Timeline for Injective maps and direct limits
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 9 at 7:35 | comment | added | Elías Guisado Villalgordo | (Here $C_i\Rightarrow C$ is the obvious natural transformation of functors $\mathcal{I}\to\mathcal{C}$ from the constant functor $j\in\mathcal{I}\mapsto C_i$ to our direct system $C:j\in\mathcal{I}\mapsto C_j$.) | |
| Jul 9 at 7:31 | comment | added | Elías Guisado Villalgordo | $\def\I{\mathcal{I}}\def\C{\mathcal{C}}\def\c{\operatorname{colim}}$Another example is when $\C$ is an abelian category in which direct limits are exact. Let $\{C_i\mid i\in \I\}$ be a direct system in $\C$ such that $C_i\to C_j$ is injective for all $i\leq j$. Let $i\in\I$. We want to see that $C_i\to \c_{j\in\I}C_j$ is injective. We may assume that $i$ is initial in $\I$ (replace $\I$ by the full subcategory $\{j\in\C\mid i\leq j\}$, which is cofinal). The components of the functor morphism $C_i\Rightarrow C$ are all injective. Hence $\c(C_i\Rightarrow C)=C_i\to\c_{j\in\I}C_j$ is injective. | |
| May 15, 2020 at 17:32 | comment | added | Andrea Ferretti | Actually it doesn't, it is just that injective objects are more commonly used in that context | |
| May 15, 2020 at 15:42 | comment | added | Andreas Blass | Where does this argument use the assumption that the category is abelian? | |
| May 15, 2020 at 15:25 | history | edited | LSpice | CC BY-SA 4.0 |
Name of the Neeman paper
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| May 15, 2020 at 13:28 | history | answered | Andrea Ferretti | CC BY-SA 4.0 |