GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 5, 1998, 475-482

COVERINGS AND RING-GROUPOIDS

OSMAN MUCUK

Abstract. We prove that the set of homotopy classes of the paths in a topological ring is a ring object (called ring groupoid). Using this concept we show that the ring structure of a topological ring lifts to a simply connected covering space.

Introduction

Let X be a connected topological space, X̃ a connected and simply connected topological space, and let p: X̃ → X be a covering map. We call such a covering simply connected. It is well known that if X is a topological group, e is the identity element of X, and ẽ ∈ X̃ such that p(ẽ) = e, then X̃ becomes a topological group such that p: X̃ → X is a morphism of topological groups. In that case we say that the group structure of X lifts to X̃. This can be proved by the lifting property of the maps on covering spaces (see, for example, [1]).

In the non-connected case the situation is completely different and was studied by R. L. Taylor [2] for the first time. Taylor obtained an obstruction class k_{X} from the topological space X and proved that the vanishing of k_{X} is a necessary and sufficient condition for the lifting of the group structure of X to X̃ as described above. In [3] this result was generalized in terms of group-groupoids, i.e., group objects in the category of groupoids, and crossed modules, and then written in a revised version in [4].

In this paper we give a similar result: Let X and X̃ be connected topological spaces and p: X̃ → X a simply connected covering. If X is a topological ring with identity element e, and ẽ ∈ X̃ such that p(ẽ) = e, then the ring structure of X lifts to X̃. That is, X̃ becomes a topological ring with identity ẽ ∈ X̃ such that p: X̃ → X is a morphism of topological rings. For this the following helps us:

In [5] Brown and Spencer defined the notion of a group-groupoid. They also proved that if X is a topological group, then the fundamental groupoid

475 π_{1}X, which is the set of all rel end points homotopy classes of paths in the topological space X, becomes a group-groupoid.

1991 Mathematics Subject Classification. 22A05, 55M99, 55R15.

Key words and phrases. Fundamental groupoid, coverings, ring-groupoid.

1072-947X/98/0900-0475$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

We introduce here the notion of a ring-groupoid, which is a ring object in the category of groupoids.

On the other hand, in [6] it was proved by Brown that if X is a semilocally simply connected topological space, i.e., each component has a simply connected covering, then the category TCov/X of topological coverings of X is equivalent to the category GpdCov/π_{1}X of groupoid coverings of the fundamental groupoid π_{1}X.

In addition to this, in [3] it was proved that if X is a topological group whose underlying space is semilocally simply connected, then the category TGCov/X of topological group coverings of X is equivalent to the category GpGpdCov/π_{1}X of group-groupoid coverings of π_{1}X.

Here we prove that if X is a topological ring, whose underlying space is semilocally simply connected, then the category TRCov/X of topological ring coverings of X is equivalent to the category RGpdCov/π_{1}X of ringgroupoid coverings of π_{1}X.

1. Ring-Groupoids

A groupoid G is a small category in which each morphism is an isomorphism. Thus G has a set of morphisms, which we call elements of G, a set O_{G} of objects together with functions α, β: G → O_{G}, ϵ: O_{G} → G such that αϵ = βϵ = 1. The functions α, β are called initial and final maps respectively. If a, b ∈ G and βa = αb, then the product or composite ba exists such that α(ba) = α(a) and β(ba) = β(b). Further, this composite is associative, for x ∈ O_{G} the element ϵx denoted by 1_{x} acts as the identity, and each element a has an inverse a^{−1} such that α(a^{−1}) = βa, β(a^{−1}) = α(a), a^{−1}a = ϵαa, aa^{−1} = ϵβa

In a groupoid G, for x, y ∈ O_{G} we write G(x, y) for the set of all morphisms with initial point x and final point y. We say G is transitive if for all x, y ∈ O_{G}, G(x, y) is not empty. For x ∈ O_{G} we denote the star {a ∈ G: αa = x } of x by G^{x}. The object group at x is G(x) = G(x, x). Let G be a groupoid. The transitive component of x ∈ O_{G} denoted by C(G)_{x} is the full subgroupoid of G on those objects y ∈ O_{G} such that G(x, y) is not empty.

A morphism of groupoids G̃ and G is a functor, i.e., it consists of a pair of functions f: G̃ → G, O_{f}: O_{G} → O_{G} preserving all the structure.

Covering morphisms and universal covering groupoids of a groupoid are defined in [6] as follows:

Let f: G̃ → G be a morphism of groupoids. Then f is called a covering morphism if for each x̃ ∈ O_{G}, the restriction G̃_{x} → G^{f_{x}} of f is bijective.

A covering morphism f: G̃ → G of transitive groupoids is called universal if G̃ covers every covering of G, i.e., if for every covering morphism a: A → G there is a unique morphism of groupoids a′: G̃ → A such that aa′ = f (and hence a′ is also a covering morphism). This is equivalent to saying that for x̃, ỹ ∈ O_{G} the set G̃(x̃, ỹ) has one element at most.

We now give

Definition 1.1. A ring-groupoid G is a groupoid endowed with a ring structure such that the following maps are the morphisms of groupoids:

(i) m: G × G → G, (a, b) ↦ a + b, group multiplication,

(ii) u: G → G, a ↦ −a, group inverse map,

(iii) e: (⋆) → G, where (⋆) is a singleton,

(iv) n: G × G → G, (a, b) ↦ ab, ring multiplication.

So by (iii) if e is the identity element of O_{G} then 1_{e} is that of G.

In a ring-groupoid G for a, b ∈ G the groupoid composite is denoted by b ∘ a when α(b) = β(a), the group multiplication by a + b, and the ring multiplication by ab.

Let G̃ and G be two ring-groupoids. A morphism f: G̃ → G from G̃ to G is a morphism of underlying groupoids preserving the ring structure. A morphism f: G̃ → G of ring-groupoids is called a covering (resp. a universal covering) if it is a covering morphism (resp. a universal covering) on the underlying groupoids.

Proposition 1.2. In a ring-groupoid G, we have

(i) (c ∘ a) + (d ∘ b) = (c + d) ∘ (a + b) and

(ii) (c ∘ a)(d ∘ b) = (cd) ∘ (ab).

Proof. Since m is a morphism of groupoids,

(c ∘ a) + (d ∘ b) = m[c ∘ a, d ∘ b] = m[(c, d) ∘ (a, b)] =
= m(c, d) ∘ m(a, b) = (c + d) ∘ (a + b).

Similarly, since n is a morphism of groupoids we have

(c ∘ a)(d ∘ b) = n[c ∘ a, d ∘ b] = n[(c, d) ∘ (a, b)]
= n(c, d) ∘ n(a, b) = (cd) ∘ (ab).

We know from [5] that if X is a topological group, then the fundamental groupoid π_{1}X is a group-groupoid. We will now give a similar result.

Proposition 1.3. If X is a topological ring, then the fundamental groupoid π_{1}X is a ring-groupoid.

Proof. Let X be a topological ring with the structure maps

m: X × X → X, (a, b) ↦ a + b,
n: X × X → X, (a, b) ↦ ab

and the inverse map

u: X → X, a ↦ −a.

Then these maps give the following induced maps:

π_{1}m: π_{1}X × π_{1}X → π_{1}X, ([a], [b]) ↦ [b + a]
π_{1}n: π_{1}X × π_{1}X → π_{1}X, ([a], [b]) ↦ [ba]
π_{1}u: π_{1}X × π_{1}X → π_{1}X, [a] ↦ [−a] = −[a].

It is known from [5] that π_{1}X is a group groupoid. So to prove that π_{1}X is a ring-groupoid we have to show the distributive law: since for a, b ∈ G a(b + c) = ab + ac we have

[a]([b] + [c]) = [a]([b + c]) = [a(b + c)] = [ab + ac] = [ab] + [ac]

Proposition 1.4. Let G be a ring-groupoid, e the identity of O_{G}. Then the transitive component C(G)_{e} of e is a ring-groupoid.

Proof. In [3] it was proved that C(G)_{e} is a group-groupoid. Further it can be checked easily that the ring structure on G makes C(G)_{e} a ring.

Proposition 1.5. Let G be a ring-groupoid and e the identity of O_{G}. Then the star G^{e} = {a ∈ G: α(a) = e} of e becomes a ring.

The proof is left to the reader.

2. Coverings

Let X be a topological space. Then we have a category denoted by TCov/X whose objects are covering maps p: X̃ → X and a morphism from p: X̃ → X to q: Ỹ → X is a map f: X̃ → Ỹ (hence f is a covering map) such that p = qf. Further for X we have a groupoid called a fundamental groupoid (see [6], Ch. 9) and have a category denoted by GpdCov/π_{1}X whose objects are the groupoid coverings p: G̃ → π_{1}X of π_{1}X and a morphism from p: G̃ → π_{1}X to q: H̃ → π_{1}X is a morphism f: G̃ → H̃ of groupoids (hence f is a covering morphism) such that p = qf.

We recall the following result from Brown [6].

Proposition 2.1. Let X be a semilocally simply connected topological space. Then the category TCov/X of topological coverings of X is equivalent to the category GpdCov/π_{1}X of covering groupoids of the fundamental groupoid π_{1}X.

Let X and X̃ be topological groups. A map p: X̃ → X is called a covering morphism of topological groups if p is a morphism of groups and p is a covering map on the underlying spaces. For a topological group X, we have a category denoted by TGCov/X whose objects are topological group coverings p: X̃ → X and a morphism from p: X̃ → X to q: Ỹ → X is a map f: X̃ → Ỹ such that p = qf. For a topological group X, the fundamental groupoid π_{1}X is a group-groupoid and so we have a category denoted by GpGpdCov/π_{1}X whose objects are group-groupoid coverings p: G̃ → π_{1}X of π_{1}X and a morphism from p: G̃ → π_{1}X to q: H̃ → π_{1}X is a morphism f: G̃ → H̃ of group-groupoids such that p = qf.

Then the following result is given in [4].

Proposition 2.2. Let X be a topological group whose underlying space is semilocally simply connected. Then the category TGCov/X of topological group coverings of X is equivalent to the category GpGpdCov/π_{1}X of covering groupoids of the group-groupoid π_{1}X.

In addition to these results, we here prove Proposition 2.3.

Let X and X̃ be topological rings. A map p: X̃ → X is called a covering morphism of topological rings if p is a morphism of rings and p is a covering map on the underlying spaces. So for a topological ring X, we have a category denoted by TRCov/X, whose objects are topological ring coverings p: X̃ → X and a morphism from p: X̃ → X to q: Ỹ → X is a map f: X̃ → Ỹ such that p = qf. Similarly, for a topological ring X, we have a category denoted by RGpdCov/π_{1}X whose objects are ring-groupoid coverings p: G̃ → π_{1}X of π_{1}X and a morphism from p: G̃ → π_{1}X to q: H̃ → π_{1}X is a morphism f: G̃ → H̃ of ring-groupoids such that p = qf.

Let X be a topological ring whose underlying space is semilocally simply connected. Then we prove the following result.

Proposition 2.3. The categories TRCov/X and RGpdCov/π_{1}X are equivalent.

Proof. Define a functor π_{1} : TRCov/X→RGpdCov/π_{1}X

as follows: Let p: X̃ → X be a covering morphism of topological rings. Then the induced morphism π_{1}p: π_{1}X̃ → π_{1}X is a covering morphism of groupgroupoids (see [3]), i.e., it is a morphism of group-groupoids and coverings on the underlying groupoids. Further the morphism π_{1}p preserves the ring structure as follows:

(π_{1}p)[ab] = [p(ab)] = [p(a)p(b)] = [p(a)][p(b)] = (π_{1}p)[a](π_{1}p)[a].

So π_{1}p: π_{1}X̃ → π_{1}X becomes a covering morphism of ring-groupoids. We now define a functor

η: RGpdCov/π_{1}X → TRCov/X

as follows: If q: G̃ → π_{1}X is a covering morphism of ring groupoids, then we have a covering map p: X̃ → X, where p = O_{q} and X̃ = O_{G}. Further p is a morphism of topological groups (see [3]). Further we will prove that the ring multiplication

ñ: X̃ × X̃ → X̃, (a, b) ↦ ab

is continuous.

By assuming that X is semilocally simply connected, we can choose a cover U of simply connected subsets of X. Since the topology X̃ is the lifted topology (see [6], Ch. 9) the set consisting of all liftings of the sets in U forms a basis for the topology on X̃. Let Ũ be an open neighborhood of ẽ and a lifting of U in U. Since the multiplication

n: X × X → X, (a, b) ↦ ab

is continuous, there is a neighborhood V of e in X such that n(V × V) ⊆ U. Using the condition on X and choosing V small enough we can assume that V is simply connected. Let Ṽ be the lifting of V. Then pñ(Ṽ × Ṽ) = n(V × V) ⊆ U and so we have ñ(Ṽ × Ṽ) ⊆ Ũ. Hence

ñ: X̃ × X̃ → X̃, (a, b) ↦ ab

becomes continuous. Since by Proposition 2.2 the category of topological group coverings is equivalent to the category of group-groupoid coverings, the proof is completed by the following diagram:

T RCov/X ⟶^{π_{1}} RGpdCov/π_{1}X
↓  ↓
T GCov/X ⟶^{π_{1}} GpGpdCov/π_{1}X.

Before giving the main theorem we adopt the following definition:

Definition 2.4. Let p: G̃ → G be a covering morphism of groupoids and q: H → G a morphism of groupoids. If there exists a unique morphism q̃: H → G̃ such that p = qq̃ we say q lifts to q̃ by p.

We recall the following theorem from [6] which is an important result to have the lifting maps on covering groupoids.

Theorem 2.5. Let p: G̃ → G be a covering morphism of groupoids, x ∈ O_{G} and x̃ ∈ O_{G} such that p(x̃) = x. Let q: H → G be a morphism of groupoids such that H is transitive and ỹ ∈ O_{H} such that q(ỹ) = x. Then the morphism q: H → G uniquely lifts to a morphism q̃: H → G̃ such that q̃(ỹ) = x̃ if and only if q[H(ỹ)] ⊆ p[G̃(x̃)], where H(ỹ) and G̃(x̃) are the object groups.

Let G be a ring groupoid, e the identity of O_{G}, and let G̃ be just a groupoid, ẽ ∈ O_{G} such that p(ẽ) = e. Let p: G̃ → G be a covering morphism of groupoids. We say the ring structure of G lifts to G̃ if there exists a ring structure on G̃ with the identity element ẽ ∈ O_{G} such that G̃ is a groupgroupoid and p: G̃ → G is a morphism of ring-groupoids.

Theorem 2.6. Let G̃ be a groupoid and G a ring-groupoid. Let p: G̃ → G be a universal covering on the underlying groupoids such that both groupoids G̃ and G are transitive. Let e be the identity element of O_{G} and ẽ ∈ O_{G} such that p(ẽ) = e. Then the ring structure of G lifts to G̃ with identity ẽ.

Proof. Since G is a ring-groupoid as in Definition 1.1 it has the following maps:

m: G × G → G, (a, b) ↦ a + b,
u: G → G, a ↦ −a,
n: G × G → G, (a, b) ↦ ab.

Since G̃ is a universal covering, the object group G̃(ẽ) has one element at most. So by Theorem 2.5 these maps respectively lift to the maps

m̃: G̃ × G̃ → G̃, (ã, b̃) ↦ ã + b̃,
ũ: G̃ → G̃, ã ↦ −ã,
ñ: G̃ × G̃ → G̃, (ã, b̃) ↦ ãb̃

by p: G̃ → G such that

p(ã + b̃) = p(ã) + p(b̃),
p(ãb̃) = p(ã)p(b̃),
p(ũ(ã)) = −(pã).

Since the multiplication m: G × G → G ↦ a + b is asociative, we have m(m × 1) = m(1 × m), where 1 denotes the identity map. Then again by Theorem 2.5 these maps m(m × 1) and m(1 × m) respectively lift to

m̃(m̃ × 1), m̃(1 × m̃): G̃ × G̃ × G̃ → G̃

which coincide on (ẽ, ẽ, ẽ). By the uniqueness of the lifting we have m̃(m̃ × 1) = m̃(1 × m̃), i.e., m̃ is associative. Similarly, ñ is associative. Further the distributive law is satisfied as follows:

Let p_{1}, p_{2} : G × G × G → G be the morphisms defined by

p_{1}(a, b, c) = ab, p_{2}(a, b, c) = bc

and

(p_{1}, p_{2}): G × G × G → G × G, (a, b, c) ↦ (ab, ac)

for a, b, c ∈ G. Since the distribution law is satisfied in G, we have n(1×m) = m(p_{1}, p_{2}). The maps n(1 × m) and m(p_{1}, p_{2}) respectively lift to the maps

ñ(1 × m̃), m̃(p̃_{1}, p̃_{2}): G̃ × G̃ × G̃ → G̃

coinciding at (ẽ, ẽ, ẽ). So by Theorem 2.5 we have ñ(1 × m̃) = m̃(p̃_{1}, p̃_{2}). That means the distribution law on G̃ is satisfied. The rest of the proof is straightforward.

From Theorem 2.6 we obtain

Corollary 2.7. Let X and X̃ be path connected topological spaces and p: X̃ → X be a simply connected covering, i.e., X̃ is simply connected. Suppose that X is a topological ring, and e is the identity element of the group structure on X. If ẽ ∈ X̃ with p(ẽ) = e, then X̃ becomes a topological ring with identity ẽ such that p is a morphism of topological rings.

Proof. Since p: X̃ → X is a simply connected covering, the induced morphism π_{1}p: π_{1}X̃ → π_{1}X is a universal covering morphism of groupoids. Since X is a topological ring by Proposition 1, π_{1}X is a ring-groupoid. By Theorem 2.6 π_{1}X̃ becomes a ring-groupoid and again by Proposition 2.3 X̃ becomes a topological ring as required.

References

1. C. Chevalley, Theory of Lie groups. Princeton University Press, 1946.

2. R. L. Taylor, Covering groups of non-connected topological groups. Proc. Amer. Math. Soc. 5(1954), 753–768.

3. O. Mucuk, Covering groups of non-connected topological groups and the monodromy groupoid of a topological groupoid. PhD Thesis, University of Wales, 1993.

4. R. Brown and O. Mucuk, Covering groups of non-connected topological groups revisited. Math. Proc. Camb. Philos. Soc. 115(1994), 97–110.

5. R. Brown and C. B. Spencer, G-groupoids and the fundamental groupoid of a topological group. Proc. Konn. Ned. Akad. v. Wet. 79(1976), 296–302.

6. R. Brown, Topology: a geometric account of general topology, homotopy types and the fundamental groupoid. Ellis Horwood, Chichester, 1988.

(Received 23.10.1996)

Author’s address:

Department of Mathematics, Erciyes University

Kayseri 38039, Turkey