diff --git a/injectives.tex b/injectives.tex index 6370c550..d94b3d5d 100644 --- a/injectives.tex +++ b/injectives.tex @@ -1276,12 +1276,14 @@ \section{Grothendieck's AB conditions} \item[AB3] $\mathcal{A}$ has direct sums, \item[AB4] $\mathcal{A}$ has AB3 and direct sums are exact, \item[AB5] $\mathcal{A}$ has AB3 and filtered colimits are exact. +\item[AB6] $\mathcal{A}$ has AB3 and small products distribute over filtered colimits: for a small diagram $J$ and a collection of $J$-indexed filtered diagrams $I_{j}$ for $j\in J$ the natural map $\colim_{(i_{j}\in I_{j})_{j}}\prod_{j\in J}M_{i_{j}}\to\prod_{j\in J}\colim_{i_{j}\in I_{j}}M_{i_{j}}$ is an isomorphism. \end{enumerate} Here are the dual notions \begin{enumerate} \item[AB3*] $\mathcal{A}$ has products, \item[AB4*] $\mathcal{A}$ has AB3* and products are exact, \item[AB5*] $\mathcal{A}$ has AB3* and cofiltered limits are exact. +\item[AB6*] $\mathcal{A}$ has AB3* and direct sums distribute over cofiltered limits: for a small diagram $J$ and a collection of $J$-indexed cofiltered diagrams $I_{j}$ for $j\in J$ the natural map $\bigoplus_{j\in J}\lim_{i_{j}\in I_{j}}M_{i_{j}}\to\lim_{(i_{j}\in I_{j})_{j}}\bigoplus_{j\in J}M_{i_{j}}$ is an isomorphism. \end{enumerate} We say an object $U$ of $\mathcal{A}$ is a {\it generator} if for every $N \subset M$, $N \not = M$ in $\mathcal{A}$ there exists a morphism