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Defects in Topos Theory + some ideas going forward

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If you have a question about this talk, please contact Tom de Jong.

I will give a little review of the research programme that I set out to investigate for my PhD – namely, exploring and developing the relationship between the adele ring and Grothendieck toposes, with a view towards applying generic reasoning in geometric logic to local-global problems in number theory.

I will then describe a roadblock that we’ve come across, and how this illustrates a fundamental defect with ordinary Grothendieck toposes. In particular, while Grothendieck toposes classify principal G-bundles when G is pro-discrete (better yet: etale complete, to use the terminology of Moerdijk), it generally fails to classify principal G-bundles when G is topological. In other words, it seems that Grothendieck toposes don’t quite capture all the information we want, and this appears to be related to how Grothendieck toposes are categories of set-valued sheaves.

The quest then is to figure out the appropriate generalisation of the Grothendieck topos – many of these ideas seem to have been developed by a naive categorification (i.e. replacing Set with CAT , and weakening the structure appropriately so that things work up to equivalence + certain coherence conditions as opposed to equality) but there are some persisting issues from the standard case of Grothendieck toposes, which suggests to me that this approach of categorification is not quite right. This appears to be connected with how there does not seem to be a developed account of higher classifying toposes (despite Lurie’s work on Higher Topos Theory having been out for a while).

The theory of stacks seem quite promising and highlight some important structural details about principal G-bundles, and I will introduce some basic results from Bunge about this issue. I will then use that as a basis to argue what the appropriate generalisation of the Grothendieck topos (and in particular, the appropriate generalisation of geometric morphism + higher geometric logic) ought to be.

This talk is part of the Bravo series.

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