THE CATEGORY OF PARTIAL ACTIONS OF A GROUP : SOME CONSTRUCTIONS

In this paper we introduce the category G-pAct of partial actions of a fixed group G. The objects or G-psets are the sets X endowed with a partial action of G on X and the morphisms, or preferably G-pmorphisms, are the maps preserving this action. As a special achievement, we extend several well-known constructions in the category G-Act, of global actions of G, to this new context. In particular, we characterize products, coproducts, equalizers and pullbacks for arbitrary G-pmorphisms. We also characterize coequalizers and pushouts for strong G-pmorphisms (category G-fpAct). Last, we prove that the category G-pAct is complete and the category G-fpAct is cocomplete. AMS Subject Classification: 18A30, 20M30


Introduction
Category theory was introduced in 1945 by Eilenberg and MacLane ([4]).However, its relevance was not recognized until the 1960s, when the works of Lawvere Received: January 13, 2014 c 2014 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author on logic, foundations of mathematics, the category of sets and closed Cartesian categories became known.During the 1970s it was finally accepted that category theory was a genuine branch of mathematics, with tools, problems and specialized techniques.The success of the theory is due mainly to the language, which generalizes many constructions that occur in many areas of mathematics.Furthermore, it shelters many types of structures that were, at first, considered dissimilar.
More recently, in 1998, Exel ([5]) introduced the notion of partial action of a group in his works on operator theory.He used collections of subsets of the base space and bijections between them (partial bijections of the space).The importance of this theory lies in its applicability to many fields of mathematics, as numerous recent publications have shown (see [3], [7] and its references).In particular, these notions have been successfully used to extend classical results on dynamical systems ( [6]), topological and metric spaces ( [8], [9]), rings ( [7]) and representation theory ( [3]), among others fields.
In the current literature there is no categorical context for partial actions nor for characterizations of objects and morphisms.In Section 2 we present first the category of global actions of a fixed group G (G-Act).Then, we construct the category of partial actions of G (G-pAct).In Section 3 we characterize the products, coproducts and equalizers in the category G-pAct.Additionally, we introduce the notion of strong G-pmorphism (category G-fpAct) and characterize the coequalizers and pushouts in the category G-fpAct.Finally, we prove that the category G-pAct is complete and the category G-fpAct is cocomplete.

The Category G-pAct
Recall that a global action of a group G, with identity element 1, on the set X is a function In this case X is called G-set.The category of global actions of a fixed group G, denoted by G-Act, is one whose objects are the sets X endowed with a global action of G on X.The morphisms between the objects X, Y , called G-morphisms, are the maps f : X → Y such that for all x ∈ X and all g ∈ G, f (g • x) = g • f (x).This category is very important because it is a topoi.Moreover, there is significant literature on applications of this theory in other sciences like physics and computer science.In mathematics, many constructions within this category as products, equalizers and pullbacks, among others concepts, are natural extensions of those well-known in the category of sets.
Global actions can be generalized in several ways, including the actions of monoids and the grupoid actions.More recently, Exel in 1998 ( [5]) in his studies on operator theory introduced the notion of partial action of a group by considering partial bijections.Definition 1.A partial action α of the group G on the set X is a collection of subsets S g , g ∈ G, of X and bijections α g : S g −1 → S g such that for all g, h ∈ G the following statements hold: 1. S 1 = X and α 1 is the identity of X.
In this case the set X is called G-pset.The partial action α, will be denoted by α = {S g , α g } g∈G .Note that item 3. implies α −1 g = α g −1 for all g ∈ G.For a fixed group G we call objets to G-psets.Now, for X and Y two G-psets, with partial actions α X = {X g , X α g } g∈ G and α Y = {Y g , Y α g } g∈ G respectively, we call G-pmorphism to maps f : X → Y such that: The usual composition between G-pmorphisms is a G-pmorphism and the identity map id X is a G-pmorphism for each object X.Thus, the G-psets together with the G-pmorphisms, the usual identity maps and the usual composition define a category, which will be called category of partial actions of G and it will be denoted by G-pAct.
Although the category G-pAct is a natural extension of the category G-Act, it has not been explicitly introduced in the current literature.Similarly, there are no studies on characterizations of objects and morphisms in this category.Since the category G-Act is a subcategory of G-pAct, it is natural to extend known results of the category G-Act to category G-pAct.

Results
In this section we extend some well known constructions in the set category and the global actions category.In particular, we characterize the products, coproducts, equalizers and pullbacks for arbitrary G-pmorphisms and the coequalizers and pushouts for strong G-pmorphisms.We assume that the reader is familiarized with the techniques and basic concepts of category theory, which can be found in [1].

Products
It is clear that the Cartesian product of a family of sets is a set and more generally the Cartesian product of a family of G-sets is also a G-set.In this first part we show that the former results can be naturally generalized to the category G-pAct.We begin by extending Affirmation 1 of [2] for an arbitrary family of G-psets.
Proof.Consider the collection of G-psets {X i } i∈I , where the partial action on X i is given by α i = { i X g , i α g } g∈G for each i ∈ I.For i∈I X i , take i∈I i X g and i∈I i α g : i∈I i X g −1 → i∈I i X g defined by i∈I i α g (x) = ( i α g (x i )) i∈I for each x ∈ i∈I i X g −1 .It is clear that i∈I i α g is a well defined function and it is a bijection.Now we verify 2. and 3. of Definition 1, since 1. is evident.
Proof.Consider the source P = ( i∈I X i π i → X i ) i∈I and suppose that there exists other source S = (S Then, w = ( i α g ((q i (s)) i∈I )) i∈I = ( i α g (q i (s))) i∈I and since for each i ∈ I, q i is a G-pmorphism then ( i α g (q i (s))) = q i ( S α g (s)) and w = (q i ( S α g (s))) i∈I = ϕ( S α g (s)).Now we must prove that the following diagram commutes for each i ∈ I: Moreover, ϕ is the unique morphism that makes the diagram commutative.In fact, if there exists and thus ϕ ′ (s) = (q i (s)) i∈I = ϕ(s) for all s ∈ S. Hence, ϕ = ϕ ′ .

Coproductos
For the following proposition we assume, without loss of generality, that each pair of different G-psets have empty intersection.The following result extends Affirmation 2 of [2] for an arbitrary family of G-psets.
Proof.Let {X i } i∈I be a collection of G-psets, where each X i , i ∈ I, has the partial action α i = { i X g , i α g } g∈G .Take i∈I X i and for each g ∈ G, define i∈I i X g .Since for each x ∈ i∈I i X g −1 there exists a unique i ∈ I such that x ∈ i X g −1 , then we define i∈I i α g : i∈I i X g −1 → i∈I i X g as i∈I i α g (x) = i α g (x) for each x ∈ i∈I i X g −1 .It is clear that i∈I i α g , g ∈ G, is a well defined function and it is a bijection.We must prove that { i∈I i X g , i∈I i α g } g∈G defines a partial action on i∈I X i .We verify 2. and 3. of Definition 1 since 1. is evident. 2 , then there exists a unique Proof.Consider the sink C = (X i c i −→ i∈I X i ) i∈I and suppose that there exists another sink S = (X i Now we must prove that the following diagram commutes for each i ∈ I: That is, φ • c i = q i for each i ∈ I. Finally, φ is the unique G-pmorphism that makes the diagram commutative.In fact, if there exists φ ′ : i∈I X i → Y such that φ ′ • c i = q i for all i ∈ I, then for each x ∈ i∈I X i we have φ

Equalizers
The G-invariant sets, those that remain fixed under the action of G, are very important in the theory of global actions.The analogous notion in partial actions is the following.
, which implies that h(j) ∈ I. So, the map φ : J → I defined by φ(j) = h(j) for all j ∈ J is a G-pmorphism and i • φ = h.Suppose that there exists a G-pmorphism φ ′ : As a direct consequence of the results above we obtain the following corollary.
1.The category G-pAct has pullbacks.
2. The category G-pAct is complete.
Proof. 1. Suppose that X, Y, Z are G-psets and f 1 : It is a direct consequence of the fact that the category G-pAct has products and pullbacks (see Theorem 12.3 of [1]).

Coequalizers
We note that if f : X → Y is a G-pmorphism, then the condition f (x) ∈ Y g for some g ∈ G and some x ∈ X does not imply that x ∈ X g .This situation is true in global actions since X g = X and Y g = Y for all g ∈ G.However, this condition is crucial to consider partial actions since it is used in the construction of coequalizers and pushouts.
Definition 11.An equivalence relation E on the G-pset X is called a G-pcongruence if the following conditions hold: x, y ∈ X g −1 and x E y imply X α g (x) E X α g (y).
Proposition 12. Let Y be a G-pset and E Y a G-pcongruence on Y .Then:

The canonical map τ E
Proof. 1. Suppose that the partial action on Y is given by, α Y = {Y g , Y α g } g∈G and consider the G-pcongruence E Y on Y .We define Finally, for the remains it is enough to apply the properties of the partial action on Y .
2. By definition we have that y We must see that ϕ is a well defined function.As a direct consequence of the results above we obtain the following corollary.Note that we can consider the category G-fpAct, in which the objects are the G-psets and the morphisms are the strong G-pmorphisms.Moreover, it is clear that the category G-fpAct has coproducts since the canonical injections are strong G-pmorphisms.
The result follows by duality since the category G-fpAct has coproducts and pushouts (see Theorem 12.3 of [1]).

2 .
The category G-fpAct is cocomplete.Proof.1. Suppose that X, Y, Z are G-psets and f 1 : X → Y and f 2 : X → Z are strong G-pmorphisms.Then, a pushout for (X, f 1 , f 2 ) is given by (C, c • c Y , c • c Z ), where (C, c) is the coequalizer of the morphisms c Y • f 1 : X → Y ⊔ Z and c Z • f 2 : X → Y ⊔ Z with c Y ,c Z the usual inclusions.This construction can be appreciate in the following diagram.