Definable additive categories: purity and model theory

Direct limits in the heart of a t-structure: The case of a torsion pair

“…First of all, note that an object Y of G is pure-injective if, and only if, for every set S, each morphism f : [Pr,Theorem 5.4]). Then lemmas 2.1, 2.3 and 2.4, together with corollary 2.2 of [op.cit] are valid here.…”

Covers in finitely accessible categories

“…Therefore, H(A) is closed under pure epimorphic images. Now let C be a finitely accessible category with products and let D be a definable subcategory of C. Note that definable subcategories need not be finitely accessible: for instance, the category of divisible abelian groups is a definable subcategory of the category Ab of abelian groups, but it is not finitely accessible [32,Example 10.3]. Nevertheless, we may establish our covering result in such a context because of the good behaviour with respect to purity, namely: purity in a definable subcategory is just the restriction of purity in the larger category (see, e.g., [32] for details).…”

“…Recall that an additive category is locally finitely presented if it is finitely accessible and cocomplete (i.e., has all colimits) or, equivalently, if it is finitely accessible and complete (i.e., has all limits) [32]. Now we are able to extend the existence of flat covers for comodules over a semiperfect coalgebra [10,Theorem 3.9] (also see [8, Corollary 2.5]) to comodules over an arbitrary coalgebra provided the category of comodules is finitely accessible Grothendieck.…”

“…In particular, A is right and left locally coherent [21,Corollary 2.11]. Then (see [32,Proposition 5.8]) the class of absolutely pure objects of C is definable, so closed under direct limits and is, moreover, closed under pure epimorphic images. Also, note that absolutely pure objects in C are in one-to-one correspondence with absolutely pure flat objects of (fp(C) op , Ab) by the proof of [9, Proposition 2.1].…”

“…The categories which arise in this way are precisely the exactly definable categories in the sense of Krause [28]. For a general treatment of these categories see [32].…”