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• CommentRowNumber1.
• CommentAuthorDavidRoberts
• CommentTimeApr 27th 2012
• (edited Apr 27th 2012)

James Wallbridge put on the arXiv a paper derived from his thesis. I’ve linked to both from his page here. Urs, in particular, was interested in seeing a copy

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 27th 2012
• (edited Apr 27th 2012)

Thanks +1! I have added the links also to the pointer to his thesis that was at the bottom of Tannaka duality. Am reading…

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 27th 2012

Ah, nonabelian ∞-gerbes in def. 4.11 done right.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 27th 2012
• (edited Apr 27th 2012)

For the convenience of anyone following this here, here is a lightning summary:

We are in the context of the $\infty$-topos $H$ over some site of formal duals of algebras. So we have a canonical $\infty$-presheaf $\mathrm{Mod}$ which sends an algebra to its collection of modules, and we consider topologies such that this is an $\infty$-sheaf. And similarly there is a $\mathrm{Perf}↪\mathrm{Mod}$ of nice objects.

The key conceptual step considered is then on p. 37: $\mathrm{Perf}$ is regarded now as a dualizing object between $H$ and the $\infty$-category of suitable monoidal $\infty$-categories:

homming any $\infty$-sheaf $X$ into $\mathrm{Perf}$ gives the tensor $\infty$-category of nice “$\infty$-vector bundles over $X$”, and dually.

(Reading the $\mathrm{Mod}$ and $\mathrm{Perf}$ as categorifications of sheaves of functions, this a categorified $𝒪⊣\mathrm{Spec}$-adjunction: from a space we form its 2-ring of 2-functions and from a 2-ring we form the space that is its 2-spectrum).

The question now is: restricted to which spaces $X\in H$ does this duality constutute an equivalence?

The answer of theorem 7.14 is: at least for those $X$ which are group objects represented in a nice way by group objects in the site.

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeApr 27th 2012

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 27th 2012
• (edited Apr 27th 2012)

Do you mean http://digital.library.adelaide.edu.au/dspace/bitstream/2440/69436/1/02whole.pdf? That works well for me.

By the way (seeing the title page of the thesis again), a) I didn’t know that Varghese Mathai is looking into such matters and b) that it an impressive thesis jury!

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeApr 27th 2012

Mathai isn’t into higher category theory, he was just James’ supervisor to start with, and at some point suggested he look into Tannaka theory when approaching geometric Langlands. Things just went from there and James moved to France to work with Toen.

• CommentRowNumber8.
• CommentAuthorTim_Porter
• CommentTimeApr 27th 2012
• (edited Apr 27th 2012)

When I download the thesis it looks as if it is trying to use fonts that are not available. Strangely it is alright on our other computer! (Edit: redirected my Firefox preferences to open in Adobe Reader and it is fine, but in Preview it failed. Weird!)