Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 11 of 11
I was fixing some Spam at generator and noticed that Grothendieck category has a link to generator, but shouldn’t this be to separator? I have fixed it so that the term generator in Grothendieck category links to separator.
I tried to clear up some formatting problems / typos at separator. (There is a query near the bottom of the page that seems to still be unanswered.) Can someone glance at the entry to check my reformatting is right as I was on autopilot when doing it!
Word generator is quite standard in the context of abelian categories, I have read it in well over 100 references. For separator, never heard.
The entry on Grothendieck categories uses generator, but clicking on that you get the dual terminology of separator / generator, which seems a reasonable compromise.
I think separator is good when thinking of sheaflike categories, but generator, although ‘classical’ is something of a misnomer as that aspect is a bit lost in a general Abelian category. A generator should have a property that corresponds to some notion of generation and that is obscured in the Abelian category usage.
Which classical references do use “separator” terminology?
I do not know of any specifically for the Abelian case. Both are mentioned in Borceux. It seems that in the Elephant Johnstone makes a distinction between them. (I am not an expert in this.)
Hey, separator and cogerentator had it interchanged. I just started reversing wrong way (not even domains fit the way it was written) in separator in the first def. (still leaving old one in the second). For generator one has implies , not for cogenerator, cf.eom. Somebody seems to use the non-Leibniz notation for composition. I would correct further but regarding abundance of wrong way in the two entries i wait for approval.
I’ve only so far looked at separator, which needs some clean-up (definition 2 in particular is a little messed up). Certainly the order of composition in def. 2 is wrong.
I did some editing at separator, including some notes surrounding the “generator” terminology. One such note was about “progenerator” which was indeed common in older texts. Therefore I am removing the following query box.
+– {: .query} Mike: The term “progenerator” seems unfortunate to me; it sounds to me like a pro-object that is a generator. Is it well-established? I’ve never heard it, though I have heard “projective generator” in the context of Morita theory.
Zoran Škoda: It is an extremely frequent term in classical algebra and in many of the standard monographs in module theory over classical rings. I personally never use the expression and mentioned it only once in a survey. But as a link to that area of mathematics I tend to behave conservatively. Notice that the terminology subsumes finite generation. =–
Zoran, I didn’t spot anything wrong with cogenerator. However, I did edit in a result that shows that every Grothendieck topos has an injective cogenerator (formulated in such a way to subsume well-pointed toposes as well).
I agree, nothing wrong with cogenerator. Th def 1 and 2 were wrong in generator entry and I corrected the 1st and then compared with cogenerator and it looked to me that it was wrong there in the same way, but this was due different notation, so I misparsed the latter one.
1 to 11 of 11