Locally type ${\text{FP}}_n$ and $n$-coherent categories

We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type ${\text{FP}}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type ${\text{FP}}_n$ categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type ${\text{FP}}_n$, called ${\text{FP}}_n$-injective objects, which will be the right half of a complete cotorsion pair. As a generalization of the category of modules over an $n$-coherent ring, we present the concept of $n$-coherent categories, which also recovers the notions of locally noetherian and locally coherent categories for $n=0,1$. Such categories will provide a setting in which the ${\text{FP}}_n$-injective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by ${\text{FP}}_n$-injective objects. Moreover, we see how $n$-coherent categories provide a suitable framework for a nice theory of Gorenstein homological algebra with respect to the class of ${\text{FP}}_n$-injective modules. We define Gorenstein ${\text{FP}}_n$-injective objects and construct two different model category structures (one abelian and the other one exact) in which these Gorenstein objects are the fibrant objects.