# nLab homotopically injective object

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

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

An object 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 $\mathcal{A}$;

• 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

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 (89.204.137.161)