Colimits of crossed modules in modified categories of interest

In this paper, we give the constructions of the coequalizer and coproduct objects for the category of crossed modules, in a modified category of interest (MCI). In other words, we prove that the corresponding category is finitely cocomplete.

definition of crossed module is adapted to modified categories of interest in [4] that unifies all crossed module structures of the algebraic structures we mentioned above. It is strongly recommended to see [6,7] for a very detailed survey of crossed modules and related structures. Some categorical properties of crossed modules are examined in [1,2,3,5,8,12,13,14,20] for various algebraic structures. In fact, some of them are examples of modified categories of interest.
As the modified category of interest is the unification of many well-known algebraic structures and their properties, it is natural to ask whether it is possible to unify some categorical properties of crossed modules via modified categories of interest. In this context, constructions of limits (of crossed modules) in a modified category of interest are given in [13] that yields the completeness of the corresponding category. Following that study, in this paper, we prove that a category of crossed modules in modified categories of interest is (finitely) cocomplete, namely, it has all finite colimits.
2. Preliminaries. We recall some notions from [4] that will be used in the sequel.
Let C be an object of C and x 1 , x 2 , x 3 ∈ C: where each juxtaposition represents an operation in Ω 2 . A category of groups with operations C satisfying conditions (a)-(f) is called a "modified category of interest", or "MCI" for short.   However, there exist other well-known algebraic categories that are not modified categories of interest. For instance, the categories of Leibniz-Rinehart algebras, Hopf algebras, racks (or quandles), etc.
As we underlined in the introduction, the following are the essential examples of modified categories of interest (which are not categories of interest), and they were the main motivation to define modified categories of interest.  Definition 2.6. Let A, B be two objects of C. An extension of B by A is a sequence where p is surjective and i is the kernel of p. We say that an extension is split if there exists a morphism s : B → E such that ps = 1 B .

Definition 2.7. The split extension induces a set of actions of B on
for all b ∈ B, a ∈ A and * ∈ Ω 2 . Actions defined by the previous equations are called derived actions of B on A. Remark that we use the notation " * " to denote both the star operation and the star action.
Definition 2.8. Given an action of B on A, the semi-direct product A B is a universal algebra, whose underlying set is A × B, and the operations are defined by Remark that, an action of B on A is a derived action, if and only if, A B is an object of C. Theorem 2.9. Denote a general category of groups with operations of a modified category of interest C by C G . A set of actions of B on A in C G is a set of derived actions, if and only if, it satisfies the following conditions: a, a 1 , a 2 ∈ A and for x, y, z, t ∈ A ∪ B whenever each side of 12 makes sense.
Without the second condition, we call it a precrossed module.
We denote the category of crossed modules by XMod, and similarly, of precrossed modules by PXMod.
The following two are the characteristic examples of crossed modules in any modified category of interest C.
Example 2.11. Let B be an object of C and A is an ideal of B. Then, the inclusion map A − → B becomes a crossed module where the action is defined via conjugation, namely for all a ∈ A and b ∈ B.
Example 2.12. Let B be an object of C. Then, we have a natural crossed module 0 → B with the trivial action. More generally, if A is an abelian object (i.e. x + y = y + x and x * y = 0, for all x, y ∈ A and * ∈ Ω 2 ), then the zero map A → B defines a crossed module with any derived action, for all B. Example 2.13. A crossed module of groups [5] is a group homomorphism ∂ : E → G, together with a group action of G on E such that X1) ∂(g e) = g + ∂(e) − g, for all e, f ∈ E and g ∈ G.
Example 2.14. A dialgebra crossed module [9] is a dialgebra homomorphism ∂ : for all e, f ∈ e and g ∈ g.
3. Colimits in XMod/C 0 . From now on, C will be a fixed MCI where E includes the identity x + y = y + x. Remark that Lie algebras, (commutative) associative algebras, dialgebras are all examples of C.
Definition 3.1. Consider the subcategory XMod/C 0 of crossed modules with a fixed codomain C 0 . Its objects will be called crossed C 0 -modules, and the morphism between (E, ∂ E ) and (D, ∂ D ) is defined by a morphism 1 µ : E → D such that the following diagram commutes for all c 0 ∈ C 0 , e ∈ E, * ∈ Ω 2 . A crossed C 0 -module ∂ : E → C 0 will be denoted by (E, ∂ E ) for short. Proof. For all d ∈ D, we have  Then we have and q (c · (d + I)) = q (c · d + I) and, similarly, for all c ∈ C 0 and d + I ∈ D/I. Hence, we obtain that for all d + I ∈ D/I, that proves q is unique and completes the proof.
is a precrossed C 0 -module.
Proof. First of all, ∂ is a morphism in C since ∂((e, d) + (e , d )) = ∂(e + d · e , d + d ) On the other hand, since ∂ E and ∂ D are crossed modules, we have for all c ∈ C 0 and (e, d) ∈ E D, which proves that (E D, ∂) is a precrossed C 0 -module. Then, ∂ : A/P −→ C 0 a + P −→ ∂(a) is a crossed C 0 -module with the action c · (a + P ) = c · a + P c * (a + P ) = c * a + P for all c ∈ C 0 , (a + P ) ∈ A/P .
Proof. Since the crossed module conditions are already satisfied, we only need to prove that the action of C 0 on A/P is well-defined.
Remark 2. We have the functor ( ) cr : PXMod → XMod which assigns the crossed module∂ : A/P → C 0 for a given precrossed module for all e ∈ E, d ∈ D.
Proof. Consider the diagram in XMod/C 0 . Define First of all, h is a morphism in C since for all c ∈ C 0 , (e, d) ∈ E D/P , that yields h is a crossed C 0 -module morphism.
On the other hand, it is easy to prove that h is unique.

Remark 3.
A category D is said to be (finitely) "cocomplete" if it has all (finite) colimits. On the other hand, the cocompleteness can be characterized in several ways as follows. For a category D, the following are equivalent: • D is finitely cocomplete.
• D has pushouts and the initial object.