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Created descent morphism.
In adding links, I discovered that Euclidean-topological infinity-groupoid and separated (infinity,1)-presheaf use the phrase “descent morphism” to refer to the comparison functor mapping into the category of descent data. If no one has any objections, I would like to change this to avoid confusion, but I’m not sure what to change it to: would “comparison functor” be good enough?
I added the example of fpqc descent for quasi-coherent sheaves…I got a little confused translating back from thinking in terms of the psuedofunctor to what the “comparison functor” is. I figured it out, but just left it kind of vague and said that fpqc morphims “induce” effective descent. There might be a better way of saying it.
If what you mean is just “QCoh(-) forms a stack over the fpqc site”, then I think the way to say that would be “every fpqc cover is effective descent for QCoh(-)”. That would be an instance of “relativization” of the notion to a fibration/indexed category other than the codomain fibration. Is that what you want?
I added descent and effective descent morphisms to the list of variations at epimorphism, and rearranged it a bit.
I see that in the “idea” section of regular epimorphism, an epimorphism is said to be regular if it “behaves like a covering”, but the page covering doesn’t tell me what that means. Clearly covering is a stub, but I can’t tell what is/was planned to go there eventually. How is a “covering” distinct from a cover?
Since the actual definition of a regular epimorphism is so short, my inclination would have been to just start out with that on the page regular epimorphism. In fact, insofar as “covering” refers to covering families in a site, I would have been more inclined to say “a cover is something which is desired or stipulated to behave like a regular epimorphism” than the other way ’round. Thoughts?
Yes. This is all I was trying to say. The part that tripped me up was figuring out whether I should try to explain how an fpqc cover actually gives a comparison functor.
Also just to clarify: every descent morphism is an effective epimorphism, but not every descent morphism is an effective descent morphism. Correct?
It is not entirely true that “Descent can also be rephrased in terms of the monadicity theorem”; only SOME (most) cases of descent reduce to (co)monadic descent, not all.
While I agree that effective descent is an unfortunately a longer name and among the practioners of the descent theory (I count also myself) we often say that we have descent and mean effective descent, Giraud had even worse terminology in which he would call fully faithful case 1-descent and the equivalence of categories case 2-descent, clashing with modern prefixes for categorical dimension involved.
I discovered that Euclidean-topological infinity-groupoid and separated (infinity,1)-presheaf use the phrase “descent morphism” to refer to the comparison functor mapping into the category of descent data. If no one has any objections, I would like to change this to avoid confusion, but I’m not sure what to change it to: would “comparison functor” be good enough?
I have changed it to “descent comparison morphism”. Thanks for catching this.
(On the other hand, while it is true that “descent morphism” at least in the form “a morphism is (effective) descent” is traditional usage, there seems to be room for arguing that this traditional usage is unfortunate. After all, it is all about the descent of one specific object, of which there is no indication in the term. )
I see that in the “idea” section of regular epimorphism, an epimorphism is said to be regular if it “behaves like a covering”, but the page covering doesn’t tell me what that means.
This was meant to allude to the plain English language idea, not to a technical term. Since we are in an “Idea”-section I find I can explain the idea of “regular epimorphism” in terms of “covering” but not the other way round (everybody has an idea of “covering”, nobody of “regular epimorphism” who hasn’t seen the formal definition of ).
every descent morphism is an effective epimorphism, but not every descent morphism is an effective descent morphism
Yes. In a category with pullbacks the descent morphisms are the stable regular epis, and regular = effective in the presence of pullbacks also.
only SOME (most) cases of descent reduce to (co)monadic descent
Well, please feel free to edit the page!
there seems to be room for arguing that this traditional usage is unfortunate.
Indeed. But “effective descent morphism for the codomain fibration” is even longer! It might be best if we had a simple adjective to stick in front of “epimorphism” to mean “effective descent morphism”. I don’t think I’ve ever heard any such usage, but one could, I guess, try to change the world.
This was meant to allude to the plain English language idea, not to a technical term.
Okay. That seems to be belied by the text at covering which says “regular epimorphism $\Leftrightarrow$ covering”. If covering is a plain English language idea, then it can’t be asserted to be equivalent to any precise mathematical concept, can it?
It’s also not clear to me that regular epimorphisms are more deserving to be identified with the plain English language idea of “covering” than are other sorts of epimorphisms. Certainly plain epimorphisms are an unlikely candidate (except when they coincide with regular ones), but what about (say) descent morphisms or effective descent morphisms?
Not sure how to best resolve what you are after here. You should maybe just go ahead and rewrite the Idea-sections in a way that you find more appropriate.
I guess I was motivated by the fact that a) every morphism that one may want to call a cover becomes an effective epi after passage to the corresponding sheaves and b) given any topos the “correct” internal notion of cover is effective epimorphism (for instance if we want to say “locally trivializable” in a topos, we say: there is an effective epi such that we have trivializability after base change along that).
I’m not exactly sure what I’m after either, but I’ll try some rewriting.
a) every morphism that one may want to call a cover becomes an effective epi after passage to the corresponding sheaves and b) given any topos the “correct” internal notion of cover is effective epimorphism
Yes… but both of those are equally true with “effective epimorphism” replaced by any other kind of epimorphism, from plain epimorphism all the way up to effective descent morphism, since all kinds of epimorphisms (except split ones) are the same in any topos. So why single out the effective epis as the ones to call “covers”?
Oh, that’s what you mean: because only for effective epis this generalizes to higher toposes! :-)
Well, certainly plain epimorphisms don’t generalize, but surely descent and effective descent morphisms do — isn’t part of the point of higher toposes that effective epis are effective descent?
And do you know for sure that strong epis aren’t necessarily effective in a higher topos? I know they are in a 2-topos.
And do you know for sure that strong epis aren’t necessarily effective in a higher topos? I know they are in a 2-topos.
Ah, that’s true. In an $(\infty,1)$-topos, too, the effective epis coincide with the strong epis, by the n-connected/n-truncated factorization system for $n = -1$.
Hm, okay, so what shall we do?
Edit: I still seem to tend to think that the definition of effective epi is the one that captures the idea of coverings: the definition of effective epi says, when translated to words: “$U \to X$ is an effective epi if $X$ is the union of the components of $U$”. Other definitions may be equivalent to this in some circumstances, but they don’t express this explicitly. What do you think?
“descent comparison morphism”
“Comparison functor” is standard. Word “morphism” here is misleading.
“effective descent morphism for the codomain fibration” is even longer
It is easy to parse and precise phrase. Some phrases are short but incomprehensible like Australian abbreviation “descent category” which was repeatedly confusing me for years, used instead of the standard and obvious “category of descent data”.
“Comparison functor” is standard. Word “morphism” here is misleading.
Only that it’s not a functor here, but an $\infty$-functor or some presentation thereof. I can’t see how it is misleading.
The (effective or not) descent morphisms, also compare (intutitively) the glued space and the disjoint union of open neighborhoods. Thus if I would see a phrase “descent comparison morphism” I would think it is about the (effective or not) descent morphism. Comparison FUNCTOR, on the other hand, has a definite and standard meaning both in descent theory and in the theory of (co)monadicity. Misleading is always with respect to wide and established tradition and system of knowledge. nLab is going often into nonstandard terminology and some of my colleagues/students sometimes avoid it for this reason when asking for a quick advice and resort to wikipedia instead.
I like “comparison functor”. If it’s not a functor in some case for some reason (although $\infty$-functors, I think, are fine to call “functors”), we could just say “comparison morphism” or “comparison map”.
$U\to X$ is an effective epi if $X$ is the union of the components of $U$
That’s a good point. I could argue that descent morphisms and strong epis also have good translations into words which might also be interpreted as matching the English word “covering” (as might plain epimorphisms, for that matter), but I won’t.
I have edited the pages regular epimorphism, effective epimorphism, strict epimorphism, effective epimorphism in an (∞,1)-category, and also created regular epimorphism in an (∞,1)-category by combining some text that was scattered at “regular epimorphism” and “effective epimorphism in an (∞,1)-category”. How do they look?
How do they look?
Great, thanks!
More basic references and links at descent.
I have added “…and coequalizers” in the first line here.
(just to implement #23, made no effort to check anything)
But this entry needs attention in any case: It should say more concretely which literature its definition and claims is following.
(For instance, currently the entry does not make clear that this definition is from Tholen’s writings, if that’s what it is.)
Added:
A criterion for categories (including quasi-abelian categories) under which effective descent morphisms and descent morphisms coincide is established in
added an original reference:
Added notes about the assumptions of different proofs of the closure under composition (I may need to come back to this to tighten up the statements).
Looking at the paper of Janelidze and Tholen, it’s not clear how it’s relevant to the theorem statement about closure properties of effective descent morphisms. In any case, the results are almost exclusively about the category of topological spaces, and the paper nowhere states the composition claim! I’m thinking of removing this reference.
Added citation to:
for the result that in a finitely complete category, effective descent morphisms are closed under pullback. Also clarified what was proved where around closure under composition, and split off the citation to Janeldize–Tholen’s CT90 paper as a reference for the new proposition that split epis are effective descent.
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