Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I started working on this but had to quit in a haste. It's the Whitehead twoer for B O(n), I'd say.
Quickly added paranthetical remarks to make clear what is meant. Will try to turn to this entry later today.
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.
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.
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:
the explicit presentation of $\frac{1}{2}\mathbf{p}_1$ by Lie integration in my article with Domenico Fiorenza and Jim Stasheff,
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 ?? ;-)
I have now written out the argument that $\mathbf{cosk}_3 \exp(\mathfrak{string})$ is indeed model for the homotopy fiber of the smooth $\frac{1}{2}\mathbf{p}_1$.
1 to 7 of 7