Fibrations of 2‐crossed modules

Abstract

In this study, we mainly show that the functor from the category X2Mod of 2‐crossed modules of groups to the category Groups of groups assigning to each 2‐crossed module {}L,M,P,∂2,∂1 the group P, and to each 2‐crossed module morphism ()f2,f1,f0, the group homomorphism f0 is a fibration. In addition, we study some related properties.

On Crossed Squares of Commutative Algebras

“…Categorical properties of these related categories such as pullback, limit and colimit are investigated in [5] [6], [7].The pullback construction is highly related with fibration. The fibration of 2-crossed modules is given in [16]. The category of pairs of crossed modules is mentioned in [7] to investigate the bifibration of crossed squares of groups.…”

On relations among quadratic modules

“…2‐crossed modules defined by Conduche 12 and crossed squares defined by Ellis 13 are also homotopy three‐type structures. Categorical properties of homotopy 3‐type structures are studied in various ways 14–16 …”

Finite coproducts in the category of quadratic modules of Lie algebras

“…It has attracted the attention of many researchers. This notion initially introduced in groups has also naturally appeared in various algebraic cases as commutative and associative algebras, Lie and Lie-Rinehart algebras, etc, [2][3][4][5][6][7][8][9]. Kassel and Loday studied the classification of central extensions of Lie algebras and crossed modules of Lie algebras in [10].…”