2
$\begingroup$

The wikipedia claims that there is a model category structure of the category of arbitrary chain-complexes of R-modules which is defined by:

weak equivalences are chain homotopy equivalences of chain-complexes;

cofibrations are monomorphisms that are split as morphisms of underlying R-modules; and

fibrations are epimorphisms that are split as morphisms of underlying R-modules.

But I have no idea on how to prove the lifting property and the factorization. More precisely, how should I verify the axioms:

MC4 Given maps f,g,i,p such that pf=gi, a lift h (i.e. ph=g, f=hi) exists in either of the following two situations: (i) i is a cofibration and p is an acyclic fibration, or (ii) i is an acyclic cofibration and p is a fibration.

and

MC5 Any map f can be factored in two ways: (i) f = pi, where i is a cofibration and p is an acyclic fibration, and (ii) f = pi, where i is an acyclic cofibration and p is a fibration?

Improve this question
$\endgroup$
2

1 Answer 1

Reset to default
3
$\begingroup$

The model structure you are looking for is developed in a paper by Christensen and Hovey called Quillen model structures for relative homological algebra. They go into the factorization systems in great detail. This paper was generalized recently by Barthel-Riehl-May, with even more on those weak factorization systems. See: https://arxiv.org/abs/1310.1159

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .