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    • CommentRowNumber1.
    • CommentAuthorGuest
    • CommentTimeOct 13th 2009
    started Whitehead tower, plus some speculative comments on versions using higher categories.

    -David Roberts
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 13th 2009

    Thanks, David!

    I added a bit to the section on "Constructions" by co-killing of homotopy groups (properly speaking it should be "co-killing" for the Whitehead tower where it is "killing" for the Postnikov tower).

    I'd be interested to see more details on your construction that you mention. I suppose I should look through your thesis.

    But I am wondering about this: the Whitehead tower construction really works in every (oo,1)-topos: there is a notion of homotopy groups of any object in an (oo,1)-topos (but currently the only entry we have is n-truncated object of an (infinity,1)-topos). Lurie discusses Postnikov towers in HTT.

    What you mention sounds like it wants to become the Whitehead tower construction in something like the (oo,1)-topos of oo-stacks on Top. Could that be?

    Notice that Dugger has the nice result, mentioned at homotopy localization, that homotopy invariant oo-stacks on Top are equivalent to topological spaces. That reminds me a bit of the remarks you made, where you approach the Whitehead construction in Top but using topological groupoids. Maybe I am wrong, i haven't really looked at your construction closely. (You should provide a direct link to page and verse to the page on Whitehead tower.)

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeOct 14th 2009
    As my thesis is not quite finished I felt like being a bit secretive as to my general idea. But since we are all friends here, I'll put a bit in when I have some time.

    A question about the n-truncated objects: is that fairly straightforward? For classical $Top$ the whole thing is a mess: killing homotopy groups - and I mean killing, not co-killing - by representatives of generators of homotopy groups. If one passes to simplicial (pre)sheaves on Top, which would be the ultimate sensible thing to do if I pursue this particular thought, then it's ok. How about other $(\infty,1)$-topoi?

    -David Roberts
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> As my thesis is not quite finished I felt like being a bit secretive as to my general idea. </blockquote> <p>Oh, I see. to clarify: all I meant to indicate is: what you say sounds interesting, I wish I knew where to learn abnout it in more detail!</p> <blockquote> A question about the <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_485d65c1e6e7757e4de10eb91bc9382b.png" title="n" style="vertical-align: -20%;" class="tex" alt="n"/>-truncated objects: is that fairly straightforward? For classical <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_d135fddf839f5858264c53bd39d0a7fc.png" title="Top" style="vertical-align: -20%;" class="tex" alt="Top"/> the whole thing is a mess: killing homotopy groups - and I mean killing, not co-killing - by representatives of generators of homotopy groups. If one passes to simplicial (pre)sheaves on Top, which would be the ultimate sensible thing to do if I pursue this particular thought, then it's ok. How about other <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_5b4e8bf1a57dcb0e372c2512e8f20c01.png" title="(\infty,1)" style="vertical-align: -20%;" class="tex" alt="(\infty,1)"/>-topoi? </blockquote> <p>I will write entries now on truncation, truncated objects and Postnikov towers in <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_5b4e8bf1a57dcb0e372c2512e8f20c01.png" title="(\infty,1)" style="vertical-align: -20%;" class="tex" alt="(\infty,1)"/>-toposes, summarizing Lurie's material.</p> <p>The general theory is rather nice and satisfactory. I suppose the mess that you are thinking of is one of actual constructions of n-truncations?</p> <p>i would like to simply "dualize" Lurie's discussion of Postnikov towers of oo-stacks to one of Whitehead towers. It feels like it should be obvious enough, but I am not nevertheless not exactly sure how to precisely state the condition of the universaility that the morphisms in the tower have to satisfy. It must have an easy answer, but I may need to think about it more...</p> <p>(by the way David: the dollar-sign latex seems to be broken here: what does work is including stuff in and html-like "latex" tag.)</p> </div>
    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeOct 14th 2009

    what does work is including stuff in and html-like "latex" tag

    You can also use double dollars (which will still be inline, not displayed), which is easier to type but will not work in the preview.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2009

    hm, last time I tried double dollars didn't work either. Okay, new attempt:

       \exp(2 \pi i) - 1 = 0
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2009

    What do you all see? I see a "formula does not parse" error.

    but let me try this

     1
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2009

    ah, then let me try this:

     \exp(2 \pi i) - 1 = 0
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2009

    Okay, that explains it. The double dollars and the content they bracket have to be on the same line!

    With double dollars I am genetically programmed to have line breaks after the initial and before the closing one. That explains my baad luck with them here...

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeOct 15th 2009

    On the subject of the Whitehead tower, I normalised the list of groups in the tower of \mathrm{O}(n) in their various articles: Fivebrane group to string group to spin group to special orthogonal group to orthogonal group. Out of curiosity, why is that first one always capitalised?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> Out of curiosity, why is that first one always capitalised? </blockquote> <p>Er, that might just be my German capitalization instinct ;-)</p> <p>Probably it shouldn't be. In fact Ithink 90 per cent of the time or so I also capitalised <a href="http://ncatlab.org/nlab/show/String+group">String group</a> and <a href="http://ncatlab.org/nlab/show/Spin+group">Spin group</a>.</p> <p>In formulas it's <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_54bb7afed08351ea1e544d0f360cbe1b.png" title=" Spin(n) " style="vertical-align: -20%;" class="tex" alt=" Spin(n) "/> and <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_17875b9d51a59d67c41a60b50ee16f69.png" title=" String(n) " style="vertical-align: -20%;" class="tex" alt=" String(n) "/> and <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_ff1fab2a7c74dcc88e068904c2db49cd.png" title=" Fivebrane(n) " style="vertical-align: -20%;" class="tex" alt=" Fivebrane(n) "/> with capitals. But I gather one doesnt write "Spin group" but "Spin group"? If so, it should probably also be "fivebrane group", too.</p> </div>
    • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeOct 15th 2009

    Er, that might just be my German capitalization instinct ;-)

    No, it's not; it's also in the references. Otherwise I'd have just changed it!

    • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeOct 15th 2009

    Also, you can definitely capitalise at the beginning of a sentence. (^_^)

    • CommentRowNumber14.
    • CommentAuthorTobyBartels
    • CommentTimeOct 15th 2009

    Wait a minute … you wrote those references! So maybe you're right, and you are wrong after all.

    Now if it were me, I'd write ‘5-brane’.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> Wait a minute … you wrote those references! </blockquote> <p>hehe. Not alone, though. And not all of (the two of) them.</p> <p>But there was some back and forth after this was accepted with the typesetting editor on such capitalization issues. Tomorrow, when I am more awake, I'll remind myself which convention made it into the published version. I forget.</p> </div>
    • CommentRowNumber16.
    • CommentAuthorGuest
    • CommentTimeOct 15th 2009
    A bit more at Whitehead tower, on the classical construction and analogues for (infinity,1)-cateories. Will get to my version soon, Urs.

    -David Roberts
    • CommentRowNumber17.
    • CommentAuthorGuest
    • CommentTimeOct 15th 2009
    More at Whitehead tower, this time on my own thoughts on how one could fancy the whole thing up the n-categorical ladder.

    In response to a question earlier, the 'mess' I was talking about was the lack of functorial Postnikov towers in Top. They are certainly available in the category of Kan complexes (and by the Ex^oo functor, on sSet, I imagine).
    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 13th 2009

    added a section "functorial constructions" to Whitehead tower by copying stuff from a MathOverflow discussion

    • CommentRowNumber19.
    • CommentAuthordomenico_fiorenza
    • CommentTimeJan 10th 2010
    • (edited Jan 10th 2010)
    edited Whitehead tower at "Construction by cokilling of homotopy groups"
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