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

]]>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.

Thanks!

]]>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! 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?

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.

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.

]]>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.

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.

]]>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 $A$.

]]>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?

]]>I was voting on whatever was on the ballot.

]]>I thought what people voted for was *moving* material away from “generator”, whereas you seemed to be voting against “separator” as such?!

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. (-: )

]]>If other people are voting for something but I can’t vote against it, that doesn’t sound like a democracy. (-:

]]>@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.

]]>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. ;-)

]]>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.

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 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?????

]]>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 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.

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