Let $\mathcal M$ be a class of arrows in a category $\mathsf C$. I would like suggestions for good notation for the following two classes.
- The smallest universal (pullback stable) subclass $\mathcal M$;
- The class of arrows "locally in $\mathcal M$", i.e those which are pulled back to $\mathcal M$ along some effective descent morphism.
In A. Carboni, G. Janelidze, G. M. Kelly, and R. Paré's On Localization and Stabilization for Factorization Systems, the respective notations $\mathcal M^\prime,\mathcal M ^\ast$ are used. In Borceux and Janelidze's Galois Theories book, the notation $\Sigma \mathcal M$ is used for the class of arrows locally in $\mathcal M$.