown in [3] that the endpoint projection p_{H} : X̃_{H}^{qtop} → X, p_{H}([α]_{H}) = α(1) induces a retraction (p_{H})_{#} : P (X̃_{H}^{qtop}, x̃_{H}) → P (X, x_{0}) on path spaces. The canonical section s : P (X, x_{0}) → P (X̃_{H}^{qtop}, x̃_{H}) is given by s(α) = α̃_{𝒮} , the standard lifts of paths starting at x̃_{H}. Thus the inclusion H ⊆ (p_{H})_{∗} (π_{1} (X̃_{H}^{qtop}, x̃_{H})) holds in general. Since a retraction is a homeomorphism if and only if it is injective, the next lemma follows immediately.

6.8. Lemma. [3, Theorem 7.7] The endpoint projection p_{H} : X̃_{H}^{qtop} → X, p_{H}([α]_{H}) = α(1) has continuous lifting of paths if and only if it has the unique path lifting property.

6.9. Lemma. Suppose Y ∈ Fan, y_{0} ∈ Y, α ∈ P (X, x_{0}), and f : (Y, y_{0}) → (X, α(1)) is a map such that f_{∗}(π_{1}(Y, y_{0})) ⊆ [α^{−}]H[α]. Then there is a continuous map f̃: (Y, y_{0}) → (X̃_{H}^{qtop}, [α]_{H}) such that p_{H} ∘ f̃ = f defined by f̃(y) = [α · (f ∘ τ)]_{H} where τ : [0, 1] → Y is any path from y_{0} to y.

Proof. As in Lemma 5.7, f̃ is well-defined; we are left to verify continuity. Consider the following commuting diagram.

\xymatrix{P (Y, y_{0}) \ar[d]_{ev} \ar[r]^{f_{#}} & P (X, f(y_{0})) \ar[r]^{λ_{α}} & P (X, x_{0}) \ar[d]^{q} \\ Y \ar[rr]_{f̃} &  & X̃_{H}^{qtop}}

The map λ_{α}(β) = α · β is left concatenation by α. Each function in the top composition is continuous and the left vertical map is a quotient map by Lemma 6.2. Thus f̃ is continuous by the universal property of quotient spaces.

6.10. Lemma. If p : X̂ → X is a Fan-covering and H = p_{∗}(π_{1}(X̂, x̂)) ⊆ π_{1}(X, x_{0}), then there is a homeomorphism p̂: X̂ → X̃_{H}^{qtop} such that p_{H} ∘ p̂ = p.

Proof. As in Lemma 5.10, there is a canonical bijection p̂: X̂ → X̃_{H}^{qtop} defined as follows. If γ is any path in X̂ starting at x_{0}, then p̂(γ(1)) = [p∘γ]_{H}. Consider the following diagram which commutes since if γ ∈ P (X̂, x̂), then p̂(ev(γ)) = p̂(γ(1)) = [p ∘ γ]_{H} = q(p_{#}(γ)).

\xymatrix{P (X̂, x̂) \ar[r]_{≅}^{p_{#}} \ar[d]_{ev} & P (X, x_{0}) \ar[d]^{q} \\ X̂ \ar[r]_{p̂} & X̃_{H}^{qtop}}

Since p has continuous lifting of paths (Corollary 6.7), the top map is a homeomorphism. Since X̂ ∈ Fan, the left vertical map is quotient by Lemma 6.2. The right vertical map is quotient by construction. Therefore the bottom horizontal map must be a continuous quotient map. Since p̂ is bijective, it is a homeomorphism.

As we did for the locally path-connected case, we can now answer the Structure Question for the category Fan: the topology of any Fan-covering space is equivalent to the natural quotient topology.

6.11. Theorem. For any subgroup H ⊆ π_{1}(X, x_{0}), the following are equivalent:

1. H is a Fan-covering subgroup of π_{1}(X, x_{0}),

2. p_{H} : X̃_{H}^{qtop} → X is a Fan-covering,

3. p_{H} : X̃_{H}^{qtop} → X has the unique path lifting property,

4. (p_{H})_{∗} (π_{1}(X̃_{H}^{qtop}, x̃_{H})) = H.

Proof. Using Lemma 6.9, the proof of (2) ⇔ (3) ⇔ (4) is identical to the proof of Lemma 5.9. (1) ⇒ (2) follows directly from Lemma 6.10. (2) & (4) ⇒ (1) is by definition.

GENERALIZED COVERING SPACE THEORIES  1161

6.12. Remark. In general, it is known that the whisker topology on X̃_{H} is finer than the quotient topology of X̃_{H}^{qtop} [26]. Our characterization of the topology of lpc_{0}- and Fancoverings indicates that if H is a Fan-covering subgroup of π_{1}(X, x_{0}), then lpc \binom{}{X_{H}^{qtop}} ≅ X̃_{H}.

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1162  JEREMY BRAZAS

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Department of Mathematics & Statistics; Georgia State University, Atlanta, Georgia 30303

Email: jbrazas@gsu.edu
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