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• CommentRowNumber1.
• CommentAuthorGuest
• CommentTimeOct 14th 2009
At string 2-group it is claimed that the sequence of classifying spaces ending --> BSO(n) --> BO(n) is the Whitehead tower of O(n). Also mentioned is the version for smooth infinity groupoids (so I assume it is Urs who put that there). It is certainly not true that the sequence of classifying spaces so stated is the Whitehead tower for O(n), but the details for groups considered as one-object infinity groupoids are open to interpretation, so I haven't changed anything. Just a heads up.

-David Roberts
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 14th 2009

I started working on this but had to quit in a haste. It's the Whitehead twoer for B O(n), I'd say.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 14th 2009

Quickly added paranthetical remarks to make clear what is meant. Will try to turn to this entry later today.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 15th 2009

Okay, now I invested time into string 2-group and feel I can officially announce it here as having been created.

various new stuff, see table of contents. Not that this is finalized of course, but I almost drop from my chair so tired am I, so that'll be it for tonight.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 6th 2010
• (edited Sep 6th 2010)

I reorganized string 2-group a bit. I made the description as a homotopy fiber the definition (since that determines all the abstract prperties) and then below that arranged the different models.

I also removed some of the aspects that were mentioned there earlier and instead added links to infinity-Lie groupoid where these are now discussed more comprehensively.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeFeb 21st 2011
• (edited Feb 21st 2011)

I am now working on finally bringing string 2-group into shape:

• using the now-available cohesive technology/language I have further stream-lined the abstract definition (looping of the homotopy fiber of the smooth refinement $\frac{1}{2}\mathbf{p}_1$ of the first fractional Pontryagin class);

• then I have now written out the statement and proof (in the new Properties-section) that this abstract definition alone implies that the geometric realization of the string 2-group is the string group in $Top$

This is essentially a direct corollary of combining two results:

1. the explicit presentation of $\frac{1}{2}\mathbf{p}_1$ by Lie integration in my article with Domenico Fiorenza and Jim Stasheff,

2. and the statement that we have been discussing at geometric realization of simplicial topological spaces, that on good simplicial spaces geometric realization preserves homotopy fibers.

All one needs to add is the observation that we may always pull back the presentation of $\mathbf{cosk}_3 \exp(\mathfrak{so}(n)) \simeq \mathbf{B}Spin$ used in FSS to a degreewise finite dimensional simplicial manifold (with countably many connected components in each degree by general nonsense, which is good enough, or even with finitely many components, using Chris Schommer-Pries’s construction, if desired).

I now want to transfer the old discussion from nactwist about the equivalence of three different relevant strict models for the String 2-group (which serves to show that the BCSS construction really does model the smooth homotopy fiber and not just the topological homotopy fiber under geometric realization), but I am not sure yet if I’ll have the energy to transform the xypic art used there into the nLab. Probably not. (read: is there by any chance a volunteer here ?? ;-)

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 21st 2011

I have now written out the argument that $\mathbf{cosk}_3 \exp(\mathfrak{string})$ is indeed model for the homotopy fiber of the smooth $\frac{1}{2}\mathbf{p}_1$.