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finally created strongly connected topos (nothing deep there, just terminology).
Today I received mild criticism for the choice of terminology “strongly connected” in strongly connected topos. I forget where this terminology originates. Googling for it by now shows tons of $n$Lab hits, but nothing else. ;-)
I forget too, if I ever knew. Did the criticizers have any particular reasons for their criticism? In this paper Johnstone uses “stably locally connected”.
Did the criticizers have any particular reasons for their criticism?
The claim was that “strong” is more commonly used for a different property of geometric morphisms. I think the claim was that it should be used for properties related to boundedness. I’ll check again.
Huh, I don’t think I’ve encountered that.
Okay, I have checked:
the commenter’s point is that “strong” in the context of adjunctions $(f_! \dashv f^*)$ suggests that it refers to the internal hom condition
$[f_! X, Y] \simeq f_* [X, f^* Y] \,.$But in discussion we decided that also in this sense “strongly connected” is not so bad after all: a connected geometric morphism $f$ such that $f_!$ exists and preserves products implies Frobenius reciprocity for $f_*$, which in turn implies the above.
(All of which is of course closely related to what we have been discussing lately.)
For the record, I have added comments to this effect here and here.
Hrm, if any locally connected geometric morphism satisfies that internal-hom condition, that seems to me to be an argument against using “strongly connected” for something more than that.
Here a vastly belated reaction to the above exchange:
Hrm, if any locally connected geometric morphism satisfies that internal-hom condition, that seems to me to be an argument against using “strongly connected” for something more than that.
Yes, true.
Does anyone feel like brainstorming what alternative terminology might be good?
My problem is that the important effects of this condition that I know of are all somewhat technical. None of them really lends itself to a crisp term here. Or so it seems.
Johnstone calls it “stably locally connected”. Presumably he has some reason, but I don’t know what it is.
Coming back to this thread almost five years later, I suggest that we switch to Johnstone’s terminology (“stably locally connected”). I still can’t see any reason for “strongly connected”, nor do I see any citations for that terminology anywhere else, and as pointed out above it’s actually somewhat misleading. Lawvere and Menni (p929) report that Johnstone says his term was chosen by analogy with “stably locally compact”, since “connected objects are closed under finite products” is similar to “compact sets are closed under finite intersections”. Lawvere and Menni also observe that for a connected locally connected geometric morphism, stable local connectedness is equivalent to the reflector $f_!$ having “stable units”, giving another justification for the terminology.
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