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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.)
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As my thesis is not quite finished I felt like being a bit secretive as to my general idea.
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<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>
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A question about the <img src="/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="/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="/extensions/vLaTeX/cache/latex_5b4e8bf1a57dcb0e372c2512e8f20c01.png" title="(\infty,1)" style="vertical-align: -20%;" class="tex" alt="(\infty,1)"/>-topoi?
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<p>I will write entries now on truncation, truncated objects and Postnikov towers in <img src="/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>
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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.
hm, last time I tried double dollars didn't work either. Okay, new attempt:
What do you all see? I see a "formula does not parse" error.
but let me try this
ah, then let me try this:
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...
On the subject of the Whitehead tower, I normalised the list of groups in the tower of 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?
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Out of curiosity, why is that first one always capitalised?
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<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="https://ncatlab.org/nlab/show/String+group">String group</a> and <a href="https://ncatlab.org/nlab/show/Spin+group">Spin group</a>.</p>
<p>In formulas it's <img src="/extensions/vLaTeX/cache/latex_54bb7afed08351ea1e544d0f360cbe1b.png" title=" Spin(n) " style="vertical-align: -20%;" class="tex" alt=" Spin(n) "/> and <img src="/extensions/vLaTeX/cache/latex_17875b9d51a59d67c41a60b50ee16f69.png" title=" String(n) " style="vertical-align: -20%;" class="tex" alt=" String(n) "/> and <img src="/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>
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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!
Also, you can definitely capitalise at the beginning of a sentence. (^_^)
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’.
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Wait a minute … you wrote those references!
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<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>
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added a section "functorial constructions" to Whitehead tower by copying stuff from a MathOverflow discussion
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