Pullback Crossed Modules in the Category of Racks

Abstract

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.

Some Remarks on MCI Crossed Modules

“…Therefore, such structures can not be included in the above theorem. In fact, some of them are already studied -for instance, see [9,11] for the case of racks.…”

Categorical properties of racks

“…Afterwards, we give some categorical properties of racks which are the constructions of product, pullback and equalizer objects. These categorical objects are defined by the universal property diagrams in [1], [9] and examined for more specific categories such as category of crossed modules of racks and (modified) categories of interest in [5], [6].…”

Colimits of crossed modules in modified categories of interest

“…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.…”