#Locally presentable category generator

Constructions realizing such triangulated categories, similarly to the conventional derived category, as quotient categories of the homotopy category of the ambient abelian category, were emphasized in the memoir, where the terminology of the derived categories of the first and second kind was suggested. The homotopy category of unbounded complexes of projective modules was first considered by Jørgensen, and the homotopy category of unbounded complexes of injective objects in a locally Noetherian Grothendieck category was first studied by Krause. The equivalence relation on the unbounded complexes then needs to be modified accordingly so the conventional quasi-isomorphism is replaced by a finer equivalence relation, making the resulting “exotic derived category” larger than the conventional one.

It took another decade or two to realize the existence and importance of an alternative point of view on unbounded complexes, in which arbitrary complexes of injective or projective objects are used as resolutions. The homotopy projective complexes of projective objects, which are more suitable for some purposes than just arbitrary homotopy projective complexes, are sometimes called “semi-projective” or “DG-projective”, and similarly for the homotopy injective complexes of injective objects (see Remark 6.4 for a terminological discussion). Nowadays some people call them “homotopy injective” or “homotopy projective” complexes we will use this terminology. So one needs to use what Spaltenstein called “K-injective” or “K-projective” resolutions for unbounded complexes. In fact, an unbounded complex of projective (or injective) modules over a ring of infinite homological dimension can be acyclic without being contractible. But these equivalences fail for unbounded derived categories, generally speaking. Such triangulated equivalences are used in the constructions of derived functors acting between bounded above or below derived categories. The problem is that the derived category D + ( A ) of bounded below complexes in an abelian category A with enough injective objects is equivalent to the homotopy category of bounded below complexes of injective objects K + ( A inj ), and similarly, the derived category D − ( B ) of bounded above complexes in an abelian category B with enough projective objects is equivalent to the homotopy category of bounded above complexes of projective objects K − ( B proj ). Subsequently they were extended to the derived categories of DG-modules by Keller (see also ). The definition of the unbounded derived category of an abelian category goes back to the work of Grothendieck and Verdier in 1960's, , but efficient techniques of working with such derived categories started to be developed only in the 1988 paper of Spaltenstein (who attributes the idea to J. In particular, the derived category of any locally presentable abelian category has Hom sets. More generally, we define the notion of an exact category with an object size function and prove that the derived category of any such exact category with exact κ -directed colimits of chains of admissible monomorphisms has Hom sets. Assuming Vopěnka's principle, we prove that the derived category D ( A ) is generated, as a triangulated category with products, by the injective cogenerator of A. For a Grothendieck abelian category A, we construct the injective derived and coderived model structures on complexes.

#Locally presentable category generator

We also show that the derived category D ( B ) is generated, as a triangulated category with coproducts, by the projective generator of B. For a locally presentable abelian category B with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects.