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nLab > Latest Changes: Dold-Kan correspondence

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    • CommentRowNumber1.
    • CommentAuthorJohn Baez
    • CommentTimeDec 22nd 2009
    I tried to slightly polish the article Dold--Kan correspondence (section: Details) to reduce my confusion concerning the concept of "normalized Moore complex" for a simplicial abelian group. Before I did this work, two different definitions of this concept were given, and it wasn't made vividly clear that these two definitions are naturally isomorphic. In fact that's made clear on the page Moore complex, but I didn't think to look there at first. So, I've tried to make this page a bit more clear and self-contained.

    This page could still use lots of work. There's a half-completed proof, and I suspect the notation and terminology regarding "normalized Moore complex", "alternating face complex", etc. is not completely consistent throughout the $n$Lab.

    I'm also quite sure that in the abelian case, about 100 times as many people have heard of the alternating face complex --- except they call it the "normalized chain complex" of a simplicial abelian group. It may be okay to force people to learn about the normalized Moore complex, which has the advantage of applying to nonabelian simplicial groups. But, it's good to explain what's going on here.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 22nd 2009
    • (edited Dec 22nd 2009)

    Thanks, John. I remember back when we started writing about this I had some discussion with Tim Porter about what the best terminology is. But it also seems to me that in this area terminology is far from being used uniformly. Compare Kathryn Hess's comment that she feels unable to compare her terminology to that on the entry on monoidal Dold-Kan correspondence, which is entirely taken from referenced literature.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeDec 24th 2009

    Maybe people prefer the term ‘alternating face complex’ because it's simpler than the alternative (which is …).

    (^_^) (^_^) (^_^)

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeDec 26th 2009
    Yes, the notation is not standardized. There's a large group of people who like getting chain complexes from simplicial abelian groups, and these people seem things like "normalized chain complex"... but then there's a smaller group of people who know about the nonabelian case, and these people say things like "normalized Moore complex". So I tried to explain a bit about both terms.

    I know that Kathryn Hess is an expert on the abelian case, but I could easily imagine that she never thinks about the nonabelian case. Tim Porter, on the other hand, has an unusual amount of expertise on the nonabelian case.

    My plan, by the way, is to take specific nLab entries and make them a bit more friendly before citing them in This Week's Finds. I like the idea of having technical details available in the nLab so I don't need to have in This Week's Finds. And I like the idea of pointing people to the nLab so more people find out about it. But before I point people to a specific nLab page, I want to make sure people will like it when they see it. Usually this means making a bit it more expository.

    So, until something changes, you can pretty much assume that when I mention an article in "Latest Changes", I'm trying to polish it up in preparation for citing it in This Week's Finds. And when I complain about something in an article, please don't be offended - I'm just wanting to list some ways it could be improved. I would love help, but of course I don't really expect it: we're all busy with our own projects.

    In the Weeks to come, I'll be talking about rational homotopy theory, starting with old-fashioned perspectives (from Quillen and Sullivan), and gradually working towards more modern ones. Luckily these topics are among those Urs has already written a lot about!
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2009
    • (edited Dec 28th 2009)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> My plan, by the way, is to take specific nLab entries and make them a bit more friendly before citing them in This Week's Finds. </blockquote> <p>Great, thanks.</p> <p>This is the way I, for one, am using the nLab, too: I add stuff to a given entry whenever I feel that I, personally, need it, for whatever I am currently doing.</p> <p>If many contributors are being selfish this way and make the nLab satisfy their personal needs, eventually a wide choice of personal needs will be satisfied by the lab entries, and they will be useful for the rest of the world. And all this, with everyone having been selfish. It's like capitalism (the working version, not the dysfunctional one that we see around us, lately.)</p> </div>
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2009

    Concerning terminology:

    the truly best term for the funtor that has so many names would be "(normalized) (co)chain geometric realization". Because it is the left adjoint of a specific case of the general mechanism of nerve and realization.

    Strikingly, this is precisely the way Kan (but not Dold, at least not initially) thought about it. He introduces Kan extensions in this context, as Todd kept emphasizing!

    • CommentRowNumber7.
    • CommentAuthorJohn Baez
    • CommentTimeJan 4th 2010
    Yes, at the end of "week287" I tried to hint at how geometric realizations were Kan extensions without actually saying the phrase "Kan extension". Actually I said another equally scary phrase: "weighted colimit". And I could have said a few other scary phrases, like "coend" and "cosimplicial object". I only recently realized - thanks to Jim Dolan - that a cosimplicial object in a category C should be thought of as "a collection of pieces, one for each simplex, which you can put together following the instructions given by a simplicial set to build an object in C". This makes them seem very intuitive and sensible, while beforehand I considered them scary.
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 19th 2015

    I have been doing some further trivial polishing at Moore complex and Dold-Kan correspondence, such as adding more links to definitions, fixing trivial typos etc.

    We should still add the original reference with page and verse for that key theorem attributed to Eilenberg-MacLane.

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