In this paper, we define the pullback crossed modules in the category of racks that are mainly based on a pullback diagram of rack morphisms with extra crossed module data on some of its arrows. Furthermore, we prove that the conjugation functor, which is defined between the category of crossed modules of groups and of racks, preserves the pullback crossed modules.

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In this work, it is shown that the category BXMod=R of braided crossed modules over a fixed commutative algebra R is an exact category in the sense of Barr.

In this paper, we prove that the category of rack crossed modules (with a fixed codomain) is finitely complete. In other words, we construct the product, pullback and equalizer objects in the category of crossed modules of racks. We therefore unify the group-theoretical analogy of the completeness property in the sense of the functor $\mathbf{Conj \colon Grp \to Rack} $.

In this paper we give the explicit construction of a pro-C completion functor which is de…ned in the category of crossed squares of commutative algebras. Afterwards, we study some functorial properties of this pro-C completion process.