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Apologies for the slew of paper related questions, but this one was bugging me too.
Given a pretopology, or more generally, a coverage $J$ on a category, and the class of arrows ($J$-epi) of arrows that admit local sections relative to $J$. This class is interesting, but I’m interested in the subclass of arrows of which all pullbacks exist and which is stable under pullback (hence forms another pretopology). I denoted this $J_{sing}$ in my paper, because it is, if you like, the singletonification of $J$. This is clearly a Bad Name (TM), but I can’t think of a good name. ’The class of pullback-stable $J$-epimorphisms’ is also too much of a mouthful. It’s a sort of saturation of $J$, but isn’t saturated as I define the notion (and I have good reason to keep the definition of saturation as is).
Any ideas?
“universally J-epic”? (To go along with “universally effective-epimorphic.”)
let me try it:
“Let $J_u$ be the class of universally $J$-epi maps.”
(where now $J_{sing} \mapsto J_u$) Hmm. Then there is a nice double meaning to the ${}_u$, as it is an abbreviation of un. How about $J_{un}$? Is it too much of a pun?
What’s the pun? Is it French? (I still don’t see the pun.)
By the way, I have seen “arrows of which all pullbacks exist” called “carrable”, which seems to be the same French word showing up on this nLab page (although not with quite the same meaning).
$un$ as in universal, and also as a singleton pretopology (of course, un = one). I’ll go with $J_{un}$ I think.
Yes, of course, I forgot the meaning behind the original symbol!
I also like the Anglo-French pun $\mathbb{F}_{un}$.
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