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I created separator, while having the nagging feeling that we already have this entry. Of course after creating it I remembered the page generator.
So we should merge the stuff. Might this be an occasion to merge away from generator? A set of “generating objects” also means other things than “separating objects” (notably colimit generation). So I’d be inclined to move all material to separator. That would also allow to drop the warning at the beginning of generator.
I have always found the terminology of ’generator’ to be confusing to the point of being a ’misnomer’. I like the term separator as it is much more descriptive.
I’ll also cast a vote of support for “separator” and “separating family”. It’s the terminology used in the Elephant, and if I remember correctly, in Lawvere and Rosebrugh’s Sets for mathematics as well.
Okay, if anyone feels energetic, feel free to move the stuff accordingly.
I started on it at separator, but the other linked entries will need attention. Whilst the dual of separator would be coseparator, is that a good term. There are a large number of entries with cogenerator in them so as a stop gap measure I will add coseparator as an alternative name in cogenerator.
Thanks, Tim. I fixed some unintended results of your search-and-replace (“a separator is sometimes called a separator” ;-). Also, we need to be careful with applying search-and-replace to query boxes, making people say things which they never said! I removed all query boxes from separator. I think that entry is fine. But generator still needs attention.
I thought I had fixed most of those bugs… oh dear! But I do like “a separator is sometimes called a separator”. We might leave that to see if someone notices … other than your good self!
I thought that I had renamed generator to become generator>History?????
I thought that I had renamed generator
Let’s wait to see what Mike says. It seems he was the author of the non-trivial content at generator, so he should have a say on this issue.
I cast a vote against “separator”. The reason for calling this a “generator” is, as explained in the section “Strengthened generators”, that there is a general notion of “$\mathcal{E}$-generator” for any kind of “epimorphism” $\mathcal{E}$. I using think a special-case terminology like “separator” for the particular case when $\mathcal{E}$ is ordinary epimorphisms is antisystematic, especially if it encourages people to use the plain word “generator” for something other than this special case of $\mathcal{E}$-generator.
You can’t vote against the term “separator”, since it is being used (and by nLab regulars, too. :-) and that won’t be undone.
What you can do is suggest ways to organize the exposition on the $n$Lab. Which, luckily, you have now done, too. ;-)
@Mike: Amusingly, that is exactly what is done in the Elephant: “generator” refers to strong generators. But personally I would rather reserve “generator” for dense generators.
While we’re on the subject, the point raised in the query box confuses me as well. In $\mathbf{Ab}$ every epimorphism is extremal, but $\mathbb{Z}$ is only a (strong) separator, not a dense generator.
If other people are voting for something but I can’t vote against it, that doesn’t sound like a democracy. (-:
If it helps, Mike: the eponymous nLab regular was me, and if you want to change that instance of ’separator’ in connected object to ’generator’, please go ahead. (We all have to come together now, on both sides of the aisle. (-: )
I thought what people voted for was moving material away from “generator”, whereas you seemed to be voting against “separator” as such?!
I was voting on whatever was on the ballot.
Can someone explain to me why ’generator’ is a good term for the concept? (I know it is the traditional one, and am rather asking for information and enlightenment than trying to change the name used in entries.) A generator presumably generates something and I have never really seen what it ‘generates’ in this context, so what is the intuition?
Tim, I’ve always understood it like this: $\mathbb{Z}$ is a generator for $Ab$ because for any abelian group $A$, there is a jointly epimorphic family of maps $\mathbb{Z} \to A$ (by the separator condition, the canonical map $\sum_f \mathbb{Z} \to A$, where $f$ ranges over all possible maps $f: \mathbb{Z} \to A$, is an epimorphism), so that such maps span or generate $A$.
My query is more that evident idea became the ‘separator’ style definition. The extension of the terminology seemed a bit forced. This is largely curiosity but also perhaps it would be useful to incorporate some motivation for the terminology in the entry. Perhaps also your wording a generator satisfies the separator condition is a step towards a good explanation.
I get a similar feeling when I see ‘locally finitely presentable’, and compare with the definitions of similar terms coming from algebra.
Todd, that’s a good explanation! I hadn’t thought of it quite like that before. More generally, I guess a set of generators for an object should be thought of as some sort of epimorphism onto that object, hence why we have different notions of “generator” for different kinds of epimorphism. That also suggests that a “dense generator” could be more reasonably called a “presenter”, as the density colimit gives not merely a generation of the object but a presentation of it.
What has happened to these entries? Currently generating set redirects me to generator>History, which can’t be right, and generator is a cache bug.
Hmm… in the absence of finite limits, it’s not clear to me that either extremal generation or strong generation, in the sense of $\epsilon$ being an extremal or strong epi, is provably equivalent to the hom-functors being jointly faithful and conservative.
Thanks! One slight issue is that in the absence of equalizers, an “extremal epimorphism” seemingly need not actually be an epimorphism, so that an “extremal generator” defined in terms of such need not actually be a generator, and similarly in the strong case. Unless, of course, one defines an “extremal epimorphism” to be an epimorphism such that blah. Currently our page strong epimorphism does this, but our page extremal epimorphism does not. Should it?
Borceux assumes that extremal epis are in particular epis, so I would say yes let’s add it. The page on extremal monomorphisms already does it.
Edit: I changed the page, feel free to roll back if you disagree.
Thanks!
This entry had long been asking (in the first line here) for a page “family fibration”. Since it looks like the intended content would be that at our page codomain fibration, I have made “family fibration” redirect to there.
Added two examples: * 2 is a separator in $$\Set^{op}$$, thus every continuous functor from Set into a locally small category has a left adjoint. * The opposite of Group does not have a separator, since there exists a non-representable continuous functor from$Set$to$Group$.
Jonas Frey
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