ABSTRACT
From complete cotorsion pairs to weak factorization systems . . . 171 From weak factorization systems to complete cotorsion pairs . . . 184
7.4 Proof of the Hovey Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.5 Abelian model structures on monoidal categories . . . . . . . . . . . . . . . 194 7.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.1 Introduction We will show that from a Hovey triple (see Definition 6.4.1) in an Abelian
category C, we can construct a particular type of model structure, known as Abelian. We will also show that from every Abelian model structure it is possible to obtain a Hovey triple formed by the classesQ,R and T of cofibrant, fibrant, and trivial objects. We start stating the Hovey Correspondence, but before giving the proof, we describe how to obtain a weak factorization system from a complete cotorsion pair, and vice versa. As a first approach to this construction, we know that if pA,Bq is a complete cotorsion pair, then for every X P ObpCq there is an exact sequence
0 Ñ B αÝÑ A βÝÑ X Ñ 0, with A P A and B P B. This gives us a factorization of the zero morphism
B X
Y
where the right factor is an epimorphism with its kernel in B, and the left factor a monomorphism with its cokernel in A. The class of all the epimorphisms (respectively, monomorphisms) with their kernels in B (respectively, with their
cokernels in A) will turn out to be the right class (respectively, left class) of morphisms of a weak factorization system.
