Actions of internal groupoids in the category of Leibniz algebras

The aim of this paper is to characterize the notion of internal category (groupoid) in the category of Leibniz algebras and investigate the properties of well-known notions such as covering groupoid and groupoid operations (actions) in this category. Further, for a fixed internal groupoid $G$, we prove that the category of covering groupoids of $G$ and the category of internal groupoid actions of $G$ on Leibniz algebras are equivalent. Finally we interpret the corresponding notion of covering groupoids in the category of crossed modules of Leibniz algebras.


Introduction
Covering groupoids have an important role in the applications of groupoids (see for example [3] and [14]). It is well known that for a groupoid G, the category GpdAct(G) of groupoid actions of G on sets, these are also called operations or G-sets, are equivalent to the category GpdCov/G of covering groupoids of G. For the topological version of this equivalence, see [6,Theorem 2].
If G is a group-groupoid, which is an internal groupoid in the category of groups, then the category GpGpdCov/G of group-groupoid coverings of G is equivalent to the category GpGpdAct(G) of group-groupoid actions of G on groups [8,Proposition 3.1]. In [2] this result has recently generalized to the case where G is an internal groupoid for an algebraic category C, acting on a group with operations. Covering groupoids of a categorical group have been studied in [21].
In [9] it was proved that the categories of crossed modules and group-groupoids, under the name of G-groupoids, are equivalent (see also [17] for an alternative equivalence in terms of an algebraic object called cat n -groups). By applying this equivalence of the categories, normal and quotient objects in the category of group-groupoids have been recently obtained in [22]. The study of internal category theory was continued in the works of Datuashvili [12] and [13]. Moreover, she developed cohomology theory of internal categories in categories of groups with operations [10] and [11] (see also [24] for more information on internal categories in categories of groups with operations). The equivalences of the categories in [9] enable us to generalize some results on group-groupoids to the more general internal groupoids for a certain algebraic category C (see for example [2], [19], [20] and [23]).
In the mid-nineteenth century, Whitehead introduced the notion of crossed module, in a series of papers [26,27,28], as algebraic models for (connected) homotopy 2-types (i.e. connected spaces with no homotopy group in degrees above 2), in much the same way that groups are algebraic models for homotopy 1-types. A crossed module consists of groups A and B, where B acts on A by automorphisms, and a homomorphism of groups α : Crossed modules can be viewed as 2-dimensional groups [4] and have been widely used in: homotopy theory, [5]; the theory of identities among relations for group presentations, [7]; algebraic K-theory [16]; and homological algebra, [15,18]. See [5, pp.49] for some discussion of the relation of crossed modules to crossed squares and so to homotopy 3-types. The equivalence between crossed modules and group groupoids, proved in [9] and has been found important in applications. It is generalised in [24]. In this paper, first we defined and investigated some properties of internal categories (and hence internal groupoids) in the category of Leibniz algebras. Further we defined coverings and actions in the category of internal groupoids in the category of Leibniz algebras and proved that the category of internal groupoid actions and the category of covering groupoids of a fixed internal groupoid G in the category of Leibniz algebras are equivalent. Finally, using the equivalence of the categories internal groupoids in the category of Leibniz algebras and crossed modules of Leibniz algebras, we interpreted the notion of covering in the category of crossed modules of Leibniz algebras.

Preliminaries
A Leibniz algebra L is a k-vector space equipped with a bilinear map for all x ∈ L, then L becomes a Lie algebra. On the other hand, every Lie algebra is a Leibniz algebra.
for all x, y ∈ L.
The category of Leibniz algebras consist of Leibniz algebras as objects and Leibniz algebra morphisms as morphisms. This category is denoted by Lbnz.

Definition 2.2. A Leibniz algebra with trivial bracket is called an Abelian (or singular) Leibniz algebra.
Definition 2.3. For any Leibniz algebras L and L ′ a Leibniz action of L on L ′ consist of two bilinear maps Λ : for all x, y ∈ L and m, n ∈ L ′ .
Let L and L ′ be two Leibniz algebras. A split extension of L by L ′ is a short exact sequence in Lbnz with a Leibniz algebra morphism s : L → E such that ps = 1 L . Here, note that p is surjective and ker p = i. Given a split extension of L by L ′ , we get derived actions of L on L ′ defined by for any x ∈ L and m ∈ L ′ . Let a split extension is given. Then by using the bijection we can define a Leibniz algebra structure on L ′ × L as follows: for all x, y ∈ L and m, n ∈ L ′ . The inverse of the function θ is defined by for all e ∈ E. Thus L ′ × L cartesian product set becomes a Leibniz algebra which is called by semi-direct product of Leibniz algebras and denoted by L ′ ⋊ L. For any Leibniz algebra L, the obvious action of L on itself corresponds to the extension where i(l) = (l, 0), p(l, l 1 ) = l 1 and s(l) = (0, l). Now, we can give the definition of crossed modules of Leibniz algebras due to Porter [24]. and (∂, 1 L 0 ) are both split extension morphisms in Lbnz.
It is more practical to have a description in terms of actions and Leibniz bracket. We recall the definitions from [1] and [25].

Proposition 2.5. A crossed module of Leibniz algebras is a Leibniz algebra morphism
Thus the category XMod(Lbnz) of Leibniz crossed modules can be constructed. The objects of this category are Leibniz crossed modules and morphisms are crossed module morphisms.
A groupoid is a category in which every morphism is an isomorphism. Let G be a groupoid. We write Ob(G) for the set of objects of G and write G for the set of morphisms. We also identify Ob(G) with the set of identities of G and so an element of Ob(G) may be written as x or 1 x as convenient. We write d 0 , d 1 : G → Ob(G) for the source and target maps, and, as usual, write G(x, y) for d −1 0 (x) ∩d −1 1 (y), for x, y ∈ Ob(G). The composition h• g of two elements of G is defined if and only if d 0 (h) = d 1 (g), and so the map 0 (x) and call the star of G at x. A groupoid G is transitive (resp. simply transitive, 1-transitive and totally intransitive) if G(x, y) = ∅ (resp. G(x, y) has no more than one element, G(x, y) has exactly one element and G(x, y) = ∅) for all x, y ∈ Ob(G) such that x = y.

Internal categories in Lbnz
Definition 3.1. Let C be an arbitrary category with pullbacks. An internal category C in C is a category in which the initial and final point maps d 0 , d 1 : C → Ob(C), the object inclusion map ε : Ob(C) → C and the partial composition • : Let G be an internal category in C. If there exist a morphism g ′ ∈ G such that g•g ′ = εd 1 (c) and g ′ • g = εd 0 (c) for all morphisms g ∈ G, then G is called an internal groupoid and g ′ is called the inverse of g which is denoted by g −1 .
Let G be an internal category in the category Lbnz of Liebniz algebras. Then G and Ob(G) are Leibniz algebras and the structural maps (d 0 , d 1 , ε, •) are Leibniz algebra morphisms. So we can give the following proposition.

Proposition 3.2. Let G be an internal category in Lbnz.
Then for all x, y ∈ Ob(G) and g, g ′ ∈ G Also note that the operation • being a Leibniz algebra morphism implies that . These identities are called interchange laws. An application of the interchange laws is that the composition can be expressed by the addition as follows: Clearly, one can see that any internal category in Lbnz is an internal groupoid. Indeed, for any g ∈ G, g −1 = εd 0 (g) − g + εd 1 (g) is the inverse morphism of g. Hence, we will use internal groupoid instead of internal category.  Proof: Since g 1 ∈ ker d 0 and g −εd 1 (g) ∈ ker d 1 , one can prove the assertion of the Lemma by using Lemma 3.6. ✷ Let G and H be two internal groupoids in Lbnz. An internal groupoid morphism (internal functor) f : G → H is a morphism of underlying groupoids and Leibniz algebra morphism on both the algebra of morphisms and the algebra of objects. So, we can construct the category of internal groupoids in Lbnz. This category may be denoted by Cat(Lbnz) or Gpd(Lbnz).
is a Leibniz algebra morphism. Moreover Ob(G) acts on ker d 0 by the maps and These are derived actions, since these are obtained from the split extension Here we note that ker d 0 ⋊ Ob(G) ∼ = G.
Also (ker d 0 , Ob(G), d 1 ) is a crossed module. Indeed, (LXM1) for all x ∈ Ob(G) and g ∈ ker d 0 and similarly and similarly for l ′ 0 = ∂(l 1 ) + l 0 and the inverse (l 1 , l 0 ) −1 = (−l 1 , ∂(l 1 ) + l 0 ). Now we need to show that these structural maps are Leibniz algebra morphisms. For all (l 1 , l 0 ), To see that the composition is a Leibniz algebra morphism, we need to verify the interchange law for bracket operation.
This shows that the composition • is a morphism of Leibniz algebras. Thus L 1 ⋊ L 0 becomes an internal groupoid on L 0 in Lbnz. Above construction also defines a functor, δ, from the category XMod(Lbnz) of crossed modules in the category of Leibniz algebras to the category Cat(Lbnz) of internal categories in the category of Leibniz algebras.

Coverings and actions of internal groupoids in Lbnz
First we will recall the definitions of coverings over groupoids from [3].
is bijective.
Assume that p : G → G is a covering morphism. Then we have a lifting function S p : So it is stated that p : G → G is a covering morphism if and only if (p, d 0 ) is a bijection [6].

Definition 4.2. An internal groupoid morphism p : G → G is a covering morphism if and only if
A covering morphism p : G → G is called transitive if both G and G are transitive. A transitive covering morphism p : G → G is called universal if for every covering morphism q : H → G there is a unique morphism of groupoids p : G → H such that q p = p (and hence p is also a covering morphism), this is equivalent to that for x, y ∈ Ob( G) the set G( x, y) has not more than one element.

Remark 4.3.
Since for an internal groupoid G in Lbnz, the star St G 0 is also a Leibniz algebra, we have that if p : G → G is a covering morphism of internal groupoids, then the restriction of p to the stars St G 0 → St G 0 is an isomorphism in Lbnz.
Let p : G → G and q : G ′ → G be two coverings of G. A morphism f : G → G ′ of coverings is a morphism of internal groupoids in Lbnz such that qf = p, i.e. following diagram is commutative.
Hence we can construct the category of covering internal groupoids of an internal groupoid G in Lbnz which has covering morphisms of G as objects and has morphisms of coverings as morphisms. This category will be denoted by Cov Cat(Lbnz) /G.
Recall that an action of a groupoid G on a set S via a function ω : S → Ob(G) is a function of such actions is a function f : S → S ′ such that w ′ f = w and f (g • s) = g • f (s) whenever g • s is defined. This gives a category GpdAct(G) of actions of G on sets. For such an action, the action groupoid G ⋉ S is defined to have object set S, morphisms the pairs (g, s) such that d 0 (g) = ω(s), source and target maps d 0 (g, s) = s, d 1 (g, s) = g • s, and the composition The projection q : G ⋉ S → G, (g, s) → s is a covering morphism of groupoids and the functor assigning this covering morphism to an action gives an equivalence of the categories GpdAct(G) and GpdCov/G. Following equivalence of the categories was given in [8].
which is called the action, satisfying whenever h • g and g • l are defined.
Note that the action being a Leibniz algebra morphism implies the following so called interchange laws: for all g, g ′ ∈ G and l, l ′ ∈ L, whenever both sides are defined.
A morphism f : (L, ω) → (L ′ , ω ′ ) of such actions is a morphism f : L → L ′ of Leibniz algebras such that ω ′ f = ω. This gives a category Act Cat(Lbnz) (G) of actions of G on Leibniz algebras.
For an action of G on a Leibniz algebra L via ω, the action groupoid G ⋉ L has a Leibniz algebra structure defined by (g, l) + (g ′ , l ′ ) = (g + g ′ , l + l ′ ), and with this operations G ⋉ L becomes an internal groupoid in Lbnz.
where g is the unique lifting of g with initial point x. It is easy to verify that this map is an action and a Leibniz algebra morphism, since p is a Leibniz algebra morphism. Conversely, let G acts on a Leibniz algebra L via ω : L → Ob(G). Then q : G ⋉ L → G, (g, l) → g is a covering morphism in Cat(Lbnz). It is straightforward to confirm that these constructions defines the intended natural equivalence. ✷ Example 4.7. Let G be an internal groupoid in Lbnz. Then 1 G : G → G is a covering morphism in Cat(Lbnz). The corresponding action to 1 G is constructed as follows: G acts on Ob(G) via 1 Ob(G) : Ob(G) → Ob(G) where the action is In this case the action groupoid is isomorph to G as an internal groupoid in Lbnz, i.e., G ⋉ Ob(G) ∼ = G.