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At present the entry double complex has a definition that say ’bla bla’!!!!! What was intended for there????
What was intended for there????
The actual definition.
Yeh! but in which form. Mac Lane Homology gives one form if I remember correctly, but there is another. In one the squares commute in the other they anti-commute.
Feel free to add all equivalent definitions that you can think of!
Commuting squares is convenient if you want to define a double complex as a chain complex in the category of chain complexes. Anticommuting squares is convenient for defining the total complex (for computing total homology).
At chain complex are the boundary morphisms differentials or codifferentials. Both seem to be used.
Tim,
it’s clearly a stub. It was written in a nanosecond years back just so that the link to it wouldn’t be gray. Just go ahead and edit it. I trust whichever way you do will be fine. After all, this is just about double complexes, isn’t it?
Or Zhen: please feel free to edit the entry!
I have started, but, Zhen, please feel free to add the other definition yourself as I may not have that much time in the near future.
I just modified your text a little and added it in.
You wrote almost exactly what I would have done. :-) I added a line to the first version. but there is a glitch in my original Tex as I forgot a space. I will fix it. I also changed $\sqcup$ to $+$ as it is more usual.
(By the way, why not start a home page on yourself. Just giving a stub if you want. It would get rid of the nasty question mark after your edits. :-)) (Edit: Re:your page: Great and welcome.)
Thanks!, to you two, for taking care of this!
added a bit more… there is still some bla bla to change!
I have edited the Idea-section at double complex a bit:
first of all I moved the picture of the double complex right to the very top, for I think this is what most immediately conveys the idea and the Idea-section should try to get the basic Idea across as quickly as possible.
but I changed the labelling a bit, compared to the diagram as it appeared earlier:
I made the first index be the horizontal one and the second the vertical one, for I think this is the pretty universally established convention (no?).
then I removed the subscripts ${}_X$ from the differentials. Since there is just a single double complex under discussion, there is no need to label its differentials this way, and it just clutters up the diagram.
I slightly edited the wording in the Idea-section where it comes to the total complex and the sign convention.
In the discussion of the bifunctors I made the notation of the three categories involved be consistent over the two displayed formulas;
I fixed the redirects: notice that it has to be
[[!redirects
instead of
[[!redirect
I hope all this is uncontroversion, but let me know if you disagree.
There is the problem that the idea section says that all squares commute but there is another which looks at the case where they anti-commute. This suggests that a note be added early on to comment on this. I have added a few words to point this out. We do discuss it further down the page.
Hey Tim,
just a few lines below your last change this point is already made.
I’d rather not say it already where you just added it, because that makes the next statement about complexes of complexes be awkward. Also I’d rather never draw a diagram that is not meant to commute.
So I’d be happier if we rolled back your last edit. But it’s not such an important point.
I will roll back, no problem, but the wording lower down seems a bit strange in places.
and the interest in double complexes is often that in these total complexes.
I suggest: Often it is such total complexes that are of interest. (with a full stop at the end of the previous sentence of course!)
The next sentence about the differential is also a bit awkward. It only becomes clear at the bottom of the page. I will make some changes on the entry but of course, feel free to change back or change further.
Double complexes also occur in algebraic topology with fibrations etc, so I changed ’usually’ to ’often’ a bit further down.
Perhaps this entry still needs tweeking, as it still reads awkwardly in places.
but the wording lower down seems a bit strange in places.
and the interest in double complexes is often that in these total complexes.
These place have been there all along as you were editing the entry up to and including the revision number 10. But feel free to change the wording. After all, this is a rather elementary entry, I am sure you’ll find a wording that does it justice.
A wording may not strike one as being awkward the first time you see it, but later it seems …. ! That will always be a problem.
Not sure, are you maybe looking for something along the lines of resolution of a chain complex ?
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