# Notation for largest universal subclass and class of arrows “locally in” a given class of arrows

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 $\mathcal M$.

• What's wrong with the notation you found on those papers?? – Fosco Aug 19 '16 at 17:58
• @FoscoLoregian that asteriks and primes are already overloaded symbols and I prefer notation from which one can often guess the meaning. I thought of just writing out $\operatorname{stab}\mathcal M$ and $\operatorname{loc}\mathcal M$ but wanted to see if there are any better suggestions. – Arrow Aug 20 '16 at 11:34