An example of homotopy theory in abelian groups
A homotopy relation associated with a unitary ring can be defined in the category of abelian groups, as follows: Let f,g: \(X\to Y\) be homomorphisms, if there exists homomorphism F: \(X\otimes R\to Y\) such that \(F(x\otimes 1)=f(x)-g(x)\), then f is homotopic to g. If X has an R- module structure, then \(id_ X\simeq 0\); however, if \(id_ X\simeq 0\), X has almost an R-module structure (associative property does not hold). In paragraphs 2 and 3, cofibrations and fibrations are studied; a homomorphism i: \(A\to X\) is said to be a cofibration if for each f: \(A\to M\), where M is an R-module, there exists \=f: \(X\to M\) such that \=fi\(=f\) (fibrations can be defined in a dual way). Let \(Cof^ A\) denote the category of cofibrations \(A\to X\), the following result is proved in paragraph 3: Let \(\alpha\),\(\beta\) : \(A\to B\) be homotopic homomorphisms, then there exists a natural equivalence \(\Lambda\) : \(\alpha_*\to \beta_*\), where \(\alpha_*\), \(\beta_*\) are functors of type \(Cof^ Ah\to cof^ Bh\), and \(Cof^ Ah\) is the homotopy category associated with \(Cof^ A.\) Finally, in paragraph 4, the category of abelian groups with the families of cofibrations, fibrations and homotopy equivalences (as weak equivalences) is proved to be a closed model category, in the sense of Quillen. The homotopy groups are defined as follows: Let S be the coker of i: \({\mathbb{Z}}\to R: i(1_{{\mathbb{Z}}})=1_ R\), and consider the tensorial product \(\otimes^ nS (\otimes^ 0S={\mathbb{Z}})\), then \(\eta_ n(X)\) is the group of homotopy classes \([\otimes^ nS,X]\). In some cases, if the homotopy groups of X vanish, X is an R-module. For example, for the rational ring if \(\eta_ 0(X)=0=\eta_ 1(X)\), it follows that X is a \({\mathbb{Q}}\)-module.
scientific article; zbMATH DE number 3842753