On totally free crossed modules

Abstract

In [10] we associate to a crossed module (T, G, მ) an invariant abelian crossed module H2(T, G, მ). The construction uses presentations by Set-free crossed modules. Now, Set-free crossed modules are special cases of totally free crossed modules, which are algebraic models of 2-dimensional CW complexes used by several authors (see [1] and [6]). The aim of this paper is to show that H2(T, G, მ) can also be constructed from presentations by arbitrary totally free crossed modules.

The non-abelian tensor product of normal crossed submodules of groups

“…In this category one can find the familiar notions of injection, surjection, (normal) subobject, kernel, cokernel, exact sequence, etc. ; most of them can be found in detail in [11,18]. Let (T, G, ∂) be a crossed module with normal crossed submodules (S, H, ∂) and (L, K, ∂).…”

“…The algebraic study of the category of crossed modules was initiated by Norrie [18] and has led to a substantial algebraic theory contained essentially in the following papers: [1,7,11,12,13,15,20,21]. In particular, Pirashvili [19] presented the concept of the tensor product of two abelian crossed modules and investigated its relation to the low-dimensional homology of crossed modules.…”

CHARACTERIZING n-ISOCLINIC CLASSES OF CROSSED MODULES

“…In this category, one can find the familiar notions of injection, surjection, (normal) subobject, kernel, cokernel, exact sequence, etc. ; most of them can be found in detail in [8,24].…”