From simplicial homotopy to crossed module homotopy in modified categories of interest

Abstract

We firstly prove the completeness of the category of crossed modules in a modified category of interest. Afterwards, we define pullback crossed modules and pullback cat 1 -objects that are both obtained by pullback diagrams with extra structures on certain arrows. These constructions unify many corresponding results for the cases of groups, commutative algebras and can also be adapted to various algebraic structures.

Graphical calculus of Hopf crossed modules

“…On the other hand, by using the properties of the functor Prim, the formula (5.1) will be turned into which is given in [9] for the case of Lie algebras.…”

“…Recall Theorem 4.7; and the groupoid structures in [9,10] for the case of groups and Lie algebras. As a result of the previous theorem we have the following.…”