homotopically injective object
Let be an abelian category with translation.
An object in the category of chain complexes modulo chain homotopy, , is homotopically injective if for every that is quasi-isomorphic to we have
be the set of morphisms in the category of chain complexes which are both quasi-isomorphisms as well as monomorphisms.
A complex is an injective object with respect to monomorphic quasi-isomorphisms precisely if
it is homotopically injective in the sense of complexes in ;
it is injective as an object of (with respect to morphisms such that is exact).
In complexes in a Grothendieck category
Proposition For a Grothendieck category with translation , every complex in is quasi-isomorphic to a complex which is injective and homotopically injective (i.e. QuasiIsoMono-injective).
Relation to derived categories
For an abelian Grothendieck category with translation the full subcategory of homotopically injective complexes realizes the derived category of :
where and has a right adjoint.
It follows that for any other triangulated category, every triangulated functor has a right derived functor which is computed by evaluating on injective replacements: for a weak inverse to , we have
Much of this discussion can be found in
The general notion of injective objects is in section 9.5, the case of injective complexes in section 14.1.
Revised on September 23, 2012 15:43:21
by Urs Schreiber