Injective objects in Mor(Ab) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:08:46Z http://mathoverflow.net/feeds/question/110309 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110309/injective-objects-in-morab Injective objects in Mor(Ab) Martin Brandenburg 2012-10-22T08:08:41Z 2012-10-22T12:17:27Z <p>Consider the abelian (Grothendieck) category $\mathcal{C} := \mathrm{Fun}({0&lt;1},\mathrm{Ab}) = \mathrm{Mor}(\mathrm{Ab})$. Objects are morphisms $(A \to B)$ of abelian groups, morphisms are commutative diagrams. Equivalently, this is the category of abelian sheaves on the Sierpinski space.</p> <p><strong>Question.</strong> How do injective objects in $\mathcal{C}$ look like?</p> <p>Since injective sheaves are stable under restriction (use extension by zero), clearly $(A \to B)$ injective implies that $A$ is injective. But is this sufficient (probably not)? When $A,B$ are injective, is the same true for $(A \to B)$?</p> http://mathoverflow.net/questions/110309/injective-objects-in-morab/110319#110319 Answer by Karol Szumiło for Injective objects in Mor(Ab) Karol Szumiło 2012-10-22T11:44:45Z 2012-10-22T12:17:27Z <p>I will use notation <code>$A_0 \to A_1$</code> for objects of <code>$\mathrm{Mor}(\mathrm{Ab})$</code>.</p> <p>EDIT: previously I claimed something stronger (that I can produce lifting properties in the functor category without factorizations), but I am not so sure about it.</p> <p>The following is a lot more general than necessary, but I think this added generality is also useful. Let <code>$(\mathcal{L}, \mathcal{R})$</code> be a weak factorization system in a category <code>$\mathcal{C}$</code> with enough colimits and limits for the following to make sense. Let $J$ be a Reedy category. Then in the functor category <code>$\mathcal{C}^J$</code> the "Reedy <code>$\mathcal{L}$</code>-cofibrations" and "Reedy <code>$\mathcal{R}$</code>-fibrations" form a weak factorization system. By "Reedy <code>$\mathcal{L}$</code>-cofibrations" I mean morphisms of diagrams <code>$X \to Y$</code> such that for every <code>$j \in J$</code> the latching morphism <code>$X_j \sqcup_{L_j X} L_j Y \to Y_j$</code> is in <code>$\mathcal{L}$</code> and dually "Reedy <code>$\mathcal{R}$</code>-fibrations" are morphisms <code>$X \to Y$</code> such that for every <code>$j \in J$</code> the matching morphism <code>$X_j \to M_j X \times_{M_j Y} Y_j$</code> is in <code>$\mathcal{R}$</code>. The proof is exactly as in the construction of the Reedy model structures and can be found for example in Hovey's <em>Model Categories</em>.</p> <p>Now we take <code>$\mathcal{C} = \mathrm{Ab}$</code>, <code>$\mathcal{L} =$</code> monomorphisms and <code>$J = [1]$</code>. Then <code>$\mathcal{R}$</code> are split epimorphisms with injective kernel. The lifting properties are easily verified while the factorizations use the fact that there are enough injectives in <code>$\mathrm{Ab}$</code>. If <code>$f : A \to B$</code> is a map in <code>$\mathrm{Ab}$</code>, pick an injective hull <code>$i : A \to \hat A$</code>, then <code>$f$</code> factors as an injection <code>$[i, f] : A \to \hat A \oplus B$</code> followed by a split surjection with injective kernel <code>$\hat A \oplus B \to B$</code>. We consider <code>$J$</code> as a Reedy category where <code>$0$</code> has degree <code>$1$</code> and <code>$1$</code> has degree <code>$0$</code>. Then "Reedy <code>$\mathcal{L}$</code>-cofibrations" are monomorphisms again, so an object <code>$X$</code> is injective if and only if the map <code>$X \to 0$</code> is a "Reedy <code>$\mathcal{R}$</code>-fibration" i.e. when both <code>$X_1 \to 0$</code> and <code>$X_0 \to X_1$</code> are split epimorphisms with injective kernel i.e. when <code>$X_0 \to X_1$</code> is a split epimorphism with injective source.</p>