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Monday, March 1, 2021

**Abstract:** Higher categories admit a notion of internal groupoids which Nikolaus et. al have shown yield a nice theory of principle bundles in any higher topos. An example of the practical use of this can be seen work of Freed-Hopkins where they define a higher topos of “generalized spaces” which then admits a universal bundle with connection. In an attempt to extend the results of Nikolaus to more kinds of categories, we inevitably end up working with the same kinds of structures as logicians. Namely, with pretoposes and logical functors, as opposed to the more homotopy theoretic grothendieck toposes and geometric morphisms. The goal of this talk will be to demystify this deep connection between model theory (in the logician’s sense) and homotopy theory. This talk will mostly operate at a conceptual level to more insightfully navigate the fact that many results that we want don’t yet have analogs proven in the higher-categorical setting, and the fact that the lower setting doesn’t quite have as nice of a picture. I assume no background in logic, and a vague awareness of the use (conceptually) of toposes in homotopy theory. Please email vb8 at illinois dot edu for the zoom details.