nLab homotopically injective object

Definition

Let $\mathcal{A}$ be an abelian category with translation.

An object $I$ in the category of chain complexes modulo chain homotopy, $K(\mathcal{A})$, is homotopically injective if for every $X \in K(\mathcal{A})$ that is quasi-isomorphic to $0$ we have

$$Hom_{K(\mathcal{A})}(X,I) \simeq 0 \,.$$

Let

$$QuasiIsoMono = \{f \in Mor(A_c) | f mono and quasiio\}$$

be the set of morphisms in the category of chain complexes $Ch_\bullet(\mathcal{A})$ which are both quasi-isomorphisms as well as monomorphisms.

Then

A complex $I$ is an injective object with respect to monomorphic quasi-isomorphisms precisely if

• it is homotopically injective in the sense of complexes in $\mathcal{A}$; and

• it is injective as an object of $\mathcal{A}$ (with respect to morphisms $f : X \to Y$ such that $0 \to X \stackrel{f}{\to} Y$ is exact).

Properties

In complexes in a Grothendieck category

Proposition For $\mathcal{A}$ a Grothendieck category with translation $T : \mathcal{A} \to \mathcal{A}$, every complex $X$ in $Ch_\bullet(\mathcal{A})$ is quasi-isomorphic to a complex $I$ which is injective and homotopically injective (i.e. QuasiIsoMono-injective).

Relation to derived categories

For $\mathcal{A}$ an abelian Grothendieck category with translation the full subcategory $K_{hi}(\mathcal{A}) \subset K(\mathcal{A})$ of homotopically injective complexes realizes the derived category $D(\mathcal{A})$ of $\mathcal{A}$:

$$Q|_{K_{hi}(A)} : K_{hi}(A) \stackrel{\simeq}{\to} D(A) \,,$$

where $Q : K(A) \to D(A)$ and $Q|_{K_{hi}(A)}$ has a right adjoint.

It follows that for $D$ any other triangulated category, every triangulated functor $F : K(\mathcal{A}) \to D$ has a right derived functor $R F : D(\mathcal{A}) \to D$ which is computed by evaluating $F$ on injective replacements: for $R : D(\mathcal{A}) \stackrel{\simeq}{\to} K_{hi}(\mathcal{A})$ a weak inverse to $Q$, we have

$$R F \simeq D(A) \stackrel{R}{\to} K_{hi}(A) \hookrightarrow K(A) \stackrel{F}{\to} D \,.$$

References

Much of this discussion can be found in

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves

The general notion of injective objects is in section 9.5, the case of injective complexes in section 14.1.