nLab
homotopically injective object

Definition

Let $𝒜$ be an abelian category with translation.

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

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

Let

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

be the set of morphisms in the category of chain complexes ${\mathrm{Ch}}_{•}(𝒜)$ 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 $𝒜$;

• it is injective as an object of $𝒜$ (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 $𝒜$ a Grothendieck category with translation $T:𝒜\to 𝒜$, every complex $X$ in ${\mathrm{Ch}}_{•}(𝒜)$ is quasi-isomorphic to a complex $I$ which is injective and homotopically injective (i.e. QuasiIsoMono-injective).

Relation to derived categories

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

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

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

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

$$RF\simeq D(A)\stackrel{R}{\to }{K}_{\mathrm{hi}}(A)\hookrightarrow K(A)\stackrel{F}{\to }D.$$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.