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The word “effective” is being used for many different things in the same context on ncatlab, and using “coeffective” can cut down on this overuse by at least half. For example,
http://ncatlab.org/nlab/show/congruence
applies the word “effective” to certain epimorphisms, as well as to certain congruences, which is confusing for two reasons: in the context of a diagram $R\rightrightarrows U \to X$,
1) neither usage implies the other, and
2) we have no word for when the diagram is “effective on both sides”.
I suggest the following revision:
A pair of parallel morphisms $R\rightrightarrows U$ (necessarily a congruence) is called effective or an effective pair if $R$ is the kernel pair of its coequalizer (and these operations all exist).
A morphism $U\to X$ (necessarily epi) is called coeffective if it is the coequalizer of its kernel pair (and these things all exist).
Then for a diagram $R\rightrightarrows U \to X$, TFAE:
1) X is the coequalizer and R is effective,
2) R is the kernel pair and X is coeffective.
So if either holds, following the usual use of “bi” in category theory, we can call the diagram “bieffective”.
I’d also recommend the phrase “h is a coeffective epimorphism” over “effective epimorphism”, because it can be shortened to “h is coeffective” with no confusion; coeffective implies epi.
Example usages:
“Gluing diagrams” are usually bieffective diagrams: X can be a scheme, U the disjoint union of a Zariski cover, and R the disjoint union of the intersections.
In the category of algebraic spaces, etale congruences are always effective, and etale surjections are always coeffective, so bieffective diagrams abound, and their bieffectiveness is what allows us to think of them intuitively as “gluing diagrams”.
In GIT, the geometric quotient of a free group action is bieffective (geometric means $G\times U\to U\times_X U$ is schematically surjective, and free means it’s an immersion, hence an isomorphism).
What do people think about this?
Having pushed Andrew to make this comment here rather than via email, now I’m going to say I disagree with the suggestion. (-: Firstly, both usages are pretty well-established, so I think we’d need a fairly good reason to depart from them.
Secondly, I think when we say that an epimorphism is “effective”, there is no ambiguity: we mean that it is the coequalizer of its kernel pair. Similarly, when we say that an equivalence relation is “effective,” there is also no ambiguity: we mean that it is the kernel pair of its coequalizer. It’s analogous to how we say both “regular epimorphism” and “regular monomorphism”; we don’t need to “co”-ify one of the words “regular,” because epi and mono are already dual.
And actually, there is a word for when the diagram is “effective on both sides”: we call it exact. (I don’t recall whether this terminology is used anywhere on the nLab, but it’s certainly in use elsewhere.) There is a close relationship to short exact sequences in abelian categories. If we want to say that a diagram $R \;\rightrightarrows\; U\to X$ has the property that $R \;\rightrightarrows\; U$ is the kernel pair of $U\to X$, I think it would be natural to call that left exact, and of course dually.
However, the exposition on the nLab can of course always be improved!
I’ve also thought of and like your usage of "left exact", "right exact", and "exact" for a diagram $R\rightrightarrows U\to X$. Unfortunately another reasonable definition of "exact" would be that "the image of R is the kernel pair of $U\to X$", i.e. that $R\to U\times_X U$ is epic or something surjective-esque.
But if your usage is already common, I’d like to see nLab make it more popular :)
I don’t think there’s much danger of misinterpretation to mean that, since it’s not really a notion that anyone’s ever considered as far as I know. If we did want to talk about that a lot, we could start saying “exact at $U$” as opposed to calling the whole diagram “exact,” which would match the abelian usage.
I’d like to see nLab make it more popular
You could contribute to that by editing some nLab pages… (-:
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