(also nonabelian homological algebra)
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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
Let
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).
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).
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 $\mathcal{A}$:
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
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.
Last revised on August 6, 2023 at 01:52:34. See the history of this page for a list of all contributions to it.