Derived categories for Grothendieck categories of enriched functors

Abstract

The derived category D[C , V ] of the Grothendieck category of enriched functors [C , V ], where V is a closed symmetric monoidal Grothendieck category and C is a small Vcategory, is studied. We prove that if the derived category D(V ) of V is a compactly generated triangulated category with certain reasonable assumptions on compact generators or K-injective resolutions, then the derived category D[C , V ] is also compactly generated triangulated. Moreover, an explicit description of these generators is given.

Recollements for derived categories of enriched functors and triangulated categories of motives

“…Inspired by Voevodsky's constructions [25], we investigate certain categorical aspects of DM e f f C (k). For this purpose, we work in the framework of Grothendieck categories of enriched functors in the sense of [1] and their derived categories [9]. We shall convert some fundamental Voevodsky's theorems into the language of enriched category theory and arrive at some categorical concepts and results which are of independent interest.…”

Rational enriched motivic spaces

“…The goal of this section is to construct a monoidal model structure on Ch(Shv(A)) that is weakly finitely generated (Definition 2.2.9), satisfies the monoid axiom [47,Definition 3.3], and in which the weak equivalences are the quasi-isomorphisms. Once we have such a model structure we can use [20,Theorem 5.5] to construct the projective model structure on the category of chain complexes Ch([C, Shv(A)]) of the Grothendieck category of enriched functors [C, Shv(A)] for any small Shv(A)-enriched category C. The model structure will be useful for proving the reconstruction theorems of the next two chapters.…”

“…Let Psh(A) be the category of Ab-enriched functors A op → Ab. We can then apply [20,Theorem 5.5] to get a weakly finitely generated monoidal model structure on Ch(Psh(A)), where weak equivalences are sectionwise quasi-isomorphisms, and the fibrations are epimorphisms. We call it the standard projective model structure on presheaves, or sometimes just the projective model structure on presheaves.…”

“…Since Shv(A) is a closed symmetric monoidal Grothendieck category, and Sm is a monoidal Shv(A)-category, we can apply [20,Theorem 5.5] to get a weakly finitely generated monoidal model structure on Ch([Sm, Shv(A)]), where the weak equivalences are the pointwise quasi-isomorphisms and the fibrations are the pointwise fibrations. We will say that…”

“…We recover the triangulated category of big A-motives DM A out of Grothendieck categories of enriched functors [B, Shv(A)] in the sense of [1], where B is either the Shv(A)-category C of the powers G ×n m or the Shv(A)-category Sm of all smooth k-schemes. To this end, we use homological algebra of enriched Grothendieck categories developed in [20,21].…”